Table 7.2: Mean-variance analysis of Nikkei 225 on 4 June 1998

Table 7.3a: FTSE 100 allocations on 6 April 1998

Table 7.3b: FTSE 100 allocations on 19 June 1998

ASDA 0.168

SmithKline Beecham 0.069

British Telecom 0.433

Misys 0.001

Railtrack 0.234

Rank 0.004

Rentokil 0.019

Sainsbury 0.073

stock. However, 6 weeks later the same model recommends a completely different optimal portfolio that is very concentrated, with 43.3% in , 23.4% in Railtrack, 18.8% in ASDA, 7.3% in Sainsbury, 8.9% in SmithKline Beecham (and a little in Misys and Rank).

Optimal portfolios will Although efficient frontiers that are based only on recent data will reflect

not stay optimal for very current market conditions more accurately, optimal portfolios will not stay

long and will require optimal for very long and will require constant rebalancing. Common methods

constant rebalancing for cutting rebalancing costs include:

> using very long-term averages to construct the covariance matrix and mean returns, so that the efficient frontier becomes more stable over time. But then, from §3.1 we know that a single extreme return far in the past will have a prolonged effect on the shape of the efficient frontier and the choice of optimal portfolio - until it drops out of the averaging period, when the efficient frontier will suddenly jump for no apparent reason.

> assigning current portfolio weights as a weighted average of current and

past optimal allocations. This has the effect of smoothing allocations over time, but the resulting portfolio may be far from optimal because it may not respond enough to current market conditions.

> setting rebalancing limits so that allocations are changed only if the weights recommended by the optimal portfolio exceed them. Depending on the range of these limits, which is an arbitrary choice, rebalancing cost can be substantially reduced, but then portfolios may be far from optimal.

> using strong priors for the values in the covariance matrix and the mean return (§8.3.3).

> doing a limited amount of rebalancing in the direction indicated by the latest mean-variance analysis.

The problems with efficient frontier analysis that have been outlined in this section have a common root. In fact there is a fundamental and insurmountable difficulty with mean-variance analysis, and this is that the basic data for mean-variance analysis consist of asset returns. Price data are detrended before the analysis even begins, but the return data are short memory processes (Granger and Terasvirta, 1993). The result is that one has to use one of the ad hoc procedures outlined above when mean-variance analysis is used for investment analysis. The question arises whether mean-variance analysis, or indeed any other analysis that is based on return data, should be used for anything other than short-term models. We shall return to this question in Chapter 12.

7.3 The RiskMetrics™ Data

J.P. Morgan launched the first version of RiskMetrics in October 1994, and over the course of the next two years made several updates to the methodology and the scope of the data. The data are downloadable from the internet at www.riskmetrics.com. They consist of large covariance matrices of the returns to many risk factors: major foreign exchange rates, money market rates, equity indices, interest rates and some key commodities. For the statistical methodology for calculating these covariance matrices, and a description of how the data should be implemented in VaR models, see the RiskMetrics Technical Document (J.P. Morgan and Reuters, 1996).

The availability of data sets that can be used to produce VaR measures is the great advantage of RiskMetrics. The disadvantage is that there are some limitations with the methodology that underpins the calculation of some of the data. The RiskMetrics data are based on weighted average models for volatility and correlation. There are three types of covariance matrix: a 1-day matrix, a 1-month (25-day) matrix and a regulatory matrix, so called because it complies with the quantitative standards for internal models for market risk requirements that were set out in the 1996 Amendment to the 1988 Basle Accord (§9.1).

The availability of data sets that can be used to produce VaR measures is the great advantage of RiskMetrics. The disadvantage is that there are some limitations with the methodology that underpins the calculation of some of the data

0 \-,-,-,-,-,-,-,-,-

May-96 Nov-96 May-97 Nov-97 May-98 Nov-98 May-99 Nov-99 May-00

- GARCH(1,1) - RiskMetrics

Figure 7.7 RiskMetrics and GARCH(1,1) volatility estimates of the S&P 500 index.

An EWMA model on squared returns is used for the 1-day matrix, with smoothing constant X = 0.94 (§3.2). Many of its elements are very similar to GARCH(1,1) 1-day estimates (§4.2). For example, the GARCH(1,1) models for the CAC and the FTSE 100 equity indices that were reported in Table 4.7 indicate that the RiskMetrics volatility data for these indices will be very close indeed to the GARCH(1, 1) 1-day volatility forecasts illustrated in Figure 4.7. On 6 October 2000, using data since January 1996, the GARCH(1,1) estimates of the volatility persistence and market reaction parameters on the FTSE 100 and CAC were very similar to the RiskMetrics values of 0.94 and 0.06.

In general, the GARCH(1,1) parameter estimates will not be as close to 0.94 and 0.06 as they were in the CAC and FTSE 100. For example, the GARCH(1,1) model estimates of the market reaction and volatility persistence parameters for the S&P 500 index were 0.089 and 0.886 (see Table 4.7). Thus there will be a difference between the volatility forecasts over the 1-day horizon, as shown in Figure 7.7.

There are a few limitations of the RiskMetrics 1-day data that should be noted:

> The same value of the exponential smoothing constant X = 0.94 must be used for all markets. This is necessary, otherwise the covariance matrix would not be positive semi-definite (see J.P. Morgan and Reuters, 1996). There is much discussion in J.P. Morgan and Reuters (1996) about the optimal value for X. However, on the basis of estimating GARCH(1,1) models it appears that for many risk factors the value X = 0.94 is a little too high, so the RiskMetrics 1-day volatility data tend to overestimate volatility