Efficient

frontier

Minimum variance portfolio having return at least

Risk

Figure 7.3. (a) Efficient frontier and minimum variance portfolios; (b) efficient frontier with no short sales.

In general, optimal capital allocations will be to portfolios that lie on the efficient frontier (illustrated by the thick line in Figure 7.3a) where it is not possible to adjust allocations to gain higher return for the same level of risk, or less risk for the same level of return. The efficient frontier will be a concave curve, the upper boundary of the opportunity set from the minimum variance portfolio. Also marked on the figure are the global minimum variance portfolio, and the minimum variance portfolio having a given minimum level of return p.

If no short sales are allowed the analytic solutions given above to the global minimum variance problem (7.4) and to the constrained minimum variance problem (7.6) are no longer valid, since the additional constraints w, 0 must be imposed. Without short sales the opportunity set cannot extend infinitely along the risk axis. In fact the efficient frontier is the envelope of all portfolios lying between the global minimum variance portfolio and the maximum return portfolio.

The efficient frontier is the envelope of all portfolios lying between the global minimum variance portfolio and the maximum return portfolio

The concavity of the boundary of the opportunity set between two assets (or portfolios) X and Y depends on their correlation, becoming more concave as

correlation decreases, as illustrated in Figure 7.3b. Note that when Zand Fare perfectly correlated the efficient frontier between them is a straight line. Since their returns are essentially the same, the majority of investors would not diversify. As their correlation decreases there is more incentive to reduce risk by diversification. To see this, consider (7.4) as the problem of choosing a proportion p for a portfolio of just two assets, with p invested in X and 1 - p invested in Y, add the constraint 0 < p < 1 and assume without loss of generality that Y is more variable than X. The portfolio variance is

V = p2cs2x + (1 - pfcsl + 2p(l - p)paxgy. (7.8)

Differentiating with respect to p and setting to zero gives the unconstrained optimal value for p as

p* = (ct? - pctvctv)/(ct? + ct? - 2paxay).

The denominator is always positive, being the variance of the portfolio X - Y, and the assumption that ct;,/ctv > 1 means that the numerator must also be positive. In fact one can put av - ct and ctv - a a into (7.8), where 0 < a < 1 is the relative volatility of X with respect to Y. Then

p* = (1 - ap)/(l +a2 - lap),

and substituting this into (7.7) gives the variance V* of the minimum variance portfolio as

V* = (1 -p2)a2o2/(\+a2-lap).

Note that this is less than 2, the variance of X, if and only if 1 + a2- 2ap > 0, that is, when the correlation is less than (1 +a2)/2a. So when p < (1 +a2)/2a the optimal portfolio has a variance reduction as shown in As correlation decreases Figure 7.3b. For example, if p = 0 then p* = 1/(1 + a2) and the minimum there is more incentive to variance portfolio has variance V* = a2a2/(\ + a2) which is less than the diversify variance 0f x. If p = -l then p* = l/(l+a) and V* - 0. But if p (1 + a2)/2a then p* - 1 and X is the minimum variance portfolio.

The analysis so far has been simplified by the assumption that no risk-free asset exists. Many other possibilities are introduced if there is a risk-free asset with return Rj and unlimited lending and borrowing at this rate. In Figure 7.4, suppose an investor forms a new portfolio Fby placing a fraction w of funds in an arbitrarily chosen portfolio Q and a fraction 1 - w in the riskless asset, where w can be greater than 1. The expected return on Y is given by the line between Q and Rf at a point that is determined by w:

R, = wRq + (1 - w)Rj. (7.9)

The variance of the new portfolio Y is

rj? = ircj2- + (1 - w)2aj + 2w(l - w)oqf = w2o2q,

Slope is IR of optimal portfolio X

Slope is IR for portfolio Q

Efficient portfolios with riskless lending and borrowing

Slope is SRfor portfolio Q

Risk

Figure 7.4 Risk-adjusted performance measures.

assuming the risk-free rate has zero variance. So w = ov/a4 and the line (7.9) becomes

This is the equation of a line through Q with slope ((R4 - Rf)/a4) that cuts the returns axis at Rf, shown in Figure 7.4.15 Efficient portfolios will lie along the thick line in Figure 7.4. They are all combinations of riskless lending or borrowing with the efficient portfolio P. Portfolios to the left of P are combinations of lending at rate Rf and portfolio P. Portfolios to the right are combinations of borrowing at rate Rf and portfolio P. The actual portfolio that is chosen along this line depends on the risk attitude of the investor: the less risk-averse, the more he will borrow at the riskless rate to achieve both higher return and higher risk.

7.2.3 Capital Allocation and Risk-Adjusted Performance Measures

Efficient capital allocation needs to consider both risk and return characteristics of portfolios. If one only considers return then too much capital would be allocated to a high-return but high-risk activity, and if one only considers the risk then too little capital would be allocated to the same activity. By taking the ratio of return to risk in a risk-adjusted performance measure (RAPM) it is possible to compare activities with different risk-return characteristics.

Figure 7.4 also illustrates two of the standard RAPMs for investment analysis. If there is unlimited riskless lending and borrowing at a risk-free rate Rf, the

15 It is worthwhile noting that (7.9) forms the basis of the Sharpe-Lintner capital asset pricing model described in §8.1.1. Without an error process the portfolio will be perfectly correlated with the market, so the CAPM (8.1) becomes v = a + PX, where a is the risk-free rate of return Rf, X is the excess return of the market over the risk-free rate and the portfolio beta is just the relative volatility <jy/ux-

Rv = Rf+((Ra-Rf)/oa)o.

(7.10)