risk model, keeping the other parameters such as the portfolio weights and the estimates of pricing model parameters constant. These are the stress tests that are explained in more detail in §9.6.3.

In linear portfolios a stress test only requires the substitution of a stress covariance matrix in place of the current covariance matrix in the quadratic form for portfolio risk (§7.1.1). This will give an estimate of portfolio volatility, or portfolio VaR, in the extreme market circumstances that are captured by the covariance matrix. However, this analysis rests on some strong distributional assumptions and the standard normality assumption for VaR models becomes even more difficult to justify when the focus is on the very extreme returns.

Standard normality assumption for VaR models becomes even more difficult to justify when the focus is on the very extreme returns

In options portfolios a stress covariance matrix may be used to generate correlated movements in the underlying risk factors in Monte Carlo simulations. But the stress test will also require scenarios for exceptional movements in the underlying asset price movements and the implied volatility movements; these scenarios must be consistent with the covariance matrix used in the Monte Carlo simulations. For example, if the covariance matrix has a negative correlation between price and volatility, then only consider scenarios where the smile shifts in the opposite direction to the price (§2.3.4, §6.3 and §9.6.2).

Risk management should have a library of stress covariance matrices containing covariance matrices that have actually been experienced historically as well as a set of covariance matrices that reflect extreme scenarios for volatility and correlation. The historical library might contain the covariance matrix for equity portfolios that pertained on Black Monday, or during the Asian crisis in 1998, and so forth. The library for extreme scenarios for volatility requires covariance matrices that are perturbations of the current covariance matrix, artificially making certain volatilities very large. Whatever the values used for volatilities, the stress covariance matrix will always remain positive semi-definite.13

Correlation stress testing is not as straightforward as volatility stress testing. It is not always clear which values for correlation will produce extreme scenarios. The reader can verify this using the stress testing functions in the VaR spreadsheets on the CD. Correlations could be set to zero, to unity, to minus unity or to some other values; it depends on the construction of the portfolio (§7.2.2). Also note that the perturbed covariance matrices for correlation stress testing may not be positive semi-definite; they will need to be checked for this, as explained in §7.1.3.

Perturbed covariance matrices for correlation stress testing may not be positive semi-definite

13 To see this, write V = DCD, where is the correlation matrix and D is the diagonal matrix with standard deviations along the diagonal. It follows that V is positive semi-definite if and only if is positive semi-definite, whatever volatilities are used in D (as long as they are positive).

7.2 Applications of Covariance Matrices in Investment Analysis

The efficient allocation of sparse resources is the fundamental problem of microeconomics. Therefore, it is not surprising that this section draws on many concepts, such as utility functions and optimization, that will be familiar to readers with an economics background. The exposition will focus on the mean-variance analysis that was introduced by Markovitz (1959) and has since formed the basis of the portfolio theory and investment analysis developed by Sharpe (1970) and others.

The long-term capital allocation problem should be viewed in terms of risk-adjusted returns

Senior managers make capital allocation decisions that require comparison of the performance of different desks throughout the organization. Simple VaR figures are based on risk alone and are only appropriate for short risk horizons. But the long-term capital allocation problem should be viewed in terms of risk-adjusted returns. The choice becomes one of selecting allocations to different parts of the organization to minimize risk while maximizing returns, and while also accounting for the preferences of senior managers.

Global asset management can be regarded as a two-stage process: first select the optimal weights to be assigned to country indices; and then allocate funds optimally to each country. Probably the first stage will be the most important, but there will also be constraints. For example, allocations may need to be restricted within a certain range, so inequality constraints such as no more than 10% of the fund invested in Japan, have to be imposed.

The mathematical concepts that are introduced in the context of asset allocation may also be applied to other financial agents that face allocation decisions

The reader should bear in mind that the mathematical concepts that are introduced in the context of asset allocation may also be applied to other problems in the efficient allocation of sparse resources. Throughout the text the opportunity set is referred to in terms of portfolios of assets, and the decisionmakers are called investors or asset managers. But exactly the same analysis can apply to other financial agents that face allocation decisions. For example, a senior manager in an investment bank faces the two-stage decision problem: first select the optimal allocations to global product lines; and then assign weights optimally at the desk level. The performance of a trader, just like the performance of a portfolio or of a product line, might be compared in terms of risk-adjusted returns (§7.2.3). If risk managers base trading limits on the risk and returns from different traders, they face an allocation problem that can be viewed from the same perspective. Trading limits cannot be negative, which is the same in mathematical terms as the constraint that no short sales are allowed.

In §7.2.4 it is shown how a portfolio managers attitude to risk will influence his choice of optimal portfolio. Similarly, the risk aversion of a manager will determine the trading limits that are optimal for his particular utility, and his choice of allocation to global product lines. Mathematically speaking, these problems are all equivalent.

This section examines the role of covariance matrices in the construction of optimal portfolios. Section 7.2.1 solves the problem of how to construct minimum risk portfolios and gives empirical examples of the application of the solution. Then in §7.2.2 the trade-off between risk and return is discussed in the context of efficient frontier analysis, both with and without short sales. The relationship between correlation and portfolio diversification is analyzed in some detail. In §7.2.3 capital allocation decisions are framed in terms of risk-adjusted performance measures. Utility theory is introduced in §7.2.4, and it is shown that the problem of maximizing expected utility of an investment cannot usually be reduced to the problem of maximizing a function of the mean and the variance of returns. One of the reasons why exponential utility functions are commonly used, despite the fact that they are limited by their constant absolute risk-aversion characteristics, is that they allow utility maximization to be viewed in a mean-variance perspective.

One of the reasons why exponential utility functions are used is that they allow utility maximization to he viewed in a mean-variance perspective

The final part of this section examines efficient portfolios in practice. There is a fundamental problem with the use of return data in mean-variance analysis. The long-run trends and any common trends in the prices will have been taken out of the data already before they are used for analysis. Return data have very short memory and it is very difficult to use them for anything other than short-term analysis. If the short-term volatilities and correlations that are generated by an EWMA model or a GARCH model are used to construct the efficient frontier, it will change considerably from day to day. In §7.2.5 the problems presented by the use of different types of covariance matrices are discussed, and practical examples are given.

7.2.1 Minimum Variance Portfolios

Assume that portfolio allocation decisions are made only on the basis of risk characteristics and that the standard deviation of portfolio returns is taken to be the appropriate measure of risk. If there are n risky assets in the portfolio having weights w - (wb . . ., w„) and if the covariance matrix of asset returns is V then the portfolio variance is wVRw, as in (7.2a). In larger linear portfolios which are best described by factor models, the portfolio variance is derived in §8.1.2. It is measured using the x covariance matrix of risk factor returns Vx, the n x matrix of the asset sensitivities to the different risk factors and the nxn specific risk covariance matrix VE. Whether portfolios are represented at the asset or the risk factor level, the minimum variance portfolio is the solution to a simple optimization problem, viz.

minwVw such that Xw>,-= 1, (7.4)

where

v J VR at the asset level,

~ \ BVXB + VE at the risk factor level.