P2.T5. Market Risk Measurement & Management. Kevin Dowd, Measuring Market Risk, 2nd Edition

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1 P2.T5. Market Risk Measurement & Management Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM

2 Dowd Chapter 3: Estimating Market Risk Measures ESTIMATE VAR USING A HISTORICAL SIMULATION APPROACH ESTIMATE VAR USING A PARAMETRIC APPROACH FOR BOTH NORMAL AND LOGNORMAL RETURN DISTRIBUTIONS ESTIMATE THE EXPECTED SHORTFALL GIVEN P/L OR RETURN DATA

3 Dowd Chapter 3: Estimating Market Risk Measures Estimate VaR using a historical simulation approach. Estimate VaR using a parametric approach for both normal and lognormal return distributions. Estimate the expected shortfall given P/L or return data. Define coherent risk measures. Estimate risk measures by estimating quantiles. Evaluate estimators of risk measures by estimating their standard errors. Interpret QQ plots to identify the characteristics of a distribution. Estimate VaR using a historical simulation approach. Historical simulation (HS) is the simplest way to estimate value at risk (VaR). The basic HS approach involves two steps: 1. Order (sort) the daily profit/loss observations, 2. Locate the loss corresponding to the specified confidence level; e.g., 95%, 99% Using Dowd s data, assume our historical window contains 200 daily P/L observations, as displayed in the second column below (but where they are unsorted). The third column simply orders (sorts) the observations from highest to lowest; e.g., the best day produced a daily gain of +5,985; the worst day produced a daily loss of -3,039. Portfolio P/L Percentile or Observation Number Unsorted Ordered Confidence Level VaR not displayed % VaR The VaRs correspond to the specified confidence level: The 99.0% VaR is 2,524 because that is the 3rd worst loss of 2,524 (the VaR is a loss but expressed as a positive typically) The 98.0% VaR is 2,466 because that is the 5 th worst loss of -2,466 3

4 If we have (n) observations, according to Dowd, the 95% VaR is the (n*5% + 1)th highest observation. For example, Assume we have n = 1,000 loss observations and we want the 95.0% confident VaR: we know there are 50 observations in the 5.0% tail, and we can assume the VaR to be the 51 st -highest loss observation. Estimate VaR using a parametric approach for both normal and lognormal return distributions. Under the assumption that profit/loss is normally distributed, the VaR at confidence level alpha (α; please note Dowd uses alpha to denote confidence whereas elsewhere we typically use alpha to denote significance!) is given by: Normal VaR VaR For example, given a mean of 10% and volatility of 20%, the 95% normal VaR is given by: z P / L P / L Mean 10% Standard Deviation 20% Confidence Level (CL) 95% Normal deviate % VaR 22.90% Lognormal VaR The lognormal VaR is given by: VaR P 1 exp t 1 R R z For example, given the same mean of 10% and volatility of 20%, the 95% lognormal VaR is given by: Mean 10% Standard Deviation 20% Confidence Level (CL) 95% Normal deviate % VaR 20.46% 4

5 Estimate the expected shortfall given P/L or return data. The expected shortfall (ES) is the probability-weighted average of tail losses. Put another way, the ES is the expected loss conditional on the loss exceeding VaR. Expected shortfall given P/L or (ordered) return data To continue using Dowd s dataset, we now include an additional (final) column for the expected shortfall (ES): In the case: Portfolio P/L Percentile or Observation Number Unsorted Ordered Confidence Level VaR Expected Shortfall (ES) not displayed The 99.0% Expected Shortfall (ES) is 3,013 because this is the average of the two worst losses: 3,013 = (2, ,039)/2 The 98.0% Expected Shortfall (ES) is 2,764 because that is the average of the four losses in the 2.0% tail Unlike Value at Risk (VaR), which is a quantile and can be ambiguous; ES is a conditional average is not ambiguous: In the above case, the 99.0% VaR can be selected as the 3 rd worst (2,524 as shown because that is Dowd s preference) or as the 2 nd worst (2,988 which is Jorion s preference) or as an interpolation between the 3 rd and 2 nd. There are actually three valid 99.0% VaRs, at least. However, the 99.0% Expected Shortfall (ES) has only one correct answer: as a conditional average, it is the average of the 1.0% tail. There is only one correct expected shortfall. Expected shortfall can also be estimated as the average of tail VaRs The fact that the ES is a probability-weighted average of tail losses implies that we can estimate ES as an average of tail VaRs. The easiest way to implement this approach is to slice the tail into a large number n of slices, each of which has the same probability mass, estimate the VaR associated with each slice, and take the ES as the average of these VaRs. 5