σP2= N2 σ2 . (2.12)
Now we ask the following question. How does the portfolio variance change as the number of assets combined in the portfolio increases towards infinity (i.e. N of). It is clear from 2.12 that, as the number of assets held increases, the first term will shrink towards zero. Also, as N increases the second term in 2.12 tends towards C. Together, these observations imply the following.
1. The portfolio variance falls as the number of assets held increases.
2. The limiting portfolio return variance is simply the average covariance between asset returns: this average covariance can be thought of as the risk of the market as a whole, with the influence of individual asset return variances disappearing in the limit.
The moral of the preceding statistical story is clear. Holding portfolios
consisting of greater and greater numbers of assets allows an investor to reduce the risk he or she bears. This is illustrated diagrammatically in Figure 2.1.
Figure 2.1
Mean–variance analysis
In the preceding two sections, we have demonstrated two important facts:
1. The expected return on a portfolio of assets is a linear combination of the expected returns on the component assets.
2. An investor holding a diversified portfolio gains through the reduction in portfolio variance, when asset returns are not perfectly correlated.
In this section, we use these facts to characterise the optimal holding of risky assets for a risk-averse agent. Our fundamental assumption is that all agents have preferences that only involve their expected portfolio return and return variance. Utility is assumed to be increasing in the former and decreasing in the latter. For illustrative purposes we begin using the assumption that only two risky assets are available. The results presented, however, generalise to the N asset case.
To begin, assume there is no risk-free aset. The investor can hence only form his or her portfolio from risky assets named X and Y. These assets have expected returns of E(Rx) and E(Ry) and return variances of ı2x and ı2y. The first question the investor wishes to answer is how the characteristics of a portfolio of these assets (i.e. portfolio expected return and variance) change as the portfolio weights on the assets change. Given equation 2.6, the answer to this question is obviously dependent on the correlation between the returns on the two assets.
First assume the assets are perfectly correlated and, further, assume asset X has lower expected returns and return variance than asset Y. We form a portfolio with weights Į on asset X and 1±Į on asset Y. Equation 2.6 then implies that the portfolio variance can be written as follows:
ı2P = (Įıx + (1±Į)ıy)2. (2.13)
Taking the square root of equation 2.13, it is clear that the portfolio standard deviation is linear in Į. As the portfolio expected return is linear in Į, the locus of expected return–standard deviation combinations is a straight line. This is shown in Figure 2.2.
Figure 2.2
If the correlation between returns is less than unity, however, the investor can benefit from diversifying his portfolio. As previously discussed, in this scenario, portfolio standard deviation is not a linear combination of ıx and ıy. The reduction of portfolio risk through diversification will imply that the mean–standard deviation frontier bows towards the y-axis. This is also shown on Figure 2.2. The final curve on Figure 2.2 represents the case where returns are perfectly negatively correlated. In this situation, a portfolio can be constructed, which has zero standard deviation.
Activities
1. Assuming asset returns are perfectly negatively correlated, use equation 2.6 to find the portfolio weights that give a portfolio with zero standard deviation.
(Hint: write down 2.6 with the correlation set to minus one and a Į and b ±Į. Then minimise portfolio variance with respect to Į.)
2. Assume that the returns on Microsoft and Bethlehem Steel have correlation of 0.5. Using the data provided earlier in the chapter, construct the mean–variance frontier for portfolios of these two assets. Start with a portfolio consisting only of Microsoft stock and then increase the portfolio weight on Bethlehem Steel by 0.1 repeatedly, until the portfolio consists of Bethlehem Steel stock only.
From here on we will assume that return correlation is between plus and minus one. The expected return–standard deviation locus for this case is redrawn in Figure 2.3. In the absence of a risk-free asset, this locus is named the mean–variance frontier. As our investor’s preferences are increasing in expected return and decreasing in standard deviation, it is clear that his or her optimal portfolio will always lie on the frontier and to the right of the point labelled V. This point represents the minimum-variance portfolio. He or she will always choose a frontier portfolio at or to the right of V, as these portfolios maximise expected return for a given portfolio standard deviation. In the absence of a risk-free asset, this set of portfolios is called the efficient set.
Figure 2.3
We can now, given a set of preferences for the investor, find his or her optimal portfolio. The condition characterising the optimum is that an investor’s indifference curve must be tangent to the mean–variance frontier.3 Two such optima are identified on Figure 2.3 at R and S. The investor locating at equilibrium point R is relatively risk-averse (i.e. his or her indifference curves are quite steep), whereas the equilibrium at S is that for a less risk-averse individual (with correspondingly flatter indifference curves). Figure 2.3 also shows sub-optimal indifference curves for each set of preferences.
3 In technical terms the optimum is characterised by the marginal rate of substitution being equal to the marginal rate of transformation (i.e. the slope of the indifference curve equals the slope of the frontier).
Hence, as Figure 2.3 demonstrates, in a world of two risky assets and no risk-free asset, the optimal portfolio of risky assets held by an investor depends on his or her preferences towards risk and return. The same is true when there are N risky assets available. Figure 2.4 depicts the same type of diagram for the N asset case.
Note that the mean–variance frontier is of the same shape as that in Figure 2.3. However, unlike the two-asset case, the interior of the frontier now consists of feasible but inefficient portfolios (i.e. those that do not maximise expected return for given portfolio risk). The mean–variance frontier now consists of those portfolios that minimise risk for a given expected return, whereas those portfolios on the efficient set (i.e. on the frontier but to the right of V) additionally maximise expected return for a given level of risk.
We now reintroduce a risk-free asset to the analysis (i.e. we assume the existence of an asset with return rf and zero return–standard deviation).
A key question to address at this juncture is as follows. Assume that we form a portfolio consisting of the risk-free asset and an arbitrary combination of risky assets. How do the expected return and return–
standard deviation of this portfolio alter as we vary the weights on the risk-free asset and the risky assets respectively?
Denote our arbitrary risky portfolio by P. We combine P with the risk-free asset using weights 1 – a and a to form a new portfolio Q. The expected return and variance of Q are given by:
E(RQ) = (1 – a)rf + aE(RP) = rf + a[E(RP) – rf ] (2.14)
ı2Q = a2ı2P . (2.15)
In order to analyse the variation in the risk and expected return of the portfolio Q with respect to changes in the portfolio weights, we construct the following expression:
dE(RQ) dE(RQ) /da
/da
dσQ = dσQ . (2.16)
Using equations 2.14 and 2.15 we find that:
dE(RQ) E(Rp) – rf
dσQ = σp . (2.17)
As this slope is independent of a, the risk–return profile of the portfolio Q is linear. This is known as the capital market line, and two such CMLs are shown in Figure 2.5 for two different portfolios of risky assets.
We now have all the components required to describe the optimal portfolio choice of an investor faced with N risky assets and a risk-free investment.
Figure 2.6 replots the feasible set of risky asset portfolios. The key question to answer is, what portfolio of risky assets should an investor hold? Using the analysis from Figure 2.5, it is clear that the optimal choice of risky asset portfolio is at K. Combining K with the risk-free asset places an investor on a capital market line (labelled rfKZ), which dominates in utility terms the CML generated by the choice of any other feasible portfolio of risky assets.4 The optimal portfolio choice and a sub-optimal CML (labelled CML2) are shown on Figure 2.6 along with the indifference curves of two investors.
Recall that we previously defined the efficient set as the group of
portfolios that both minimised risk for a given level of expected return and
4 That is, choosing portfolio K places an investor on a CML with greater expected returns at each level of return variance than does any
maximised expected return for a given level of risk. With the introduction of the risk-free asset, the efficient set is exactly the optimal CML.
Figure 2.5
The key result that is depicted in Figure 2.6 is known as two-fund separation. Any risk-averse investor (regardless of his or her degree of risk-aversion) can form his or her optimal portfolio by combining two mutual funds. The first of these is the tangency portfolio of risky assets, labelled K, and the second is the free asset. All that the degree of risk-aversion dictates is the portfolio weights placed on each of the two funds.
The investor with the optimum depicted at X on Figure 2.6, for example, is relatively risk-averse and has placed positive portfolio weights on both the risk-free asset and K. An investor locating at Y, however, is less risk-averse and has sold the risk-free asset short in order to invest more in K.5
Figure 2.6
Two-fund separation is the result that underlies the capital asset pricing model (CAPM), which is developed in the next section.
5 A short sale is the sale of an asset that one does not actually own.
One borrows the asset in order to complete the transactions and imme-diately receives the sale price. Subsequently, one uses the proceeds from the sale to repurchase a unit of the asset, and deliver it to the creditor.
If the price of the asset has dropped in the interim, one makes a ECUJRTQƂV