Value at Risk and Expected Shortfall (12)

Question Being Asked in VaR

"What loss level is such that we are X% confident it will not be exceeded in N business days?"

Back-testing

- (of a VaR calculation) involves looking at how often exceptions (loss > VaR) occurs - Alternatives a. Compare VaR with actual change in portfolio value b. Compare VaR with change in portfolio value assuming no change in portfolio composition - Suppose that the theoretical probability of an exception is p (= 1-X) **Probability of m or more exceptions in n days is: Summation from k=m to n of (n!/k!(n-k)!) x p^k x (1-p)^n-k

Coherent Risk Measures

- A coherent risk measure is 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 lambda should result in the risk measure being multiplied by lambda > 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

Properties of Component VaR

- Approximately the same as incremental VaR - Total VaR is the sum of the component VaR's (Euler's Theorem) - Component VaR therefore provides a sensible way of allocating VaR to different activities

Spectral Risk Measures

- Assigns weights to quantiles of the loss distribution - VaR assigns all weight to the x-th percentile of the loss distribution - ES assigns equal weight to all percentiles greater than the x-th percentile - For coherent risk measure weight must be a non-decreasing function of the percentiles

Advantages of VaR

- Captures important aspect of risk in a single number - Easy to understand (...) - Asks simple question: "how bad can things get" (...)

VaR and Regulatory Capital

- Regulators have traditionally used VaR to calculate the capital they require banks to keep - The market-risk capital has been based on a 10-day VaR estimation where the confidence level is 99% - Credit risk and operational risk capital are based on a one-year 99.9% VaR

Choice of VaR Parameters

- Time horizon should depend on how quickly portfolio can be unwound. **Regulators are planning to move toward a system where ES is used and the time horizon depends on liquidity - Confidence level depends on objectives. **Regulators have used 99% for market risk and 99.9% for credit/operational risk - A bank wanting to maintain an AA credit rating might use confidence levels as high as 99.97% for internal calculations

VaR vs. Expected Shortfall

- VaR is the loss level that will not be exceeded with a specified probability - Expected shortfall (ES) is the expected loss given that the loss is greater than the VaR level (also called C-VaR and Tail Loss) - Regulators have indicated that they plan to move from using VaR to using ES for determining market risk capital - Two portfolios with the same VaR can have very different expected shortfalls

VaR and ES and Conditions of Coherent Risk Measures

- VaR satisfies the first three conditions but not the fourth one - ES satisfies all conditions

Changing the Time Horizon

If losses in successive days are independent, normally distributed, and have a mean of zero T-day VaR = 1-day VaR x sqrt(T) T-day ES = 1-day ES x sqrt(T)

Extension

If there is autocorrelation rho between the losses (gains) on successive days, we replace sqrt(T) by sqrt(T+2(T-1)rho+2(T-2)rho^2+2(T-3)rho^3+...+2rho^T-1) in these equations (where rho is first autocorrelation of daily changes)

Normal Distribution Assumption

Losses (and gains) are normally distributed with mean mew and standard deviation sigma VaR = mew + sigma x Y ES = mew + sigma x [(e^-(Y^2/2))/(sqrt(2 x pi))x(1-X)] where X = confidence level and Y = N^-1(X) (which is calculated in Excel with NORMSINV and X as a decimal)

VaR Measures for a Portfolio Where an Amount x(i) is Invested in the i-th Component of the Portfolio

Marginal VaR = curly d(VaR) / curly d(x(i)) = Delta VaR / Delta x(i) Incremental VaR: Incremental effect of the i-th component on VaR Component VaR = x(i) x curly d(VaR)/curly d(x(i)) = (Delta VaR / Delta x(i)) x x(i)

Bunching

Occurs when exceptions are not evenly spread throughout the back testing period

Altering VaR and ES for new confidence levels given old VaRs and ESs

VaR(X*) = VaR(X) x (N^-1(X*)) / (N^-1(X)) ES(X*) = ES(X) x ((1-X)xe^-((Y*-Y)x(Y*+Y))/2)) / (1-X*)

Aggregating VaRs

sqrt(Summation(i)Summation(j)VaR(i)Var(j)rho(ij)) VaR(i) = VaR for i-th segment rho(ij) = coefficient of correlation between losses from the i-th and j-th segment