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### Enterprise Risk Management Formula Book 13. Miscellaneous

13.1        Combining solvency capital requirements using correlations

A correlation based combination of individual solvency capital requirements involves a formula along the lines of:

13.2        Credit risk modelling

A single factor credit portfolio model generally assumes  where  is the (standardised) factor return/movement,  is the exposure of the ’th obligor to that factor and  is the idiosyncratic noise term for that obligor.

If  is the same for all instruments (e.g. all are assumed to have same correlation with market plus only an idiosyncratic term) then . In such circumstances and if all obligors have the same probability of default  say then probability  that  out of  default is:

If  is the cumulative probability that the percentage loss on the portfolio does not exceed  then in the well diversified limit (Vasicek’s loss distribution):

13.3        GARCH models

Risk models may cater for heteroscedasticity by including GARCH features, e.g. the model might involve formulae along the lines of (in practice  will slowly evolve as additional data is received):

where for, say, a GARCH(1,1) model

RiskMetrics typically uses the following approach for estimating  (often using  for some suitably chosen decay factor ) which can be viewed as an example of a GARCH approach and/or using weighted moments.

13.4        Linear algebra and principal components

Suppose we have  data series (e.g. returns) each with  observations, , that are coincident in time across the different data series. Suppose the  covariance matrix of the (empirical) covariances between the different series is . The eigenvalues and eigenvectors of  are the values of  (scalar) and associated  (vector) for which . An  matrix has  (not necessarily distinct) eigenvalues and associated eigenvectors. Eigenvectors associated with distinct eigenvalues are orthogonal, i.e.  for . Orthonormal eigenvectors have  and  for .  For any distinct eigenvalue the associated orthonormal eigenvector is unique up to a change of sign. If  eigenvalues all take the same value then it is possible to find  orthogonal eigenvectors corresponding to all of these eigenvalues. For empirical covariance matrices,  is symmetric non-negative definite (and positive definite if no two data series are perfectly correlated) and all of its  eigenvalues, , are greater than or equal to zero. One way of telling if a matrix is positive definite is to test whether it is possible to apply a Cholesky decomposition to it.

The eigenvalues and associated eigenvectors of an empirical covariance matrix may be sorted so that . The first principal component is the mixture of the underlying (de-meaned) series, i.e. the , that corresponds to the orthonormal eigenvector, , corresponding to the largest eigenvalue of . This choice of  maximises  subject to ,  Other (lesser) principal components correspond to orthonormal eigenvectors corresponding to smaller eigenvalues.

13.5        Central limit theorem

Suppose we have a series of independent random variables  each with finite (bounded) expected value  and finite (bounded) standard deviation . Suppose  and  are defined as:

Then subject to certain regularity conditions the distribution of  tends asymptotically to  (it is exactly  if each of the  is normally distributed).

13.6        Cornish-Fisher asymptotic expansion

The (4th moment) Cornish-Fisher asymptotic expansion approximates a standardised QQ-plot via the following function:

where  and  are the skew and excess kurtosis of the data.

13.7        Euler capital allocation principle

A function  where  is said to be homogenous of order  if:

Suppose we have  business lines, the outcome (loss) to each business line given its current size is  (a random variable) so the total loss is  where for the current business portfolio the business line allocation is  . Suppose the risk measure used to determine economic capital is  and that it is homogeneous of order 1, i.e. . Then the Euler capital allocation principle (and, in effect, the Marginal VaR or Internal beta approach to setting RAROC rates) allocates total economic capital,  (technically a function of the business portfolio allocation, ) into capital for each business line, , as follows:

13.8        Equiprobable outcomes for a multivariate normal distribution

If  where  then equiprobable scenarios (i.e. contours where  is constant) are ellipsoids defined by  for some constant value of . The probability that  lies within this ellipsoid is given by a chi-squared with  degrees of freedom:

13.9        RAROC, EIC, SHV and SVA

Risk adjusted return on capital (RAROC) is usually defined as follows, where  = Adjusted earnings = Earnings – Interest costexpected lossfunding costother costs and  = capital:

Economic income created (EIC) is usually defined as where  = per unit cost of equity (i.e. hurdle rate):

Shareholder value (SHV) and shareholder value added (SVA) (also known as economic value added, EVA) translate current period return contribution to overall economic value. Given suitable assumptions about future growth prospects for a business, , these are: