Covariance Matrices

The covariance matrix of a set of time series Xu . . ., Xk is a square, symmetric matrix of the form1

In financial analysis the series Xu . . ., Xk will be returns (for short holding periods these are often taken to be the first difference in log prices) on the different assets in a portfolio, or on the risk factors of a portfolio. Note that the covariance matrix is a concise and convenient form for the information on all the volatilities and correlation in a system: volatilities and correlations can be obtained through simple operations on the elements of the covariance matrix, as described in Chapter 1.

This chapter begins with the applications of covariance matrices in financial analysis. A covariance matrix lies at the heart of risk management: statistical forecasts of the variances and covariances of asset or risk factor returns are summarized in such a matrix to obtain a forecast of portfolio risk (§7.1). Covariance matrices are also a cornerstone of investment analysis, and §7.2 shows how they are used to diversify investments and obtain minimum risk portfolios, and to generate the efficient frontier for optimizing linear portfolios.

Typically hundreds of risk factors, such as all yield curves, interest rates, equity indices, foreign exchange rates and commodity prices, need to be encompassed by a very large-dimensional covariance matrix. Covariance matrices must always be positive semi-definite, for reasons that are explained in §7.1.3 and it has become quite a challenge to generate large, meaningful, positive semi-definite covariance matrices that are necessary to net the risks across all positions in a firm. Simplifying assumptions may be necessary. For example the RiskMetrics methodology designed by J.P. Morgan uses either simple equally weighted

1 The series should be not only stationary but also jointly stationary, otherwise the covariances will not be stable over time, as explained in §11.4.2.

/ V(X,) cov(Xl,X2) ... cov¹,2) V(X2) cov(Xu X3) cov(X2, X3) V(X3)

cov(X„ Xk)\ cov(X2, Xk) cov(X3, Xk)

\cov(XuXk)

V(Xk) J

moving averages, or exponentially weighted moving averages with the same smoothing constant for all volatilities and correlations of returns. There are some limitations with the RiskMetrics methodology; these are discussed in §7.3.

Section 7.4 describes a new method for generating large covariance matrices. It is easy to implement and will produce matrices that have very desirable characteristics (§7.4.5), so I believe that many large banks will eventually adopt this approach. The basic principles of the method are explained in §7.4.1. Then the orthogonal EWMA method is described in §7.4.2; it provides a simple but computationally efficient way to apply EWMA to generate a covariance matrix that will always be positive semi-definite. Moreover, the degree of smoothing on each variable will be determined by its correlation in the system.

The orthogonal approach can also be applied with GARCH models. The flexibility and accuracy of GARCH forecasting techniques place them in a unique position to fulfil many of the needs of back office risk management and front office trading systems, but without a feasible method for computing large covariance matrices using GARCH techniques, this potential will not be realized. Given the insurmountable problems in direct estimation of large GARCH covariance matrices (§4.5.3), but given also the need for mean-reverting covariance forecasts for measuring portfolio risk, the orthogonal GARCH model presented in §7.4.3 is of great significance.

7.1 Applications of Covariance Matrices in Risk Management

The first section of this chapter sets out some fundamental concepts for the measurement of portfolio risk. The exposition centres on the risk models that are used to measure the variance of portfolio returns for a volatility analysis, or the variance of portfolio P&L for value-at-risk analysis. It provides a background for Chapters 8 and 9.

7.1.1 The Variance of a Linear Portfolio

Suppose the return on a linear portfolio2 can be described as a sum of individual asset returns, say

Rw + .-. + w.R,,, (7.1)

where Ew, = 1. So there are n risky assets in the portfolio, asset i has return Rt and the proportion of capital that is invested in it is >,. Using the rule for variance of a sum of random variables,3 the variance of the portfolio, VP, is

-A linear portfolio is one whose pay-off function is a linear function of underlying risk factors. Option and bond portfolios are therefore non-linear but cash or simple future positions are linear.

3K(vr,«i + u2R2) = n1V(R,) + ,ijV(R2) + 2H-ln2cov(Rl, )

It is easy to implement and will produce matrices that have very desirable characteristics, so I believe that many large banks will eventually adopt this approach

Fp = >2 V(Rt) + ... + wk V(Rn) + 2W] w2cov(W, w2) + 2w\ w3cov(W, w3) + 2win„cov(wb w„) + ... + 2w„ ,w„cov(w„ ,, w„).

It is more convenient to use matrix notation. Denote the portfolio weights by w = (wu . . ., w„) and the asset returns - (Ru . . ., Rn), so that (7.1) may be written

RP = v/r. (7.1a)

Denote the covariance matrix of asset returns by VR. Then the portfolio variance (7.2) is more succinctly expressed as

VP = wVRw. (7.2a)

Thus the variance of a linear portfolio is a quadratic form.4

If a portfolio has very many assets it may be appropriate to use a factor model to describe its returns by those of a smaller set of risk factors (Chapter 8). For

example, a large international equity portfolio may be represented as a The variance of a linear weighted sum of equity index returns where the weights correspond to the net portfolio is a quadratic portfolio betas with respect to that index (§8.1.1). Alternatively, a multi-factor form model may be used that is based on the arbitrage pricing theory (§8.1.2) or on statistical factors using principal component analysis.

When a factor model is used, let X], ., Xk denote returns to the different risk factors and be the x sensitivity matrix, whose (/, /)th element is the sensitivity of the /th asset to the jth risk factor.5 In this case the portfolio variance due to market risk factors is the quadratic form

VP = wBVxB w, (7.3)

where w is the n x 1 vector of portfolio weights and Vx is the x risk factor covariance matrix (see equation (8.6) in §8.1.2).

So far we have shown how the covariance matrix is used to compute the variance of a portfolio return as a quadratic form; but the variance of the portfolio P&L is also a quadratic form.6 We can obtain the variance of the portfolio P&L simply by multiplying the portfolio weights vector by the nominal amount invested in the portfolio. This gives the vector p of nominal amounts invested in each of the assets, and then the variance of portfolio

4 A quadratic form is a scalar quantity (that is, a single number, not a matrix or vector) that can be written as the product of a row vector, a square matrix and the column vector (transpose of the row vector), for example xAx. It is called a quadratic form because every term in the expansion has a square or cross product of elements of x.

5 For example, the risk factors for international equity portfolios are normally taken to be the relevant equity indices in different countries and the exchange rates.

6 For example, suppose a portfolio of bonds or loans is represented by a cash flow map = { 1]. PV2, - -, PVk), where PV, denotes the present value of the income flow at time t for fixed maturitv dates t = I, . . k. Then the variance of the portfolio P&L is pVp, where V is the x covariance matrix of interest rates at maturity dates r = 1.....A. For more details, see §9.3.2.