Risk management Exercises Session 5 «Value at Risk»

Transcription

1 Risk maagemet Exercises Sessio 5 «Value at Risk»

2 «Small Questios» «Problems»

3 Exercise Time horizo & Impact of correlatio Questio Suppose that the chage i the value of a portfolio over a -day time period is ormal with a mea of zero ad a stadard deviatio of $2 millio. ) What is (a) the -day 97.5% VaR, (b) the 5-day 97.5% VaR ad (c) the 5-day 99% VaR? 2) What for aswers b ad c if there is first-order daily autocorrelatio with correlatio parameter equal to 0.6?

4 Exercise Time horizo & Impact of correlatio Theory VaR aswers the followig statemet : We are X percet certai that we will ot lose more tha VaR euros i the ext N Days. The formula of the VaR is (if we suppose losses are ormally distributed) : VaR N X p R c ou

5 Exercise Time horizo & Impact of correlatio Solutio What is (a) the -day 97.5% VaR, (b) the 5-day 97.5% VaR ad (c) the 5-day 99% VaR? The -day 97.5% VaR VaR N 0, , The 5-day 97.5% VaR If you make the assumptio that the daily chages i portfolio value is idepedet idetically distributed, the N-Day VaR is equal to -Day VaR times the square root of N. Which gives : N Day VaR VaR N N Day VaR VaR N N Day VaR N 0,99 N

6 Exercise Time horizo & Impact of correlatio Solutio What for aswers b ad c if there is first-order daily autocorrelatio with correlatio parameter equal to 0.6? Imagie that the correlatio betwee P i ad Pi is for all i. 2 Suppose that the variace of P i is for all i. The usual formula for the variace of the sum of ad is : j The correlatio betwee ad is. N The variace of i is : I our case it gives : As : ij i i j i P P i Pi j N 2 2 i i i j ij i j 2 N 2 N 2 N N P P i i

7 Exercise Time horizo & Impact of correlatio Solutio The variace of the retur of the portfolio is equal to : ,6 230, ,6 2 0, The solutio to questios B) ad C) are ow : b) c) N Day VaR N , , N Day VaR N , ,

8 Exercise 2 Back testig Questio & Theory Suppose that we back test a VaR model usig.000 days of data. The VaR cofidece level is 99% ad we observe 7 exceptios. Should we reject the model at the 5% cofidece level? Use a oe-tailed test. Oe-tailed test : H0 : probability of a exceptio is p H : probability of a exceptio is higher tha p

9 Exercise 2 Back testig Questio & Theory From the properties of the biomial distributio, the probability of the VaR limit beig exactly m is :! k Pr X k p p k! k! From the properties of the biomial distributio, the probability of the VaR limit beig exceeded o m or more days is : k! k Pr X k p p k! k! km k X K X K Pr Pr K! k p k! k! k p k

10 Exercise 2 Back testig Solutio Suppose that we back test a VaR model usig.000 days of data. The VaR cofidece level is 99% ad we observe 7 exceptios. Should we reject the model at the 5% cofidece level? Use a oe-tailed test. I our case it gives : X K X K Pr Pr K! k p k! k! k k 6! k p k! k! 2, 64% 5% p p k k Which meas that the probability of 7 or more exceptios should be rejected at the 5% level.

11 Exercise 3 Back testig Questio & Theory Suppose that we back test a VaR model usig.000 days of data. The VaR cofidece level is 99% ad we observe 5 exceptios. Should we reject the model at the 5% cofidece level? Use Kupiec s two-tailed test. Oe-tailed test : H0 : probability of a exceptio is p H : probability of a exceptio is differet of p

12 Kupiec s two tailed test? Test statistic Exercise 3 Back testig Solutio Follows a Khi-Squared distributio with degree of freedom. If the umber you obtai is higher tha 3,84 you reject the model (at a 95% cofidece level). I the exercise: 000 p % m 5 Which meas : m m m m m m 2l p p 2l m m m m m m 2l p p 2l 2,89 3,84

13 Exercise 4 Back testig Questio Suppose that a -day 97.5% VaR is estimated as $3 millio from observatios. The observatios o the -day chages are approximately ormal with mea 0 ad stadard deviatio $6 millio. Estimate a 99% cofidece iterval for the VaR estimate. The 0,975 quatile of a ormal distributio with mea 0 ad std. deviatio 6 millio is,760 millio. The likelihood to have a value equal to,760 is 0,0097.

14 Exercise 4 Back testig Questio The stadard error of the estimate is: I the exercise : q 0,975 0,975 q f x 0, q q f x 0,358 Kowig that 0,995 quatiles of the ormal stadard distributio are 2,576 ad -2,576 it gives : 3 2,576 0,358 ;3 2,576 0,358

15 «Small Questios» «Problems»

16 Problem Portfolio & VaR Questio Suppose that each of two ivestmets has a 4% chace of loss of $0 millio, a 2% chace of a loss of $ millio, ad a 94% chace of a profit of $ millio. They are idepedet of each other. ) What is the VaR for oe of the ivestmets whe the cofidece level is 95%? 2) What is the expected shortfall whe the cofidece level is 95%? 3) What is the VaR for a portfolio cosistig of the two ivestmets whe the cofidece level is 95%? 4) What is the expected shortfall for a portfolio cosistig of the two ivestmets whe the cofidece level is 95%? 5) Show that, i this example, VaR does ot satisfy the subadditivity coditio whereas expected shortfall does.

17 Problem Portfolio & VaR Solutio ) What is the VaR for oe of the ivestmets whe the cofidece level is 95%? A loss of $ millio exteds from the 94th percetile poit of the loss distributio to the 96th poit. The 95% VaR is therefore $ millio.

18 Problem Expected Shortfall Theory Expected shortfall is the expected loss give that the loss is greater tha the VaR level (also called C-VaR ad Tail Loss) Two portfolios with the same VaR ca have very differet expected shortfalls

19 Problem Portfolio & VaR Solutio 2) What is the expected shortfall whe the cofidece level is 95%? Expected shortfall: "if thigs do get bad, what is the expected loss? The expected shortfall for oe of the ivestmets is the expected loss coditioal that the loss is the 5% tail. Give that we are i the tail there is a 20% (/5) chace that the loss is $ millio ad a 80% chace that the loss is $0 millio. The expected loss is therefore $8,2 millio.

20 Problem Portfolio & VaR Solutio 3) What is the VaR for a portfolio cosistig of the two ivestmets whe the cofidece level is 95%? For a portfolio cosistig of the two ivestmets there is a 0,04*0,04 = 0,006 chace that the loss is $20 millio; There is a 2*0,04*0,02=0,006 chace that the loss is $millio; There is a 2*0,04*0,94=0,0752 chace that the loss is $9millio; There is a 0,02*0,02=0,0004 chace that the loss is $2millio; There is a 2*0,02*0,94=0,0376 chace that the loss is $0; There is a 0,94*0,94=0,8836 chace that the profit is $2millio. VaR = 9 millio

21 Problem Portfolio & VaR Solutio

22 Problem Portfolio & VaR Solutio 4) What is the expected shortfall for a portfolio cosistig of the two ivestmets whe the cofidece level is 95%? Expected shortfall for a portfolio cosistig of the two ivestmets is the expected loss coditioal that the loss is the 5% tail. Give that we are i the tail there is a 0,006/0,05=0,032 chace that the loss is $20 millio; 0,006/0,05=0,032 chace that the loss is $ millio; ad 0,936 chace of a loss of $9millio. Why 93.6%? Loss Probability Coditioal Probability ,006 0, ,006 0, ,0468 0, Expected Shortfall

23 Problem Portfolio & VaR Solutio 5) Show that, i this example, VaR does ot satisfy the subadditivity coditio whereas expected shortfall does. Subadditivity coditio : The risk measure for 2 portfolios after they have bee merged should be o greater tha the sum of their risk measures before they were merged. VaR does ot satisfy the subadditivity coditio because 9>+ However, expected shortfall does because 9,46 < 8,2+8,2

24 Problem 2 VaR approaches Questio Use the values for the NASDAQ composite idex durig the 500 days precedig March 0, Calculate the -day 99% VaR o March 0, 2006, for a $0 millio portfolio ivested i the idex usig: ) The basic historical simulatio approach, 2) The expoetial weightig scheme with λ=0.995, 3) Extreme value theory with u=0.03, 4) A model where daily returs are assumed to be ormally distributed (use both a approach where observatios are give equal weights ad the EWMA approach with λ= 0.94 to estimate the stadard deviatio of daily returs). Discuss the reasos for the differeces betwee the results you get.

25 Problem 2 VaR approaches Solutio ) The basic historical simulatio approach

26 Problem 2 VaR approaches Solutio Whe we use historical simulatio to determie the VaR, we simply take the value of % quatile of the historical cumulative distributio of returs. The VaR is so : VaR 0,

27 Problem 2 VaR approaches Theory 2) The expoetial weightig scheme with λ=0.995 A EWMA is of the form : Which meas : Kowig that : ) ( u ( ) m i m i m i u 0 0 k k k k x x x x x x

28 Problem 2 VaR approaches Theory The total weigths attributed to the last 500 days are : k k0 Which meas that the weigth of the day i is : i

29 Problem 2 VaR approaches Solutio The weights are assiged to each day ad the daily chages are listed from the worst to the best Startig at the worst outcome, weights are summed util the required quatile of the distributio is reached. The weights for daily chages that are worse tha -2,3% add up to just uder 0,0 The -day 99% VaR is therefore VaR 0,

30 3) Extreme value theory with u=0.03 Problem 2 VaR approaches Questio & Theory Suppose F x is the cumulative distributio fuctio of a variable xad that u is a value of x i the right-had tail of the distributio. The probability that x lies betwee u ad u y is F u y F u. Defie F y as F u y u Fu F u y F u gree blue gree A result from Gedeko states that, for a wide class of distributios, coverges to a geeralized Pareto distributio as u is icreased. The geeralized Pareto cumulative distributio is : y G, y Fu y

31 Problem 2 VaR approaches Theory Here is the Pareto distributio fuctio: The formula for the probability desity fuctio is equal to : g, y y

32 Problem 2 VaR approaches Theory If we wat to fid the values of parameters ad, we maximize : u i u i Whe we have the parameters, we wat to estimate the tail of the distributio. The probability thatx u y coditioal that x u is G., y The probability that x u is Fu. The ucoditioal probability that x u y is therefore : x i u xi u l F u G, y

33 Problem 2 VaR approaches Theory If the total umber of observatios is, a estimate of F u is. The ucoditioal probability that x u y is u u y G, y gree bluegree This meas that our estimator of the tail of the cumulative distributio of x is : u x u Fx Now, to compute the VaR it is ecessary to compute F VaR q Which meas : u u VaR u q u VaR u q

34 Problem 2 VaR approaches Solutio 4) A model where daily returs are assumed to be ormally distributed (use both a approach where observatios are give equal weights ad the EWMA approach with λ =0.94 to estimate the stadard deviatio of daily returs). If observatios are give equal weights: Simply compute the stadard deviatio of the returs : VaR is equal to : If we use the EWMA approach with λ =0.94 : Use the coditioal stadard deviatio computed i part 3) : VaR is equal to : VaR ,76% VaR %