Value at Risk and Expected Shortfall

Value at Risk and Expected Shortfall

1. 12.2 Calculation of VaR

   • Can be calculated as a continuous distribution or a discrete distribution. VaR asks the question: “How bad can things get?”

2. 12.1 VaR (Value at Risk) Definition

   • Statement: “We are X percent certain that we will not lose more than V dollars in time T.” It is a function of two parameters: time horizon (T), and confidence level, X percent.

3. 12.3 Drawback of VaR-

   • When VaR is used in order to limit the risks taken by a trader it can show undesirable results

4. 12.4 Expected Shortfall

   • Expected shortfall is also referred as: conditional value at risk, conditional tail expectation or expected tail loss. ES asks: “If things do get bad, what is the expected loss?” It is a two parameter function: T (time horizon) and X (confidence level). You need VaR in order to calculate ES. Setting a ES limit is better than a VaR limit. ES has better properties in that it always recognizes the benefits of diversification

5. 12.5 Coherent Risk Measures

   • Monotonicity: If a portfolio produces a worse result than another portfolio for every state of the world, its risk measure should be greater.
   • Translation Invariance: If an amount of cash K is added to a portfolio, its risk measure should go down by K.
   • Homogeneity: Changing the size of a portfolio by a factor λ while keeping the relative amounts of different items in the portfolio the same should result in the risk measure being multiplied by λ.
   • Subadditivity: The risk measure for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were merged. (diversification)
   • VaR satisfies the first three conditions. However, it does not always satisfy the fourth one

6. 12.5.1 Exponential Spectral Risk Measures

   • A risk measure can be characterized by the weights it assigns to percentiles of the loss distribution.3 VaR gives a 100% weighting to the Xth percentiles and zero to other percentiles. ES gives equal weight to all percentiles greater than the Xth percentile and zero weight to all percentiles below the Xth percentile

7. 12.6.1 The Time Horizon

   • An appropriate choice for the time horizon, when VaR or ES is calculated, depends on the application, very liquid and actively traded positions are done in a short time horizon, while the manager of a pension fund will calculate VaR or ES with a longer time horizon.

8. 12.6.2 Impact of Autocorrelation

   • In practice, the changes in the value of a portfolio from one day to the next are not always totally independent. The ratio of the T-day VaR (ES) to the one day VaR (ES) does not depend on the daily standard deviation,σ,or the confidence level.

9. 12.6.3 Confidence Level

   • The confidence level that is actually used for the first VaR or ES calculation is sometimes much less than the one that is required.This is because it is very difficult to estimate a VaR directly when the confidence level is very high.
     • If daily portfolio changes are assumed to be normally distributed with zero mean, we can use equations (shown in time horizon part) to convert a VaR or ES calculated with one confidence level to that with another confidence level.

10. 12.7 Marginal, Incremental, and Component Measures

    • Consider the situation where a portfolio has a number of subportfolios. The marginal value of a risk measure (VaR or ES) with respect to the ith subportfolio is the partial derivative of the risk measure with respect to the size of the subportfolio. The incremental VaR (ES) with respect to a particular subportfolio is the incremental effect of that subportfolio on VaR (ES). There is a formula that can be used for dividing VaR (ES) into components that correspond to the positions taken in the subportfolios. The component VaRs (ESs) sum to VaR (ES), and each component is, for a large portfolio of relatively small positions, approximately equal to the corresponding incremental VaR (ES).

11. 12.8 Euler´s Theorem

    • Euler’s theorem allows risk to be decomposed into its components. It is a useful tool in determining risk in what is referred to as risk budgeting. This is concerned with the amount of risk that should be allocated to different components of a portfolio.

12. 12.9 Aggregating VaRs and ESs

    • Sometimes a business has calculated VaRs, with the same confidence level and time horizon, for several different segments of its operations and is interested in aggregating them to calculate a total VaR. A formula for doing this is

13. 12.10 Back-Testing

    • It is a test of how well the current procedure for calculating the measure would have worked in the past. VaR is easier to back-test than ES. Back-testing may indicate weaknesses in a VaR model if the percentage of exceptions (that is, the percentage of times the actual loss exceeds VaR) is much greater or much less than that expected

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