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1 Confidential – Highly Restricted Copula Representation of Joint Risk Driver Distribution Why Copulas? Copulas provide a method of joining together individual (marginal) distributions along with a dependence structure. Copulas provide a way to generalize multivariate distributions, but they have the advantage that the input marginals are not all required to belong to the same distribution family. For example: –Suppose we are modeling the dependence between the risk factors OAS and volatility. If the distributions of OAS and volatility are both normal, we can model the joint distribution of OAS and Volatility by a bivariate normal distribution and evaluate the VaR as the tail risk. –This method also works if the risk drivers are from, say, t-distributions having the same degree of freedom (df, or tail weight). –But what if the distributions of the risk drivers are not the same? If, for example, OAS is distributed according to a normal distribution and volatility as a t-distribution? Multivariate distribution theory fails us here. Copulas allow the flexibility of using different marginals to fit the empirical data. The distribution fits the data, instead of forcing the data to fit the distribution just for model tractability.

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2 Example: Meta t Copula Representation of a Multivariate Distribution The linear correlation matrix for the market risk drivers is shown below, with detail of a bivariate meta t copula for the risk drivers OAS and Debt Spread (DS). Construction of the optimal copula requires the following: 1)The inverse distributions resulting from the marginal distributions 2)Specification of the optimal form of the copula, based on variance minimization techniques 3)Estimates for the required parameters of the optimal copula, such as correlation matrix and degree of freedom. As an example, the bivariate meta-t copula was constructed for the risk drivers OAS and DS. Since both risk drivers OAS and DS were distributed as t distributions (17 df for OAS, 6 df for DS), a meta-t copula was constructed using the correlation matrix below. General d-dimension Meta-t copula function: Here d=2, Σ is the covariance matrix and df is a degrees of freedom parameter. In instances where the marginal variables have been standardized, is similar to a correlation matrix

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3 Confidential – Highly Restricted Implementation of the Copula Approach Individual marginal distributions are computed for each risk driver (no change from present approach, but much more complex distributions can now be fit) The inverse distribution of each marginal and correlation matrix are computed (new) The copula is constructed incorporating marginals, fitted parameters and dependence structure (new) The Maximum Likelihood Estimation (MLE) method is used to determine remaining parameters of the optimal copula best fitting the data to describe the joint distribution (new) N random draws (N a sufficiently large number) resulting in d-dimensional realizations of correlated risk drivers are drawn from copula (new) These random draws are transformed back to draws on the original risk drivers and multiplied by the risk weight factors to derive the market value sensitivity (no change from present approach) The joint distribution is simulated and tail risk evaluated at the th percentile (no change from present approach)

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4 Confidential – Highly Restricted New Estimates of Risk Driver Marginal Distributions and their Dependence Structure (starting values for fitting of the copula)

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5 Confidential – Highly Restricted Statistical Significance Tests for Risk Driver Correlation Matrix The significance of the estimated correlation between risk drivers can be tested. Since the joint distribution of the risk drivers is not elliptical, we apply tests based upon Spearman-type correlations. In the following table, Spearman pairwise correlation coefficients appear on and above the diagonal, while p values for testing the null hypothesis (the correlations are equal to zero) appear below the diagonal. The p value is computed by transforming the correlation to a t statistic having n-2 df, where n is the number of data points. The p value may be interpreted as the probability of getting a correlation as large as the observed value by chance, when the true correlation is zero. If p is small (say, less than 0.05), then the correlation i,j is significant.. All of the correlation coefficients are found to be significant at the 95% confidence level, with the exception of PC1 and PC2, which are orthogonal by construction.

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6 Simulated Aggregate Value Change Distribution The aggregate market value change distribution computed via Monte Carlo simulation is shown below. The th percentile is selected as the worst case required economic capital.

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7 Aggregation via Copula Representation of the Multivariate Risk Distribution In the presence of correlation amongst the risk drivers, the aggregate portfolio value change cannot simply be determined based upon the standalone marginal value change distributions. To determine economic capital, we must: –estimate and model the joint realization of risk drivers, and –simulate the aggregate portfolio value change associated with joint realizations of risk driver values. The problem of estimating the total portfolio loss distribution is actually one of estimating the multivariate (with respect to risk type) probability distribution F(r 1,r 2,…,r d ) = Pr(R 1 <=r 1,…,R d <=r d ) Most modeling approaches estimate the marginal risk distributions with respect to individual risk types F 1 (r 1 ), F 2 (r 2 ), etc. The representation of the joint distribution using a copula – a multivariate distribution with support over the unit intervals – promotes direct estimation of risk factor correlations. Copula Representation F(r 1,…,r d | ) = C R (F 1 (r 1 | ),…,F d (r d | )| )

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8 Example Fitting of an Optimal Copula by MLE Likelihood function for Meta t copula The maximum likelihood method is used to determine the copula parameters best fitting the data. The log likelihood function (below) is minimized with respect to the parameters df and. For this example, the df parameter was estimated to be 7, and the estimated covariance matrix is given by

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9 Confidential – Highly Restricted Distribution Plots for Market Risk Distributions

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10 Confidential – Highly Restricted Comparative Plots of Marginal Market Risk Drivers to Fitted Distributions

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11 Example: Meta-t copula, graphed with uniform marginals

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12 Calculation of Expected Shortfall While VaR provides an estimate of the loss most likely to occur at a given confidence level, it provides no insight into how bad that loss might be. For example, one might say The portfolio is not expected to lose more than $2 million 9,997 days out of 10,000, but this statement does not answer the question How bad could the expected loss be, given that a loss does occur? (Note that VaR is not designed to measure the worst loss, in fact, the VaR should be exceeded percent of the time.) The expected shortfall provides the answer to this question. This measures the average of the loss conditional on the fact that the loss is greater than that predicted by VaR. If the VaR number is q, the expected shortfall (also referred to as conditional VaR or CVaR) is: where f(x) is the probability density function of the distribution.

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13 Expected Shortfall Enjoys Desirable Risk Measure Properties 1.Monotonicity If a portfolio has lower returns than another portfolio for every state of the world, its risk measure should be greater. 2.Translation Invariance If we add (subtract) an amount of cash K to a portfolio, the portfolio risk measured in the same units as the cash should decrease (increase) by the amount K. 3.Positive Homogeneity changing the size of a portfolio by a factor ( ) while holding the weighting of the portfolio constituents constant should result in the risk measure being multiplied by. (That is, for all 0, ( X) = (X).) 4.Sub-additivity 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, that is, Mergers do not create additional risk. This is a very important property for a risk measure that is to be used as the basis for allocation of capital to sub- portfolios. The fourth condition states that diversification helps reduce risks. When two risks are aggregated, the total of the risk measures corresponding to the risks should either decrease or stay the same. VaR satisfies the first three conditions, but it does not always satisfy the fourth. Coherent risk measures: Risk measures satisfying all four of the conditions are referred to as coherent. A risk measure can also be characterized by the weights it assigns to quantiles of the loss distribution. –VAR gives a 100% weighting to the Xth quantile and zero to other quantiles. –Expected shortfall gives equal weight to all quantiles greater than the Xth quantile and zero weight to all quantiles below the Xth quantile. –We can define what is known as a spectral risk measure by making other assumptions about the weights assigned to quantiles. A general result is that a spectral risk measure is coherent (that is, it satisfies the sub- additivity condition) if the weight assigned to the qth quantile of the loss distribution is a non-decreasing function of q. Expected shortfall satisfies this condition. –VAR, however, does not because the weights assigned to quantiles greater than X are less than the weight assigned to the Xth quantile.

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