Christopher Cloke-Browne managed the team that developed a ground breaking analytical risk modelling framework. He was co-author on a series of papers that were published in the peer reviewed industry magazine: RISK.
Portfolio modelling and management has come of age. Institutions compute the loss distribution for both credit and operational risk by application of internal models or off-the-shelf commercial products. Most of these rely on simulation techniques. In the first of a series of articles, Richard Martin, Kevin Thompson and Christopher Browne take portfolio modelling a step further, proposing an analytical technique to construct the loss distribution of correlated events and exploring the insights that it brings.
The rapid and accurate construction of the loss distribution, and with it the calculation of Value at Risk (VaR) is at the heart of modern portfolio and capital management. In a previous RISK paper (ABGM 1998) two of the authors mentioned the saddle-point method for constructing a fast and accurate analytical approximation to the tail of the loss distribution for a portfolio of assets that have credit default risk. Other applications of the technique have been suggested in catastrophe, operational and insurance risk. Some rather restrictive assumptions were made in that paper, the most restrictive of which was that default events were assumed to be independent. In this paper we describe the technique in more depth and show how to extend it using a general procedure for modelling dependent events. The saddle-point method is based on the construction of the moment generating function (mgf), which is an ideal tool for analysing the distribution of sums of individual losses, and the way we model correlation allows the mgf to be computed easily.
Richard Martin, Kevin Thompson and Christopher Browne discuss default-rate volatility models in a conditional independence framework. They show that discretisation of the latent variable facilitates comparison of models and makes it clear what is being assumed about the frequency and severity of clusters of default events.
Recent advances in credit risk modelling have led to the development of a variety of credit portfolio risk models: for reviews see Crouhy, Galai & Mark (2000) and Gordy (2000). These have received much public attention from practitioners and generated much academic study, including KMV’s PortfolioManager, JP Morgan’s CreditMetrics, McKinsey & Co’s CreditPortfolioView, and CSFP’s CreditRisk+. Koyluoglu & Hickman (1998) have shown that the existing default correlation models are very similar and can be put in a conditional-independence framework in which default events are independent conditionally on an underlying or latent variable. In this paper we show that discretisation of the latent variable does not significantly alter the results that the models produce, and has the benefit of making clear what is being assumed about the probability and severity of clusters of default events. We also show how to calibrate a discrete latent variable model from historical data. In a companion paper we have shown that the saddle-point method, a fast analytical approximation scheme that is particularly effective in the tail of the loss distribution, can easily be applied to the discrete latent variable model of dependence. Here we provide sample results to illustrate that it provides good approximation without having to run extensive Monte Carlo simulations. Later papers will show that it can be used to analytically derive the risk contributions. Finally we discuss different ways of measuring correlation, and in particular what it means to impose constant correlation across rating classes, and show that there are profound differences between using default correlations and using the asset return correlations of the Merton model.
Understanding the risk of an asset in a portfolio is fundamental to managing and pricing risk. Richard Martin, Kevin Thompson and Christopher Browne extend their work on saddle-point techniques for portfolio loss distributions by analytically deriving the sensitivity of VaR to asset allocation. Their method is fast, easily-implemented (without simulation) and holds for non-Normal loss distributions.
Portfolio and business managers need to know, as a first step towards understanding the risk of their portfolio, and towards optimising it, how each component of the portfolio contributes to the overall risk. A common measure of risk contribution is the sensitivity of the risk to an infinitesimal fractional change in asset allocation; these have the convenient property that their sum is the overall risk . The simplest measure of risk is the mean-variance framework in which the risk is equal to the mean plus a given number of standard deviations (or simply the standard deviation). This is sufficient if the assets in the portfolio have a joint distribution that is multivariate Normal. It is easy to derive the sensitivities of the mean and standard deviation to asset allocation. But real loss distributions, and particularly those in credit, operational and insurance risk, are not Normal, and the standard deviation does not adequately capture the risk of large losses. The VaR (Value at Risk), which is simply a quantile of the distribution at some specified level of confidence, better captures extreme risks. It has some undesirable properties, such as lack of subadditivity , which are not shared by other risk measures such as expected shortfall. However, VaR is currently an industry standard often used for setting bank capital and it is the one that we shall study here. Leibowitz & Henriksson (1989) have discussed the use of VaR constraints in portfolio optimisation, and Litterman (1996) discusses risk contributions in the VaR-based framework. Most recently Gouriéroux et al. (2000) give a general expression for sensitivity of VaR to portfolio allocation, but to put this into practice still requires Monte Carlo simulation (except in the Gaussian case).
Classical portfolio theory uses mean-variance to derive an optimal portfolio allocation. However, for non-Normally-distributed losses such as credit portfolios this theory is inadequate and the VaR is often used to measure risk and set bank capital. However VaR is considerably less analytically tractable, which makes portfolio optimisation difficult. In this paper Richard Martin, Kevin Thompson and Christopher Browne use saddle-point methods, which can give an analytical risk contribution of each asset in a portfolio, to obtain the optimal allocation. Their method is fast (no simulation) and easy to implement.
A central problem in portfolio theory is finding the portfolio that gives the best return-on-risk characteristics. This requires a measure of how much each asset contributes to the risk. For example, a loan portfolio manager wants to know which risks should be reduced (by buying default protection or by securitising parts of the portfolio) and which could be increased (by lending more or by selling default protection). To answer this question the portfolio manager needs to know her price of risk of each asset, and that depends on the risk contribution of the asset in her portfolio. It will in general differ from the market price, and the difference can be used to increase the portfolio efficiency by trading in the market.