If we assume that weights can be negative or zero then (7.4) has a straightforward solution for the optimal weights w* = (w*, . . ., w*) of the minimum variance portfolio. The solution is given by

wf= (7.5)

where is the sum of the elements in the ith column of V-1, and is the sum of all the elements of V-1. The variance of the global minimum variance portfolio, V*, is then

To illustrate the method, consider a portfolio of the following US stocks: American International, American Express, Boeing, Citigroup, General Electric, Coca-Cola and Merrill Lynch. Suppose that the minimum variance portfolio of these stocks is computed using long-term equally weighted average estimates of each element in VR, the covariance matrix of the returns to these stocks (§3.1). These estimates, and so also the minimum variance portfolio weights, will depend on the sample period chosen, and in Table 7.1 the results of applying (7.5) are given for three different samples. All three samples are daily data up to 6 October 2000, but they start at the beginning of January 1997, 1998 and 1999, respectively.

Table 7.1a: Minimum variance portfolio weights (%)

Table 7.1b: Volatility estimates of stocks and the minimum variance

portfolio (%)

The weights in the minimum variance portfolios in Table 7.1a change relatively little because long-term equally weighted averages are used to compute the covariance matrix. Table 7.1b shows that a considerable risk reduction is obtained with this method. But suppose one requires the minimum variance portfolio to respond more rapidly to changes in market conditions, aiming at a greater reduction in risk through more frequent rebalancing. In this case it is possible to use GARCH or EWMA estimates of the variances and covariances in VR (§7.3 and §7.4).

Figure 7.2a shows the weights that should be allocated to a portfolio of just three US stocks, to minimize the variance when daily estimates of VR are obtained using EWMAs with X = 0.94. They change considerably during the data period, particularly in July 1999 when most US stocks were very volatile. This type of reaction to market conditions is impossible to capture with long-term equally weighted averages. Figure 7.2b shows that American Express was particularly volatile and so it is not surprising that the minimum variance portfolio weight allocated to American Express is much smaller at this time. Around July 1999 the minimum variance portfolio had a volatility of between 30% and 40%, much higher than during other periods; the rest of the time it was possible to find a portfolio of just these three stocks that had a volatility of between 10% and 20%.14

This type of reaction to market conditions is impossible to capture with long-term equally weighted averages

7.2.2 The Relationship between Risk and Return

The unconstrained problem (7.4) ignores the portfolio return characteristics. Rather than always seeking to minimize risk, the view may be taken that more risk is perfectly acceptable if it is accompanied by higher returns. In fact managers are in danger of under-utilizing resources if insufficient capital is allocated to high-risk, high-return activities.

The portfolio return is wr, where r = (Ru . . ., Rn) is the vector of asset returns, or, for large portfolios that are represented by a risk factor model, - Bx + £ where x is the x 1 vector of risk factor returns, = (P,y) is the n x matrix of factor sensitivities and £ is the n x 1 vector of specific returns (an alternative matrix form will be given in §8.1.2). The capital allocation problem of finding minimum variance portfolios having a given minimum level of return p becomes:

Managers are in danger of under-utilizing resources if insufficient capital is allocated to high-risk, high-return activities.

min wVw such that £ >, = 1 and wr p,

(7.6)

where V is defined in (7.4). In this case the optimal weights wf = (w\, are given by

"«)

14 One would not normally wish to rebalance daily to maintain the absolute minimum variance portfolio at all times. Usually one sets rebalancing bounds and uses the daily chart analysis such as Figure 7.2 to monitor when the bounds have been exceeded.

Jan-99 Apr-99 Jul-99 Oct-99 Jan-00 Apr-00 Jul-00 (a) -Weight on American Intl ---Weight on American Express - -Weight on Boeing

Jan-99 Apr-99 Jul-99

Oct-99 Jan-00 Apr-00

Jul-00

-Volatility of American Intl Volatility of Boeing

Volatility of American Express Volatility of Minimum Variance Portfolio

Figure 7.2 (a) Minimum variance portfolio weights; (b) volatility of minimum variance portfolio compared to individual stocks.

w] = ((ay,. - + yi(V% - bv,M(.V*b - a2), (7.7)

where V* is the variance of the global minimum variance portfolio, v/; is the sum of the elements in the ;th column of V-1, is the returns-weighted sum of the rth column of V"1, a - \/r and b = rV"r.

Figure 7.3a shows a stylized plot of portfolio returns wr against portfolio risk (wTw)12 for all possible portfolio allocations w among n risky assets with returns vector r and covariance matrix V. This is called the opportunity set. It is the convex hull of the points in risk-return space defined by the n risky assets.