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### Enterprise Risk Management Formula Book 9. Risk measures

9.1          Value-at-Risk (VaR)

If  is a (continuous) random variable (e.g. an outcome) with pdf  then the Value-at-Risk at confidence level  (e.g. 95%, 99%, 99.5%) is defined as:

If  has cdf  with an inverse cdf, i.e. quantile function  then . Sometimes signs are inverted and  and  are swapped around when defining .

The relative VaR of  relative , e.g. of an active equity portfolio versus a benchmark portfolio is usually taken to mean the VaR of the random variable . However, for relative returns there are several alternative ways in which we can define the equivalent of , see definition of tracking error.

9.2          Tail Value-at-Risk (TVaR)

The Tail Value-at-Risk (also called the conditional Value-at-Risk, CVaR) is generally defined as the value of the loss conditional on it being worse than the VaR at confidence level , so is defined as:

A coherent risk measure is one that satisfies subadditivity, monotonicity, homogeneity and translational invariance. If losses follow a continuous probability distribution then TVAR is a coherent risk measure.

Occasionally TVaR (less commonly CVaR) is ascribed the same meaning as expected shortfall, in which case the  factor is ignored, or is defined relative to some specific limit  that in effect defines the  to be used in the above formula.

9.3          Expected shortfall (ES)

The expected shortfall, ES, and expected policyholder deficit, EPD are usually defined as follows:

Expected policyholder deficit:

where  and  is often but not always the policyholder wealth

Expected shortfall:

Or more generally the expected shortfall below some trigger level  is

Sometimes expected shortfall is ascribed the same meaning as is given above for TVaR.

9.4          Expected worst outcome (EWO)

The expected value of the worst outcome in  (non-overlapping) observations is:

where the integral is -dimensional,  and the joint distribution  involves  independent marginal distributions each with pdf . This type of risk measure can also be extended to, say, ’th worst outcome,  with  as defined above being the special case where .

9.5          Tracking error (TE)

If  is a random variable (e.g. a portfolio return) with (assumed forward looking) pdf  then its ex-ante tracking error (if it exists) is  where

Nearly always  is here the relative return of a portfolio  of exposures versus a benchmark portfolio  and the tracking error is then normally expressed as a percentage of the total portfolio value. The tracking error of   versus  is then  (more precisely,  for a time period indexed by ) where if the future returns on the ’th instrument during this are  and relative returns are calculated arithmetically (i.e. using  an arithmetic difference):

where  is the vector of active positions.

If the  have covariance matrix  with elements  then .

However, returns compound rather than add through time so for non-infinitesimal time period lengths there are alternative and potentially preferable ways of defining relative returns, including (if we are trying to calculate the return  relative to , each expressed as fractions) using geometric relative returns, i.e. , or logarithmic relative returns, i.e.  rather than arithmetic relative returns .

If a factor structure is assumed for the  then this normally involves assuming that:

where  is the exposure (beta) of the ’th instrument to the ’th factor and  are residual (idiosyncratic) components.

A portfolio described by a vector of (active) weights  then has an expected return of  and an (expected) future tracking error as follows, where  is the matrix formed by  and  is the covariance matrix between the factors

If  is the length of time (time horizon) to which the ex-ante tracking error relates and returns in individual time periods of length  are assumed to be independent of each other then (assuming e.g. we measure returns logarithmically and that the portfolio and benchmark remain unchanged through time) we can apply the square-root of time adjustment to derive ex ante tracking errors applicable to different time periods, i.e.:

A portfolio’s ex-post tracking error is derived from past observed values of its returns and might then be either a sample standard deviation or (perhaps less accurately, but slightly lower) the corresponding ‘population’ standard deviation.

9.6          Drawdown

If a portfolio has exhibited past returns  over the previous  time periods (which could be days, weeks, months, years etc., where  is earlier than  etc.) then the portfolio’s drawdown at time  is usually defined to be  (if negative). Its maximum drawdown is usually defined as  (if negative). Its cumulative maximum drawdown (i.e. peak-to-trough) at time  is usually defined by creating an index  such that  and then determining at time  the maximum of  for all  and .

9.7          Marginal VaR

The overall outcome of a portfolio of exposures containing a (constant) amount  of exposure to the ’th risk where each risk involves an random outcome  (technically  is the value ascribed to the random outcome) is . Strictly speaking combining exposures in this manner requires that the way in which we ascribe a financial value to an outcome satisfies the axioms of uniqueness, additivity and scalability, i.e. that , the value we ascribe to an outcome should be unique and should satisfy .

The VaR of such a portfolio with confidence level  is .

The marginal VaR with confidence level  of the ’th exposure in such a portfolio is:

The contribution to overall VaR of the ’th exposure is then .

If the outcomes are Gaussian (i.e. multivariate normal, say  exposures with  then where:

Here  and . Given the properties of the normal distribution

As tracking error is a special case of VaR (with assumed normal underlying distribution and , ) we can likewise define the marginal tracking error and contribution to tracking error from an individual exposure.

9.8          Incremental VaR

The incremental VaR with confidence level  of the ’th exposure in such a portfolio is:

9.9          Estimating VaR

If the observations are normally distributed then  may be estimated approximately in a parametric manner using . Alternatively it can be estimated (approximately) in a non-parametric manner (if the data does not exhibit temporal dependencies) by taking the observations , say, reordering them so that , say, identifying the ’th order statistic, where  is an integer between 1 and , and estimating the VaR using the ’th order statistic where . Using a binomial distribution, the variance of the ’th order statistic is approximately as follows (where  is the pdf at  and  is the probability of outcome) meaning that estimating the standard error of this non-parametric statistic requires us to estimate :