金融风险管理课件：Chapter 12 Value at Risk and Expected Shortfall

1、Chapter 12Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 2015Value at Risk and Expected Shortfall1The Question Being Asked in VaR“What loss level is such that we are X% confident it will not be exceeded in N business days?”Risk Management and Financial Institutions

2、 4e, Chapter 12, Copyright John C. Hull 20152VaR and Regulatory CapitallRegulators have traditionally used VaR to calculate the capital they require banks to keeplThe market-risk capital has been based on a 10-day VaR estimated where the confidence level is 99%lCredit risk and operational risk capit

3、al are based on a one-year 99.9% VaRRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 20153Advantages of VaRlIt captures an important aspect of riskin a single numberlIt is easy to understandlIt asks the simple question: “How bad can things get?” Risk Management and F

4、inancial Institutions 4e, Chapter 12, Copyright John C. Hull 20154Example 12.1 (page 257)lThe gain from a portfolio during six month is normally distributed with mean $2 million and standard deviation $10 millionlThe 1% point of the distribution of gains is 22.3310 or $21.3 millionlThe VaR for the p

5、ortfolio with a six month time horizon and a 99% confidence level is $21.3 million. Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 20155Example 12.2 (page 258)lAll outcomes between a loss of $50 million and a gain of $50 million are equally likely for a one-year pr

6、ojectlThe VaR for a one-year time horizon and a 99% confidence level is $49 millionRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 20156Examples 12.3 and 12.4 (page 258)lA one-year project has a 98% chance of leading to a gain of $2 million, a 1.5% chance of a loss

7、of $4 million, and a 0.5% chance of a loss of $10 millionlThe VaR with a 99% confidence level is $4 million lWhat if the confidence level is 99.9%?lWhat if it is 99.5%?Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 20157Cumulative Loss Distribution for Examples 12.

8、3 and 12.4 (Figure 12.3, page 258)Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 20158VaR vs. Expected Shortfall lVaR is the loss level that will not be exceeded with a specified probabilitylExpected shortfall (ES) is the expected loss given that the loss is greate

9、r than the VaR level (also called C-VaR and Tail Loss)lRegulators have indicated that they plan to move from using VaR to using ES for determining market risk capitallTwo portfolios with the same VaR can have very different expected shortfallsRisk Management and Financial Institutions 4e, Chapter 12

10、, Copyright John C. Hull 20159Distributions with the Same VaR but Different Expected Shortfalls Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 2015VaRVaR10Coherent Risk Measures (page 260)lDefine a coherent risk measure as the amount of cash that has to be added to

11、 a portfolio to make its risk acceptablelProperties of coherent risk measurelIf one portfolio always produces a worse outcome than another its risk measure should be greaterlIf we add an amount of cash K to a portfolio its risk measure should go down by KlChanging the size of a portfolio by l should

12、 result in the risk measure being multiplied by llThe risk measures for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were mergedRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201511VaR vs Expected S

13、hortfalllVaR satisfies the first three conditions but not the fourth onelES satisfies all four conditions.Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201512Example 12.5 and 12.7lEach of two independent projects has a probability 0.98 of a loss of $1 million and

14、0.02 probability of a loss of $10 millionlWhat is the 97.5% VaR for each project?lWhat is the 97.5% expected shortfall for each project?lWhat is the 97.5% VaR for the portfolio?lWhat is the 97.5% expected shortfall for the portfolio?Risk Management and Financial Institutions 4e, Chapter 12, Copyrigh

15、t John C. Hull 201513Examples 12.6 and 12.8lA bank has two $10 million one-year loans. Possible outcomes are as followslIf a default occurs, losses between 0% and 100% are equally likely. If a loan does not default, a profit of 0.2 million is made.lWhat is the 99% VaR and expected shortfall of each

16、projectlWhat is the 99% VaR and expected shortfall for the portfolioRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201514OutcomeProbabilityNeither Loan Defaults97.5%Loan 1 defaults, loan 2 does not default1.25%Loan 2 defaults, loan 1 does not default1.25%Both loans

17、 default0.00%Spectral Risk Measures1.A spectral risk measure assigns weights to quantiles of the loss distribution2.VaR assigns all weight to Xth percentile of the loss distribution3.Expected shortfall assigns equal weight to all percentiles greater than the Xth percentile4.For a coherent risk measu

18、re weights must be a non-decreasing function of the percentilesRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201515Normal Distribution AssumptionlWhen losses (gains) are normally distributed with mean m and standard deviation s Risk Management and Financial Instit

19、utions 4e, Chapter 12, Copyright John C. Hull 201516)1 (2ES)(VaR212XeXNYsmsmChanging the Time HorizonlIf losses in successive days are independent, normally distributed, and have a mean of zeroRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201517TTTESday -1 ESday -

20、T VaRday -1VaRday -ExtensionlIf there is autocorrelation r between the losses (gains) on successive days, we replace by in these equationsRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201518T1322)3(2)2(2) 1(2TTTTTrrrrRatio of T-day VaR to 1-day VaR (Table 12.1, pa

21、ge 266)Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 2015T=1T=2T=5T=10T=50T=250r=01.01.412.243.167.0715.81r=0.05 1.01.452.333.317.4316.62r=0.11.01.482.423.467.8017.47r=0.21.01.552.623.798.6219.3519Choice of VaR ParameterslTime horizon should depend on how quickly

22、portfolio can be unwound. Regulators are planning to move toward a system where ES is used and the time horizon depends on liquidity. (See Fundamental Review of the Trading Book)lConfidence level depends on objectives. Regulators use 99% for market risk and 99.9% for credit/operational risk.lA bank

23、wanting to maintain a AA credit rating might use confidence levels as high as 99.97% for internal calculations.Risk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201520Aggregating VaRsAn approximate approach that seems to works well iswhere VaRi is the VaR for the ith

24、segment, VaRtotal is the total VaR, and rij is the coefficient of correlation between losses from the ith and jth segmentsRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201521rijijjiVaRVaRVaRtotalVaR Measures for a Portfolio where an amount xi is invested in the it

25、h component of the portfolio (page 268-270)lMarginal VaR: lIncremental VaR: Incremental effect of the ith component on VaRlComponent VaR:ixVaRiixxVaRRisk Management and Financial Institutions 4e, Chapter 12, Copyright John C. Hull 201522Properties of Component VaRlThe component VaR is approximately the same as the incremental VaRlThe total VaR is the sum of the component VaRs (Eulers theorem)lThe component VaR therefore provides a sensible way of allocating VaR to different activitiesRisk Management and Fi