interrelations amongst liquidity, market and credit risks

7/21/2019 Interrelations amongst Liquidity, Market and Credit Risks

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Balance Sheet Items Requiring Statistic-Financial ModelsLiquidity and Counterparty Risks

Interrelations amongst Liquidity, Market and

Credit Risks

-some proposals for integrated approaches

Antonio Castagna

www.iasonltd.com

28th February 2012

Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved


2/59


Index

1 Balance Sheet Items Requiring Statistic-Financial Models

Deposits

Prepayment of Fixed Rate Mortgages

Credit Lines

2 Liquidity and Counterparty Risks

Introduction

Liquidity Risk Pricing in OTC DerivativesHaircut Setting



3/59


DepositsPrepayment of Fixed Rate MortgagesCredit Lines

Index


Deposits


Credit Lines


Introduction




4/59



Index


Deposits


Credit Lines


Introduction



D i


5/59



The Relevance of Deposits in the Banking Activity

Non-maturing deposits represent a significant source of funding for a financialinstitution. Yet, there is still no commonly accepted framework for valuationand for interest rate and liquidity risk management.Such a framework is needed even more in the current environment:

non-maturing deposits are a low-cost source of funding, compared withother sources, so that in a funding-mixthey contribute to abate the totalcost of funding (although a consistent model is needed to manage them,such as the one proposed by Iason);

the interest rate and the liquidity risks have to be properly identified,measured and hedged, in order to make possible the insertion of the

deposits volumes in the funding management;liquidity crisis have to be modelled, allowing for stress-testing activityinvolving both idiosyncratic and market specific extreme scenarios.


Deposits


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Overview of the Proposed Model

We assume that the bank offers different deposits, i.e.: transactionaccount and different savings accounts.

A transaction account is mainly used by the customer to fulfill his shortterm liquidity needs while he uses the savings accounts as investmentopportunities with very small risk.

The framework models three factors:market rates,deposits ratesdeposits volumes.

These factors determine the customer behaviour, modelled as reasonable

rules or strategies to set the total level of deposits, and then theallocation amongst the different types of deposits.

Once this is done, we can calculate the value of the deposits, theirsensitivities to market rates and the amounts available, thus allowing forthe design of hedging and liquidity management strategies.


Deposits


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Market Rates

We can use any available model for market rates. We choose an extended Cox,

Ingersoll and Ross model (CIR), with a time dependent deterministic shiftparameter (to perfectly match the starting term structure of market rates). Theinstantaneous rate r(t) is defined as

r(t) =x(t) +(t)

where x(t) has a CIR dynamics:

dxt=[ xt]dt+

xtdZt

and (t) is a deterministic function of time.

An example of curves generated with:x0 = 2%, = 0.5, = 4.5%, =7.90%, (t) = 0 for any tZero-rate curves and implied 6 monthforward rates are shown in the picture.

TermStructure

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15 16 17 18 19 20

Zero

6M


S SDeposits


8/59



Deposit Rates

We assume that deposit rates are a function of the (short term) market interest

rates and volumes:dn(t) =d(r(t),V(t))

where dn is the rate for the deposit of type n, r(t) is the instantaneous rateand V(t) is the amount deposited. The function is given by the banks pricingpolicy.Examples may be:

constant spread below market rates:

dn(t) = max[r(t) , 0]a proportion of market rates:

dn(t) =r(t)

a function as the two above dependent on the amount deposited

dn(t) =m

j=1

dj(r(t))1{vj,vj+1}(V(t))

where vj and vj+1 are ranges of the volume Vproducing different levels of

the deposits rate.Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved

B l Sh t It R i i St ti ti Fi i l M d lDeposits


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Deposits Volumes

We make the following assumptions to model the customers behaviourdetermining the deposits volumes:

A customer modifies his balance on the transaction account targeting agiven fraction of its average monthly income. This level can be interpretedas the amount he needs to cover his short time liquidity needs.

There is an interest rates strike level, specific for the customer, such that,when the market rate is above it, then the customer reconsiders the targetlevel and redirects a higher amounts to other investments.

A customer has a savings account policy targeting to save a fraction ofhis income.

Again, there is a strike level, specific for the customer, such that when themarket rate is above it then the customer increases the fraction saved.

A customer may reconsider his savings policy and may decide to close oneor more of his savings accounts. If he takes such actions he reallocates allof his money to one of the other savings accounts offered by the bank.


Balance Sheet Items Requiring Statistic Financial ModelsDeposits


10/59


pPrepayment of Fixed Rate MortgagesCredit Lines

Deposits Volumes

On an aggregate basis, we consider the average customer to model the totaldeposits volume.To take into account the degree of heterogeneity among the customers,concerning the level of the market rate at which the customer changes hispolicy for the transaction account and for the total savings volume, we adopt aGamma distribution:

(x/)1

exp(x/)()

The Gamma function is very flexible.If we, for example, set = 1.5 and= 0.05 we have a distribution labeled

as 1 in the figure. If = 30 and = 0.002 we have a distribution 2.It is possible to model the aggregatedcustomers behaviour, making it more orless concentrated around specific levels. 0

5

10

15

20

25

30

35

40

0

0.

02

0.

04

0.

06

0.

08

0.1

0.

12

0.

14

0.

16

0.

18

0.2

0.

22

0.

24

0.

25

0.

27

0.

29

0.

31

0.

33

0.

35

0.

37

Disitribution 1

Distribution 2


Balance Sheet Items Requiring Statistic Financial Models Deposits


11/59


pPrepayment of Fixed Rate MortgagesCredit Lines

Evolution of Deposits Volumes: Second Example

We present the market and deposits in-terests rates, the evolution of the vol-umes of the transaction account, of thetotal volume of the savings accounts andthe split between the savings 1 and

savings 2 accounts, in a different eco-nomic cycle.At the beginning market rates are stable,then they sharply increase. The transac-tion volume experiences a decline due tothe reallocation of the total savings inother investments, since they offer higher

returns.Also the composition of the total savingsaccount volume shows that the savings2 account (yielding higher rates) is pre-ferred to the savings 1 account.

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

10.00%

-

0.83

1.67

2.50

3.33

4.17

5.00

5.83

6.67

7.50

8.33

9.17

10.00

Ref. rate

Trans. Depo

Savings2

Savings2

5400.0

5600.0

5800.0

6000.0

6200.0

6400.0

6600.0

6800.0

7000.0

-

0.83

1.67

2.50

3.33

4.17

5.00

5.83

6.67

7.50

8.33

9.17

10.00

Trans. Depo

0.0

20000.0

40000.0

60000.0

80000.0

100000.0

120000.0

140000.0

-

0.83

1.67

2.50

3.33

4.17

5.00

5.83

6.67

7.50

8.33

9.17

10.00

Savings Tot

Savings1

Savings2


Balance Sheet Items Requiring Statistic-Financial Models Depositsf


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Balance Sheet Items Requiring Statistic Financial ModelsLiquidity and Counterparty Risks

Prepayment of Fixed Rate MortgagesCredit Lines

Liquidity Risk

The behavioural modelling of the amounts of the different deposits allows us to

manage the liquidity risk, which is based on the concept of the term structureof liquidity.

Generate a number Sof scenarios.

For each 0 t Tdefine the process of minima by:

Mj(t) = min0st V(s)

In each scenario the stochastic process Mj(t) specifies the minimal volumebetween [0, t]. This is the amount available for investment over the entireperiod in the given scenario.

Define Lts(t, q) as the p-quantile ofMj(t). The probability that thevolume V drops below level Lts(t, q) in the time interval [0, t] is q.Lts(t, q) is the amount available for investment with probability q. Wename it Term Structure of Liquidity.

The Term Structure of Liquidity can therefore be used to implementliquidity policies.


Balance Sheet Items Requiring Statistic-Financial Models DepositsP t f Fi d R t M t


13/59

Balance Sheet Items Requiring Statistic Financial ModelsLiquidity and Counterparty Risks


Liquidity Risk: a Practical Example

The table indicates the Term Structureof Liquidity up to 10 years for transactiondeposits, given the assumptions statedabove, with a probability of 99%.

Time Amnt1 59052 58083 57964 57985 58566 5919

7 60538 61519 6325

10 6430

The Term Structure of Liquidity up to 10years for the transaction deposits is alsoshown in the figure.

5400.00

5500.00

5600.00

5700.00

5800.00

5900.00

6000.00

6100.00

6200.006300.00

6400.00

6500.00

1 2 3 4 5 6 7 8 9 10

Trans. Depo


Balance Sheet Items Requiring Statistic-Financial Models DepositsPrepayment of Fixed Rate Mortgages


14/59

q gLiquidity and Counterparty Risks


Liquidity Risk: a Practical Example

We now show in the table the TermStructure of Liquidity for the two typesof savings deposits, given the assump-tions stated above, with a probability of

99%.

Time Amnt Sav 1 Amnt Sav 21 47657 508822 47087 520143 47301 532144 47305 543725 47810 562326 47990 57455

7 48743 596208 48979 610439 48496 62477

10 48572 63865

Also for savings deposits, a visual repre-sentation of the Term Structure of Liq-uidity is also produced.

0.0

10000.0

20000.0

30000.0

40000.0

50000.060000.0

70000.0

1 2 3 4 5 6 7 8 9 10

Savings1

Savings2


Balance Sheet Items Requiring Statistic-Financial Models DepositsPrepayment of Fixed Rate Mortgages


15/59

Liquidity and Counterparty RisksPrepayment of Fixed Rate MortgagesCredit Lines

Valuation of Deposits

The value of the deposits is different from their amount. It is possible to showthat the value to the bank, assuming that volumes are deposited up to time T,is:

V(0,T) =Nn

j=1

T0

EQ[r(t) dj(t)Vj(t)P(0, t)]dt

where:Nis the number of average customers;

n is the number of types of deposits;

Vj(t) and dj(t) are, respectively, the amount and the rate for deposit j attime t;

P(0, t) is the price, at time 0, of a pure discount bond expiring in T.

The value of the deposits can be considered as the value of an exotic swap,paying the floating rate dj(t) and receiving the floating rate r(t), on thestochastic principalVj(t), for the period between 0 and t.


Balance Sheet Items Requiring Statistic-Financial ModelsC

DepositsPrepayment of Fixed Rate Mortgages


16/59


Valuation and Interest Rate Risk: a Practical Example

The table shows the value V(0, T) attime 0 of the transaction deposits. Weassume we have 10 swap contract, ex-

piring from 1 year to 10 years, as hedg-ing insturments. We identify 10 scenarioswhere each swap rate is tilted up 10 bps.The table also shows the values corre-sponding to 10 scenarios and the sensi-tivities to each swap rate. Since we lim-ited our time horizon to 10 years, the

bulk of the exposure is to the 10-yearswap, all other sensitivity being negligi-ble.

Value Sens +10bps0 1994.0571 1994.058 0.00

2 1994.052 - 0.013 1994.044 - 0.014 1994.033 - 0.025 1994.023 - 0.036 1994.013 - 0.047 1994.002 - 0.068 1993.977 - 0.089 1993.911 - 0.15

10 1998.343 4.29


Balance Sheet Items Requiring Statistic-Financial ModelsLi idit d C t t Ri k



17/59


Adding the Risk of Bank Run

The framework can be extended so as to include a possible bank run causing asudden drop in the deposits volumes. This is useful to compute a LiquidityVaR and to operate stress-testing. To that end:

we assume that customers may loose confidence in the financialinstitution, so that they withdraw in a very short time a large percentage

of the deposited volumes;we use as an indicator of the confidence the CDS spread (or a similarspread extracted from bonds issued by the bank), which is modelled by aCIR dynamics, similarly to the instantaneous interest rate;

the bank run is triggered by a high level of the spread, indicating a high

perceived default probability of the bank;the heterogeneity in the customers behaviour is modelled by a Gammadistribution, as for the behaviour determining the allocation amongstdeposits and other investments.





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Liquidity and Counterparty Risksp y g g

Credit Lines

Risk of Bank Run: a Practical Example

The figure plots the evolution of the in-stantaneous zero-spread, assuming a CIRdynamics for the default intensity and arecovery rate of 40% of the face valueon default.The trigger rate has been placed in a nar-

row range around 800 bps of the CDSspread; we assume customers withdraw85% of the deposited volumes in casethey loose confidence in the bank. Thetransaction deposits experience a suddendrop in volumes as the spread climbs to-wards the trigger level.We make a similar assumption of with-drawal for the savings accounts. Alsotheir volumes quickly decreases in abank run scenario.

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

-

0.83

1.67

2.50

3.33

4.17

5.00

5.83

6.67

7.50

8.33

9.17

10.00

CDS spread

0.0

1000.0

2000.0

3000.0

4000.0

5000.0

6000.0

7000.0

-

0.83

1.67

2.50

3.33

4.17

5.00

5.83

6.67

7.50

8.33

9.17

10.00

Trans. Depo

0.0

20000.0

40000.0

60000.0

80000.0

100000.0

120000.0

-

0.83

1.67

2.50

3.33

4.17

5.00

5.83

6.67

7.50

8.33

9.17

10.00

Saving Tot

Savings1

Savings2





19/59

Liquidity and Counterparty RisksCredit Lines

Index

1 Balance Sheet Items Requiring Statistic-Financial ModelsDepositsPrepayment of Fixed Rate Mortgages

Credit Lines

2 Liquidity and Counterparty RisksIntroductionLiquidity Risk Pricing in OTC Derivatives

Haircut Setting



DepositsPrepayment of Fixed Rate MortgagesC di Li


20/59


Facts about Prepayments

Empirical features commonly attributed to mortgage prepayment include:

some mortgages are prepaid even when their coupon rate is below currentmortgage rates;

some mortgages are not prepaid even when their coupon rate is abovecurrent mortgage rates.

prepayment appears to be dependent on a burnout factor.

The model we propose takes into account these features:

mortgagees decide whether to prepay their mortgage at random discreteintervals. The probability of a prepayment decision taken on interest ratereasons, is commanded by a hazard function : the probability that thedecision is made in a time interval of length dt is approximately dt;

besides refinancing for interest rate reasons, the mortgagees may alsoprepay for exogenous reasons (e.g.: job relocation, or sale of the house).The probability of exogenous prepayment is described by a hazard function: this represents a baseline prepayment level, the expected prepaymentlevel when no optimal (interest-driven) prepayment should occur.



DepositsPrepayment of Fixed Rate MortgagesC dit Li


21/59


The Probability of Prepayment

We model the interest rate based prepayment within a reduced form approach.This allows us to include consistently the prepayments into the pricing, theinterest rate risk management (ALM) and the liquidity management.

We adopt a stochastic intensity of prepayment , assumed to follow a CIRdynamics:

dt=[ t]dt+ttdZt

the intensity is common to all obligors and provides the probability thatthe mortgage rationally terminates over time;

the intensity is correlated to the interest rates, so that when rates move tolower levels, more rational prepayments occur;

the framework is stochastic and it allows for a rich specification of theprepayment behaviour;

The exogenous prepayment is also modelled in a reduced form fashion, by aconstant intensity.





22/59

q y p yCredit Lines

The Probability of Prepayment

Consider a mortgage with coupon rate cexpiring at time T:each period, given the current interest rates, the optimal prepaymentstrategy determines whether the mortgage holder should refinance;

for a given coupon rate c, considering also transaction costs, there is acritical interest rates level r such that if rates are lower (rt


23/59

Credit Lines

Prepayment Probability Curves: An Example

The figure plots the prepayment proba-bilities for different times up to the (fixedrate) mortgages expiry, assumed to be in10 years. The three curves refer to:

the exogenous prepayment, given

by a constant intensity = 3.5%;

the rational (interest driven)prepayment, produced assuming0 = 10.0%, = 27%, =50.0% = 10.0%.

the total prepayment when it is

rational to prepay the mortgage(rt < r).

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

1 2 3 4 5 6 7 8 9 10

Exogenous

Rational

Total





24/59

Credit Lines

Modelling Losses upon Prepayment

It is possible to compute the Expected Loss on Prepayment, defined as

the Expected Loss at a time tkgiven that the decision to prepay (forwhatever reason) is taken between tk1 and tk:

ELoP(tk) =E

PP(tk1, tk)max

j

P(tk, tj)jAj1(c c); 0.

the computation of the ELoP is unfortunately not possible in a closedform formula: this can be a problem since the ELoP has to be computedfor all the possible exercise dates for a mortgage, and this for a likely largenumber of mortgages;

considering the fact that we are interested also in the computation of thesensitivities of the ELoP, for hedging purposes, the computation via

numerical techniques, such as Montecarlo, can be very time-consumingand unfeasible;

for this reason we developed an analytical approximation to model:

the future fair mortgage rate c,the correlation between the fair rate c and the rational prepaymentintensityt;





25/59

C

Modelling Losses upon Prepayment

We are now able to measure which is the expected loss occurring to thebank upon prepayment, at each possible prepayment date tk;

We define the current value, at time t0, of all the expected losses is theTotal Prepayment Cost (TPC) related to a mortgage:

TPC(t0) =

kP(t0, tk)ELoP(tk)

the TPC is the quantity to be hedged. It is a function of:

the Libor forward rates,the volatilities of the Libor forward rates,

the stochastic rational prepayment intensity tand constantexogenous intensity ;

the TPC can be also included in the mortgage pricing when calculatingthe fair rate c.





26/59

Hedging Prepayment Exposures

The model allows also to compute sensitivities to the main underlying riskfactors:

sensitivity to Libor rates can be computed by tilting each forward a givenamount (e.g.: 10 bps), and then recomputing the TPC;

Libor exposures can be translated to swap rates exposures, since these arethe most liquid and easily tradable hedging tools;

by the same token, we can compute also the sensitivities to Liborvolatilities: these exposures can be hedged by trading caps&floors;

the sensitivity to the prepayment, both rational and exogenous, can be

derived, but no market instrument exists to hedge this exposure. In thiscase, a VAR-like approach can be adopted and the unexpected costincluded into the fair rate, or economic capital posted to cover this risk.





27/59

A Practical Example

Assume 1Y Libor forward rates and theirvolatilities are those in the table besides.Assume also that the exogenous prepay-ment intensity is 3% p.a. and the rationalprepayment intensity has the same dy-namics parameters as presented above.

We consider a 10Y mortgage, with afixed rate paid annually of 3.95%. Thefair rate has been computed wihout tak-

ing into account any prepayment effect(also credit risk is not considered, al-though it can be included within theframework). The amortization scheduleis in the table besides.

Yrs Fwd Libor Vol1 3.50% 18.03%2 3.75% 18.28%3 4.00% 18.53%4 4.25% 18.78%5 4.50% 18.43%6 4.75% 18.08%7 5.00% 17.73%

8 5.25% 17.38%9 5.50% 17.03%10 5.75% 16.78%

Yrs Notional1 100.002 90.003 80.004 70.005 60.006 50.007 40.008 30.009 20.00

10 10.00





28/59

A Practical Example: The EL and the ELoP.

Given the market and contract dataabove, we can derive theEL at each pos-sible prepayment date, which we assumeoccurs annually. It is plotted in the fig-ure. The closed form approximation hasbeen employed to compute the EL.In a similar way it is possible to calculate

the ELoP. We use also in this case ananalytical approximation that allows fora correlation between interest rates andthe rational prepayment intensity.In the figure the ELoP is plotted for the0 correlation case and for a negative cor-relation set at 0.8. This value impliesthat when interest rates decline, the de-fault intensity increases.Since the loss for the bank is bigger whenthe rates are low, the ELoP in this caseis higher than the uncorrelated case.

-

0.10000

0.20000

0.30000

0.40000

0.50000

0.60000

0.70000

0.80000

0.90000

1 2 3 4 5 6 7 8 9

Expected loss

-

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

0.0700

0.0800

0.0900

1 2 3 4 5 6 7 8 9

ELoP Zero Corr

ELoP Negative Corr





29/59

A Practical Example: Hedging the Prepayment Risk

The hedging of the prepayment risk (or,of the TPC) is possible with respect tothe interest rates and to the Libor for-ward volatilities. The TPC is 48 bps.The first table shows the sensitivity ofthe TPC to a tilt of 10 bps for eachforward rate. Those sensitivities are then

translated in an equivalent quantity ofswaps, with an expiry form 1 year to 10years, needed to hedge them.

The second table shows the Vega of theTPC with respect to the volatilities ofeach Libor forward rate. Those exposurescan hedged with caps&floors, or swap-tions in the Libor Market Model settingwe are working in (by calibrating the Li-bor correlation matrix to the swaptionsvolatility surface).

Yrs Sensitivity Hedge Qty1 0.02 16.822 0.02 9.073 0.02 5.584 0.01 3.455 0.01 2.196 0.01 1.387 0.01 0.92

8 0.00 0.649 0.00 0.40

Yrs Vega1 0.082 0.203 0.33

4 0.455 0.536 0.547 0.488 0.339 0.12





30/59

Liquidity Management

We have shown how the framework canbe used to measure and hedge the pre-payment risk in its form of replacementcost.

The model can be also use to project ex-pected cash flows due to the prepaymentactivity.Since the rational prepayment is stochas-tic and correlated to the level of the in-terest rate, a VaR-like approach can beadopted also in this case to calculate the

maximum and minimum amount of cashflows;

Expected Cash Expected Amort.Cash Flows Flows Amort.

22.34 13.95 81.61 90.0020.96 13.56 64.08 80.0017.59 13.16 49.44 70.0014.15 12.77 37.78 60.0011.28 12.37 28.55 50.009.13 11.98 21.08 40.007.56 11.58 14.81 30.006.39 11.19 9.35 20.00

5.48 10.79 4.46 10.00

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

1 2 3 4 5 6 7 8 9

Amort.Exp. Amort.





31/59

Index

1 Balance Sheet Items Requiring Statistic-Financial ModelsDepositsPrepayment of Fixed Rate Mortgages

Credit Lines


Haircut Setting





32/59

Introduction

Loan commitments or credit lines are the most popular form of banklending representing a high percent of all commercial and industrial loansby domestic banks.

Loan commitments allow firms to borrow in the future at terms specifiedat the contracts inception.

The model we propose simple, analytically tractable approach thatincorporates the critical features of loan commitments observed inpractice:

random interest rates;multiple withdrawals by the debtor;

impacts on the cost of liquidity to back-up the withdrawals;interaction between the probability of default and level of usage ofthe line.





33/59

The model

The bank has a portfolio ofm different credit lines, each one with a givenexpiry Ti (i= 1, 2, ...,m).

Each credit line can be drawn within the limit Liat any time tbetweentoday (time 0) and its expiry.

We assume the there is a withdrawal intensity indicating which is the used

percentage of the total amount of the line Liat a given time t:from each credit line i, the borrower can withdraw an integerpercentage of its nominal: 1%, 2%, . . . , 100%;each withdrawal is modelled as a jump from a Poisson distribution(one specific to each credit line). This distribution can not have morethan 100 jumps. As an example, a 3% withdrawal is equivalent forthe Poisson process to jump 3 times;the withdrawal intensity i(t) determines the probability of thejumps during the life of the credit line. This intensity is stochastic(so we have a doubly stochastic Poisson process).





34/59

Stochastic Withdrawal Intensity

The stochastic intensity allows to model:

the documented correlation between the worsening of thecreditworthiness of the debtor and the higher usage of the amount Liof the credit line;the correlation between the probabilities of default of the debtors of

the m credit lines.

Both effects have a heavy impact on the single and joint distributions ofthe usage of the credit lines.

The liquidity management at a portfolio level is enhanced, since the bankcan properly take into account the joint distribution

The correlation between debtors determines in a more precise fashion thevalue of credit lines and the credit VaR, relying on a framework morerobust and consistent than simple credit conversion factors.





35/59

Stochastic Intensity

The stochastic withdrawal intensity i(t), for the i-th debtor, is a combinationof three terms:

i(t) =i(t)( i(t) +Di (t))

A multiplicative factor i(t) of a deterministic function of time i(t),which can be used to model the withdrawals of the credit line independentfrom the default probability of the debtor.

A multiplicative factor i(t) of the stochastic default intensity Di (t)

(default is modelled as a rare event occurring with this intensity).

We model the correlation between debtor by assuming that the defaultintensity is the sum of two separate components:

Di (t) =Ii(t) +pi

C(t)




S C


36/59

An Example: Usage Distribution of a Single Credit Line

The default intensity parametersare chosen so that the debtorsprobability of default is about 2%in 1 year, and it declines onaverage in the future toward along term average of 1.5%.

The deterministic intensity isconstant and equal to 2%.

The multiplying factor is alsoconstant and equal to = 1000.Given this, we have an averagewithdrawal intensity at time 0equal to 1000 (2% + 2%) = 40

or 40% of the total amount of theline (since each jump correspondsto 1% of usage).

The credit line nominal is equal to5 mln, the expiry is in 1 year.

The figure shows the usage distribution ofthe credit line over a period of 1 year. The

total amount of the credit line is Euro 5mln. The average usage is 2,058,433. Weadded the 99%-percentile of thisdistribution which has the following value:4,050,000.




J i U Di ib i f S l C di Li


37/59

Joint Usage Distribution of a Several Credit Line

In the present framework, the correlation amongst different debtors play agreat role in determining the joint usage distribution.

The pis coefficient weights the dependence of the default intensity on thecommon factor, and then it is an indication of the correlation amongstdebtors.

The higher the correlation (i.e.: pis), the larger th expected usage and the99%-percentile unexpected usage.

Even if for different scenarios (created by different pis), the expectedusage is equal for the single credit lines, the expected joint usage and thepercentile changes with the choice ofpi.




A E l U Di ib i f S l C di Li


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An Example: Usage Distribution of Several Credit Lines

We choose 3 credit lines, 5 mln, each and

two opposite scenarios:First Scenario: we choosepi = 0.999 and calibrate and theparameters of the default intensityso as to have an expected usageof 50%:

Second Scenario: we choosepi = 0.001 and calibrate again and the parameters of the defaultintensity so as to have anexpected usage of 50%:

The default intensity starts at 2%

for each debtor in both scenarios,but the correlation amongst theprobability of defaults and ofdefault events is very different:almost perfect in the first caseand almost nil in the second.

The figure shows the joint usagedistribution, and the 99% precentile, forfirst scenario (in red) and for the secondscenario (in blue). In both scenarios theexpected usage is almost 7.5 mln. The99%-percentile usage is 14, 750, 000 in thefirst scenario and 9, 150, 000 in the secondscenario.



IntroductionLiquidity Risk Pricing in OTC DerivativesHaircut Setting

I d


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Index

1 Balance Sheet Items Requiring Statistic-Financial ModelsDepositsPrepayment of Fixed Rate MortgagesCredit Lines


Haircut Setting




I d


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Index



Haircut Setting




C dit Miti ti A ts


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Credit Mitigation Agreements

Agreements between counterparties aim at limiting and reducing thecounterparty risk

1 Netting (e.g.: ISDA Master Agreement)

a contract that allows aggregation of transactions;in the event of default of one of the counterparties, the entireportfolio included in a netting agreement is considered as a singletrade.

2 Collateral Agreements (e.g.: ISDAs Credit Support Annex (CSA))collateral is required if unsecured exposure is above a given threshold;threshold and frequency depend on counterpartys credit quality.

3 Early termination clauses:

Termination clause: trade-level agreement that allows one (or both)counterparties to terminate the trade at fair market value at a

predefined set of dates;Downgrade provision: portfolio-level agreement that forces thetermination of the entire portfolio at fair market value the first timethe credit rating of one (or either) of the counterparties falls below apredefined level.

4

Certain contracts, like Contingent CDS (CCDS).Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved



Collateral and Margin Agreements


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Collateral and Margin Agreements

Collateral agreement is a contract between two counterparties that

requires one or both counterparties to post collateral (typically cash orhigh quality bonds) under certain conditions.

Margin agreement is a legally binding collateral agreement with specificrules for posting collateral, which include:

1 Minimum transfer amount: defines the minimum amount of

collateral that can be exchanged. If the exposure entails a collateralposting below the minimum, amount, no collateral is provided;

2 A threshold, defined for one (unilateral agreement) or both (bilateralagreement) counterparties. If the difference between the net portfoliovalue and already posted collateral exceeds the threshold, thecounterparty must provide collateral sufficient to cover this excess

(subject to minimum transfer amount);3 Frequency: defines the periodicity of the exposure calculation and of

the determination of the collateral to post.

The terms of the rules depend mainly on the credit qualities of thecounterparties involved.




Index


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Index



Haircut Setting




Pricing OTC Derivatives with CSA


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In a very general fashion, the price at time 0 of a derivatives contractwhich is not subject to counterparty risk is:

V0 =EQ

e

T0 rsdsVT

where

VTis the terminal pay-off of the contract;rt is the (possibly time dependent) risk-free interest rate.

When counterpaty risk is considered, then we have to include the so calledCVA (the expected losses we suffer when on default of the counterparty)the and DVA(the expected losses the counterparty suffers on our default):

VCCP

0 =EQ

e

T0 rsdsVT

CVA+DVA

The terminal value of the contract is still discounted ad the risk-free ratert, but then the price is adjusted with the net effect due to the lossesupon default of the two counterparties involved in the trade.






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Assume now we have a CSA agreement operating between the twocounterparties. The CSA provides for a daily margining mechanism of thefull variation of the NPV (nowadays a very common form of the CSA).The party that owns a positive balance on the collateral account(corresponding to a positive NPV of the contract) pays the rate ct to theother party.

The pricing of the contract can be now be operated by excluding thedefault risk (there is still a very small residual risk between two dailymargining).

It can be shown that the pricing formula is very similar to the standardcase we have seen above, but with the collateral rate ct replacing the

risk-free rate rt.:

VCSA

0 =EQ

e

T0 csdsVT

(3)






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This result is very convenient, since we have a well defined rate that has

to be paid on the collateral balance (set within the contract), whereas therisk-free rate is very difficult to determine in the current marketenvironment (it used to be the Libor in the interbank market).

Usually the daily margined CSA agreements set the remuneration of thecollateral at the EONIA for contracts in euro (or some equivalent OIS rate

for other currencies). EONIA (OIS) rates can be considered the bestapproximation of a risk-free rate.

Nevertheless there is still one assumption that is made when deriving thepricing formula with the CSA:

The rate at which the bank can lend money is the same of the one itcan borrow money.

This assumption can be easily accepted when we price contracts whoseNPV can be replicated by a dynamic strategy.

When the NPV of the contract cannot be replicated (e.g.: forward andswap contracts) then relaxing the assumption is trickier.






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Assume we have a contract whose value during the life of the contract at

any time 0< t


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A Practical Example

We show an example, assuming the following market data for interest rates:

Time Eonia Fwd Spread Fwd Libor

0 0.75% 0.65% 1.40%0.5 0.75% 0.64% 1.39%

1 1.75% 0.64% 2.39%1.5 2.00% 0.63% 2.63%

2 2.25% 0.63% 2.88%2.5 2.37% 0.62% 2.99%

3 2.50% 0.61% 3.11%3.5 2.65% 0.61% 3.26%

4 2.75% 0.60% 3.35%4.5 2.87% 0.60% 3.47%

5 3.00% 0.59% 3.59%5.5 3.10% 0.59% 3.69%

6 3.20% 0.58% 3.78%6.5 3.30% 0.58% 3.88%

7 3.40% 0.57% 3.97%7.5 3.50% 0.57% 4.07%

8 3.60% 0.56% 4.16%8.5 3.67% 0.56% 4.23%9 3.75% 0.55% 4.30%

9.5 3.82% 0.55% 4.37%10 3.90% 0.54% 4.44%

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Euribor Fwd

Eonia Fwd



Introduction


A Practical Example


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A Practical Example

Market data for caps&floors and swaptions volatilities are:

Caps&FloorsExpiry Volatility

0.5 30.00%1 40.00%

1.5 45.00%2 40.00%

2.5 35.00%

3 32.00%3.5 31.00%4 30.00%

4.5 29.50%5 29.00%

5.5 28.50%6 28.00%

6.5 27.50%

7 27.00%7.5 26.50%8 26.00%

8.5 25.50%9 25.50%

9.5 25.50%10 25.50%

SwaptionsExpiry Tenor Volatility

0.5 9.5 27.95%1 9 28.00%

1.5 8.5 27.69%2 8 27.09%

2.5 7.5 26.61%

3 7 26.32%3.5 6.5 26.16%4 6 26.02%

4.5 5.5 25.90%5 5 25.79%

5.5 4.5 25.68%6 4 25.57%

6.5 3.5 25.46%

7 3 25.37%7.5 2.5 25.28%8 2 25.22%

8.5 1.5 25.21%9 1 25.34%

9.5 0.5 25.50%10 0



Introduction


A Practical Example: Collateralized Swap


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We price under a CSA agreement areceiver swap whereby we we pay theLibor fixing semi-annually (set at theprevious payment date) and we re-ceive the fixed rate annually. Withmarket data shown above, the fair

rate can be easily calculated (we areusing the new market standard ap-proach to employ the EONIA/OIScurve for discounting and the 6M Li-bor curve to project forward rates).We assume also that we have to pay

a funding spread of 15bps over theEONIA/OIS curve. This is applied tothe ENE plotted beside.

FVA -0.0512%Fair Swap rate 3.3020%Swap Rate + Coll. Fund 3.3079%Difference 0.0059%

ENE

(7.0000)

(6.0000)

(5.0000)

(4.0000)

(3.0000)

(2.0000)

(1.0000)

-

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5



Introduction




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act ca a p e Co ate a ed S ap

We may be interested in calculat-

ing the impact of the liquidity of acollateralized swap with respect to amore conservative measure than theENE, similarly to what happens inthe counterparty risk management.We choose the Potential Future Ex-

posure, which is the expected nega-tive NPV of the swap at a given levelof confidence, set in this example atthe 99% and computed with marketvolatilities.The funding spread is still 15bps overthe EONIA/OIS curve. The PotentialFuture Exposure (blue line), and theENE(purple line, same as before) forcomparison, are plotted beside.

FVA -0.2156%Fair Swap rate 3.3020%Swap Rate + Coll. Fund 3.3264%Difference 0.0244%

(30.0000)

(25.0000)

(20.0000)

(15.0000)

(10.0000)

(5.0000)

-

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

PFE

ENE



Introduction


Index


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Haircut Setting



Introduction


Bonds as Collateral


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Collateral can be also made of bonds, usually of good credit quality.

In this collateralization transforms counterparty credit risk into market riskand issuers credit risk, which should be relatively small for the collateralto be effective.

Even when the collateral itself can be defaultable (e.g.: in corporatebonds or emerging currencies sovereign bonds) the counterparty credit risk

is strongly mitigated.

Common practices have appeared to manage the intrinsic risk ofcollateral: marking to market and haircuts.

The problem is now to determine the fair level of the haircut on a bond,given a chosen mark to market period.

We outline a simplified framework, with no mark-to market periods awhere the haircut can be set only once.

The approach can be extended to a portfolio of bonds posted as collateral,but we will not pursue the analysis that further in the current discussion.



Introduction


A Framework to Set Haircuts


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Assume that at time ta bank has a fixed exposure E to a givencounterparty.

The bank asks the counterparty for units of a bond expiring in Tb withprice B(t,Tb) as collateral.

We assume that the time to maturity of the collateral is greater than theexpiration date of the contract, originating the exposure (e.g.: loan),between the bank and the counterparty. The contract expiry is Ta

For the collateral pledged, there exists a haircut h.Let us divide the interval [t,Ta] in m periods of equal extension.

Assume now that our counterparty goes defaulted in tm, the end of one ofthe m periods, in which case the bank has to recover the loss on theexposure by selling the collateral bond in the market. Since the collateral

bond can go defaulted also, the loss the bank still suffers after the sellingis:LGD(tm) =E [B(tm,Tb)1B>tm +RB1tm1Btm ]

where B is the default time of the collateral bond and RB is its recoveryon default. We also assume that the default of the counterparty isindependent from the default of the bonds issuer.



Introduction




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The expected loss can be computed by considering the PD of both thecounterparty and the bonds issuer.

We want to compute the maximum loss given a confidence level, say 99%,if the counterparty goes defaulted.

We assume that the interest rates are modelled by a CIR model of thekind:

dr=r(r r)dt+rr dZt

and thePDis produced by a stochastic intensity still with a CIR dynamics:

d= ( )dt+dWt

At the end of each period tm the minimum value of the bond isdetermined after deriving the maximum level of the interest rate and ofthe default intensity at the 99% c.l., given it is still alive in tm.

In case the collateral bond is in default, the recovery value must beconsidered. The collateral value in tm is then the expected value in thetwo states of the world.



Introduction




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In the case we consider we have to set the level of the haircut once for all

at the beginning of the contract, so that we do not revise the quantity ofcollateral bond after its market value changes.

Since counterpartys default can happen anytime during the periodbetween [t,Ta], we will weight the possible values of the collateral bondsat the several times tiby the probability of default between [tm1, tm] overthe total default probability over the entire contracts period:

wm = PDC(tm1, tm)

PDC(t,Ta)

The expected maximum loss over [t,Ta] is

ELGD(tm) =m

i=1

E [B(ti,Tb)(1 PDB(t, ti))

+RB(PDB(t, ti1) PDB(t, ti))]wm



Introduction


A Practical Example


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Assume we have an exposer E originatedby a contract expiring in Ta = 6M and

the bank requires as collateral a bondexpiring in Tb= 10Y.The term structure of the interest ratesand thePDs of the issuers are generatedby a CIR short rate and a CIR default in-tensity processes with the following pa-rameters:

r0 1.00%r 0.75r 4.50%r 25.00%

0 0.50% 0.75 1.00% 25.00%

Years 1Y Libor PD1 2.05% 0.65%2 3.34% 1.47%

3 3.91% 2.38%4 4.17% 3.33%5 4.28% 4.31%6 4.33% 5.30%7 4.35% 6.30%8 4.36% 7.31%9 4.36% 8.34%

10 4.37% 9.37%



Introduction


A Practical Example


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The collateral bond has the characteris-tics shown in the table below so that itsmarket price at time t0 is 100.4675.We divide the contact period of 6months in 6 monthly intervals and com-pute the minimum value of the bond atthe end of each period, considering alsothe occurrence of default and the recov-

ery. We weight these values as describedabove so that we derive the quantity capable to match the exposure, andthen we can set the haircut on the mar-ket value of the bond. The table besideshows all calculations.

Face Value 100Expiry 10

Coupon 4.50%Frequency 1Recovery 40%

PD Bond

Month Cpt Price Recov.

0 0.00% 100.47 0.001 0.25% 97.94 0.042 0.52% 96.56 0.133 0.79% 95.38 0.154 1.07% 94.20 0.135 1.35% 93.42 0.08

6 1.64% 92.93 0.02

Min Coll Wed Coll

Month Value Weight Value

0 100.471 97.99 15.49% 15.182 96.69 16.03% 15.503 95.53 16.51% 15.774 94.33 16.95% 15.995 93.49 17.34% 16.21

6 92.95 17.69% 16.44Av. Coll. Value 95.08

1.05Fair Haircut 5.66%



Introduction


About Iason


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Iason is a company created by market practitioners, financial quants and programmers

with valuable experience achieved in dealing rooms of financial institutions.Iason offers a unique blend of skills and expertise in the understanding of financialmarkets, in the pricing of complex financial instruments and in the measuring and themanagement of banking risks. The companys structure is very flexible and grants afully bespoke service to our Clients.Iason believes that the ability to develop new quantitative finance approaches throughresearch as well as to apply those approaches in practice, is critical to innovation in riskmanagement and derivatives pricing. It brings into all the areas of the risk managementa new and fresh approach based on the balance between rigour and efficiency Iasonspeople aimed at when working in the dealing rooms.Besides tailor made services, Iason offers software applications to calculate and monitorcredit VaR and conterparty VaR, fund transfer pricing and loan pricing, liquidity-at-risk.

cIason - 2012

This is a Iasons creation.

The ideas and the model frameworks described in this presentation are the fruit of the intellectual efforts and of the skills of the peopleworking in Iason. You may not reproduce or transmit any part of this document in any form or by any means, electronic or mechanical,including photocopying and recording, for any purpose without the express written p ermission of Iason ltd.


interrelations amongst liquidity, market and credit risks

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