Value at risk and expected shortfall

Value at risk and expected shortfall

1. VaR

   • “We are X percent certain that we will not lose more than V dollars in time T.”
   • Can be calculated from either the probability distribution of gains during time T or the probability distribution of losses during time T.
   • VaR Calculation
     • Gain
       • Loss
   • "How bad things can get?"
     • Drawback of VaR
       • • Many trading strategies give a high probability of good returns and a small probability of a huge loss

2. 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.

3. Expected shortfall

   • ES is a measure that can produce better incentives for traders than VaR
     • “If things do get bad, what is the expected loss?”
     • To calculate ES first we need to know VaR
   • Benefits
     • ES has better properties than VaR in that it always recognizes the benefits of diversification
   • Disadvatages
     • it does not have the simplicity of VaR and as a result is more difficult to understand.
       • It is more difficult to back-test a procedure

4. Coherent Risk measures

   • Suppose we are trying to design a risk measure that will equal the capital a financial institution is required to keep. Is VaR (with an appropriate time horizon and an appro- priate confidence level) the best measure?
     • Properties that risk measure should have:
       • 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.
       • VaR satisfies the first three conditions. However, it does not always satisfy the fourth one.

5. Parameters for VaR and ES

   • 2 parameters:
     • 1. Time horizon
       • 2. Confidence Level
         • Impact of autocorrelation
           • the changes in the value of a portfolio from one day to the next are not always totally independent.
         • depend on a number of factors
           • The confidence level that is actually used for the first VaR or ES calculation is some- times 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.
       • When positions are very liquid and actively traded, it makes sense to use a short time horizon.
         • When VaR or ES is being calculated by the manager of a pension fund, a longer time horizon is likely to be used
   • When the loss in the portfolio value has a mean of μ and a standard deviation of σ,
     • N^-1 : is the inverse cumulative normal distribution
       • NORMSINV in Excel

6. Marginal, Incremental and Component Measures

   • Marginal
     • The marginal value at risk for the ith subportfolio is the sensitivity of VaR to the amount invested in the ith subportfolio
   • Incremental
     • It is the difference between VaR with the subportfolio and VaR without the subportfolio.
   • Component

7. Euler 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.

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