Example: Probability ($1 million in S&P 500 Index will decline by more than 20% within a

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1 Value at Risk For a give portfolio, Value-at-Risk (VAR) is defied as the umber VAR such that: Pr( Portfolio loses more tha VAR withi time period t) <!. all this give: - amout of time t, ad - probability level! (cofidece level) - all this uder ormal market coditios! Example: Probability ($1 millio i S&P 500 Idex will declie by more tha 20% withi a year) < 10% meas that VAR = $200,000 (20% of $1,000,000) with! = 0.10, t = 1 year (typically time period is much shorter, expressed i days). VAR is typically a dollar amout, ot %. Value at Risk is oly about Market Risk uder ormal market coditios. VAR is importat because it is used to allocate capital to market risk for baks, uder their Risk Based Capital requiremets. More precisely: The 1988 Bak for Iteratioal Settlemets (BIS) Accord defies how capital held for credit risk is calculated. The 1996 Amedmet distiguishes the followig: - Tradig book: loas ot revalued o a regular basis. - Bakig book: differet istrumets (stocks, bods, swaps, forwards, optios, etc.) that are usually revalued daily. Capital for tradig book calculated usig VaR with N = 10 (tradig days), ad! = 0.01, but usual otatio used is X = 1! " = Resultig VaR is multiplied by a coefficiet k, where k varies by bak, but is at least 3. Coditioal VaR (C-VaR) is defied as the expected loss durig a N-day period, coditioal that we are i the (100 - X)% left tail of the distributio. This cocept overcomes problems with distributios with two peaks, oe of which is i the left tail. But it is much less popular tha VaR (after all, regulators require VaR). A short ote from K.O.: Basic VAR methodologies: - Parametric; - Historical; - Simulatio. How is parametric doe? - Estimate historical parameters: asset returs, variaces ad covariaces, for all asset classes, or assets comprisig the portfolio; - Calculate portfolio expected retur ad stadard deviatio;

2 - Estimate VAR assumig ormal distributio of portfolio retur. Typically assumes ormality ad serial idepedece. Wrog theoretically, but practitioers do ot care. Problems with estimatig parameters, especially volatility. How is historical doe? - Assemble ad maitai historical database; - Use historical data as the future distributio. Also wrog: What if the future is t what it used to be? But if geeralized to the oparametric method of bootstrap (resamplig), may be the best there is. Of course, bootstrap is ot used i practice, because practitioers geerally do ot kow what it is. How is simulatio doe? - Specify distributios of model iput factors, - Use Mote Carlo simulatio for factors, - Combie them ito global outcome, get a probability distributio. - Assumptios o factors crucial. If oe ca get that distributio ideally, this may be a ideal method. Ed of ote. Volatility per year versus volatility per day! yr =! day 252,! day =! yr 252. There are 252 tradig days, ad studies idicate that volatility o o-tradig days is miimal if oexistet. Cosider $10 millio i IBM stock, N = 10 days (two tradig weeks), ad X = 99% cofidece level. Assume daily volatility of 2%, i.e., daily stadard deviatio (SD) of $200,000. Assume successive days returs are idepedet, the over 10 days SD is 10! $200,000 " $632, 456. It is customary to assume i VAR calculatios that the expected retur over period cosidered is 0% (because i practice calculatios are doe over very short periods). It is also customary to assume ormal distributio of returs. Because the 1st percetile of the stadard ormal distributio is (ad the 99 th percetile is 2.33), VAR of this IBM stock portfolio is: 2.33! $632, 456 = $1,473,621. Now cosider a $5 millio i AT&T stock (symbol T). Assume its daily volatility is 1%. The its 10 day SD is $50, Its VAR is 2.33!$50, " $368, 405. Copyright 2007 by Krzysztof Ostaszewski

3 Now combie the two assets i a portfolio ad assume that the correlatio of their returs is 0.7. We have! X +Y =! X 2 +! Y 2 + 2"! X! Y, ad we get! X +Y = $751,665. Thus 10-day 99% VAR of the combied IBM + T portfolio is 2.33! $751,665 = $1, 751, 379. The amout ( $1,473,621 + $368,405)! $1, 751, 379 = $90,647. is the VAR beefit of diversificatio. A liear model Cosider a portfolio of assets such that the chages i the values of those assets have a multivariate joit ormal distributio. Let be the chage i value of asset i i oe day, ad! i be the allocatio to asset i, the for the chage i value of the portfolio!p = #" i i=1 is ormally distributed because of the multivariate joit ormal distributio. Sice E ( ) = 0 is assumed for every i, E (!P) = 0. We also have:! i = Var ("x i ),# ij = Corr "x i,"x j! 2 P = ## " ij $ i $ j! i! j i=1 j =1 ( ) ad the 99% VAR for N days is: 2.33! P N. How bods/iterest rates are hadled Duratio gives!p = "D # P #!y. Let! y be the yield volatility per day. What is it? Oe way to look at it: SD of!y. The! P = D " P "! y. Aother way of lookig at it: SD of!y, where y is the zero coupo bod yield for y maturity D. The!P = "D # P # y #!y y, so that! P = D " P " y "! y. Ca iclude covexity i this approach (i additio to duratio), but this still does ot accout for oparallel yield curve shifts. Copyright 2007 by Krzysztof Ostaszewski

4 Cash flow mappig Alterative approach i hadlig iterest rates, dealig with the problem of ot havig data o volatility of most bods, as most bods are rarely traded. But data o certai Treasuries of stadard maturities is available due to their frequet tradig, especially othe-ru Treasuries. I this approach, we use prices of zeros with stadard maturities (1/12 year, 0.25 year, 1, 2, 5, 7, 10, 30 year) as market variables. Note that ay Treasury ca be stripped ito a packet of zeros. The mappig procedure is illustrated by this example: Cosider a $1 millio Treasury maturig i 0.8 years, with 10% semi-aual coupo. It ca be viewed as a 0.3 year $50K zero plus 0.8 year $1050K zero. Suppose that the rates ad zero prices are as follows: 3 mos. 6 mos. 1 year zero yield 5.50% 6.00% 7.00% zero price volatillity 0.06% 0.10% 0.20% (% per day) Assume the followig daily returs correlatios 3 mos. 6 mos. 1 year 3 mos mos year What is the rate for 0.80 years? We iterpolate betwee 0.5 ad 1.0 years ad get yield of 6.60%. If you iterpolate daily volatility, you get 0.16%. We ow try to replicate volatility of the 0.8 year zero with 0.5 year zero ad 1 year zero, by a positio of! i the 6 mos. zero ad 1 -! i the 1 year zero. Matchig variaces we get the followig equatio: = ! ( 1"! ) # 0.7 # 0.001# #! ( 1"! ) This is a quadratic equatio which gives! = The 0.8 year zero is worth $1,050K = $997, Copyright 2007 by Krzysztof Ostaszewski

5 ! is the portio of this amout, i.e., ($997,662) = $319,589 which is allocated to the six moths zero, ad the rest, i.e., ($997,662) = $678,073 is allocated to the oe year zero. Do the same calculatios for the 0.3 year $50K zero. This is how that calculatio goes. The 0.25-year ad 0.50 year rates are 5.50% ad 6.00%, respectively. Liear iterpolatio gives the 0.30-year rate as 5.60%. The preset value of $50,000 received at time 0.30 years is $50,000 = $49, The volatilities of 0.25-year ad 0.50-year zero-coupo bods are 0.06% ad 0.10%per day, respectively. Usig liear iterpolatio we get the volatility of a 0.30-year zerocoupo bod as 0.068% per day. Assume that! is the value of the 0.30-year cash flow allocated to a 3-moth zero-coupo bod, ad 1 -! is allocated to a six-moth zerocoupo bod. We match variaces obtaiig the equatio: = ! ( 1"! ) # 0.9 # # 0.001#! ( 1"! ) which simplifies to 0.28! 2 " 0.92! = 0. This is a quadratic equatio, ad its solutio is:! = " " 4 # 0.28 # = # 0.28 This meas that a value of ! $49, = $37,397 is allocated to the three-moth bod ad a value of !$49, = $11,793 is allocated to the six-moth bod. This way the etire bod is mapped ito positios i stadard maturity zero Treasury bods. Sice we are give the volatilities ad correlatios of those bods, ad sice we are assumig zero retur i a short period of time, we just take the portfolio as mapped, ad calculate its stadard deviatio. The portfolio cosists of $678,073 i 1-year bod, $11,793 + $319,589 = $331,382 i 0.5-year bod, $37,397 i 0.25-year bod. The 10 day 99% VAR of the bod is the 2.33 times 10 times the SD calculated. Whe the liear model ca be used The liear model starts with the equatio!p = #" i i=1 Copyright 2007 by Krzysztof Ostaszewski

6 which meas that the chage i the value of the portfolio is a liear fuctio of the chages i the values of the uderlyig. This is ot the case for may derivatives, especially optios. Is there a case whe derivatives ca be hadled with the liear model? Here are some examples. Assets deomiated i foreig currecy ca be accommodated, by measurig them i U.S dollars. Forward cotract o a foreig currecy ca be regarded as a exchage of a foreig zero coupo bod maturig at cotract maturity for a domestic zero maturig at the same time. Iterest rate swap: it ca be viewed as the exchage of a floatig rate bod for a fixed rate bod. The floater ca be regarded as a zero with maturity equal the ext reset date. Thus this is a bod portfolio ad ca be hadled by the liear model. Whe the portfolio cotais optios, the liear model ca be used as a approximatio. Cosider a portfolio of optios o a sigle stock with price S. Suppose the delta of the portfolio is!, so that:! " #P #S!S. Defie!x =. The!P " S#!x. If the portfolio S cosists of may such istrumets, we get!p " # S i $ i, which is essetially the liear model. Example. A portfolio cosists of optios o IBM, with delta of 1,000 ad optios o T, with delta of 20,000. You are give IBM share price of $120 ad T share price of $30. The!P = 120 "1000 "!x " 20,000 "!x 2 = 120,000!x ,000!x 2. If daily volatility of IBM is 2% ad daily volatility of T is 1%, with correlatio 0.70, the stadard deviatio of!p is ( 120! 0.02) 2 + ( 600! 0.01) 2 + 2!120! 0.02! 600! 0.01! 0.7 = 7,869. The 5th percetile of stadard ormal distributio is -1.65, ad so the 5 day 95% VAR is 1.65! 5! 7,869 = $29,033. A quadratic model Whe a portfolio icludes optios, its gamma! ( ) should be icluded i the aalysis. We have!p = "!S # (!S)2, ad with!x =!S S, we ca write this as!p = S"!x S2 # (!x) 2. i=1 Copyright 2007 by Krzysztof Ostaszewski

7 The problem is that!p is ot ormally distributed, although we assume s,!x ~ N 0," 2 ( ).The momets of!p are ad E (!P) = 1 2 S2 "# 2, E ((!P) 2 ) = S 2 " 2 # S 4 $ 2 # 4, E ((!P) 3 ) = 9 2 S 4 " 2 #$ S6 # 3 $ 6. We ca preted that!p is ormal ad fit a ormal distributio to the first two momets. The alterative is to use Corish-Fischer expasio. For a portfolio i which each istrumet depeds o oe market variable, we get 1!P = " S i # i + " S 2 i $ i ( ) 2. i=1 i=1 2 If pieces of the portfolio ca deped o more tha oe variable the we get a more complicated picture: 1!P = " S i # i + "" S i S j $ ij!x j, 2 i=1 i=1 i=1 where! ij = "2 P "S i S j. This ca be used to estimate momets of!p. Mote Carlo Simulatio Oe day VAR calculatio 1.Value the portfolio today i the usual way usig the curret values of market variables. 2. Sample oce from the multivariate ormal probability distributio of the s. 3. Use the values of the s that are sampled to determie the value of each market variable at the ed of oe day. 4. Revalue the portfolio at the ed of the day i the usual way. 5. Subtract the value calculated i step oe from the value i step four to determie a sample!p. 6. Repeat steps two to five may times to build up a probability distributio for!p. VAR is the calculated as the appropriate percetile of the probability distributio so obtaied. A alterative approach, lowerig the umber of calculatios is to assume that 1!P = " S i # i + "" S i S j $ ij!x j, i=1 i=1 i=1 2 ad skip steps 3, 4 above. This is called a partial simulatio. Copyright 2007 by Krzysztof Ostaszewski

8 Historical Simulatio Create a database of daily movemets of all market variables for several years. Use the database as the probability distributio (the book says first day i database is first day i your simulatio, ad so o, that s ot really true, you simulate from the empirical distributio give by the sample).!p is the calculated for each simulatio trial, ad the empirical distributio of!p has the percetiles determiig VAR. Stregth: o artificial assumptio of ormal distributio Weakesses: - Limited by data set available (that s ot really true, you ca do bootstrap/resamplig, but the book says so, thus you must remember), both i legth of time ad data availability. - Sesitivity aalysis difficult. - Ca t use volatility updatig schemes (volatility chages over time, but schemes have bee desiged to accout for that). Stress testig ad back testig - Stress testig: Estimatig how the portfolio would have performed uder the most extreme market coditios. For example: five stadard deviatio move i a market variable i a day. This is ext to impossible uder ormal distributio assumptio (happes oce i 7000 years) but i reality it happes about oce every 10 years (ad some researchers say that oe such move i some market happes every year). - Back testig: checkig how well our model did i predictig thigs i the past. For example: how ofte did a oe day loss exceed 1-day 99% VAR. If this happes roughly oe percet of the time, well the we are i busiess. Pricipal compoet aalysis Dealig with correlated market variables the most importat example beig iterest rates at various maturities beig correlated, but ot perfectly correlated. The example i text deals with the yield curve ad factors (variables) developed to model its chages. Variables are called PC1 (Pricipal Compoet 1) through PC10. Iterest rate chages observed o a give day are expressed as a liear combiatio of the factors by solvig a set of te equatios. Iterest rate move for a particular factor is kow as factor loadig. Factor scores are the amouts of the factors i the rate movemet. Importace of the factor is measured by SD of the factor score. Sum of squares of SD of factor scores is the total variace. How does oe use PC aalysis to calculate VAR? I this illustratio, oly two factors are assumed (PC1 ad PC2). Suppose that we have a portfolio with the followig exposures to the iterest rates movemets (chage i portfolio value for a 1 bp rate move i $millios): 1 year rate: +10, 2 year rate: +4, Copyright 2007 by Krzysztof Ostaszewski

9 3 year rate: -8, 4 year rate: -7, 5 year rate: +2 The first factor PC1 has loadigs for these Treasury rates: 1 year: 0.32, 2 year: 0.35, 3 year: 0.36, 4 year: 0.35, 5 year: 0.36 The secod factor PC2 has loadigs for these Treasury rates: 1 year: -0.32, 2 year: -0.10, 3 year: 0.02, 4 year: 0.14, 5 year: 0.17 PC1 measures parallel shifts i the curve, PC2 measures twist of the yield curve (steepeig or becomig flatter). Exposure to PC1 is: 10! ! 0.35 " 8! 0.36 " 7! ! 0.36 = "0.08, Exposure to PC2 is: 10!("0.32) + 4!("0.10) " 8! 0.02 " 7! ! 0.17 = "4.40. If f 1, f 2 are factor scores (i bps) the the chage i the portfolio value is approximated by!p " #0.08 f 1 # 4.40 f 2. The factor scores are assumed ucorrelated ad SD s of factors are give as for PC1 ad 6.05 for PC2. We get the SD of!p as ! ! = The 1 day 99% VAR is calculated as 26.66! 2.33 = This example was chose itetioally to have relatively low exposure to parallel shifts (PC1) versus twists (PC2) so that usig stadard duratio aalysis would result i sigificat error. Key ideas i PC aalysis is to replace depedet variables drivig returs (such as Treasury rates) by ucorrelated pricipal compoet factors. Cosider a positio cosistig of $1,000,000 ivestmet i asset X ad $1,000,000 ivestmet i asset Y. Assume that the daily volatilities of both assets are 0.1% ad that the correlatio coefficiet betwee their returs is What is the 5-day 95% Value at Copyright 2007 by Krzysztof Ostaszewski

10 Risk for this portfolio, assumig a parametric model with zero expected retur? The 95 th percetile of the stadard ormal distributio is The stadard deviatio of the daily dollar chage i the value of each asset is $1,000. The variace of the portfolio s daily chage is: ! 0.3!1000!1000 = 2,600,000. The stadard deviatio of the portfolio s daily chage i value is the square root of 2,600,000, i.e., $1, The stadard deviatio of the five-day chage i the portfolio value is: $1,612.45! 5 = $3, The 95 th percetile of the stadard ormal distributio is Therefore (assumig zero mea), the five-day 95% Value at Risk is: 1.645! $3, = $5,931. A pesio pla has a positio i bods worth $4 millio. The effective duratio of the portfolio is 3.70 years. Assume that the yield curve chages oly i parallel shifts ad that the volatility of the yield (stadard deviatio of the daily shift size) is 0.09%. Use the duratio model for estimatig volatility of the portfolio ad estimate the 20-day Value at Risk for the portfolio. The 90 th percetile of the stadard ormal distributio is 1.282, assume the parametric VAR model. The duratio model says!b = "D # B #!y, where B is the bod portfolio value, D is the effective duratio, ad y is the yield. We kow that D = 3.70, ad that the stadard deviatio of!y is 0.09%. Thus the stadard deviatio of the retur of the portfolio!b = "D #!y is (0.09%)(3.70) = %. The portfolio value is $4 millio. The B stadard deviatio of its daily chage i value is $4,000,000(0.3332%) = $13,320. The 90 th percetile of the stadard ormal distributio is 1.282, ad thus our estimate of the 20-day 90% Value at Risk is: $13,320! 20!1.282 = $76,367. A bak ows a portfolio of optios o the U.S. dollar Poud Sterlig exchage rate. The delta of the portfolio is give as Curret exchage rate is $1.50 per Poud Sterlig. You are give that the daily volatility of the exchage rate is 0.70%. What is the approximate liear relatioship betwee the chage i the portfolio value ad the proportioal chage i the exchage rate? Estimate the 10-day 99% Value at Risk. Give the value of delta, the approximate relatioship betwee the daily chage i the portfolio value,!p, ad the daily chage i the exchage rate,!s, is!p = 56!S. Let Copyright 2007 by Krzysztof Ostaszewski

11 !x be the proportioal daily chage i the exchage rate. The!x =!S 1.5. Therefore!P = 56 "1.5!x = 84!x. The stadard deviatio of!x equals the daily volatility of the exchage rate, i.e., 0.70%. The stadard deviatio of!p therefore is 84(0.70%)= The 10-day 99% Value at Risk is thus estimated as: 0.588! 2.33! 10 = We kow that optio portfolios are ot easily represeted by a liear model. I fact, the portfolio gamma for the previous problem is How does this chage the estimate of the relatioship betwee the chage i the portfolio value ad the proportioal chage i the exchage rate? Calculate a update of the 10-day 99% Value at Risk based o estimate of the first two momets of the chage i the portfolio value. Based o the Taylor series expasio!p = 56 "1.5 "!x "1.52 "16.2 "(!x) 2. This simplifies to!p = 84!x (!x) 2. The first two momets of!p are E!P # $ % ad ( ) = E 1 2 "1.52 "16.2 "(!x) 2 & ' ( = 1 2 "1.52 "16.2 " = E ((!P) 2 ) = " 56 2 " "1.54 " " = The stadard deviatio of!p is 0.346! = We use the mea ad stadard deviatio so calculated ad preted that!p has ormal distributio, fittig a ormal distributio with the same mea ad variace. The te-day 99% Value at Risk is calculated as: 10! 2.33! " 10! = Assume that the daily chage i the value of a portfolio is well approximated by a liear combiatio of two factors calculated from a pricipal compoets aalysis. The delta of the portfolio with respect to the first factor is 6 ad the delta of the portfolio with respect to the secod factor is - 4. The stadard deviatios of the two factors are 20 ad 8, respectively. What is the 5-day 90% Value at Risk? The factors used i a pricipal compoets aalysis are assumed to be ucorrelated. Therefore, the daily variace of the portfolio is: Copyright 2007 by Krzysztof Ostaszewski

12 6 2! ("4) 2! 8 2 = 15,424. The daily stadard deviatio is the square root of that, i.e., $ Sice the 90 th percetile of the stadard ormal distributio is 1.282, the 5-day 90% value at risk is estimated as: ! 5!1.282 = $ Suppose a compay has a portfolio cosistig of positios i stocks, bods, foreig exchage ad commodities. Assume that there are o derivatives i the portfolio. Explai the assumptios uderlyig (a) the liear model, ad; (b) the historical simulatio model, for calculatig Value at Risk. The liear model: - It assumes that the percetage daily chage i each market variable has a ormal probability distributio. The historical simulatio model: - It assumes that the probability distributio observed for the percetage daily chages i the market variables i the past is the probability distributio that will rule over the ext day (or whatever period is uder cosideratio). Explai how a iterest rate swap is mapped ito a portfolio of zero-coupo bods with stadard maturities for the purposes of Value at Risk calculatios. Whe a fial exchage of pricipal is added i, the floatig side of a swap is equivalet to a zero coupo bod with a maturity date equal to the date of the ext paymet. The fixed side is a regular coupo-bearig bod, which is equivalet to a portfolio of zerocoupo bods. The swap ca therefore be mapped ito a portfolio of zero-coupo bods with maturity dates correspodig to the paymet dates. Each of the zero-coupo bods ca the be mapped ito positios i the adjacet stadard-maturity zero-coupo bods. Explai why the liear model ca provide oly approximate estimates of Value at Risk for a portfolio cotaiig optios. The chage i the value of a optio is ot liearly related to the chage i the value of the uderlyig variables. Whe the chage i the values of uderlyig variables is ormal, the chage i the value of the optio is o-ormal. The liear model assumes Copyright 2007 by Krzysztof Ostaszewski

13 that the distributio cosidered is ormal. Usig it for optios produces oly a approximatio, ad ca produce potetially sigificat error. Suppose that the 5-year rate is 6%, the seve year rate is 7%, both expressed with aual compoudig. Also assume that the daily volatility of a 5-year zero-coupo bod is 0.5%, ad the daily volatility of a 7-year zero-coupo bod is 0.58%. The correlatio coefficiet betwee daily returs of the two bods is Map a cash flow of $1,000 received at time 6.5 years ito a positio i a five-year bod ad a positio i a seveyear bod. The 6.5-year cash flow is mapped ito a 5-year zero-coupo bod ad a 7-year zerocoupo bod. The 5-year ad 7-year rates are 6% ad 7%, respectively. Usig liear iterpolatio we get the 6.5-year rate as 6.75%. The preset value of $1,000 received i 6.5 years is $1, 000 = The volatility of 5-year ad 7-year zero-coupo bods are 0.50% ad 0.58% per day, respectively. We iterpolate the volatility of a 6.5-year zero-coupo bod as 0.56% per day. Assume that the fractio! is allocated to a 5-year zero-coupo bod ad 1 -! is allocated to a 7-year zero-coupo bod. To match variaces we solve the equatio: = ! ( 1 "! ) # 0.6 # 0.5 # 0.58! ( 1"! ) which simplifies to: ! 2 " ! = 0. This is a quadratic equatio with solutio:! = " " 4 # # = # This meas that amout of ! $ = $48.56 is allocated to the 5-year bod ad amout of ! $ = $ is allocated to the 7-year bod. Note that the equivalet 5-year ad 7-year cash flows are $48.56! = $64.98 ad $605.49! = $ A compay has etered ito a six-moth forward cotract to buy 1 millio Poud Sterlig for $1.5 millio. The daily volatility of a six-moth zero-coupo Poud Sterlig bod (whe its price is traslated to U.S. dollars) is 0.06% ad the daily volatility of the sixmoth zero-coupo dollar bod is 0.05%. The correlatio betwee returs from the two bods is Curret exchage rate is Calculate the stadard deviatio of the Copyright 2007 by Krzysztof Ostaszewski

14 chage i the dollar value of forward cotract i oe day. What is the 10-day 99% Value at Risk? Assume that the six-moth iterest rate i both Poud Sterlig ad dollars is 5% per aum with cotiuous compoudig. The cotract if a log positio i a Poud Sterlig bod combied with a short positio i a dollar bod. The value of the Poud Sterlig bod is $1.53! e "0.05!0.5 millio = $1.492 millio. The value of the dollar bod is $1.5! e "0.05!0.5 millio = $1.463 millio. The variace of the chage i the value of the cotract i oe day is, based o the formula forvar( X! Y ), ! ! " " 2! 0.8!1.492! !1.463! = The square root of this quatity, , is the stadard deviatio, i millios of dollars. Therefore, the 10-day Value at Risk is ! 10! 2.33 = $ millio. Casualty Actuarial Society May 2005 Course 8 Examiatio, Problem No. 37 Cosider a ivestmet portfolio cosistig of the followig Asset Market Value Daily Volatilities Alumium $100, % Zic $400, % i) The coefficiet of correlatio is ii) The 99% oe-tailed Z-value is a) Calculate the 15-day, 99% value at risk (VaR) of the portfolio. Assume the chage i portfolio value is ormally distributed. b) Calculate the impact of diversificatio o the portfolio VaR. c) Suppose this portfolio also icludes optios ad the gamma of the portfolio is 10. Without doig ay calculatios, state a alterative method oe could use to estimate VaR. Let us write A ad Z for the two assets, ad use these subscripts to idicate the parameters of the distributios of returs of these two assets. Let us also write P for items referrig to the portfolio. We have! P =! 2 A +! 2 Z + 2 " 0.80 "! A "! B = " 0.80 " 700 " 800 # Therefore, the portfolio 15-day, 99%, VaR is 2.33! ! 15 " 12, O the other had, the idividual VaR s are: Copyright 2007 by Krzysztof Ostaszewski

15 2.33! 700! 15 " for A, ad 2.33! 800! 15 " for Z. Therefore, the diversificatio gai for VaR is , = This is the iterpretatio of diversificatio gai i Hull s book. The CAS model solutio used a differet iterpretatio. Istead, it asked the questio: what would VaR be if the two assets were perfectly correlated? It would be 2.33! ( )! 15 " 13, The gai is therefore, 13, , = , same as before. If we were to icorporate the gamma of the optio i the portfolio, we would fit a quadratic model with!p = " #!S + 1 #$ # (!S 2 )2. Copyright 2007 by Krzysztof Ostaszewski