Value at Risk (VaR) and Expected Shortfall (ES)

Value at Risk (VaR) and Expected Shortfall (ES)

1. Calculation of measures such as delta, gamma and vega to quantify and manage risks .

   • Risk measures do not provide an indication of the total risk to which you are exposed.
   • Value at risk (VaR) and expected shortfall (ES) are attempts to provide a single number that summarizes the total risk in a portfolio.

2. Value at Risk (VaR)

   • Statement: “We are X percent (confidence level) certain that we will not lose more than V dollars in time T (time horizon).”
   • “How bad can things get?”

3. Expected Shortfall (ES)

   • Also referred to as conditional value at risk, conditional tail expectation, or expected tail loss.
   • “If things do get bad, what is the expected loss?”
   • ES is the expected loss during time T conditional on the loss being greater than the VaR.
   • Advantage: it recognises diversification.
   • Disadvantages: doesn't have the simplicity of VaR and it is more difficult to back-test a procedure for calculating ES than it is to back-test a procedure for calculating VaR.
     • Back Testing
       • It is a test of how well the current procedure for calculating the measure would have worked in the past.
         • The difficulty of back-testing a VaR model increases as the VaR confidence level increases.
       • If daily portfolio changes are independent, exceptions should be spread evenly throughout the period used for back testing. In practice, they are often bunched together, suggesting that losses on successive days are not independent.
         • Bunching

4. Coherent Risk Measures

   • Properties that such a risk measure should have:
     • 1) Monotonicity: If a portfolio produces a worse result than another portfolio, its risk measure should be greater. Meaning if one portfolio always performs worse than another portfolio, it clearly should be viewed as more risky and require more capital.
     • 2) Translation Invariance: If an amount of cash K is added to a portfolio, its risk measure should go down by K. because the cash provides a buffer against losses and should reduce the capital.
     • 3) 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.
     • 4) 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. Refers to the fact that diversification helps reduce risks.When we aggregate two portfolios,the total risk measure should either decrease or stay the same.
       • A risk measure can be characterized by the weights it assigns to percentiles of the loss distribution.
         • VaR gives a 100% weighting to the Xth percentiles and 0% to other percentiles.
           • The weights assigned to percentiles greater than X are less than the weight assigned to the Xth percentile.
             • To this end, measures have been proposed in which the weights assigned to the q-percentile of the loss distribution increase relatively rapidly with q. This is done by making the weight assigned to the q-percentile proportional to e-(1-q)∕γ; where γ is a constant.
               • Exponential Spectral Risk Measures
         • ES gives equal weight to all percentiles greater than the Xth percentile and zero weight to all percentiles below the Xth percentile.
           • The weight assigned to the qth percentile of the loss distribution is a nondecreasing function of q.
   • Risk measures satisfying all four conditions given above are referred to as coherent.
     • ES measure is always coherent.
     • VaR measure is not coherent. It satisfies the first three conditions. However, it does not always satisfy the fourth one

5. Choice of Parameters for VaR and ES

   • When the loss in the portfolio value has a mean of μ and a standard deviation of σ,
     • 1. VaR = μ + σN^(−1)*(X)
       • Where μ (mean), σ (standard deviation), N−1 (inverse cumulative normal distribution), and X (confidence level)
     • For relatively short time horizons, μ is often assumed to be zero. VaR for a particular confidence level is then proportional to σ.
   • When the loss is assumed to be normally distributed with mean μ and standard deviation σ, ES with a confidence level of X is given by
     • 2. ES = μ + σ[(e−Y2∕2) / 2π^(1/2)*(1 − X)]
       • Where Y is the Xth percentile point of the standard normal distribution.
     • This shows that when μ is assumed to be zero, ES, like VaR, is proportional to σ.
   • Time Horizon
     • Liquid and actively traded options = short time horizon
     • When market risks are being considered, analysts often start by calculating VaR or ES for a time horizon of one day. The usual assumption is:
       • T -day VaR = 1-day VaR* T^(2/1)
       • T -day ES = 1-day ES* T^2
   • Impact of Autocorrelation
     • The changes in the value of a portfolio from one day to the next are not always totally independent. Define ΔPi as the change in the value of a portfolio on day i.
     • The variance of ΔPi−1 + ΔPi is: σ^2 +σ^2 +2pσ^2 = 2(1+p)*σ^2
     • The ratio of the T-day VaR (ES) to the one- day VaR (ES) does not depend on the daily standard deviation or the confidence level.
   • Confidence Level
     • 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 1 and 2.
     • VaR(X*) = VaR(X)* [N^(-1)(X*) / N^(-1)(X)]

6. Marginal, Incremental, and Component Measures

   • Analysts sometimes calculate measures of the contribution of each subportfolio to VaR or ES.
   • The marginal value at risk for the ith subportfolio is the sensitivity of VaR to the amount invested in the ith subportfolio.
     • ∂VaR / ∂xi ; where xi= mount invested in the subportfolio
   • Marginal VaR
     • ΔVaR∕Δxi ; where ΔVaR is the increase in VaR
     • For a well-diversified investment portfolio, marginal VaR is closely related to the capital asset pricing model’s beta. If an asset’s beta is high, its marginal VaR will tend to be high, and viceversa. In some circumstances, marginal VaR is negative indicating that increasing the weighting of a particular subportfolio reduces the risk of the portfolio.
     • The incremental value at risk for the ith subportfolio is the incremental effect of the ith subportfolio on VaR. It is the difference between VaR with the subportfolio and VaR without the subportfolio.
       • The component value at risk for the ith subportfolio is
         • Ci = [∂VaR / ∂xi]*xi
           • approximated as
             • [ΔVaR /Δxi]* x i

7. Euler’s Theorem

   • Suppose that V is a risk measure for a portfolio and xi is a measure of the size of the ith subportfolio (1 ≤ i ≤ M).Assume that,when xi is changed to λxi for all xi (so that the size of the portfolio is multiplied by λ), V changes to λV
     • The total VaR for a portfolio is the sum of the component VaRs for the subportfolios.
   • 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.

Images and files in the map