The VaR Measure Chapter 8 Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 1 The 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 2e, Chapter 8, Copyright John C. Hull 2009 2 VaR and Regulatory Capital Regulators

base the capital they require banks to keep on VaR The market-risk capital is k times the 10day 99% VaR where k is at least 3.0 Under Basel II, capital for credit risk and operational risk is based on a one-year 99.9% VaR Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 3 Advantages of VaR It captures an important aspect of risk in a single number It is easy to understand It asks the simple question: How bad can things get?

Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 4 Example 8.1 (page 159) The gain from a portfolio during six month is normally distributed with mean $2 million and standard deviation $10 million The 1% point of the distribution of gains is 22.3310 or $21.3 million The VaR for the portfolio with a six month time horizon and a 99% confidence level is $21.3 million. Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 5

Example 8.2 (page 159) All outcomes between a loss of $50 million and a gain of $50 million are equally likely for a one-year project The VaR for a one-year time horizon and a 99% confidence level is $49 million Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 6 Examples 8.3 and 8.4 (page 160) A one-year project has a 98% chance of leading to a gain of $2 million, a 1.5%

chance of a loss of $4 million, and a 0.5% chance of a loss of $10 million The VaR with a 99% confidence level is $4 million What if the confidence level is 99.9%? What if it is 99.5%? Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 7 Cumulative Loss Distribution for Examples 8.3 and 8.4 (Figure 8.3, page 160) Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 8 VaR vs. Expected Shortfall VaR

is the loss level that will not be exceeded with a specified probability Expected shortfall is the expected loss given that the loss is greater than the VaR level (also called C-VaR and Tail Loss) Two portfolios with the same VaR can have very different expected shortfalls Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 9 Distributions with the Same VaR but Different Expected Shortfalls VaR VaR Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009

10 Coherent Risk Measures (page 162) Define a coherent risk measure as the amount of cash that has to be added to a portfolio to make its risk acceptable Properties of coherent risk measure If one portfolio always produces a worse outcome than another its risk measure should be greater

If we add an amount of cash K to a portfolio its risk measure should go down by K Changing the size of a portfolio by l should result in the risk measure being multiplied by l The risk measures for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were merged Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 11 VaR vs Expected Shortfall VaR satisfies the first three conditions but not the fourth one Expected shortfall satisfies all four conditions.

Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 12 Example 8.5 and 8.7 Each of two independent projects has a probability 0.98 of a loss of $1 million and 0.02 probability of a loss of $10 million What is the 97.5% VaR for each project? What is the 97.5% expected shortfall for each project? What is the 97.5% VaR for the portfolio?

What is the 97.5% expected shortfall for the portfolio? Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 13 Examples 8.6 and 8.8 Two $10 million one-year loans each has a 1.25% chance of defaulting. All recoveries between 0 and 100% are equally likely. If there is no default the loan leads to a profit of $0.2 million. If one loan defaults it is certain that the other one will not default. What is the 99% VaR and expected shortfall of each

project What is the 99% VaR and expected shortfall for the portfolio Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 14 Spectral Risk Measures A spectral risk measure assigns weights to quantiles of the loss distribution VaR assigns all weight to Xth quantile of the loss distribution Expected shortfall assigns equal weight to all quantiles greater than the Xth quantile For a coherent risk measure weights must be a non-decreasing function of the

quantiles Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 15 Normal Distribution Assumption The simplest assumption is that daily gains/losses are normally distributed and independent with mean zero It is then easy to calculate VaR from the standard deviation (1-day VaR=2.33s) The T-day VaR equals T times the one-day VaR Regulators allow banks to calculate the 10 day VaR as 10 times the one-day VaR Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009

16 Independence Assumption in VaR Calculations (Equation 8.3, page 166) When daily changes in a portfolio are identically distributed and independent the variance over T days is T times the variance over one day When there is autocorrelation equal to r the multiplier is increased from T to T 2(T 1)r 2(T 2)r 2 2(T 3)r 3 2rT 1 Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 17 Impact of Autocorrelation: Ratio of N-day

VaR to 1-day VaR (Table 8.1, page 204) T=1 T=2 T=5 T=10 T=50 T=250 1.0 1.41 2.24

3.16 7.07 15.81 r=0.05 1.0 1.45 2.33 3.31 7.43 16.62 r=0.1 1.0

1.48 2.42 3.46 7.80 17.47 r=0.2 1.0 1.55 2.62 3.79

8.62 19.35 r=0 Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 18 Choice of VaR Parameters Time horizon should depend on how quickly portfolio can be unwound. Bank regulators in effect use 1-day for market risk and 1-year for credit/operational risk. Fund managers often use one month Confidence level depends on objectives. Regulators use 99% for market risk and 99.9% for credit/operational risk.

A bank wanting to maintain a AA credit rating will often use confidence levels as high as 99.97% for internal calculations. Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 19 VaR Measures for a Portfolio where an amount xi is invested in the ith component of the portfolio (page 168-169) Marginal VaR: VaR xi

Incremental VaR: Incremental effect of the ith component on VaR Component VaR: VaR xi xi Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 20 Properties of Component VaR The

component VaR is approximately the same as the incremental VaR The total VaR is the sum of the component VaRs (Eulers theorem) The component VaR therefore provides a sensible way of allocating VaR to different activities Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 21 Back-testing (page 169-171)

Back-testing a VaR calculation methodology involves looking at how often exceptions (loss > VaR) occur Alternatives: a) compare VaR with actual change in portfolio value and b) compare VaR with change in portfolio value assuming no change in portfolio composition Suppose that the theoretical probability of an exception is p (=1X). The probability of m or more exceptions in n days is n n! k n k p ( 1 p )

k m k!( n k )! Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 22 Bunching Bunching occurs when exceptions are not evenly spread throughout the back testing period Statistical tests for bunching have been developed by Christoffersen (See page 171) Risk Management and Financial Institutions 2e, Chapter 8, Copyright John C. Hull 2009 23