research article basel iii and asset...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 439305 19 pageshttpdxdoiorg1011552013439305

Research ArticleBasel III and Asset Securitization

M Mpundu M A Petersen J Mukuddem-Petersen and F Gideon

Faculty of CommerceampAdministration North-West University (Mafikeng Campus) Private Bag x2046Mmabatho 2735 South Africa

Correspondence should be addressed to M A Petersen markpetersennwuacza

Received 14 June 2013 Revised 28 September 2013 Accepted 30 September 2013

Academic Editor Oswaldo Luiz do Valle Costa

Copyright copy 2013 M Mpundu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Asset securitization via special purpose entities involves the process of transforming assets into securities that are issued to investorsThese investors hold the rights to payments supported by the cash flows from an asset pool held by the said entity In thispaper we discuss the mechanism by which low- and high-quality entities securitize low- and high-quality assets respectivelyinto collateralized debt obligations During the 2007ndash2009 financial crisis asset securitization was seriously inhibited In responseto this for instance new Basel III capital and liquidity regulations were introduced Here we find that we can explicitly determinethe transaction costs related to low-quality asset securitization Also in the case of dynamic and static multipliers the effects ofunexpected negative shocks such as rating downgrades on asset price and input debt obligation price and output and profit will bequantified In this case we note that Basel III has been designed to provide countercyclical capital buffers to negate procyclicalityMoreover we will develop an illustrative example of low-quality asset securitization for subprime mortgages Furthermorenumerical examples to illustrate the key results will be provided In addition connections between Basel III and asset securitizationwill be highlighted

1 Introduction

Asset securitization involves the process by which securitiesare created by a special purpose entity (SPE)mdashhereaftersimply known as an entitymdashand then issued to investors witha right to payments supported by the cash flows from a poolof financial assets held by the entity There is broad-basedusage of entities by financial institutions of many types invarious jurisdictions and for many purposes (see eg [1])Securitization has been popular as an alternative fundingsource for consumer and asset lending in market economiesIts main objective is to improve credit availability by con-verting hard-to-trade and nontradable assets into securitiesthat can be traded on capital markets The categorization ofthe payment rights into ldquotranchesrdquo paid in a specific orderand supported by credit enhancement mechanisms providesinvestors with diversified credit risk exposure to particularinvestor risk appetites (see eg [2 3]) Immediately prior tothe securitization market collapse in 2007-2008 structuredasset products (SAPs) such as asset-backed securities (ABSs)and collateralized debt obligations (CDOs) as well as coveredbonds provided between 25 and 65 of the funding for newresidential assets originated in the US and Western Europe

(see eg [4]) In most developed economies SAP growthpeaked by 2007 before declining rapidly due to a lack ofliquidity in secondary markets and decreases in primaryissuance (see [5] formore details) For example SAP issuancein the US decreased from about US $2 trillion in 2007 toaround US $400 billion in 2008 The impact of the financialcrisis on securitization in emergingmarketswasmoremodestas initial growth had been more subdued

The contribution of securitization to the financial crisisnecessitated changes to banking regulation In this regardthe introduction of Basel III capital and liquidity regulationincludes elements that will potentially affect the incentivesfor banks to securitize assets (see eg the Basel documents[2 5ndash7]) There are several Basel III provisions that addressareas of concern that were highlighted during the financialcrisis and which supervisors determined as not adequatelyaddressed under the previous framework (see eg [6])More specifically in July 2009 the Basel Committee forBanking Supervision (BCBS) published enhancements to theBasel II framework that were intended to strengthen theframework and respond to lessons learned from the financialcrisis (see [8] for more details) For instance because of thehigher degree of inherent risk in resecuritization exposures

2 Discrete Dynamics in Nature and Society

the BCBS significantly increased the risk weights applicableto such exposures under both the standardized (SA) andinternal ratings based (IRB) approaches relative to the riskweights for other securitization exposures (see eg [3 910]) As a result the capital requirements for resecuritizationhave risen dramatically In addition to address the lack ofappropriate due diligence on the part of investing institutionsand deter them from relying solely on external credit ratingsthe Basel framework now requires banks to meet specificoperational criteria in order to use the risk weights specifiedin the Basel II securitization framework During the financialcrisis credit rating agencies (CRAs) downgraded the ratingsof many securitization tranches including senior tranceshighlighting deficiencies in credit rating agency modelsoriginally used to determine the ratings Capital requirementsassigned to highly rated (eg AAA) senior and mezzaninesecuritization exposures were too low and this was illustratedby the poor performance of these securities (see eg [9 10])As CRAs downgraded highly rated securitization exposuresbelow investment grade regulatory capital requirementsincreased rapidly and significantly due to the presence ofcliff effects within the securitization framework (in thiscontext cliff effects refer to significant increases in capitalrequirements resulting from a change to a factor used toassign regulatory capital) (see eg [9 10]) The BCBS alsorevised the market risk rules to increase the level of capitalthatmust bemaintained against securitization exposures heldin the trading book In addition the BCBS is reviewingwhether the risk weights for all securitization exposuresshould be recalibrated which could lead to higher capitalrequirements (see for instance the Basel papers [9 10])

The main motivation for this paper is that securitizationhas to be reestablished on a sound basis in order to supportcredit provision to the real economy and enhance banksrsquoaccess to funding globally (see eg [4]) Basel III would liketo ensure an appropriate risk-sensitive and prudent capital-ization of risks arising from securitization exposures whilereducing cliff effects and mitigating mechanistic reliance onexternal credit ratings (see [3 9]) The interplay betweenBasel III and asset securitization will be the central themeof this paper As far as the contribution of this paper isconcerned we investigate the securitization of low- and high-quality assets (denoted by LQAs andHQAs resp) into CDOsvia low-quality entities (LQEs) and high-quality entities(HQEs) respectively (see eg the BCBSpublications [9 10])

11 Literature Review about Asset Securitization and Basel IIIIn this subsection we provide literature reviews about assetsecuritization and Basel III capital and liquidity regulation(see [5] for more details)

111 Literature Review about Asset Securitization Motivatedby the 2007ndash2009 financial crisis there is an ever-growingbody of literature on LQA-related issues such as shocks toLQEs investors and CDOs via for instance rating changesThe contribution [11] studies the pricing of LQAs and relatedSAPs on the basis of data for the ABXHE family of indices(see [12] for further details) This of course is a recurringtheme in our contribution where we consider asset and CDO

pricing during the financial crisis Moreover [4] addressesthe impact of speculative asset funding on the pricing ofLQAs (measured by risk premia) and securities backed bythese assets (measured by ABXHE indices) In addition thepaper [13] extends a Kiyotaki-Moore-type model that showshow relatively small shocks might suffice to explain businesscycle fluctuations if credit markets are imperfect (see [14] formore details) Our work has a connection with this paper viathe consideration of the effect of shocks on asset parametersalthough we do not emphasize the imperfection of creditmarkets (see [15] for additional analysis) The paper [16]studies the impact of penalties on LQ-loans (see also [17])Here asset prices and penalties are chosen simultaneouslywith the latter being associated with lower asset prices (see[15] for more details) The paper also contains discussionson prices and penalties and their relationship with loan-to-value ratios (see also [18] for more on house equity) In ourcontribution we will use the framework introduced in [16] toshow how a change in profit subsequent to a negative shockis influenced by subprime mortgage features (see also [19])

112 Literature Review about Basel III In July 2009 the BCBSintroduced enhancements to the Basel II framework via [8]This was done in order to address deficiencies identifiedduring the 2007ndash2009 financial crisis These measures pri-marily addressed immediate concerns over resecuritizationand was part of reforms known as ldquoBasel 25rdquo (see the Baseldocuments [8 9] for more details about securitization) TheBCBS subsequently agreed to conduct a more fundamentalreview of the securitization framework including its relianceon external ratings (see eg [20]) The performance of androle played by securitization exposures during the financialcrisis were a key motivation for studying this category of thecapital framework Subsequently in [2] the BCBS noted thatit was ldquoconducting amore fundamental review of the securiti-zation framework including its reliance on external ratingsrdquoThe BCBS has now performed a broader review of thesecuritization framework for regulatory capital requirementswith objectivesmotivated by events during the financial crisis(see also [7]) Furthermore the consultative document [6]reflects the BCBSrsquos proposal to revise the Basel frameworkrsquostreatment of securitization exposures In developing thisproposal the BCBS seeks to make capital requirements moreprudent and risk sensitivemitigate reliance on external creditratings and reduce cliff effects (see eg [3 6]) In particularour paper adds to the debate about rating downgrades andshocks associated with them The policy directions set outin [6] form part of the BCBSrsquos broader agenda of reformingbank regulatory capital standards to address the lessons ofthe crisis These proposals build on a series of reforms thatthe BCBS has delivered through Basel III and explain theapproaches under consideration by the BCBS to revise thesecuritization framework (see [6 21] for further discussion)As in our paper the features of SPEs are discussed in somedetail in [1]

12 Preliminaries about Asset Securitization In this subsec-tion we provide preliminaries about low- and high-quality

Discrete Dynamics in Nature and Society 3

Table 1 Defining features of LQAs and HQAs source [5]

Feature LQAs HQAsExternal information

External ratings Low HighMarket data Poor RichAnalystsrsquo reports Negative Positive

Internal informationCredit risk High LowLiquidity risk High LowSecuritization exposure Not understood UnderstoodFinancial obligations Not easily met Easily met

classification as well as LQA and HQA securitization (seeTable 1 for more information)

121 Preliminaries about Low- and High-Quality Classifi-cation As indicated in the BCBS documents [2 7] thecharacterization of ldquohigh qualityrdquo is based both on availableexternal information such as external ratings market dataand analystrsquos reports and the entityrsquos own assessment of creditand liquidity risk whereby the entity should demonstrate itsunderstanding of the terms of the securitization exposure andthe risks of the underlying collateral (see eg [1 5]) Theentity would be required to demonstrate that the creditquality of the position is strong with very low default riskand is invulnerable to foreseeable events implying thatfinancial commitmentswould bemet in a timelymannerwitha very high probability (see [6 21] for further discussion)Where this determination could not be made the positionwould be assumed to be ldquolow qualityrdquo In particular in theBasel III document [1] the Joint Forum Working Groupon Risk Assessment and Capital (JFWGRAC) consisting ofthe BCBS International Organization of Securities Commis-sions (IOSC) and the International Association of InsuranceSupervisors (IAIS) under the support of the Bank for Interna-tional Settlements (BIS) make recommendations about suchentities

122 Preliminaries about LQA and HQA Securitization Inthis paper we have that the reference asset portfolios are botha means of generating CDOs and collateral for interentitysponsoring (see eg [1]) Our paper quantifies the effects ofunexpected negative shocks such as rating downgrades onasset price and input CDO price and output and profit ina Basel III context (see also [19]) For instance the afore-mentioned result demonstrates how the proportional changein profit subsequent to a rating downgrade is influenced byLQA features such as asset rates Finally we present examplesthat illustrate that asset price is most significantly affectedby unexpected negative shocks from asset rates while forCDO price shocks to speculative asset funding investorrisk characteristics and prepayment rate elicit statisticallysignificant responses (compare with [3])

LQAswere financed by securitizing these assets into SAPssuch as ABSs and CDOs The lower-rated tranches of low-quality ABSs formed 50 to 60 of the collateral for CDOs

during 2007These were extremely sensitive to a deteriorationin asset credit quality Housing went through a classic inven-tory cycle with a worsening of the inventory-to-sales cyclebeing evident in the midst of the 2007ndash2009 financial crisisWhen this inventory situation worsened the risk that pricewould fallmore rapidly deepenedThemore substantial fall inprices accelerated the delinquency and foreclosure rater andspelt doom for the CDOmarket which was further revised inBasel III (see eg the BCBS paper [3 9]) We briefly describethe aforementioned SAPs in turn

LQ-ABSs are quite different from other securitizationsbecause of the unique features that differentiate low-qualityassets from other assets Like other securitizations LQ-ABSsof a given transaction differ by seniority But unlike othersecuritizations the amount of credit enhancement for and thesize of each tranche depend on the cash flow coming into thedeal in a very significant way The cash flow comes largelyfrom prepayment of the reference asset portfolios throughrefinancing What happens to the cash coming into the dealdepends on triggerswhichmeasure (prepayment and default)performance of the reference asset portfoliosThe triggers canpotentially divert cash flows within the structure In somecase this can lead to a leakage of protection for higher-rated tranches Time tranching in low-quality transactions iscontingent on these triggers The structure makes the degreeof credit enhancement dynamic and dependent on the cashflows coming into the deal

In our case a CDO issues debt and equity and uses theincome to invest in financial assets such as assets and ABSsIt distributes the cash flow from its asset portfolio to holdersof various liabilitiesmdashusually a capital structure consistingof equity or preferred shares subordinated debt mezza-nine debt and AAA-rated senior debt and borrowingsmdashinset ways taking into account the relative seniority of theaforementioned liabilities A key feature of such CDOs isthat they are mainly constituted by ABS portfolios that arerated according to their prevailing credit risk and put intotranches (compare with [3]) CDO tranches include LQ-and Alt-A deals and consist of three categories namelysenior mezzanine and equity or low tranches according toincreasing credit risk This risk is spread to investors whoinvest in these risky CDO tranches It is difficult for investorsto locate the risk exposure of these CDOs because of theircomplex design structureDespite this CDOshave additionalstructural credit protection which can be characterized aseither cash flow or market value protection Finally all CDOsare created to fulfill a given purpose that can be classified asarbitrage balance sheet or origination

13 Main Contributions and Outline Themain contributionsabout Basel III regulation and asset securitization in thispaper are constituted by the answers to the questions listedbelow

Question 1 (LQA securitization) How can we characterizethe securitization of LQAs into CDOs by the LQE Forinstance canwe determine the LQE cash flow constraint (seeLemma 3 in Section 21)


Question 2 (HQA securitization) How can we character-ize the securitization of HQAs into CDOs by HQEs Forinstance can we determine the HQEsrsquo cash flow constraint(see Lemma 5 in Section 22)

Question 3 (LQE at steady state) How can we characterizethe behavior of an LQE at steady state (see Theorem 7 inSection 23)

Question 4 (dynamic multiplier negative shocks to assetsecuritization) For a dynamic multiplier how do negativeshocks affect asset price and input CDO price and outputand profit (see Theorem 8 in Section 31)

Question 5 (static multiplier temporary shocks to asset secu-ritization) For a static multiplier how do negative shocksaffect asset price and input (see Corollary 9 in Section 32)

Question 6 (example of subprime mortgage securitization)How canwe provide an example to illustrate themain featuresof LQA securitization specifically for subprime mortgages(see Corollary 10 in Section 41)

Question 7 (numerical examples of asset securitization) Howcan we provide numerical examples to illustrate the mainresults obtained in this paper (see Section 42)

Question 8 (Basel III and asset securitization in general) Ingeneral how does Basel III capital and liquidity regulationassist in eliminating flaws in existing asset securitizationpractice in banking (see Sections 2 3 and 4 formore details)

This paper is arranged as follows Section 2 describesHQA and LQA securitization Also Section 3 sheds lighton the effect of negative shocks like rating downgrades onthe asset securitization process in a Basel III context whileSection 4 provides numerical examples of the aforemen-tioned Finally Section 5 provides some concluding remarksand possible topics for future research

2 Low- and High-Quality Entities

In this section we consider LQEs and HQEs and theirequilibrium features We study an economy consisting ofLQAs with a fixed total supply of 119860 and CDOs that cannotbe retained by the entity In the sequel for the sake of argu-ment we assume that the CDOs correspond to senior CDOtranches In this model CDOs are taken as the numeraireThere is a continuum of infinitely lived LQEs andHQEs withpopulation sizes 1 and 119899 respectively Both these entities takeone period to securitize assets into CDOsmdashthe LQEs andHQEs produce CDOs from HQAs and LQAs respectivelymdashbut they differ in their securitization technologies (refer toTable 1) At each date 119905 there is a competitive spot marketin which assets for CDOs are purchased by entities at a priceof 119901119860119905 The only other market is a one-period credit market

in which one CDO unit at date 119905 is exchanged for a claimto 1 + 119903B

119905units of CDOs at date 119905 + 1 These markets are

opaque and are dominated by a handful of interests During

Risk profile

GoodBad xLow

High

y

High-gradeLQ ABS

LQ ABSbonds

Low-

assetsMezzanineABS CDO

Senior

Mezzanine

Equity

Senior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality

x-axis Investor credity-axis Investor down payment

Figure 1 Chain of LQAs and their SAPs source [4]

the 2007ndash2009 financial crisis because CDOs were lightlyregulated their details often went undisclosed This createdmajor problems in the monitoring of these credit derivativesand the new regulatory framework in Basel III focuses oncorrecting such problems (see [9] for further explanation)

21 The LQE Figure 1 illustrates the securitization of assetsinto ABSs and ABS CDOs by the LQE

We notice from Figure 1 that LQAs are securitized intoABSs that in turn get securitized into ABS CDOs As faras the latter is concerned it is clearly shown that seniorABS bonds rated AAA AA and A constitute the high-gradeABS CDO portfolio On the other hand the mezzanine-ratedABS bonds are securitized into mezzanine ABS CDOs sinceits portfolio is based on BBB-rated ABSs and their trancheswhich expose the portfolio to an increase in credit risk (seeeg [3]) From Figure 1 it is clear that the LQEs (and anyother entities) rather than banks hold assets and ABSs As aresult there are reductions in the incentives of banks to playtheir traditional monitoring function The fundamental roleof banks in financial intermediation according to Basel IIIin the document [22] (see also [21]) makes them inherentlyvulnerable to liquidity risk of both an institution-specific andmarket nature (see [5] for more details) Financial marketdevelopments have increased the complexity of liquidity riskand its management During the early ldquoliquidity phaserdquo ofthe financial crisis that began in 2007 many banksmdashdespiteadequate capital levelsmdashstill experienced difficulties becausethey did not manage their liquidity in a prudent manner(see [5] for further discussion) The difficulties experiencedby some banks which in some cases created significantcontagion effects on the broader financial system were dueto lapses in basic principles of liquidity risk measurementand management (see eg [5] for more details) During the2007ndash2009 financial crisis systemic risk from CDOs wasproblematic In this case the default of one or more collateral


assets or bond classes generated a ripple effect on CDOdefaults Figure 1 suggest how this may have happened TheLQE is risk neutral with its expected utilities being

E119905(

infin

sum

119904=0

120573119904119909119905+119904) E

119905(

infin

sum

119905=0

120573119905119909119905) (1)

where119909119905+119904

and119909119905are their respective LQECDOconsumption

at dates 119905 + 119904 and 119905 with E119905denoting the expectation

formed at date 119905These entities have constant returns on scalesecuritization function of

119862119905+1

= 119878 (119860119905) equiv (120583 + ]) 119860

119905 ] = 119888 + 119903119891

(120583

120583 + ]) lt 120573

(2)

where 119860119905are the input assets securitized at date 119905 and 119862

119905+1is

the CDO output at date 119905 + 1 Also 119903119891 is the fraction of assetsthat have refinanced and 119888 the fraction of CDOs consumedHowever only 120583119860

119905of the CDO output is marketable Here

]119860119905is nonmarketable and can be consumed by the LQE We

introduce ]119860119905in order to avoid the situation in which the

LQE continually postpones consumptionThe ratio 120583(120583+])minus1may be thought of as a technological upper bound on theLQErsquos retention rate Since 120573 is near 1 the inequality in (2)amounts to a weak assumption We shall see later that thisinequality ensures that in equilibrium the LQE will not wantto consumemore than illiquid CDOs as observed in Basel III(see eg [21]) The overall return from investment 120583 + ] ishigh enough that all its marketable CDO outputs are used forinvestment There is a further critical assumption we makeabout investing

Assumption 1 (LQE CDO technology and labor) We assumethat each LQErsquos CDO technology is distinct in the sense thatonce securitization has started at date 119905 with assets 119860

119905 only

the LQE has the skill necessary for securitizing assets intoCDOs at date 119905 + 1 subject to the availability of appropriatetechnology and labor Secondly we assume that an LQEalways has the option to withdraw its labor

In other words if the LQE were to withdraw its laborbetween dates 119905 and 119905 + 1 there would be no CDO outputat 119905+1 Assumption 1 leads to the fact that if an LQE is highlyleveraged it may find it advantageous to threaten the HQEsby withdrawing its labor and repudiating its debt contractHQEs as interentity lenders protect themselves from thethreat of repudiation by collateralizing the LQAs Howeverbecause assets yield no SAPs without the LQErsquos labor theasset liquidation value (outside value) are less than whatthe assets would earn under its control (inside value) Thusfollowing a repudiation it is efficient for the LQE to persuadethe sponsoring HQE into letting it keep the assets In effectthe LQE can renegotiate a smaller loan HQEs know of thispossibility in advance and so take care never to allow thesize of the debt (gross of interest) to exceed the value of thecollateral as in the following assumption

Assumption 2 (credit limit) If at date 119905 the LQE possessesthe assets 119860

119905 then it can borrow B

119905in total as long as the

repayment does not exceed the market value of assets at date119905 + 1 given by

(1 + 119903B) B119905le 119901119860

119905+1119860119905 (3)

where 119901119860119905+1

represents the asset price in period 119905 + 1 while 119860119905

represents the LQErsquos asset holdings in period 119905 In this casegiven rational expectations agents have perfect foresight offuture asset prices

As noted in the Basel III document [21] the objectiveof the LCR is to promote the short-term resilience of theliquidity risk profile of banks (see [5] for additional) Itdoes this by ensuring that banks have an adequate stock ofunencumbered high-quality liquid assets (HQLA) that can beconverted easily and immediately in privatemarkets into cashto meet their liquidity needs for a 30-calendar-day liquiditystress scenario Of course during the 2007ndash2009 financialcrisis when monitoring incentives was reduced it is unlikelythat the HQE monitored the LQE closely The LQErsquos balancesheet consists of assets and marketable securities (assets)as well as borrowings and capital (liabilities) Therefore anLQErsquos balance sheet constraint can be represented at time 119905 as

119901119860

119905119860119905minus1+ 119861119905= B119905+ 119870119905 (4)

where 119901119860 119860 119861 B and 119870 represent the LQErsquos asset priceasset holdingsmarketable securities borrowings and capitalrespectively As we have mentioned before the entitiesrsquocapital structure consisting of equity or preferred sharessubordinated debt and mezzanine debt LQE enforces a pricecap (PC) with the weighted average PC being denoted by 119901(see eg [4] for more details) In this case we have that theCDO price is given by

119901119862

119905= min [119901119860

119905 119901119905] (5)

where 119862119905minus1

denotes the quantity of CDOs in period 119905 minus 1Hedge funds and other sophisticated investors have incen-tives to manipulate the pricing and structuring of CDOsSome studies suggest that CDO managers manipulate col-lateral in order to shift risks among various tranches Thepotential for this can be clearly seen in (5) where the PCoffers ameans of changing collateral features that is importantin determining the CDO price 119901119862

119905 During the 2007ndash2009

financial crisis collateral according to Basel III was alsomanipulated via the violation of restrictions on asset portfoliocomposition rating category weighted average life weightedaverage weighting factor correlation factors and the numberof obligors (see [2]) Nevertheless in our case the value ofassets in period 119905 can be represented as

119901119860

119905119860119905minus1

= B119905+ 119870119905minus 119861119905 (6)

In general the asset rate 119903119860 for profit maximizing entities(see also [19]) may be represented as

119903119860

119905= 119903119871

119905+ 984858119905 (7)


where 119903119871 is for instance the 6-month LIBOR rate and 984858 is therisk premium that is indicative of asset price (see eg [3])Next the LQErsquos profit may be expressed as

Π119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905 (8)

where 119903119860 119888119901 119903119891 119903119877 119903119878 119903119861 and 119903B represent the asset

rate prepayment costs fraction of assets that refinancerecovery rate default rate returns on marketable securitiesand borrowing rate in period 119905 respectively In this case assetvalue can be represented by

119901119860

119905119860119905minus1

=Π119905minus 119903119861119861119905+ 119903

BB119905

119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905

(9)

From (9) it is clear that as recognized by Basel III regulationeven a relatively small default rate can trigger a crisis Theunwinding of contracts involving the securitization of suchassetsmdashsuch as CDO contractsmdashcreated serious liquidityproblems during the 2007ndash2009 financial crisis as detailedin Basel III (see [5 21 22]) Since the CDO market wasquite large the crisis caused convulsions throughout globalfinancial markets By considering the above we can deducean appropriate LQE cash flow constraint in the followingresult

Lemma 3 (LQE cash flow constraint) Suppose that the creditconstraint (3) as well as (6) to (9) holds In this case the LQErsquoscash flow is subject to the constraint

Π119905ge (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905

minus 119901119860

119905+1119860119905+ B119905

(10)

Proof The proof follows from taking constraint (3) fromAssumption 2 and (10) into consideration

The LQE can expand its scale of securitization by invest-ing in more assets Consider an LQE that holds 119860

119905minus1assets

at the end of date 119905 minus 1 and incurs a total debt of B119905minus1

Atdate 119905 the LQE harvests 120583119860

119905minus1marketable CDOs which

together with a new loan B119905 is available to cover the cost

of purchasing new assets to repay the accumulated debt(1 + 119903

B)B119905minus1

(which includes interest) and to meet any addi-tional consumption 119909

119905minus ]119860119905minus1

that exceeds the normal con-sumption of non-marketable output ]119860

119905minus1 The LQErsquos flow-

of-funds constraint is thus

119901119860

119905(119860119905minus 119860119905minus1) + (1 + 119903

B) B119905minus1+ 119909119905minus ]119860119905minus1

= 120583119860119905minus1+ B119905

(11)

22 HQEs For HQEs Figure 2 shows the chain formed byHQAs ABSs and ABS CDOs

As we proceed from left to right in Figure 2 HQAs aresecuritized into ABSs that in turn get securitized into ABSCDOs Only the higher-grade ABS bonds rated AAA AAandA are securitized thatmake out the high-gradeABSCDOportfolio Figure 2 also suggests that HQE ABSs and CDOsare not as risky as those of the LQE since the reference asset

Risk profile

GoodBad

Low

High

y

High-gradeABS CDO

High-qualityABS

bondsHigh-

assetsSenior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality

x


Figure 2 Chain of HQAs and their SAPs source [4]

portfolios have higher credit quality HQE capital levels willalso be greater than those of the LQE in the sense that theLQE used its capital to provision for low-quality default Inthis regard we have the secondary effect of securitizationwhere credit risk is transferred to investors Basel III identifiesa number of shortcomings within the current securitizationframework some of which were categorised broadly as toolow-riskweights for highly rated securitizations and too high-riskweights for low-rated senior securitization exposures (see[9] for detailed explanation) Furthermore we assume thatHQEs are risk neutral with expected utilities

E119905(

infin

sum

119904=0

12057310158401199041199091015840

119905+119904) E

119905(

infin

sum

119905=0

12057310158401199051199091015840

119905) (12)

where 1199091015840119905+119904

and 1199091015840119905are their respective consumptions of CDOs

at dates 119905 + 119904 and 119905 For the discount factors 120573119904 and 1205731015840119904we have that 0 lt 120573

119904 1205731015840119904

lt 1 and suppose that 120573 lt

1205731015840 This inequality ensures that in equilibrium the LQEs

will not want to postpone securitization because they arerelatively impatient (compare with [13] and the referencescontained therein) The following assumption is made forease of computation

Assumption 4 (HQE price asset default and borrowing rate)For HQEs suppose that 119901119860

1015840

1199031198601015840

and 119903B1015840

are the asset priceasset rate and borrowing rate respectively For all 119905 weassume that

1199011198601015840

119905= 119901119860

119905 119903

1198601015840

119905= 119903119860

119905 119903

B1015840

119905= 119903

B119905 (13)

where 119901119860 119903119860 and 119903B are as before for HQEs Also we assumethat the assets held by HQEs do not default or refinance

In reality this assumptionmay be violated since LQAs aremore expensive thanHQAs However this adjustment can becatered for in the sequel We shall see that in equilibrium theLQE borrows from HQEs and that the rate of interest alwaysequals the HQEsrsquo constant rate of time preference so that

119903B= 119903

B119905equiv1

1205731015840minus 1 (14)

All HQEs have an identical securitization function thatexhibits decreasing returns to scale In a Basel III context the


additional quantitative information that HQEs may considerdisclosing could include customised measurement tools ormetrics that assess the structure of HQEsrsquo balance sheets aswell as metrics that project cash flows and future liquiditypositions taking into account off-balance sheet risks whichare specific to that HQE (see the BCBS publications [5 2122]) In this case per unit of population an asset input of 1198601015840

119905

at date 119905 yields an output of 119862119905+1

marketable CDOs at date119905 + 1 according to

119862119905+1

= 119875 (1198601015840

119905) where 1198751015840 gt 0

11987510158401015840lt 0 119875

1015840(119860

119899) lt 120583 (1 + 119903

B) lt 1198751015840(0)

(15)

The last two inequalities in (15) are included to ensure thatboth the LQE and HQEs are producing in the neighborhoodof the steady-state equilibrium HQEs securitization doesnot require any specific skill nor do they produce any non-marketable CDOs As a result no HQE is credit constrainedas noted in Basel III (see [9]) At date 119905 such entitiesrsquo budgetconstraint can be expressed as

119901119860

119905(1198601015840

119905minus 1198601015840

119905minus1) + (1 + 119903

B) B1015840

119905minus1+ 1199091015840

119905= 119875(119860

1015840

119905minus1)119905minus1+ B1015840

119905

(16)

where 1199091015840119905is secondary securitization at date 119905 (1 + 119903B)B1015840

119905minus1

is debt repayment and B1015840119905is new interbank sponsoring The

HQEsrsquo balance sheet constraint

119901119860

1199051198601015840

119905minus1+ 1198611015840

119905= B1015840

119905+ 1198701015840

119905 (17)

is the same as in the case for an LQE but the ratios of thesevariables will differ from those of the LQEs with much lowerrisk (compare with (4)) In this regard assets held by HQEsare less risky long-term loans with fixed rates (compare with[3]) Next the HQEsrsquo profit may be expressed as

Π1015840

119905= 119903119860

119905119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905 (18)

where 119903119860 119903119861 and 119903B represent the asset rate returns on

marketable securities and borrowing rate in period 119905 respec-tively Notice that the prepayment cost is zero in the case forHQEs (see (10)) Thus the value of HQAs is represented by

119901119860

1199051198601015840

119905minus1=Π1015840

119905minus 1199031198611198611015840

119905+ 119903

BB1015840119905

119903119860 (19)

We provide an appropriate HQE cash flow constraint inthe following result

Lemma 5 (HQE cash flow constraint) Suppose that the creditconstraint (3) as well as (16) to (19) holds In this case theHQEsrsquo cash flow constraint is given by

Π1015840

119905ge 119903119860119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119901119860

119905+11198601015840

119905+ B1015840

119905 (20)

23 Market Equilibrium In equilibrium B1015840119905minus1

and B1015840119905are

negative reflecting the fact that HQEs lend the LQE For ourpurposes market equilibrium is defined as follows

Definition 6 (market equilibrium) Market equilibrium is asequence of asset prices and allocations debt and securitiza-tion by the LQE and HQEs given by

119901119860

119905 119860119905 1198601015840

119905 B119905 B1015840

119905 119909119905 1199091015840

119905 (21)

such that each LQE chooses (119860119905 B119905 119909119905) to maximize the

expected discounted utilities of the LQE andHQEs subject tothe securitization function sponsoring constraint and flow-of-funds constraint given by (2) (3) and (11) respectively Onthe other hand each HQE chooses (1198601015840

119905 B1015840119905 1199091015840

119905) to maximize

the above expected discounted utilities subject to the securi-tization function (15) and budget constraint (16) Also in thecase of the HQE we have that the markets for assets CDOsand debt clear

231 LQE at Equilibrium In the sequel we assume that theasset price bubble does not burst during securitization Inthis case it turns out that there is a locally unique perfect-foresight equilibrium path starting from initial values 119860

119905minus1

and B119905minus1

in the neighborhood of the steady state In thisstate the LQErsquos marketable output 120583119860lowast is just enoughto cover the interest on their debt 119903BBlowast Equivalently therequired screening costs per asset unit 119906lowast equal the LQErsquossecuritization of marketable output 120583 As a result entitiesneither expand nor shrink To further characterize entityequilibrium we provide the following Kiyotaki-Moore-typeresult (see [14] for further discussion)

Theorem 7 (LQE behavior at steady state) Assume that theasset bubble does not burst during the securitization process Inthe neighborhood of the steady state the LQE prefers to borrowup to the maximum and invest in assets consuming no morethan its current output of non-marketable CDOs In this casethere is a unique steady-state (119901119860lowast 119860lowast Blowast) with the associatedtransaction cost 119906lowast being given by

119906lowast=

119903B

1 + 119903B119901119860lowast

=1

1 + 119903B1198751015840[1

119899(119860 minus 119860

lowast)] = 120583 (22)

Blowast=120583

119903B119860lowast (23)

Proof The result follows by considering the LQErsquos marginalunit of marketable CDOs at date 119905 This entity can invest in1119906119905assets which yields 119888119906

119905non-marketable CDOs and 119886119906

119905

marketable CDOs at date 119905 + 1 The former are consumedwhile the latter are reinvested This in turn yields

119886

119906119905

119888

119906119905+1

(24)

non-marketable CDOs and119886

119906119905

119886

119906119905+1

(25)

marketable CDOs at date 119905 + 2 and so on


Next we appeal to the principle of unimprovability whichstates that we need to only consider single deviations at date 119905to show that this investment strategy is optimalThere are twoalternatives open to the LQE at date 119905 Either it can save themarginal unitmdashequivalently reduce its current borrowingby onemdashand use the return 119877 to commence a strategy ofmaximum levered investment from date 119905 + 1 onward orthe LQE can simply consume the marginal unit Its choiceis equivalent to choosing one of the following consumptionpaths

Invest 0 119888119906119905

119886

119906119905

119888

119906119905+1

119886

119906119905

119886

119906119905+1

119888

119906119905+2

(26)

Save 0 0 119903B 119888119906119905

119903B 119886

119906119905

119888

119906119905+2

(27)

Consumption 1 0 0 0 (28)

at dates 119905 119905 + 1 119905 + 2 119905 + 3 respectivelyTo complete the proof of the result we need to confirm

that given the LQErsquos discount factor 120573 consumption path(26) offers a strictly higher utility than (27) or (28) in theneighborhood of the steady-state We shall be in a positionto show this once we have found the steady-state value of 119906

119905

in (34) belowSince the optimal 119886

119905and 119887119905are linear in 119886

119905minus1and 119887119905minus1

wecan aggregate across entities to find the equations of motionof the aggregate asset holding and borrowing 119860

119905and B

119905

respectively of entities may be given by

119860119905=1

119906119905

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1] (29)

B119905=

1

1 + 119903B119901119860

119905+1119860119905 (30)

Next we consider market clearing Since all HQEshave identical securitization functions their aggregate assetdemand equals 1198601015840

119905times their population 119899 The sum of the

aggregate demand for assets by the LQE and HQEs is equalto the total supply given by

119860 = 119860119905+ 1198991198601015840

119905 (31)

In this case from (35) we obtain the asset market (clearing)equilibrium condition

119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B= 119906 (119860

119905)

where 119906 (119860) equiv 1

1 + 119903B1198751015840[1

119899(119860 minus 119860)]

(32)

Next it is useful to look at the steady-state equilibriumFrom (29) (30) and (32) it is easily shown that there is aunique steady-state (119901lowast 119860lowast Blowast) with associated steady-statetransaction cost 119906lowast where (22) and (23) hold

Theorem 7 postulates that at each date 119905 the LQErsquosoptimal choice of (119860

119905 B119905 119909119905) satisfies 119909

119905= ]119860119905minus1

in (11) andthe borrowing constraint (3) is binding so that

B119905=

119901119860

119905+1

1 + 119903B119860119905

119860119905=

1

119901119860

119905minus 1 (1 + 119903B) 119901

119860

119905+1

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1]

(33)

Here the term (120583 + 119901119860

119905)119860119905minus1

minus (1 + 119903B)B119905minus1

is the LQErsquos nettworth at the beginning of date 119905This corresponds to the valueof its marketable CDOs and assets held from the previousperiod nett of debt repayment In effect (33) says that the LQEuses all its nett worth to finance the difference between theasset price 119901119860

119905 and the amount the entity can borrow against

each asset unit 119901119860119905+11 + 119903

B This difference is given by

119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B(34)

and can be thought of as the screening costs required topurchase an asset unit The equations of motion of the aggre-gate asset holding and borrowing 119860

119905and B

119905 respectively of

entities may be given by (29) and (30) Notice from (29) thatif for example present and future asset prices 119901119860

119905and 119901119860

119905+1

were to rise then the LQErsquos asset demand at date 119905would alsorisemdashprovided that leverage is sufficient that debt repayments(1 + 119903

B)B119905minus1

exceed current output 120583119860119905minus1

which holds inequilibrium The usual notion that a higher asset price 119901119860

119905

reduces the LQErsquos demand is more than offset by the factsthat they can borrow more when 119901119860

119905+1is higher and their

nett worth increases as 119901119860119905rises Even though the required

screening costs 119906119905 per asset unit rises proportionately with

119901119860

119905and 119901119860

119905+1 the LQErsquos nett worth is increasing more than

proportionately with 119901119860119905because of the leverage effect of the

outstanding debt

232 HQEs at Equilibrium Next we examine the HQEsrsquobehavior at equilibrium Such entities are not credit con-strained and so their asset demand is determined at the pointat which the present value of the marginal product of assetsis equal to the transaction fee associated with assets (refer toBasel III in [1]) In this case we have that

119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B=

1

1 + 119903B1198751015840(1198601015840

119905) (35)

In themodel119906119905is both theHQEsrsquo opportunity cost of holding

an asset unit and the required screening costs per unit ofassets held by the LQE

233 LQE and HQE Market Clearing In this subsection weconsider market clearing that refers to either a simplifyingassumption made that markets always go to where the assetssupplied equal the assets demanded or the process of gettingthere via price adjustment Amarket clearing price is the price


of goods or a service at which assets supplied are equal toassets demanded also called the equilibrium price Anothermarket clearing price may be a price below equilibrium priceto stimulate demand

Since all HQEs have identical securitization functions(see Basel III and revision to securitization in [9]) theiraggregate asset demand equals 1198601015840

119905times their population 119899

The sum of the aggregate demand for assets by the LQE andHQEs is equal to the total supply given by (31) In this casefrom (35) we obtain the asset market (clearing) equilibriumcondition (32) The function 119906(sdot) is increasing This arisesfrom the fact that if the LQErsquos asset demand119860

119905 goes up then

in order for the asset market to clear the HQEsrsquo demand hasto be stymied by a rise in the transaction fee 119906

119905 Given that the

HQEs have linear preferences and are not credit constrainedin equilibrium they must be indifferent about any path ofconsumption and debt (or credit) In this case the interestrate equals their rate of time preference so that

119903B=1

1205731015840minus 1 (36)

Moreover given (32) the CDO markets and credit are inequilibrium

We restrict attention to perfect-foresight equilibria inwhich without unanticipated shocks the expectations offuture variables realize themselves For a given level of theLQErsquos asset holding and debt at the previous date 119860

119905minus1and

B119905minus1

an equilibrium from date 119905 onward is characterized bythe path of asset price LQE asset holding and interbank bor-rowings given by

(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)

satisfying (29) (30) and (32) at dates 119905 119905 + 1 119905 + 2

24 LQE and HQE Equilibrium Summary Figure 3 displaysthe main features of market equilibrium for the LQE andHQEs

The horizontal axis represents LQA and HQA demandfrom the left-hand side and right-hand-side respectively Wenote that the total asset supply is denoted by 119860 The verticalaxis represents the marginal products of assets for LQE andHQEs given by 120583 + ] and 1198751015840(1198601015840119899) respectively The HQEsrsquomarginal product decreases with asset use If there are nocredit limits then 1198640 would be the best allocation for wherethe LQE and HQEmarginal products are in equilibriumTheasset price would then be 1199010 = (120583 + ])(119903B)minus1 On the otherhand when credit limits exist then the equilibrium is at point119864lowast where the marginal product of the LQE is greater than

that of the HQE In this case we have

120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)

This means that the LQErsquos asset use is not enoughThe outputof CDOs per period in equilibrium is represented by thelight gray area under the thick line whereas the gray trianglerepresents the CDO loss per period In this case CDO outputincreases relative to LQA holdings If 119860

119905increases then the

CDO output will also increase in period 119905 + 1

LQE HQEs0 Alowast At A0 A

Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)

Figure 3 LQE and HQE market equilibrium

3 Asset Securitization Shocks

In this section we describe the effect of unexpected negativeshocks on LQA price and input CDO price in a Basel IIIcontext and output and profit (see eg [19]) In this regardtwo kinds of multiplier processes are considered The firstis the within-period or static multiplier process Here theshock such as a ratings downgrade reduces the nett worthof the constrained LQE and compels it to reduce its assetdemand In this case by keeping the future constant thetransaction fees decrease to clear the market and the assetprice drops by the same amount In turn this lowers the valueof the LQErsquos existing assets and reduces their nett worth evenmore Since the future is not constant this multiplier missesthe intuition offered by the more realistic intertemporal ordynamic multiplier In this case the decrease in asset pricesresults from the cumulative decrease in present and futureopportunity costs stemming from the persistent reductionsin the constrained LQErsquos nett worth and asset demand whichare in turn exacerbated by a decrease in asset price and nettworth in period 119905

31 Dynamic Multiplier Response to Temporary Shock Inorder to understand the effect of unexpected inter-temporalshocks on the economy suppose at date 119905minus1 that it is in steadystate with

119860lowast= 119860119905minus1 B

lowast= B119905minus1 (39)

311 Dynamic Multiplier Shock Equilibrium Path We intro-duce an unexpected inter-temporal shock where the CDOoutput of the LQE and HQEs at date 119905 are 1 minus Σ timestheir expected levels In order for our model to resonatewith the 2007ndash2009 financial crisis we take Σ to be positiveEventually the LQE and HQEsrsquo securitization technologiesbetween dates 119905 and 119905+1 (and thereafter) return to (2) and (15)


respectively Combining the market-clearing condition (32)with LQA demand under a temporary shock and borrowingconstraint given by (29) and (30) respectively we obtain

119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)

The formulae (40) and (41) imply that at each date the LQEcan hold assets up to the level 119860 at which the required costof funds 119906(119860)119860 is covered by its nett worth Notice that in(41) at each date 119905 + 119904 119904 ge 1 the LQErsquos nett worth is just itsambient output of marketable CDOs 120583119860

119905+119904minus1 In this case

from the borrowing constraint at date 119905 + 119904 minus 1 the value ofthe LQErsquos assets at date 119905 + 119904 is exactly offset by the amountof debt outstanding From (40) subsequent to the shock wesee that the LQErsquos nett worth at date 119905 is more than only theircurrent output given by

(1 minus Σ) 120583119860lowast (42)

because 119901119860119905changes in response to the shock and unexpected

capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)

result in their asset holdings In this case the asset value heldfrom date 119905 minus 1 is now 119901119860

119905119860lowast while the debt repayment is

(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)

To find closed-form expressions for the new equilibriumpath we take Σ to be small and linearized around the steadystate In the sequel we let the proportional changes in119860

119905 119901119860119905

and Π119905relative to their steady-state values 119860lowast 119901119860lowast and Πlowast

respectively be given by

119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast

Π119905=Π119905minus Πlowast

Πlowast

(45)

respectively For our purpose assume that steady-state profitΠlowast

119905 represents profit when the asset value and borrowings are

in steady state (compare with [19]) Thus steady-state profitfor the LQE and HQEs are represented by

Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)

respectively Then by using the steady state transaction feeand (22) we have from (40) and (41) that

(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)

where 120578 gt 0 denotes the elasticity of the residual asset supplyto the LQE with respect to the transaction fee at the steadystate Here we have

1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast

119860 minus 119860lowast

(49)

The right-hand side of (47) divides the change in the LQErsquosnett worth at date 119905 into two components the direct effectof the securitization shock Σ and the indirect effect of thecapital gain arising from the unexpected rise in price 119901119860

119905 In

order to compute (47) from (22) (40) and (45) we have thatthe RHS of (47) is given by

1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)

Also from (22) (40) and (45) we have that the LHS of (47)is given by

(1 +1

120578)119860119905

= (119860119905minus 119860lowast

119860lowast) + (minus

119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast

119860 minus 119860lowast)(

119860119905minus 119860lowast

119860lowast)

(51)

Crucially the impact of 119901119860119905is scaled up by the factor (1 +

119903B)(119903

B) because of leverage Furthermore the factor 1 + 1120578

on the left-hand sides of (47) and (48) reflects the fact thatas LQA demand rises the transaction fee must rise for themarket to clear and this in turn partially chokes off theincrease in the LQErsquos demandThekey point to note from (48)is that except for the limit case of a perfectly inelastic supply120578 = 0 the effect of a shock persists into the futureThe reasonis that the LQErsquos ability to invest at each date 119905+119904 is determinedby how much screening costs they can afford from their nettworth at that date which in turn is historically determined bytheir level of securitization at the previous date 119905 + 119904 minus 1

312 Dynamic Multiplier Asset Price and Input CDO Priceand Output and Profit We will determine the size of theinitial change in the LQErsquos asset holdings 119860

119905 which from

(47) can be jointly determined with the change in assetprice 119901119860

119905 Also we would like to compute the proportional

change in CDO output and profit denoted by 119862119905+1

and Π119905

respectively (see eg [19])

Theorem 8 (dynamic multiplier shocks to asset price andinput CDO price and output and profit) Assume that theasset bubble does not burst during the securitization processand that 119901119860

119905le 119901119905 for all 119905 in (5) In this case one has that the


proportional change in asset price and input CDO price andoutput and profit subject to a negative shock is given by

119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861

119905minus 119903BBlowast119905

minus 1

(56)

respectively

Proof Since there are no bursting bubbles (32) intimatesthat the asset price 119901119860

119905 is the discounted sum of future

opportunity costs given by

119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)

Linearizing around the steady state and then substitutingfrom (48) given by

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)

We have to verify that

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)

is standard for infinite series The dynamic multiplier

[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)

in (59) captures the effects of persistence in entitiesrsquo referenceasset portfolio holdings and has a dramatic effect on the sizesof119901119860119905and119860

119905 In order to find119901119860

119905and119860

119905in terms of the size of

the shock Σ we utilize (47) and (59) The calculations aboveverify that (52) and (53) as well as (54) hold

Next we prove that (55) holds As we saw in Figure 3aggregate CDO outputmdashthe combined harvest of the LQEand HQEsmdashis positively correlated to the LQA holdingssince such entitiesrsquomarginal product is higher than theHQEsrsquoSuppose that the proportional change in aggregate output119862119905+119904 is given (compare with 119901119860

119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)

In this case we can verify that at each date 119905 + 119904 the propor-tional change in aggregate output 119862

119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)

The RHS of (63) yields

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)

In order to verify (63) we have to show that

119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)

This of course is true since

119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)

Theproportional change in profit Π119905 given by (56) is a direct

consequence of its definition

The proportional changes in CDO output 119862 and profitΠ given by (55) and (56) respectively have importantconnections with the 2007ndash2009 financial crisis and Basel IIIFor mortgage loans this relationship stems from the termsinvolving the asset and prepayment rates refinancing andhouse equity

At date 119905 (52) tells us that in percentage terms the effecton the asset price is of the same order of magnitude as thetemporary securitization shock As a result the effect of theshock on the LQA holdings at date 119905 is large In this case themultiplier in (53) exceeds unity and can do so by a sizeablemargin thanks to the factor (1 + 119903B)(119903B)minus1 In terms of (47)the indirect effect of 119901119860

119905 scaled up by the leverage factor (1 +

119903B)(119903

B)minus1 is easily enough to ensure that the overall effect on

119860119905 is more than one-for-one


32 Static Multiplier Response to Temporary Shocks At thebeginning of this section we made a distinction betweenstatic and dynamic multipliers Imagine hypothetically thatthere was no dynamic multiplier In this case suppose 119901119860

119905+1

were artificially pegged at the steady-state level119901119860lowast Equation(47) would remain unchanged However the right-hand sideof (59) would contain only the first term of the summationmdashthe term relating to the change in transaction fee at date 119905mdashso that the multiplier (61) would disappear Combining themodified equation we have

119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)

The following result follows from the above

Corollary 9 (static multiplier shocks to asset price andinput) For the static multiplier suppose that the hypothesis ofTheorem 8 holds Then one has that

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)

Proof Weprove the result by considering (47) and (59) wherethe changes in the asset price and the LQA holdings can besolely traced to the static multiplier

Subtracting (68) from (52) we find that the additionalmovement in asset price attributable to the dynamic mul-tiplier is (1 + 119903B)minus1 times the movement due to the staticmultiplier And a comparison of (53) with (69) shows that thedynamic multiplier has a similarly large proportional effecton LQA holdings The term

120583 + ] minus (1 + 119903B) 120583

120583 + ](70)

reflects the difference between LQA (equal to 120583 + ]) andHQA securitization (equal to (1 + 119903B)120583 in the steady state)The ratio (120583 + ])119860lowast119862lowastminus1 is the share of the LQErsquos output Ifaggregate securitization was measured by 119862

119905+119904119860minus1 it would

be persistently above its steady-state level even thoughthere are no positive securitization shocks after date 119905 Theexplanation lies in a composition effect In this regard thereis a persistent change in asset usage between the LQE andHQEs which is reflected in increased aggregate output

4 Illustrative Examples of Asset Securitization

In this section we present examples of asset securitizationFirstly we consider a LQA securitization example involvingsubprime mortgages Next we illustrate LQE and HQEequilibrium from Section 2 as well as the effects of shocks toasset and CDO prices as discussed in Section 3

41 Example of Subprime Mortgage Securitization In thissubsection we provide a specific example of LQA securi-tization involving subprime mortgages (see eg [4 15])For such securitization we bring into play the main resultscontained in [16] Subprime mortgages are usually adjustableratemortgages (ARMs) where high step-up rates are chargedin period 119905+1 after low teaser rates apply in period 119905 Secondlythis higher step-up rate causes an incentive to refinance inperiod 119905+1 Refinancing is subject to the fluctuation in houseprices When house prices rise the entity is more likely torefinance This means that investors could receive furtherassets with lower interest rates as house prices increaseThirdly a high prepayment penalty is charged to dissuadeinvestors from refinancing (see [4] for more details)

In subprime mortgage context the paper [16] providesa relationship between the mortgage rate 119903119872 loan-to-valueratio (LTVR) 119871 and prepayment cost 119888119901 by means of thesimultaneous equations model

119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)

Investors typically have a choice of 119903119872 and 119871 while thechoice of 119888119901 triggers an adjustment to the mortgage rate119903119872 Thus 119871 and 119888

119901 are endogenous variables in the 119903119872-equation There is no reason to believe that 119871 and 119888119901 aresimultaneously determined Therefore 119888119901 does not appear inthe 119871-equation and 119871 does not make an appearance in the 119888119901-equation From [16]119883 comprises explanatory variables suchas asset characteristics (owner occupation asset purposeand documentation requirements) investor characteristics(income and Fair Isaac Corporation (FICO) score) anddistribution channel (broker origination) The last term ineach equation 119885119903

119872

119885119871 or 119885119888

119901

comprises the instrumentsexcluded from either of the other equations Reference [16]points out that the model is a simplification with otherterms such as type of interest rate the term to maturity anddistribution channel possibly also being endogenous

Corollary 10 (dynamic multiplier shocks to profit for sub-prime mortgages) Suppose that the hypothesis of Theorem 8holds Then the relative change in profit may be expressed interms of 119903119872 119888119901 and 119871 as

Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)


respectively Here one has in (72) (73) and (74) that

119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively

The most important contribution of the aforementionedresult is that it demonstrates how the proportional changein profit subsequent to a negative shock is influenced byquintessential low-quality asset features such as asset andprepayment rates refinancing and house equity given by 119903119872119888119901 119903119891 and 119871 respectively The default rate is also implicitlyembedded in formulas (72) to (74) in Corollary 10 In thisregard by consideration of simultaneity in the choice of 119903119872and 119888119901 it is possible to address the issue of possible bias inestimates of the effect of 119888119901 on 119903119872

42 Numerical Examples of Asset Securitization In this sub-section we provide numerical examples to illustrate LQEand HQE equlibrium as described in Section 2 as well as theeffects of shocks on asset and CDO prices as in Section 3Initial asset securitization parameter choices are given inTable 2

The financial crisis has exposed the limits of liabilitymanagement and the proposed regulation will make theretreat from liability management permanent To a muchgreater extent than at any time since the 1970s banks willbe forced back towards ldquoasset managementrdquo in other wordstowards a business model in which balance sheet size isdetermined from the liabilities side of the balance sheet bythe amount of funding which the bank can raise and inwhich asset totals have to be adjusted to meet the availableliabilities This amounts to a ldquomacroprudentialrdquo policymdashthatis a policy designed to prevent credit creation from spirallingout of control as it did in the run-up to the recent crisis asexpressed in Basel III on the the macroprudential overlay(see eg [21 22])

421 Numerical Example LQE and HQE Equilibrium Sup-pose that the LQErsquos and HQEsrsquo deposits borrowings mar-ketable securities and capital are equal In this case noticethat the LQErsquos and HQEsrsquo asset holdings A and 1198601015840 are a

Table 2 Asset securitization parameter choices

Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000

proportion120572 and 1minus120572 of the aggregate assets119860 respectivelyThus we have that

119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)

We compute the LQErsquos debt obligation output in period t + 1by considering the securitization function (2) Therefore theCDO output can be computed by

119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)

Next the upper bound of the LQErsquos retention rate should beless that the discount factor 120573 thus

120573(0002

0002 + 02) = 00099099 (78)

The value of LQAs in period t is computed by using (6)Thuswe have

119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)

The asset price in period t is therefore

119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)


The LQErsquos profit is computed by considering the cash flowconstraint (10)

Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903119863119863119905minus 119903119861119861119905

Π119905= (0061 + 003 times 02) times 4000 + 0205 times 5000 minus 0205

times 1200 minus 02 times 4800 = 87

(81)

Furthermore the LQErsquos profit is subject to the constraint (15)thus

Π119905= (0061 + 003 times 02) 4000 + 0205 times 5000 minus 0205

times 1200 minus 0113 times 03 times 720000 + 4800 = minus18561

(82)

We compute the LQErsquos additional consumption 119909119905minus]119860119905minus1

byconsidering the flow of funds constraint given by

0002 times 03 times 460000 + 8000 minus (1 + 02) times 2600

minus 0011904761 (03 times 720000 minus 03 times 460000)

= 42274286

(83)

Next we concentrate onHQE constraints In this case HQEsrsquosecondary securitization at date 119905 is computed by using thebudget constraint (16) thus

1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000

minus (1 minus 03) times 460000 minus (1 + 02) times 2600

= minus4631999954

(84)

Thus 119909119905= 5410364210 Next we consider the HQErsquos profit

(18) at face value to compute profit at date 119905 thus

Π1015840

119905= 0061 times 4000 + 0205 times 5000 minus 0205 times 1500

minus 02 times 4800 = 1235

(85)

In this regard the value of assets can be computed as

119901119860

1199051198601015840

119905minus1=1235 minus 0205 times 5000 + 0205 times 1200 + 02 times 4800

0061

=49918

(86)

The HQEsrsquo cash flow constraint (20) is given by

Π1015840

119905ge 0061 times 4000 + 0205 times 5000 minus 0205 times 1200 minus 0113

times (1 minus 03) times 720000 + 4800 = minus51007

(87)

The screening cost an LQE has to pay to purchase an assetunit is financed by the HQErsquos nett worth This screening costis represented by

119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)

Themotion of the aggregate asset holding and borrowing119860119905

and 119861119905 of the entity may be computed as

119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)

The sum of the aggregate asset demand from asset originatorsby the LQE and HQEsrsquo is computed by

119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)

The steady-state asset price and borrowings for the LQE are

119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)

Notice that the required screening costs per asset unit equalsthe LQErsquos securitization of marketable output 119906lowast = 120583 =

0001 Also we have 119860119905minus1

= 119860lowast and 119861

119905minus1= 119861lowast

422 Numerical Example Negative Shocks to LQAs andTheirCDOs LQA demand and borrowings under a temporaryshock at date 119905 are computed by

119860119905=

1

minus0082261905[(0002 minus 0002 times 0002 + 0011904761)

times03 times 460000 minus (1 + 02) times 2600]

= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)

respectively In this regard we compute the cost of funds inperiod 119905 as

119906 (119860119905) 119860119905= (0002 minus 0002 times 0002 + 0011904761 minus 0022)

times 03 times 460000 = minus111769

(93)

Also we see that the LQErsquos nett worth at date 119905 is more thantheir current output just after the shock thus

(1 minus Σ) 119906119860lowast= (1 minus 0002) 0002 times 03 times 460000 = 275448

(94)

With unexpected capital gains

(119901119860

119905+ 119901119860lowast)119860lowast= (0011904761 + 0022) times 03 times 460000

= 4679

(95)


While the debt repayment is given by

(1 + 119903119861) 119861lowast= 119901119860lowast119860lowast= (1 + 02) 2600 = 3120 (96)

Proportional change in 119860119905and 119901119860

119905can be computed as

119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)

The steady-state profit for LQE is

Πlowast

119905= (0061 + 003 times 02) 0022 times 03 times 460000 + 0205

times 5000 minus 0205 times 1200 minus 02 times 2600 = 462412

(98)

and steady-state profit for HQEs is

Π1015840lowast

119905= 0061 times 0022 times (1 minus 03) times 460000 + 0205 times 5000

minus 0205 times 1200 minus 02 times 2600 = 691124

(99)

Thus the proportional changes in Π119905and Π1015840

119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)

Elasticity of the residual asset supply to the LQE with respectto the monitoring cost at the steady-state at date 119905 is

120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)

The proportional changes for 119901119860119905and 119860

119905in terms of the size

of the shock Σ are computed by

119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)

respectively By considering (47) we see from (63) and (71)that 119901119860

119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901

119860lowast= minus0002

(103)

Table 3 Computed asset securitization parameters

Parameter Value119862119905+1

43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905

minus0082261905Aggregate B

119905$20 340

119901119860lowast 0012119860119905under shock $14 60815893

119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905

$minus21118120578 0126718931119860119905in terms of shock 001

119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007

Aggregate 119860119905

14 60144865119860 720 000Blowast $2 600B119905under shock $20 340

(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860

119905in terms of shock minus0015782961

119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869

respectively The proportional change in aggregate output119862119905+1

represented by (54) is given by

119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)

We provide a summary of computed asset securitizationparameters in Table 3

An analysis of the computed shock parameters in Table 3shows that the aggregate output 119862

119905+1 increases to $43632

The value of LQAs 119901119860119905119860119905minus1

increases to $5 000 LQAdemand119860

119905 has declined to $14 60815893 while the borrow-

ings 119861119905have increased to $20 340 This implies that HQAs


were more sensitive to changes in market conditions andthat asset transformation may have been a greater priorityThe proportional negative change in profit for the LQEsubsequent to a temporary shock is higher than that of theHQEs In addition the steady-state asset price 119901119860

lowast

and theborrowings 119861lowast for the LQE increased to $0012 and $2 600respectively In summary this example shows that whenparameter choices are altered and the size of shock increasedthe LQE suffers bigger losses on their asset holdings and therate of borrowing to refinance increases This explains whymost people could not pay back their loans during the 2007ndash2009 financial crisis

423 Numerical Example Financial Intuition A finan-cial motivation for the numerical example discussed inSection 421 is outlined in the current paragraph Accordingto Basel III capital and liquidity regulation LQEs and HQEshave a couple of obligations in terms of transaction costs(refer to (22))The first is towards the originator for acquiringthe assets while the second is for using the assets forsecuritization into CDOs a type of user cost In Section 421we compute LQE and HQE equilibrium variables to addcredence to the aforementioned statements

The financial intuition behind the numerical examplediscussed in Section 422 is given below Investors soughtSAPs because they met certain prudential standards such asrestrictions to purchase only investment grade debt (see theBasel documents [8 9] for more details about securitization)Investors could meet relative ldquosafety requirementsrdquo sincesecuritization is essentially a form of bankruptcy-remotesecured lending (as assets are legally isolated in a SPE) withcredit enhancement (or government guarantees for RABSin some jurisdictions) that often resulted in the securitiesbeing highly rated (eg AAA AA or A) However as notedbefore the reliance on credit ratings sowed the seeds of laterdifficulties as many investors relied too heavily on creditratings and effectively ldquooutsourcedrdquo their due diligence to thecredit rating agencies (see the BCBS papers [8 9] for furtherdiscussion) It is also worth noting that investor requirementsfor HQLAs created liquidity concerns during the crisis whenpolicies or requirements forced investors to sell assets afterratings downgrades (see [5] for more details) This tested theliquidity assumptions about securitized products whenmanyinvestors were simultaneously forced to sell driving pricesever lower in a down market (see eg [5 21 22]) Investorscould avoid exceeding concentration limits both regulatoryrestrictions and internal limits on exposures to a single nameby purchasing SAPs Further investors could manage riskin the entire portfolio by holding securitized assets that hada low correlation with other components of the investorsrsquoportfolios such as equities and corporate bonds (see [3]for further discussion) Also investors could meet theirinternal portfolio diversification requirements by increasingthe types of assets as well as the geographical location ofthe assetsrsquo origination Synthetic securitizations also allowedinvestors to increase the variety and volume of instrumentsthey could acquire without funding the credit exposure (see[8 9] for further discussion on asset securitization) However

as the crisis unfolded investors learned that their expecteddiversification benefits were partially false and that in somecases the underlying assets were highly correlated SAPshelped investors achieve higher yield thresholds since therisk-adjusted return on ABS was typically higher relative toa similarly rated nonsecuritization investment In the periodbefore the crisis investors were seeking higher yields at thesame time spreads in the broader fixed income markets werenarrowing (see the Basel documents [8 9])

5 Conclusions about Basel III andAsset Securitization

The main conclusions about LQA and HQA securitizationan LQE at equilibrium negative shocks to asset securitizationfor dynamic and static multipliers and illustrative examplesinvolving asset securitization are summarized in this section

51 Conclusions about LQA Securitization We answeredQuestion 1 in Section 21 by characterizing the securitizationof LQAs by an LQE In this case the cash flow constraintis given by (10) The discussion on LQE cash flow hasconnections with Basel III capital and liquidity regulation(see [5] for further discussion) In this regard it is clear thatinvestors may be motivated to purchase securities issued byLQEs to avert Basel III regulatory limits such as those relatingto LQAs In the case of synthetic transactions investors mayfind it beneficial that they would not have to fund creditexposures at the outset (see eg the Joint Forum paper [1])In cases where parties to entities possessed a comprehensiveunderstanding of the associated risks and possible structuralbehaviors of these LQEs under various scenarios they haveeffectively engaged in and reaped benefits from their LQEactivities (see [1 3 6] for more details) However it isunclear that LQAs sold into the LQE can be attributed to theexistence of these structures that were simply the legal forminwhich such assets were held to issue bonds backed by themNonetheless it is important for Basel III to address why someof the recent failures of LQE usage occurred (see eg theBasel publications [1 6])

52 Conclusions about HQA Securitization Next we an-sweredQuestion 2 in Section 22 by describing the securitiza-tion of HQAs by HQEs In this case the HQE cash flow con-straint is expressed as in (20) In Section 22 the discussionon HQEs has relationships with Basel III capital and liquidityregulation (see [5] for more details) For example a bankmight view the Basel III risk-based capital charge appliedto HQAs as being excessive and therefore might engage insecuritization activities of these receivables to benefit from aslightly lower capital charge resulting from other aspects ofthe risk-based capital rules (see eg [1]) In the bankrsquos viewthis slightly lower Basel III charge might be more rationalin terms of the true risks and capital needed for corporatelending activities (see eg the Basel documents [1 6])53 Conclusions about an LQE at Steady State Theorem 7in Section 23 provides the answer to Question 3 Here acharacterization of the behaviour of an LQE in equilibrium


is given According to the Basel III securitization frameworkLQEs and HQEs have at least two obligations in terms oftransaction costs (refer to (22)) The first is towards theoriginator for acquiring the assets while the second is forusing the assets for securitization into CDOs a type ofuser cost The latter fee involves for instance an externalcredit rating agency (see Section 24 for more details) Alsobecause of information asymmetry and regulatory dysfunc-tion CDOs open up opportunities for arbitrage a point thathas been considered in Basel III for balanced informationdistribution (see eg [9]) In this regard sophisticated CDOentities often circumvent regulatory constraintsThis type ofarbitrage is accompanied by high costs with originators andother financial intermediaries earning huge transaction feesand eroding value for entities and investors (see eg the Baselpapers [9 10])

54 Conclusions about Negative Shocks to Asset Securitization(Dynamic Multiplier) In the presence of a dynamic multi-plier in Theorem 8 from Section 31 we quantify changes toasset price and holdings CDO output and profit subsequentto negative shocks Also we quantify changes to profitin terms of asset and prepayment rates as well as equitysubsequent to negative shocks (refer to Question 4) Weconclude that the proportional change in asset price 119901119860

119905

and input 119860119905 as well as CDO price 119901119862

119905 is negative in

response to a negative shock like a ratings downgrade BaselIII endeavours to make provision for these negative changesin a countercyclical way (see eg [1 6 23]) The sameobservation can bemade about CDO output119862

119905+1 and profit

Π119905 where a ratings downgrade has a negative impact (see eg

Basel IIIrsquos (see eg the BCBS documents [1 6])

55 Conclusions about Temporary Shocks to Asset Securitiza-tion (Static Multiplier) When a static multiplier is presentin Corollary 9 in Section 32 we determine changes to assetprice 119901119860

119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1

given by (68) and (69)respectively subsequent to a negative shock such as a ratingsdowngrade (refer to Question 5) As before Basel III capitalregulation attempts to negate the deleterious effects of ratingsdowngrades on price and input in a countercyclical manner(see eg [23])

56 Conclusions about Example of Subprime Mortgage Secu-ritization Question 6 is answered in Corollary 10 fromSection 41 Here an example of LQA securitization forsubprime mortgages is provided Of course the subprimemortgage crisis preceded the 2007ndash2009 financial crisis andas such its connection with Basel III is very important (seeeg [4 15]) The BCBS reports in Basel III documents thatrisk weights for low-rated senior subprime securitizationexposures were too high Many senior securitization expo-sures were downgraded during the crisis (compare with[3]) While some of these exposures resulted in total loss toinvestors most of these exposures have resulted in recoveryof some principal (see the Basel documents [8 9])

57 Conclusions about Example of Numerical Examples ofAsset Securitization Section 42 produces a numerical exam-ple to illustrate the impact of negative shocks on asset secu-ritization by the LQE and HQEs In particular the exampleillustrates the impact of negative shocks on assets and CDOsIt shows that when parameter choices are altered and thesize of the shock increased the LQE suffers significant lossesto their asset holdings and the rate of borrowing increases(compare with Question 7) Also we illustrated that assetprice is most significantly affected by unexpected negativeshocks from asset rates while for CDO price shocks tospeculative asset funding investor risk characteristics andprepayment rate elicit statistically significant responses (com-pare with [3]) Problems from the financial crisis relate to ourmodels for assets and CDOs with respect to the reductionin incentives for banks to monitor entities transaction costsmanipulation of CDO price and structure CDO marketopacity self-regulation systemic risks associated with CDOsand the mispricing of debt (see the BCBS papers [3 8 9])

According to [8 pages 9ndash11] the role of Basel III in thenumerical example from Section 42 can be considered fromtwo perspectives which are the

(i) quantitative perspectivemdashthe amount of HQLAs thatthe banks will have to amass in the next few yearsboth to meet the new requirements and to repayspecial facilities provided by governments and cen-tral banks which is assumed to not be renewedThere is clearly a potential transition problem whichneeds to be considered in determining the timingof implementation of new regulations (see the BCBSpublication [8] for more details)

(ii) structural perspectivemdashthis refers to the ways inwhich the funding and liquidity management notonly of banks but also of their customers will bepermanently affected by the proposed regulations (see[5] for further details) What may appear to be ratherdetailed aspects of the proposed regulations mayin fact have very serious implications for financialstability in the future (see the Basel document [8] forfurther discussion)

58 General Conclusions about Basel III and Asset Secu-ritization In answering Question 8 we note that in theBasel III document [1] the JFWGRAC makes the followingrecommendations about the supervision of the LQE andHQEs discussed in previous subsections

581 LQE and HQE Economic Risks and Purposes Firstlysupervisors should make sure that market participants assessall economic risks and business purposes of LQEs and HQEsduring the lifespan of a transaction distinguishing betweenrisk transfer and transformationWe note that over time thenature of these risks can change Supervisors should ensurethat such assessment is ongoing and that management hassufficient understanding of the risks (see the Basel document[1 3] for more details)


582 LQE and HQE Transaction Complexity Secondly mar-ket participants should be able to assess and risk manage fac-tors that increase transaction complexity such as structuralfeatures of the LQE andHQEs including triggers and the rolesof parties involved (see eg the Basel documents [1 6])

583 LQE andHQEGovernance Processes In the third placebanks and supervisors should ensure the governance processof LQEs and HQEs commensurate with the degree of activeintervention and discretion of the parties participating inthese entities (see [1] for more discussions)

584 LQE and HQE Portfolio Risk Profiles The fourthrecommendation is that banks should continually monitorthe quality of transferred exposures in relation to the riskprofile of LQEsrsquo and HQEsrsquo remaining portfolios and theimpact on its balance sheet components and supervisorsshould where appropriate assess systemic implications of riskdispersion to transferees (see the Basel documents [1 3 6] formore details)

585 LQE and HQE Risk Exposures Fifthly banks shouldhave the capability of aggregating assessing and reporting alltheir LQE and HQE risk exposures in conjunction with allother bank-wide risks (see eg [1 3])

586 LQE and HQE Support In the sixth place if atinception or at any point during LQE and HQE existencethere is a likelihood or evidence of support by the bankincluding noncontractual support then the activities andrisks of those entities should be aggregated with those ofthe bank for both supervisory assessment and internal riskmanagement purposes (see the Basel documents [1 6] formore discussions)

587 LQE and HQE Transaction Standardization The sev-enth recommendation is that supervisors should supportmarket participantsrsquo efforts towards greater standardizationof definitions documentation and disclosure requirementsof LQE and HQE transactions and provide for the commu-nication of any material divergence from these standardsto investors in individual transactions (see [1 3] for moredetails)

588 LQE and HQE Weakness Contagion or ProcyclicalityFinally supervisors should regularly oversee andmonitor theuse of all LQE and HQE activities and assess the implicationsfor regulated banks of the activities of such entities in orderto identify developments that can lead to systemic weaknessand contagion or that can exacerbate procyclicality (see egthe Basel discussion documents [1 6])

59 Future Directions Related to Basel III and Asset Securitiza-tion A topic of future research may be the effects of shockson high-quality liquid assets (HQLA) as in [21] In this regardwe recall that an asset is considered to be a HQLA if it canbe easily and immediately converted into cash at little or no

loss of value Asset liquidity depends on the underlying stressscenario the volume to be monetized and the time-frameconsidered (see [5] for more details) Nevertheless there arecertain assets that are more likely to generate funds withoutincurring large discounts in sale or repurchase agreement(repo) markets due to fire-sales even in times of stress (seeeg [6 21] for more details) While [6] lays out proposedrevisions to the securitization framework it does not includeproposed rules text that would effectuate these changes TheBCBS is seeking industry feedback on some key elementsof the proposed changes and will in the coming monthsconduct a quantitative impact study (QIS) of the proposalsbefore deciding on a definitive way forward (see eg [1])The BCBS will consider all comments received along withthe results of the QIS before determining the appropriatenext steps in the process of moving forward with revisionsto the securitization framework This would mean that thecontents of the current paper would have to be refined quiteconsiderably

Conflict of Interests

The authors of this paper do not have a direct financialrelationship with the commercial entities mentioned in thispaper that might lead to a conflict of interests

References

[1] Basel Committee on Banking Supervision The Joint ForumReport on Special Purpose Entities Bank for InternationalSettlements (BIS) Publications 2009 httpwwwbisorgpubljoint23pdf

[2] Basel Committee on Banking Supervision Basel III A GlobalRegulatory Framework for More Resilient Banks and BankingSystems Bank for International Settlements (BIS) Publications2010 httpwwwbisorgpublbcbs189htm

[3] F Gideon J Mukuddem-Petersen and M A Petersen ldquoMin-imizing banking risk in a Levy process settingrdquo Journal ofAppliedMathematics vol 2007 Article ID 32824 25 pages 2007

[4] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova Science New York NY USA2012

[5] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013

[6] Basel Committee on Banking Supervision ConsultativeDocument Revisions to the Basel Securitisation FrameworkBank for International Settlements (BIS) Publications 2012httpwwwbisorgpublbcbs236htm


[8] Basel Committee on Banking Supervision Enhancements to theBasel II Frame-Work Bank for International Settlements (BIS)Publications 2009 httpwwwbisorgpublbcbs157htm

[9] Basel Committee on Banking SupervisionRevisions to the BaselSecuritization Framework Bank for International Settlements(BIS) Publications 2012 httpwwwbisorgpublbcbs158htm


[10] Basel Committee on Banking Supervision Report on AssetSecuritization Incentives Bank for International Settlements(BIS) Publications 2011 httpwwwbisorgpublbcbs174htm

[11] I Fender andM Scheicher ldquoThe pricing of subprime mortgagerisk in good times and bad evidence from theABXHE indicesrdquoWorking Paper Series No 1056 European Central Bank 2009

[12] Housing Derivatives ABX Subprime Mortgage Index StillMaking New Lows Housing Derivatives Applications andEconomics for the US Housing Market 2008 httphous-ingderivativestypepadcomhousing derivatives200802abx-subprime-mohtml

[13] M Iacoviello ldquoHouse prices borrowing constraints and mon-etary policy in the business cyclerdquo American Economic Reviewvol 95 no 3 pp 739ndash764 2005

[14] N Kiyotaki and J Moore ldquoCredit cyclesrdquo Journal of PoliticalEconomy vol 105 no 2 pp 211ndash248 1997

[15] J Mukuddem-Petersen M A Petersen T Bosch and Bde Waal ldquoSpeculative funding and its impact on subprimemortgage product pricingrdquoApplied Financial Economics vol 21no 19 pp 1397ndash1408 2011

[16] G Elliehausen M E Staten and J Steinbuks ldquoThe effect ofprepayment penalties on the pricing of subprime mortgagesrdquoJournal of Economics and Business vol 60 no 1-2 pp 33ndash462008

[17] R S J Koijen O van Hemert and S van NieuwerburghldquoMortgage timingrdquo Journal of Financial Economics vol 93 no2 pp 292ndash324 2009

[18] G Elliehausen and M Hwang Mortgage Contract Choice inSubprime Mortgage Markets Finance and Economics Discus-sion Series Working Paper No 2010-53 Divisions of Researchamp Statistics and Monetary Affairs Federal Reserve BoardWashington DC USA 2010

[19] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007

[20] Basel Committee on Banking SupervisionRevisions to the BaselII Market Risk Framework Bank for International Settlements(BIS) Publications 2009 httpwwwbisorgpublbcbs158htm

[21] Basel Committee on Banking Supervision Basel III The Liq-uidity Coverage Ratio and Liquidity Risk Monitoring ToolsBank for International Settlements (BIS) Publications 2013httpwwwbisorgpublbcbs238htm

[22] Basel Committee on Banking Supervision Liquidity CoverageRatio Disclosure Standards Bank for International Settlements(BIS) Publications 2013 httpwwwbisorgpublbcbs144htm

[23] M Drehmann C Borio L Gambacorta G Jimenez and CTrucharte ldquoCountercyclical capital buffers exploring optionsrdquoBIS Working Papers No 317 Monetary and Economic Depart-ment Bank for International Settlements (BIS) Publications2010

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


the BCBS significantly increased the risk weights applicableto such exposures under both the standardized (SA) andinternal ratings based (IRB) approaches relative to the riskweights for other securitization exposures (see eg [3 910]) As a result the capital requirements for resecuritizationhave risen dramatically In addition to address the lack ofappropriate due diligence on the part of investing institutionsand deter them from relying solely on external credit ratingsthe Basel framework now requires banks to meet specificoperational criteria in order to use the risk weights specifiedin the Basel II securitization framework During the financialcrisis credit rating agencies (CRAs) downgraded the ratingsof many securitization tranches including senior tranceshighlighting deficiencies in credit rating agency modelsoriginally used to determine the ratings Capital requirementsassigned to highly rated (eg AAA) senior and mezzaninesecuritization exposures were too low and this was illustratedby the poor performance of these securities (see eg [9 10])As CRAs downgraded highly rated securitization exposuresbelow investment grade regulatory capital requirementsincreased rapidly and significantly due to the presence ofcliff effects within the securitization framework (in thiscontext cliff effects refer to significant increases in capitalrequirements resulting from a change to a factor used toassign regulatory capital) (see eg [9 10]) The BCBS alsorevised the market risk rules to increase the level of capitalthatmust bemaintained against securitization exposures heldin the trading book In addition the BCBS is reviewingwhether the risk weights for all securitization exposuresshould be recalibrated which could lead to higher capitalrequirements (see for instance the Basel papers [9 10])

The main motivation for this paper is that securitizationhas to be reestablished on a sound basis in order to supportcredit provision to the real economy and enhance banksrsquoaccess to funding globally (see eg [4]) Basel III would liketo ensure an appropriate risk-sensitive and prudent capital-ization of risks arising from securitization exposures whilereducing cliff effects and mitigating mechanistic reliance onexternal credit ratings (see [3 9]) The interplay betweenBasel III and asset securitization will be the central themeof this paper As far as the contribution of this paper isconcerned we investigate the securitization of low- and high-quality assets (denoted by LQAs andHQAs resp) into CDOsvia low-quality entities (LQEs) and high-quality entities(HQEs) respectively (see eg the BCBSpublications [9 10])

11 Literature Review about Asset Securitization and Basel IIIIn this subsection we provide literature reviews about assetsecuritization and Basel III capital and liquidity regulation(see [5] for more details)

111 Literature Review about Asset Securitization Motivatedby the 2007ndash2009 financial crisis there is an ever-growingbody of literature on LQA-related issues such as shocks toLQEs investors and CDOs via for instance rating changesThe contribution [11] studies the pricing of LQAs and relatedSAPs on the basis of data for the ABXHE family of indices(see [12] for further details) This of course is a recurringtheme in our contribution where we consider asset and CDO

pricing during the financial crisis Moreover [4] addressesthe impact of speculative asset funding on the pricing ofLQAs (measured by risk premia) and securities backed bythese assets (measured by ABXHE indices) In addition thepaper [13] extends a Kiyotaki-Moore-type model that showshow relatively small shocks might suffice to explain businesscycle fluctuations if credit markets are imperfect (see [14] formore details) Our work has a connection with this paper viathe consideration of the effect of shocks on asset parametersalthough we do not emphasize the imperfection of creditmarkets (see [15] for additional analysis) The paper [16]studies the impact of penalties on LQ-loans (see also [17])Here asset prices and penalties are chosen simultaneouslywith the latter being associated with lower asset prices (see[15] for more details) The paper also contains discussionson prices and penalties and their relationship with loan-to-value ratios (see also [18] for more on house equity) In ourcontribution we will use the framework introduced in [16] toshow how a change in profit subsequent to a negative shockis influenced by subprime mortgage features (see also [19])

112 Literature Review about Basel III In July 2009 the BCBSintroduced enhancements to the Basel II framework via [8]This was done in order to address deficiencies identifiedduring the 2007ndash2009 financial crisis These measures pri-marily addressed immediate concerns over resecuritizationand was part of reforms known as ldquoBasel 25rdquo (see the Baseldocuments [8 9] for more details about securitization) TheBCBS subsequently agreed to conduct a more fundamentalreview of the securitization framework including its relianceon external ratings (see eg [20]) The performance of androle played by securitization exposures during the financialcrisis were a key motivation for studying this category of thecapital framework Subsequently in [2] the BCBS noted thatit was ldquoconducting amore fundamental review of the securiti-zation framework including its reliance on external ratingsrdquoThe BCBS has now performed a broader review of thesecuritization framework for regulatory capital requirementswith objectivesmotivated by events during the financial crisis(see also [7]) Furthermore the consultative document [6]reflects the BCBSrsquos proposal to revise the Basel frameworkrsquostreatment of securitization exposures In developing thisproposal the BCBS seeks to make capital requirements moreprudent and risk sensitivemitigate reliance on external creditratings and reduce cliff effects (see eg [3 6]) In particularour paper adds to the debate about rating downgrades andshocks associated with them The policy directions set outin [6] form part of the BCBSrsquos broader agenda of reformingbank regulatory capital standards to address the lessons ofthe crisis These proposals build on a series of reforms thatthe BCBS has delivered through Basel III and explain theapproaches under consideration by the BCBS to revise thesecuritization framework (see [6 21] for further discussion)As in our paper the features of SPEs are discussed in somedetail in [1]

12 Preliminaries about Asset Securitization In this subsec-tion we provide preliminaries about low- and high-quality





























Risk profile

GoodBad xLow

High

y

High-gradeLQ ABS

LQ ABSbonds

Low-


Senior

Mezzanine

Equity

Senior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality








E119905(

infin

sum

119904=0

120573119904119909119905+119904) E

119905(

infin

sum

119905=0

120573119905119909119905) (1)

where119909119905+119904




119862119905+1

= 119878 (119860119905) equiv (120583 + ]) 119860

119905 ] = 119888 + 119903119891

(120583

120583 + ]) lt 120573

(2)


119905+1is







119905 only







(1 + 119903B) B119905le 119901119860

119905+1119860119905 (3)

where 119901119860119905+1




119901119860

119905119860119905minus1+ 119861119905= B119905+ 119870119905 (4)


119901119862

119905= min [119901119860

119905 119901119905] (5)





119901119860

119905119860119905minus1

= B119905+ 119870119905minus 119861119905 (6)


119903119860

119905= 119903119871

119905+ 984858119905 (7)



Π119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905 (8)



119901119860

119905119860119905minus1

=Π119905minus 119903119861119861119905+ 119903

BB119905

119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905

(9)



Π119905ge (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905

minus 119901119860

119905+1119860119905+ B119905

(10)



119905minus1assets






B)B119905minus1


119905minus ]119860119905minus1




119901119860

119905(119860119905minus 119860119905minus1) + (1 + 119903


= 120583119860119905minus1+ B119905

(11)



Risk profile

GoodBad

Low

High

y

High-gradeABS CDO

High-qualityABS

bondsHigh-

assetsSenior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality

x




E119905(

infin

sum

119904=0

12057310158401199041199091015840

119905+119904) E

119905(

infin

sum

119905=0

12057310158401199051199091015840

119905) (12)

where 1199091015840119905+119904



119904 1205731015840119904





1015840

1199031198601015840

and 119903B1015840


1199011198601015840

119905= 119901119860

119905 119903

1198601015840

119905= 119903119860

119905 119903

B1015840

119905= 119903

B119905 (13)



119903B= 119903

B119905equiv1

1205731015840minus 1 (14)




119905



119862119905+1

= 119875 (1198601015840

119905) where 1198751015840 gt 0

11987510158401015840lt 0 119875

1015840(119860

119899) lt 120583 (1 + 119903

B) lt 1198751015840(0)

(15)


119901119860

119905(1198601015840

119905minus 1198601015840

119905minus1) + (1 + 119903

B) B1015840

119905minus1+ 1199091015840

119905= 119875(119860

1015840

119905minus1)119905minus1+ B1015840

119905

(16)


119905minus1



119901119860

1199051198601015840

119905minus1+ 1198611015840

119905= B1015840

119905+ 1198701015840

119905 (17)


Π1015840

119905= 119903119860

119905119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905 (18)



119901119860

1199051198601015840

119905minus1=Π1015840

119905minus 1199031198611198611015840

119905+ 119903

BB1015840119905

119903119860 (19)



Π1015840

119905ge 119903119860119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119901119860

119905+11198601015840

119905+ B1015840

119905 (20)


and B1015840119905are



119901119860

119905 119860119905 1198601015840

119905 B119905 B1015840

119905 119909119905 1199091015840

119905 (21)



119905 B1015840119905 1199091015840

119905) to maximize



119905minus1

and B119905minus1



119906lowast=

119903B

1 + 119903B119901119860lowast

=1

1 + 119903B1198751015840[1

119899(119860 minus 119860

lowast)] = 120583 (22)

Blowast=120583

119903B119860lowast (23)



119905


119886

119906119905

119888

119906119905+1

(24)


119906119905

119886

119906119905+1

(25)




Invest 0 119888119906119905

119886

119906119905

119888

119906119905+1

119886

119906119905

119886

119906119905+1

119888

119906119905+2

(26)

Save 0 0 119903B 119888119906119905

119903B 119886

119906119905

119888

119906119905+2

(27)




119905


119905and 119887119905are linear in 119886

119905minus1and 119887119905minus1


119905and B

119905


119860119905=1

119906119905

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1] (29)

B119905=

1

1 + 119903B119901119860

119905+1119860119905 (30)




119860 = 119860119905+ 1198991198601015840

119905 (31)


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B= 119906 (119860

119905)

where 119906 (119860) equiv 1

1 + 119903B1198751015840[1

119899(119860 minus 119860)]

(32)



119905 B119905 119909119905) satisfies 119909

119905= ]119860119905minus1


B119905=

119901119860

119905+1

1 + 119903B119860119905

119860119905=

1

119901119860

119905minus 1 (1 + 119903B) 119901

119860

119905+1

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1]

(33)

Here the term (120583 + 119901119860

119905)119860119905minus1

minus (1 + 119903B)B119905minus1



each asset unit 119901119860119905+11 + 119903


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B(34)


119905and B



119905and 119901119860

119905+1


B)B119905minus1



119905





119901119860

119905and 119901119860



outstanding debt


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B=

1

1 + 119903B1198751015840(1198601015840

119905) (35)









119905 goes up then




119903B=1

1205731015840minus 1 (36)



119905minus1and

B119905minus1


(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)





120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)





Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)





119860lowast= 119860119905minus1 B





119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Risk profile

GoodBad xLow

High

y

High-gradeLQ ABS

LQ ABSbonds

Low-


Senior

Mezzanine

Equity

Senior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality








E119905(

infin

sum

119904=0

120573119904119909119905+119904) E

119905(

infin

sum

119905=0

120573119905119909119905) (1)

where119909119905+119904




119862119905+1

= 119878 (119860119905) equiv (120583 + ]) 119860

119905 ] = 119888 + 119903119891

(120583

120583 + ]) lt 120573

(2)


119905+1is







119905 only







(1 + 119903B) B119905le 119901119860

119905+1119860119905 (3)

where 119901119860119905+1




119901119860

119905119860119905minus1+ 119861119905= B119905+ 119870119905 (4)


119901119862

119905= min [119901119860

119905 119901119905] (5)





119901119860

119905119860119905minus1

= B119905+ 119870119905minus 119861119905 (6)


119903119860

119905= 119903119871

119905+ 984858119905 (7)



Π119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905 (8)



119901119860

119905119860119905minus1

=Π119905minus 119903119861119861119905+ 119903

BB119905

119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905

(9)



Π119905ge (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905

minus 119901119860

119905+1119860119905+ B119905

(10)



119905minus1assets






B)B119905minus1


119905minus ]119860119905minus1




119901119860

119905(119860119905minus 119860119905minus1) + (1 + 119903


= 120583119860119905minus1+ B119905

(11)



Risk profile

GoodBad

Low

High

y

High-gradeABS CDO

High-qualityABS

bondsHigh-

assetsSenior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality

x




E119905(

infin

sum

119904=0

12057310158401199041199091015840

119905+119904) E

119905(

infin

sum

119905=0

12057310158401199051199091015840

119905) (12)

where 1199091015840119905+119904



119904 1205731015840119904





1015840

1199031198601015840

and 119903B1015840


1199011198601015840

119905= 119901119860

119905 119903

1198601015840

119905= 119903119860

119905 119903

B1015840

119905= 119903

B119905 (13)



119903B= 119903

B119905equiv1

1205731015840minus 1 (14)




119905



119862119905+1

= 119875 (1198601015840

119905) where 1198751015840 gt 0

11987510158401015840lt 0 119875

1015840(119860

119899) lt 120583 (1 + 119903

B) lt 1198751015840(0)

(15)


119901119860

119905(1198601015840

119905minus 1198601015840

119905minus1) + (1 + 119903

B) B1015840

119905minus1+ 1199091015840

119905= 119875(119860

1015840

119905minus1)119905minus1+ B1015840

119905

(16)


119905minus1



119901119860

1199051198601015840

119905minus1+ 1198611015840

119905= B1015840

119905+ 1198701015840

119905 (17)


Π1015840

119905= 119903119860

119905119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905 (18)



119901119860

1199051198601015840

119905minus1=Π1015840

119905minus 1199031198611198611015840

119905+ 119903

BB1015840119905

119903119860 (19)



Π1015840

119905ge 119903119860119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119901119860

119905+11198601015840

119905+ B1015840

119905 (20)


and B1015840119905are



119901119860

119905 119860119905 1198601015840

119905 B119905 B1015840

119905 119909119905 1199091015840

119905 (21)



119905 B1015840119905 1199091015840

119905) to maximize



119905minus1

and B119905minus1



119906lowast=

119903B

1 + 119903B119901119860lowast

=1

1 + 119903B1198751015840[1

119899(119860 minus 119860

lowast)] = 120583 (22)

Blowast=120583

119903B119860lowast (23)



119905


119886

119906119905

119888

119906119905+1

(24)


119906119905

119886

119906119905+1

(25)




Invest 0 119888119906119905

119886

119906119905

119888

119906119905+1

119886

119906119905

119886

119906119905+1

119888

119906119905+2

(26)

Save 0 0 119903B 119888119906119905

119903B 119886

119906119905

119888

119906119905+2

(27)




119905


119905and 119887119905are linear in 119886

119905minus1and 119887119905minus1


119905and B

119905


119860119905=1

119906119905

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1] (29)

B119905=

1

1 + 119903B119901119860

119905+1119860119905 (30)




119860 = 119860119905+ 1198991198601015840

119905 (31)


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B= 119906 (119860

119905)

where 119906 (119860) equiv 1

1 + 119903B1198751015840[1

119899(119860 minus 119860)]

(32)



119905 B119905 119909119905) satisfies 119909

119905= ]119860119905minus1


B119905=

119901119860

119905+1

1 + 119903B119860119905

119860119905=

1

119901119860

119905minus 1 (1 + 119903B) 119901

119860

119905+1

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1]

(33)

Here the term (120583 + 119901119860

119905)119860119905minus1

minus (1 + 119903B)B119905minus1



each asset unit 119901119860119905+11 + 119903


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B(34)


119905and B



119905and 119901119860

119905+1


B)B119905minus1



119905





119901119860

119905and 119901119860



outstanding debt


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B=

1

1 + 119903B1198751015840(1198601015840

119905) (35)









119905 goes up then




119903B=1

1205731015840minus 1 (36)



119905minus1and

B119905minus1


(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)





120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)





Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)





119860lowast= 119860119905minus1 B





119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E119905(

infin

sum

119904=0

120573119904119909119905+119904) E

119905(

infin

sum

119905=0

120573119905119909119905) (1)

where119909119905+119904




119862119905+1

= 119878 (119860119905) equiv (120583 + ]) 119860

119905 ] = 119888 + 119903119891

(120583

120583 + ]) lt 120573

(2)


119905+1is







119905 only







(1 + 119903B) B119905le 119901119860

119905+1119860119905 (3)

where 119901119860119905+1




119901119860

119905119860119905minus1+ 119861119905= B119905+ 119870119905 (4)


119901119862

119905= min [119901119860

119905 119901119905] (5)





119901119860

119905119860119905minus1

= B119905+ 119870119905minus 119861119905 (6)


119903119860

119905= 119903119871

119905+ 984858119905 (7)



Π119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905 (8)



119901119860

119905119860119905minus1

=Π119905minus 119903119861119861119905+ 119903

BB119905

119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905

(9)



Π119905ge (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905

minus 119901119860

119905+1119860119905+ B119905

(10)



119905minus1assets






B)B119905minus1


119905minus ]119860119905minus1




119901119860

119905(119860119905minus 119860119905minus1) + (1 + 119903


= 120583119860119905minus1+ B119905

(11)



Risk profile

GoodBad

Low

High

y

High-gradeABS CDO

High-qualityABS

bondsHigh-

assetsSenior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality

x




E119905(

infin

sum

119904=0

12057310158401199041199091015840

119905+119904) E

119905(

infin

sum

119905=0

12057310158401199051199091015840

119905) (12)

where 1199091015840119905+119904



119904 1205731015840119904





1015840

1199031198601015840

and 119903B1015840


1199011198601015840

119905= 119901119860

119905 119903

1198601015840

119905= 119903119860

119905 119903

B1015840

119905= 119903

B119905 (13)



119903B= 119903

B119905equiv1

1205731015840minus 1 (14)




119905



119862119905+1

= 119875 (1198601015840

119905) where 1198751015840 gt 0

11987510158401015840lt 0 119875

1015840(119860

119899) lt 120583 (1 + 119903

B) lt 1198751015840(0)

(15)


119901119860

119905(1198601015840

119905minus 1198601015840

119905minus1) + (1 + 119903

B) B1015840

119905minus1+ 1199091015840

119905= 119875(119860

1015840

119905minus1)119905minus1+ B1015840

119905

(16)


119905minus1



119901119860

1199051198601015840

119905minus1+ 1198611015840

119905= B1015840

119905+ 1198701015840

119905 (17)


Π1015840

119905= 119903119860

119905119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905 (18)



119901119860

1199051198601015840

119905minus1=Π1015840

119905minus 1199031198611198611015840

119905+ 119903

BB1015840119905

119903119860 (19)



Π1015840

119905ge 119903119860119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119901119860

119905+11198601015840

119905+ B1015840

119905 (20)


and B1015840119905are



119901119860

119905 119860119905 1198601015840

119905 B119905 B1015840

119905 119909119905 1199091015840

119905 (21)



119905 B1015840119905 1199091015840

119905) to maximize



119905minus1

and B119905minus1



119906lowast=

119903B

1 + 119903B119901119860lowast

=1

1 + 119903B1198751015840[1

119899(119860 minus 119860

lowast)] = 120583 (22)

Blowast=120583

119903B119860lowast (23)



119905


119886

119906119905

119888

119906119905+1

(24)


119906119905

119886

119906119905+1

(25)




Invest 0 119888119906119905

119886

119906119905

119888

119906119905+1

119886

119906119905

119886

119906119905+1

119888

119906119905+2

(26)

Save 0 0 119903B 119888119906119905

119903B 119886

119906119905

119888

119906119905+2

(27)




119905


119905and 119887119905are linear in 119886

119905minus1and 119887119905minus1


119905and B

119905


119860119905=1

119906119905

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1] (29)

B119905=

1

1 + 119903B119901119860

119905+1119860119905 (30)




119860 = 119860119905+ 1198991198601015840

119905 (31)


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B= 119906 (119860

119905)

where 119906 (119860) equiv 1

1 + 119903B1198751015840[1

119899(119860 minus 119860)]

(32)



119905 B119905 119909119905) satisfies 119909

119905= ]119860119905minus1


B119905=

119901119860

119905+1

1 + 119903B119860119905

119860119905=

1

119901119860

119905minus 1 (1 + 119903B) 119901

119860

119905+1

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1]

(33)

Here the term (120583 + 119901119860

119905)119860119905minus1

minus (1 + 119903B)B119905minus1



each asset unit 119901119860119905+11 + 119903


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B(34)


119905and B



119905and 119901119860

119905+1


B)B119905minus1



119905





119901119860

119905and 119901119860



outstanding debt


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B=

1

1 + 119903B1198751015840(1198601015840

119905) (35)









119905 goes up then




119903B=1

1205731015840minus 1 (36)



119905minus1and

B119905minus1


(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)





120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)





Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)





119860lowast= 119860119905minus1 B





119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Π119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905 (8)



119901119860

119905119860119905minus1

=Π119905minus 119903119861119861119905+ 119903

BB119905

119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905

(9)



Π119905ge (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905

minus 119901119860

119905+1119860119905+ B119905

(10)



119905minus1assets






B)B119905minus1


119905minus ]119860119905minus1




119901119860

119905(119860119905minus 119860119905minus1) + (1 + 119903


= 120583119860119905minus1+ B119905

(11)



Risk profile

GoodBad

Low

High

y

High-gradeABS CDO

High-qualityABS

bondsHigh-

assetsSenior

Mezzanine

Equity

Senior

Mezzanine

Equity

quality

x




E119905(

infin

sum

119904=0

12057310158401199041199091015840

119905+119904) E

119905(

infin

sum

119905=0

12057310158401199051199091015840

119905) (12)

where 1199091015840119905+119904



119904 1205731015840119904





1015840

1199031198601015840

and 119903B1015840


1199011198601015840

119905= 119901119860

119905 119903

1198601015840

119905= 119903119860

119905 119903

B1015840

119905= 119903

B119905 (13)



119903B= 119903

B119905equiv1

1205731015840minus 1 (14)




119905



119862119905+1

= 119875 (1198601015840

119905) where 1198751015840 gt 0

11987510158401015840lt 0 119875

1015840(119860

119899) lt 120583 (1 + 119903

B) lt 1198751015840(0)

(15)


119901119860

119905(1198601015840

119905minus 1198601015840

119905minus1) + (1 + 119903

B) B1015840

119905minus1+ 1199091015840

119905= 119875(119860

1015840

119905minus1)119905minus1+ B1015840

119905

(16)


119905minus1



119901119860

1199051198601015840

119905minus1+ 1198611015840

119905= B1015840

119905+ 1198701015840

119905 (17)


Π1015840

119905= 119903119860

119905119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905 (18)



119901119860

1199051198601015840

119905minus1=Π1015840

119905minus 1199031198611198611015840

119905+ 119903

BB1015840119905

119903119860 (19)



Π1015840

119905ge 119903119860119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119901119860

119905+11198601015840

119905+ B1015840

119905 (20)


and B1015840119905are



119901119860

119905 119860119905 1198601015840

119905 B119905 B1015840

119905 119909119905 1199091015840

119905 (21)



119905 B1015840119905 1199091015840

119905) to maximize



119905minus1

and B119905minus1



119906lowast=

119903B

1 + 119903B119901119860lowast

=1

1 + 119903B1198751015840[1

119899(119860 minus 119860

lowast)] = 120583 (22)

Blowast=120583

119903B119860lowast (23)



119905


119886

119906119905

119888

119906119905+1

(24)


119906119905

119886

119906119905+1

(25)




Invest 0 119888119906119905

119886

119906119905

119888

119906119905+1

119886

119906119905

119886

119906119905+1

119888

119906119905+2

(26)

Save 0 0 119903B 119888119906119905

119903B 119886

119906119905

119888

119906119905+2

(27)




119905


119905and 119887119905are linear in 119886

119905minus1and 119887119905minus1


119905and B

119905


119860119905=1

119906119905

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1] (29)

B119905=

1

1 + 119903B119901119860

119905+1119860119905 (30)




119860 = 119860119905+ 1198991198601015840

119905 (31)


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B= 119906 (119860

119905)

where 119906 (119860) equiv 1

1 + 119903B1198751015840[1

119899(119860 minus 119860)]

(32)



119905 B119905 119909119905) satisfies 119909

119905= ]119860119905minus1


B119905=

119901119860

119905+1

1 + 119903B119860119905

119860119905=

1

119901119860

119905minus 1 (1 + 119903B) 119901

119860

119905+1

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1]

(33)

Here the term (120583 + 119901119860

119905)119860119905minus1

minus (1 + 119903B)B119905minus1



each asset unit 119901119860119905+11 + 119903


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B(34)


119905and B



119905and 119901119860

119905+1


B)B119905minus1



119905





119901119860

119905and 119901119860



outstanding debt


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B=

1

1 + 119903B1198751015840(1198601015840

119905) (35)









119905 goes up then




119903B=1

1205731015840minus 1 (36)



119905minus1and

B119905minus1


(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)





120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)





Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)





119860lowast= 119860119905minus1 B





119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905



119862119905+1

= 119875 (1198601015840

119905) where 1198751015840 gt 0

11987510158401015840lt 0 119875

1015840(119860

119899) lt 120583 (1 + 119903

B) lt 1198751015840(0)

(15)


119901119860

119905(1198601015840

119905minus 1198601015840

119905minus1) + (1 + 119903

B) B1015840

119905minus1+ 1199091015840

119905= 119875(119860

1015840

119905minus1)119905minus1+ B1015840

119905

(16)


119905minus1



119901119860

1199051198601015840

119905minus1+ 1198611015840

119905= B1015840

119905+ 1198701015840

119905 (17)


Π1015840

119905= 119903119860

119905119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905 (18)



119901119860

1199051198601015840

119905minus1=Π1015840

119905minus 1199031198611198611015840

119905+ 119903

BB1015840119905

119903119860 (19)



Π1015840

119905ge 119903119860119901119860

1199051198601015840

119905minus1+ 1199031198611198611015840

119905minus 119901119860

119905+11198601015840

119905+ B1015840

119905 (20)


and B1015840119905are



119901119860

119905 119860119905 1198601015840

119905 B119905 B1015840

119905 119909119905 1199091015840

119905 (21)



119905 B1015840119905 1199091015840

119905) to maximize



119905minus1

and B119905minus1



119906lowast=

119903B

1 + 119903B119901119860lowast

=1

1 + 119903B1198751015840[1

119899(119860 minus 119860

lowast)] = 120583 (22)

Blowast=120583

119903B119860lowast (23)



119905


119886

119906119905

119888

119906119905+1

(24)


119906119905

119886

119906119905+1

(25)




Invest 0 119888119906119905

119886

119906119905

119888

119906119905+1

119886

119906119905

119886

119906119905+1

119888

119906119905+2

(26)

Save 0 0 119903B 119888119906119905

119903B 119886

119906119905

119888

119906119905+2

(27)




119905


119905and 119887119905are linear in 119886

119905minus1and 119887119905minus1


119905and B

119905


119860119905=1

119906119905

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1] (29)

B119905=

1

1 + 119903B119901119860

119905+1119860119905 (30)




119860 = 119860119905+ 1198991198601015840

119905 (31)


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B= 119906 (119860

119905)

where 119906 (119860) equiv 1

1 + 119903B1198751015840[1

119899(119860 minus 119860)]

(32)



119905 B119905 119909119905) satisfies 119909

119905= ]119860119905minus1


B119905=

119901119860

119905+1

1 + 119903B119860119905

119860119905=

1

119901119860

119905minus 1 (1 + 119903B) 119901

119860

119905+1

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1]

(33)

Here the term (120583 + 119901119860

119905)119860119905minus1

minus (1 + 119903B)B119905minus1



each asset unit 119901119860119905+11 + 119903


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B(34)


119905and B



119905and 119901119860

119905+1


B)B119905minus1



119905





119901119860

119905and 119901119860



outstanding debt


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B=

1

1 + 119903B1198751015840(1198601015840

119905) (35)









119905 goes up then




119903B=1

1205731015840minus 1 (36)



119905minus1and

B119905minus1


(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)





120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)





Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)





119860lowast= 119860119905minus1 B





119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Invest 0 119888119906119905

119886

119906119905

119888

119906119905+1

119886

119906119905

119886

119906119905+1

119888

119906119905+2

(26)

Save 0 0 119903B 119888119906119905

119903B 119886

119906119905

119888

119906119905+2

(27)




119905


119905and 119887119905are linear in 119886

119905minus1and 119887119905minus1


119905and B

119905


119860119905=1

119906119905

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1] (29)

B119905=

1

1 + 119903B119901119860

119905+1119860119905 (30)




119860 = 119860119905+ 1198991198601015840

119905 (31)


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B= 119906 (119860

119905)

where 119906 (119860) equiv 1

1 + 119903B1198751015840[1

119899(119860 minus 119860)]

(32)



119905 B119905 119909119905) satisfies 119909

119905= ]119860119905minus1


B119905=

119901119860

119905+1

1 + 119903B119860119905

119860119905=

1

119901119860

119905minus 1 (1 + 119903B) 119901

119860

119905+1

[(120583 + 119901119860

119905)119860119905minus1minus (1 + 119903

B) B119905minus1]

(33)

Here the term (120583 + 119901119860

119905)119860119905minus1

minus (1 + 119903B)B119905minus1



each asset unit 119901119860119905+11 + 119903


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B(34)


119905and B



119905and 119901119860

119905+1


B)B119905minus1



119905





119901119860

119905and 119901119860



outstanding debt


119906119905= 119901119860

119905minus119901119860

119905+1

1 + 119903B=

1

1 + 119903B1198751015840(1198601015840

119905) (35)









119905 goes up then




119903B=1

1205731015840minus 1 (36)



119905minus1and

B119905minus1


(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)





120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)





Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)





119860lowast= 119860119905minus1 B





119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119905 goes up then




119903B=1

1205731015840minus 1 (36)



119905minus1and

B119905minus1


(119901119860

119905+119904 119860119905+119904 B119905+119904) 119904 ge 0 (37)





120583 + ] gt 1198751015840 [(119860 minus 119860

lowast)

119899] = 120583 (1 + 119903

B) (38)





Elowast

Et

E0

A A998400

P998400(A998400n)

120583 +

120583(1 + rB)





119860lowast= 119860119905minus1 B





119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (119860119905) 119860119905= (120583 minus 120583Σ + 119901

119860

119905minus 119901119860lowast

)119860lowast (date 119905) (40)

119906 (119860119905+119904) 119860119905+119904= 120583119860119905+119904minus1

(dates 119905 + 1 119905 + 2 ) (41)




(1 minus Σ) 120583119860lowast (42)


capital gains of

(119901119860

119905+ 119901119860lowast

)119860lowast (43)



(1 + 119903B) Blowast= 119901119860lowast

119860lowast (44)


119905 119901119860119905



119860119905=119860119905minus 119860lowast

119860lowast 119901

119860

119905=119901119860

119905minus 119901119860lowast

119901119860lowast


Πlowast

(45)




Πlowast

119905= (119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861119905minus 119903

BBlowast

119905

Π1015840lowast

119905= 119903119860119901119860lowast

1199051198601015840lowast

119905minus1+ 1199031198611198611015840

119905minus 119903

BB1015840

119905

lowast

(46)


(1 +1

120578)119860119905=1 + 119903

B

119903B119901119860

119905minus Σ (date 119905) (47)

(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1 (dates 119905 + 1 119905 + 2 ) (48)


1

120578=119889 log 119906 (119860)119889 log119860

10038161003816100381610038161003816100381610038161003816119860= 119860lowast

= minus119889 log1198751015840(1198601015840)119889 log1198601015840

1003816100381610038161003816100381610038161003816100381610038161198601015840= 1119899(119860minus119860lowast)

times119860lowast


(49)


119905 In


1 + 119903B

119903B119901119860

119905minus Σ =

119906 (119860119905) 119860119905

120583119860lowastminus 1 (50)


(1 +1

120578)119860119905

= (119860119905minus 119860lowast


119889 log1198751015840 (1198601015840)119889 log1198601015840

100381610038161003816100381610038161003816100381610038161003816100381610038161198601015840=1119899(119860minus119860lowast)

times119860lowast


119860119905minus 119860lowast

119860lowast)

(51)


119903B)(119903




119905 which from




and Π119905






119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901119860

119905= minus

1

120578Σ (52)

119860119905= minus

1

1 + 1120578(1 +

1 + 119903B

120578119903B)Σ (53)

119901119862

119905= minus

1

120578Σ (54)

119862119905+1

=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905 (55)

Π119905=

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

(119903119860

119905+ 119888119901

119905119903119891

119905minus (1 minus 119903

119877

119905) 119903119878

119905) 119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1

(56)

respectively




119906119905+119904= 119906 (119860

119905+119904) 119904 ge 0 (57)


(1 +1

120578)119860119905+119904= 119860119905+119904minus1

for 119904 ge 1

(dates 119905 + 1 119905 + 2 ) (58)

we obtain

119901119860

119905=1

120578

119903B

1 + 119903B

infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904

=1

120578

119903B

1 + 119903B

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

(59)


infin

sum

119904=0

(1 + 119903B)minus119904

119860119905+119904=

1

1 minus 120578 (1 + 119903B) (1 + 120578)119860119905

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578119860119905

(60)


[1 minus120578

(1 + 119903B) (1 + 120578)]

minus1

=

(1 + 119903B) (1 + 120578)

(1 + 119903B) (1 + 120578) minus 120578

(61)


119905 In order to find119901119860

119905and119860




119905and 119860

119905above) by

119862119905=119862119905minus 119862lowast

119862lowast 119862

119905= (119862119905+ 1)119862

lowast 119862

lowast=

119862119905

119862119905+ 1

(62)


119905+119904 is given by

119862119905+119904=

120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

for 119904 ge 1

(63)


120583 + ] minus (1 + 119903B) 120583

120583 + ]

(120583 + ]) 119860lowast

119862lowast119860119905+119904minus1

=

119862119905+119904minus [(1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast]

119862lowast

(64)


119862lowast= (1 + 119903

B) 120583119860119905+119904minus1

+ (120583 + ] minus (1 + 119903B) 120583)119860lowast

= [(1 + 119903B) 120583119860119905+119904minus1

+ 120583 + ]] 119860lowast(65)


119862lowast= (120583 + ]) 119860lowast (1 + 119903

B) 120583119860119905+119904minus1

= (1 + 119903B) 120583119860lowast

or 119860119905+119904minus1

= 119860lowast

(66)






119903B)(119903





119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905+1


119901119860

119905= [

119903B

120578 (1 + 119903B)]119860119905 (67)



119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast = minus

119903B

120578 (1 + 119903B)Σ (68)

119860119905|119901119860

119905+1=119901119860lowast = minusΣ (69)



120583 + ] minus (1 + 119903B) 120583

120583 + ](70)


119905+119904119860minus1 it would






119903119872

119905= 1205720119871119905+ 1205721119888119901

119905+ 1205722119883119905+ 1205723119885119903119872

119905+ 119906119905

119871119905= 1205951119903119872

119905+ 1205952119883119905+ 1205953119885119871

119905+ V119905

119888119901

119905= 1205741119903119872

119905+ 1205742119883119905+ 1205743119885119888119901

119905+ 119908119905

(71)



119872

119885119871 or 119885119888

119901



Π119905(119903119872) =

119865119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119865119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (72)

Π119905(119888119901) =

119866119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119866119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (73)

Π119905 (119871) =

119867119905119901119860

119905119860119905minus1+ 119903119861119861119905minus 119903

BB119905

119867119905119901119860lowast

119905119860lowast

119905minus1+ 119903119861119861


minus 1 (74)



119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865119905= 119903119872

119905(1 + 120574

1119903119891

119905) + (120574

2119883119905+ 1205743119885119888119901

119905+ 119908119905) 119903119891

119905

minus (1 minus 119903119877

119905) 119903119878

119905

119866119905= 119888119901

119905(1

1205741+ 119903119891

119905) minus

1

1205741(1205742119883119905+ 1205743119885119888119901

119905+ 119908119905)

minus (1 minus 119903119877

119905) 119903119878

119905

119867119905= [12057411

1205951119871119905minus1205952

1205951119883119905minus1205953

1205951119885119871

119905minus1

1205951V119905 + 1205742119883119905+ 1205743119885119888119901

119905

+119908119905] [1205741+ 119903119891

119905] + 1205720119871119905+ 1205722119883119905+ 1205723119885119903119872

119905

+ 119906119905minus (1 minus 119903

119877

119905) 119903119878

119905

(75)

respectively






Parameter Value120583 0002119901119905+1

0113119860119905

720 000119903119877 05B119905minus1

$2 600119903119861 0205Σ 0002] 02119888119901 003119860119905minus1

460 000119903119878 015119903B 02119870119905

$3 000119862lowast 240 000

120572 03119903119891 02119903119860 0061B119905

$4 800119861119905

$5 000119899 1119875(1198601015840

119905minus1) 240 000


119860 = 120572119860 = 03 times 720000 = 216000

1198601015840= (1 minus 120572)119860 = (1 minus 03) 720000

= 504000

(76)


119862119905+1

= (120583 + ]) 119860119905= (120583 + ]) 120572119860

119905

= (0002 + 02) times 03 times 720000 = 43632

(77)


120573(0002

0002 + 02) = 00099099 (78)


119901119860

119905119860119905minus1

= 119863119905+ 119861119905+ 119870119905minus 119861119905= 1200 + 4800 + 3000 minus 5000

= 4000

(79)


119901119860

119905=4000

119860119905minus1

=4000

120572119860119905minus1

=4000

(03 times 460000)

= 00289855072

(80)



Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Π119905= (119903119860+ 119888119901

119905119903119891

119905) 119901119860




(81)




(82)





= 42274286

(83)


1199091015840

119905= 280000 + 4800 minus 0011904761 (1 minus 03) 720000


= minus4631999954

(84)



Π1015840


minus 02 times 4800 = 1235

(85)


119901119860

1199051198601015840


0061

=49918

(86)


Π1015840



(87)


119906119905= 119901119860

119905minus

119901119860

119905+1

1 + 119903119861= 0011904761 minus

0113

1 + 02

= minus0082261905

(88)



119860119905=

1

0082261905[(0002 + 0011904761) times 03 times 460000

minus (1 + 02) times 2600 = 1460144865]

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(89)


119860=119860119905+ 1198991198601015840

119905=03 times 720000 + 1 (1 minus 03) 720000=720000

(90)


119901119860lowast

=00021 + 02

02=0012 119861

lowast=0002

02260000=2600

(91)



= 119860lowast and 119861



119860119905=

1



= 1460815893

119861119905=

1

1 + 020113 times 03 times 720000 = 20340

(92)




(93)



(94)


(119901119860


= 4679

(95)






119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119860119905=03 (720000 minus 460000)

03 times 460000= 0565217391

119901119860

119905=0011904761 minus 0022

0022= minus04588745

(97)


Πlowast



(98)


Π1015840lowast



(99)


119905are

Π119905=87 minus 462412

462412= minus081186

Π1015840

119905=1235 minus 691124

691124= minus082131

(100)


120578 = [((2 + 02) 02) 04588745 minus 0002

0565217391minus 1]

minus1

= 0126718931

(101)




119901119860

119905= minus

1

01267189310002 = minus0015782961

119860119905= minus

1

1 + 10126718931(1 +

1 + 02

0126718931 times 02) 0002

= minus0010875327

(102)


119905and 119860

119905become

119901119860

119905

10038161003816100381610038161003816119901119860119905+1=119901119860lowast= minus

02

0126718931 (2 + 02)0002

= minus0001434814

119860119905

10038161003816100381610038161003816119901119860119905+1=119901


(103)



43 632119901119860

119905119860119905minus1

$4 000Π119905ge $minus18 561

1199091015840

119905$minus46 3199995

119901119860

1199051198601015840

119905minus1$4 9918

119906119905


119905$20 340


119906(119860119905)119860119905

minus1 11769(119901119860

119905+ 119901119860lowast

)119860lowast $4 679

119860119905

0565217391Πlowast

119905$462412

Π119905


119860119905where 119901119860

119905+1= 119901119860lowast

minus0002120573 gt 00099099Π119905

$87119909119905

$54 1036421Π1015840

119905$1235

Π1015840

119905ge $minus51 007



(1 minus Σ)120583119860lowast 275448

(1 + 119903B)Blowast $3 120

119901119860

119905= 119901119862

119905minus04588745

Πlowast1015840 $27531

Π1015840

119905$691124

119901119860


119901119860

119905where 119901119860

119905+1= 119901119860lowast

minus0001434814119862119905+1

0064869



119862119905+1

=0002 + 02 minus (1 + 02) 0002

0002 + 02

times(0002 + 02) 03 times 460000

2400000565217391

= 0064869

(104)



119905+1 increases to $43632







lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











lowast












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119905




119905+1 and profit




119905|119901119860

119905+1

and input 119860119905|119901119860

119905+1





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article basel iii and asset...

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