ie590 project team 14 final revised

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Measures of Financial Riskand their usage in Risk Management

Final Project Report

IE590: Financial Engineering

Seongjin Shin | Jish BhattacharyaTae Yong Yoon | Luis Zertuche

August 05, 2014



Measures of Financial Risk and their usage in Risk ManagementTeam 14

08/05/14IE590 Financial Engineering

Abstract:Conditional Value at Risk, an extension of Value at Risk, is nancial mathematical

tool aimed at quantifying and reducing p ossible investment losses. The purpose ofthis p roject is t o study Value a t Risk and Conditional Value a t Risk as a portfolio riskmetrics in management and asset selection. A practical approach was t aken tounderstand the t opic. First a b asket of stocks w as sel ected with diversication inmind: stocks f rom different types of industries, veried to be n egative-to-posivelycorrelated . F or the sel ected stocks, a st atistical validation of the ex pected return waspreformed based on a two-year data se t. On different one-year d ata se t for the sameassets, validated by the p revious step, a Monte Carlo one-day p rice s imulation wasperformed. I n order to est imate price uctuations, the Geometric Brownian Motionmodel was used as the Monte Carlo sampled function. Value at Risk andConditional Value at Risk were es timated and compared from the simulation data;these r esults were used convert the p ortfolio sel ection into a linear p rogrammingoptimization problem, with the effi cient portfolio frontier as t he nal result.

1. Introduction

Events l ike t he g reat r ecession of 2008, the d ot-com bubble o f 2000, a politicalupheaval or weather-related disasters are f orcing investors to give m ore at tention to ri skmanagement. M any investors do not know exactly w hat is in their portfolios, and moreimportant, how those as sets work — or do not work — together by virtue of theircovariance st ructure. For example, consider al l the u nintended risk a n investor i sincurring when assembling a p ortfolio composed from a few different bu t correlatedmutual. What the i nvestor m ay not realize is t hat all funds contained a co nsiderableamount of the sam e st ocks. Instead of diversifying risk, the i nvestor h as co ncentrated it.

A key and intuitive way to mitigate ri sk i ncurred in investment is by diversifying a

portfolio by including a l arge n umber of assets in it, while en suring that we d on’t incurpunitive co sts of maintaining such a large p ortfolio. F inding out the p roportions of our

budget to invest in ewith the aid of quantitative tools. A nother equally important and challenging p roblemis h ow to characterize t he ri sk o f such kind of portfolio. T he ideas s tated above are w hat

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we con sidered an interesting for an a pplied project: We were keen on applying anylearned concepts to real data u sing both quantitative a nd computational tools.

2. Data Collection

We started by assembling a tentative p ortfolio with ve technology stocks an d fourstocks f rom energy and soft-drinks s ectors, aiming at diversication. We v eried suchdiversication by means of ensuring the corr elation coefficients acr oss our 9 st ocksranged from negative t o positive. Daily closing prices w ere co llected for these 9 st ocks i ntwo separate d ata set s: A two-year d ata set for s tatistical parameter va lidation and, one-year d ata set for t he si mulation and risk o ptimization; year 2011- 2012 a nd year 2013

respectively. T he collected st ocks are comprised of: AAPL, INTC, MSFT, SAP, COKE,WMT, XOM, BRK-A, and DDD.

3. Statistical Validation

The (1 – α) * 100% condence interval estimate for t he m ean with standard deviation σunknown is : - tn-1 S/√(n) ≤ µ ≤ + tn+1 S/√ (n) where t n-1 is the cr iticalvalue of the t distribution with n – 1 d egrees of freedom for an area of α /2 in the

upper tail. Two-sided 95% condence intervals on the m ean daily return for theportfolio of 9 stocks w ere com puted in Minitab.

As p er t he Figure 1, the b askets of 9 st ocks h ave a mix o f positive and negativelycorrelated stocks for a w ell-balanced portfolio. As n oted from the m atrix, the t ech stocksare s omewhat positively correlated which raises the overall Value at Risk f or theportfolio. Having energy and Retail stocks i n the p ortfolio, hedges t he o verall risk tosome extent.

From the Figure 2 w ith 95% condence, we can conclude that the mean d ailyreturn falls w ith the 95% Condence i nterval and 0 is i ncluded in the r ange. Moreover,the p -value f or al l the st ocks i n t-tests i s g reater t han a value o f 0.05 or l evel ofsignicance. With a sam ple si ze o f 252, the n ormality assumption is no t overlyrestrictive a nd the u se o f t distribution is l ikely to be a ppropriate. Since t he d ata i snormally d istributed, the exp ected value of sample Standard Deviation follows:

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E(s 2) ~ σ 2

We u se t he exp ected return of 9 stocks from this val idation and supply as aparameter to Monte Carlo Simulation program.

Figure 1: Correlation Matrix o f the P ortfolio (Daily Closing Price 2013 )

Figure 2: Sample Data Analysis with 95% Condence Interval

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4. Theoretical Background and Methodology

We es tablished a d ata p rocessing stream that takes as n number of stocks input for any

time interval and produces an efficient frontier of optimal asset weights for a t arget rate

of return. Before d etailing the i mplementation steps w e w ill describe t he q uantitative

tools an d concepts t hat went into the scr ipting of our data p ipeline.

Geometric Brownian Motion for Price Estimates

First conceived in early twentieth-century France by nancial theory pioneer an d

mathematician Louis Bachelier, Geometric Brownian Motion is a formalism that

attempts t o model uctuations i n asset prices as a s tochastic p rocess, akin to the ran dom

motion of gas p articles. I n essence t he Brownian Motion equation tries t o account for

change in price over a given period of time, by mixing w hat is currently known about

the stock p rice w ith a random process. The ‘known’ or certain component is called drift

(d) and is cal culated using the ex pected value o f the st ock’s p rice h istorical data ( μ data ),

eroded by half of the h istorical variance ( σ2data ):

d = μ data – σ2data /2

The random or uncertain component of the price change is modeled as a sample

from a t arget distribution of prices. What distribution to choose, its i mplications an d

how to properly characterize it, is a w hole o ther t opic b eyond the scop e o f this p roject;

we w ill briey touch back o n it when we el aborate on some of the limitations of this

model. For t his r eport and data p ipeline, we ch oose t o use n ormally-distributed asset

prices as a w orking and still-useful assumption. When put together, both the cer tain and

uncertain components r esult in the GBM equation for t he d aily price:

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Where is the number of time-steps to be simulated, and the function

generates a pseudo random number sampled from the Standard Normal Distribution.

Value at Risk and Conditional Value at Risk

Value at Risk ( VaR ) is a m etric of the incurred risk o f an investment expressed as

the most likely maximum loss of a p ortfolio or an asset give a co ndence interval (CI)

and time horizon. Put in other words, VaR multiplied by the cap ital in the investment

allows investors to calculate the most probable am ount of money they would lose

within the dened time horizon. A n intuitive to way to think about VaR is in terms of

its p osition as i n the d istribution function of returns; a distribution of rate o f returns

serves t his p urpose t oo, if we are i nterested in the n egative rat e o f return instead of the

actual amount of lost dollars. In such scenario of a d istribution of prices o r r ate o f

returns, VaR will be t he f rontier l ine t hat delimits t he C I expressed as t he area u nder t he

curve o f the p robability density function.

Fig. 1) Value at risk f or a C ondence I nterval of 95%

It is i mplied by gure 1 t hat if we w ere t o numerically recreate t he d istribution as

histogram, produced either by a M onte Carlo simulation (as i n this r eport), or from

measured price d ata, the VaR can be interpreted as t he quantile in which t he

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accumulated histogram area-under-the-curve h as r eached the CI percentage. This is true

if we a pproach the VaR line f rontier f rom the ri ght, or i f approached from the l eft, it

corresponds to t he (1 – C I) value. This ki nd formulation of the VaR allows for an easy

computation when following a numeric app roach; one simply needs to sum the areas of

the ext reme h istogram bins an d stop when this val ue is equ al to either CI or (1 – C I).

This i s ex pressed analytically as:

With as probability density function or histogram of the rate of return.

From this VaR denition of risk one apparent problem becomes obvious: A lthough the

VaR frontier q uintile i s t he m ost likely monetary loss or negative rat e o f return, there are

other l ess l ikely scenarios w ith much higher l osses; basically any point to the l eft of the

VaR frontier f alls i n such category of less l ikelihood coupled with higher l oss. T his

specic limitation can partially surpassed if somehow we take into account information

about the w hole “l oss tail” a fter t he VaR frontier t o obtain a m easure o f risk.

This is w hat is achieved by the Conditional Value at Risk (CVaR) metric. CVaR can beconceptualized as t he ex pected value of the p ortfolio or asset losses gi ven that we h ave

surpassed the VaR frontier i n the d istribution. F ormally, is t he co nditional probability

of the ex pected value o f the ‘loosing tail’ given that we h ave exceeded the VaR quantile

in the d istribution:

Where, r is t he p ortfolio or asset rate o f return. From the p oint of view of thehistogram of the probability function, and relevant to the com putational

implementation, CVaR can be se en as t he inner product between t he vector of number

of elements in each bin of the h istogram losing tail, an d the m atching vector of rate o

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histogram; put another way, CVaR is t he w eighted sum of rate o f returns from the

negative innity to -VaR, each one w eighted by its r espective probability.

Consequently, since C VaR takes al l the p ossible l osing-scenarios, it is a m ore

conservative m etric t han VaR, as i llustrated in gure 2 .

Fig. 2) VaR and CVaR for a Condence Interval of 95%

One can realize i ntuitively that CVaR is m ore sen sitive t o extreme ev ents. For example if

we w ere com puting a d istribution with high kurtosis (with ‘fat tails’) our CVaR value

would move m ore and more t o the left of the –VaR quantile line.

Summary of Data Pipeline

Using the v alidated statistical parameters f or t he st ock average d aily return andstandard deviation, a M onte C arlo simulation was u sed to estimate d aily stock prices.Using such simulated data, Value at risk and Conditional Value at Risk were es timatedfor each stock; subsequently mean-C-VAR was u sed as ob jective function in a p ortfoliooptimization problem to select optimal weights an d plot the efficient frontier.

5. Simulation Results

Below we see a c omparison between individual-stock Value at Risk and ConditionalValue at Risk as n egative rate o f returns. Under the VaR and CVaR frontiers w e p lot the

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Figure 4: Individual Value at Risk v s. Conditional Value at Risk RRs

We t hen formulated our portfolio weight selection problem as an optimization

scheme that minims -CVaR. We used l inear programming over various target portfolio

rate o f returns to obtain the efficient frontier of minimum portfolio CVaR. The t arget

portfolio rate of returns ranged from the m inimum to the m aximum individual rates of

return for ou r basket of stocks. Such process i s s ummarized in the following linear

programming in standard form:

minimize [–CVaR] Tw i

subject to [r i] T w i = r p-target

∑w i = 1

w i ≥ 0

Where r p-target is t he portfolio target rate of return, w i is t he v ector o f portfolio weights

and ri is t he v ector of individual rate o f returns. Then, using this f ormula, we rep eated

the o ptimization step varying rp-targe t from min (r i) to max (r i) to construct an efficient

frontier ( Figure 5 ).

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Figure 5: C-VaR Optimization and Efficient Frontier

We learned from reviewing the literaturethat CVaR has superior mathematicalproperties vers us VAR. C-VAR is a so -called “coherent risk measure”; for i nstance, theC-VAR of a p ortfolio is a con tinuous an d convex function with respect to positions in

instruments, whereas the VAR may be even a d iscontinuous function. This is what madethe ab ove op timization step easy a nd convenient.

6. Discussion

Ways t o Improve S tatistical Validation

When evaluating parameters such a s mean and sigma for estimators, there is still a r iskof over tting on the t est set because t he p arameters can be t weaked until the est imators

performs op timally. This w ay the k nowledge about the t est set can interfere w ith modeland evaluation metrics. To address this p roblem, yet another p art of dataset can be h eldout as v alidation set; training proceeds o n the t raining set, after w hich evaluation isdone on the va lidation set, and when the exp eriment seems to be succes sful, nalevaluation can be d one o n the t est set.

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A diversied portfolio that is constructed to have ze ro sensitivity to eachmacroeconomic factor is essen tially risk-free rat e o f interest . If the p ortfolio had a h igherreturn, investors cou ld make a risk-free p rot by borrowing to buy the p ortfolio. If itoffered a l ower return, one can make an arbitrage p rot by selling the d iversied zero-sensitivity portfolio and invest the proceeds t owards T-bills.

In our s tudy, we n oticed that some st ocks w ill be m ore sen sitive to a p articularfactor t han other s tocks. Exxon Mobil would be m ore sen sitive to an oil factor t han say,Coco-Cola w hich is m ore d ependent on underlying corn prices and supplies. If oil factorpicks u p unexpected changes i n oil prices, β will be h igher for Exxon Mobil. To mitigaterisk, we d iversied our portfolio initially from technology to energy and soft drink

stocks an d noticed that the VaR and average r eturn got adjusted with positive a ndnegatively correlated stocks. Ideally, it would have been better t o add more st ocks t o theportfolio from additional sectors s uch as a lternative energy, retail, construction andautomotive sect ors as such. However, to keep our study in scope, the p ortfolio w asrestricted to nine st ocks.

Distribution Assumptions for VaR and CVaR

From inspecting the h istogram plots w ith VaR and CVaR, it’s eas y to realize t hatthese m easures ar e v ery sensitive to the d istribution we ass ume o ur price d ata t o have.One can imagine how the ku rtosis and skewness of the distribution have considerableimpact, as bo th would change t he am ount of area u nder the tails of our histogram plot.

As t he t ail-ends o f the d istribution become l arger, we w ill consistently see o urCVaR increase. This i s al so a good way to illustrate w hy investors conceived CVaR overVaR. Consider the d istributions bel ow in which both have the same VaR but differentCVaR:

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Figure 6: Same VaR, different C-VaR due t o n ature of di stributions.

The m ost striking implication that follows f rom such idea i s t hat we m ight consistentlyunder es timate o r overes timate t he ri sk i n our p ortfolio just by assuming aninappropriate price d istribution.

7. Conclusions

We cr eated a d ata ow that takes d aily stock prices as r aw data i nput and validates

their statistical parameters based on a larger d ata set for the sam e st ocks. Then we u sedthe validated parameters t o simulate o ne-day uctuations in price u sing a M onte Carlosimulation, based o n the Geometric Brownian Motion model of pricing. Fr om theMonte Carlo sam ples, we es timated the future on e-day p rice of our stocks. We al soestimated VaR and CVaR risk m etrics and optimized CVaR to obtain portfolios that liein an efficient frontier.

We learned about the importance of risk m easures as key quantitative informationfor any investor. Specically about VaR and CVaR, we l earned that a st riking limitationof such models i s d istribution. A valuable i nsight was t he real ization that risk m etricsare con tingent upon the p arameters and data that go into them. Two d ifferent indicatorsmight be in disagreement, or be too liberal or t oo conservative, depending on whatinformation we feed to the m odels. Knowing what is inside t he “b lack-box” o f a r isk

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metric m odel is t he best way a rational investor can make t he m ost informed decisionsabout a p ortfolio or asset. .

As p arallel outcome o f this p roject, we i ncreased our kn owledge about robabilisticreasoning, linear op timization and nancial computational tools, which are ex tremelyvaluable t ools for investors gi ven the en ormous grow th of information availablenowadays.

8. References

Valdemar Antonio, D. F. Portfolio management Using Value at Risk: A ComparisionBetween G enetic Algorithms and Particle Swarm Optimization. Erasmus U niversiteit

Rotterdam , 67.Luenberger, D. G. (1998). Investment science . New York: Oxford U niversity Press.

Shephard, N. Markov chain Monte Carlo methods for stochastic vo latility models. Journal of Econometrics , 281-316.

Uryasev, S. Portfolio optimization by minimizing conditional Value at Risk v ianondifferentiable optimization . Computational Optimization and Applications , 391-415.

9. Appendix

Appendix A: Data Pipeline MATLAB script with commentary:%Simulation ParametersstockNames= { 'AAPL' 'INTC' 'MSFT' 'SAP' 'COKE' ' MT' '!OM' '"#K$A' ' ' &loa ata(matnsim=)****+ %num,er o- simulations.con-=*(/0 %con-i ence 1alue 2* 34n,ins=3*** % num,er o- ,ins -or simulation 5isto6ram%com.ute co1ariance matri7 an 1ector o- st e1iationsco1Prices= co18 ail9Prices:+1arPrices = 8 ia68co1Prices::+%com.ute a1era6e ail9 return-or i=3;si<e8 ail9Prices ): -or >=3; len6t58 ail9Prices:$3 ail9##8> i:=8 ail9Prices8>?3 i:$ ail9Prices8> i: : @ ail9Prices8> i:+ enen%com.ute a1era6e ail9## rate o- return 1ector e7.##=mean8 ail9##:+% constant ri-t 1ector = mu $ 1ar@) B5ere mu is E7.( 1alue o- 9earl9 rate o- return% an 1ar is t5e 1ariance on ##

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% 6enerate 3*k sam.les o- a ail9 .rice -or t5e / stocks-or i=3;nsim simPrices8i ;:= ail9Prices8en ;:( e7.8 ri-t ec?st ##( ran n83 /::+en%simulate rate o- return -or simPrices-or i=3;nsim sim##8i ;:=8 ail9Prices8en ;: $ simPrices8i ;::(@ ail9Prices8en ;:+en%.lots simulate .rices o tte' -or i=3;len6t58e7.##: 2nelem8i ;: centers8i ;:4=5ist8sim##8; i: n,ins:+en%-in A# -or esire con-i ence inter1al.1al=3$.con-+.Area=*+ k=*+-or i=3;len6t58e7.##: ,inLen6t5=a,s8centers8i 3:$centers8i )::+ totArea=nsim ,inLen6t5+ B5ile .Area .1al %start -rom t5e le-tmost ,in an a u. area until ci% is reac5e k=k?3+ .Area=sum8nelem8i 3;k:: ,inLen6t5 @totArea+ en %k is t5e in e7 o- t5e last "in ## %com.ute c ar as t5e E7.ecte 1alue o- t5e G A# <oneG un er t5e cur1e %eac5 5isto6ram ## 8centers8:: is Bei65et ,9 its .ro,a,ilt9 an a e rrC ar8i:= centers8i 3;k: 8nelem8i 3;k:'@sum8nelem8i 3;k:::+ % t5is .er-orms t5e Bei65te sum rr ar8i:=centers8i k:+ %5ere Be sim.l9 use K to retrie1e t5e 7 a7is 1alue -or t5e A# rate o- return k=*+ .Area=*+ %reset B5ile con itionsen% noB .lot ar Line -or CI o tte'%c5oose .ort-olio Bei65ts t5at minimi<e C ar

A=$e9e8len6t58e7.##::+,=<eros83 len6t58e7.##::+rtar6et=2min8e7.##:;8ma78e7.##:$min8e7.##::@H*;ma78e7.##:4

Ae =2e7.##+ ones83 len6t58e7.##::4+co1##=co18 ail9##:+-or i=3;len6t58rtar6et: ,e =2rtar6et8i: 34+B8i ;:=lin.ro68$rrC ar A , Ae ,e :+r.8i:=B8i ;: e7.##'+.C ar8i:=B8i ;: $rrC ar'+si6maP8i:=B8i ;: co1## B8i ;:'en% .lot e--icient -rontier o tte'

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ie590 project team 14 final revised

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