MEAN-VARIANCE PORTFOLIO THEORY AND THE CAPM Minimum Variance Efficient Portfolios Suppose that the investor chooses the proportion of his total wealth to invest in each asset in order to minimise portfolio risk. He is not, at this stage, allowed to borrow or lend or place any of his wealth in a risk-free asset. Suppose there are only two assets available with random returns R 1, R 2 and variance σ 1 or σ 2. Suppose the investor chooses to hold a proportion w 1 of his wealth in asset-1 and a proportion w 2 = 1 w 1 in asset-2. The actual return on this diversified portfolio (which will not be revealed until one period later) is R p = w 1 R 1 + w 2 R 2 (4) 1
The expected return on the portfolio (formed at the beginning of the period) is defined as ER p = w 1 ER 1 + w 2 ER 2 = w 1 μ 1 + w 2 μ 2 (5) The variance of the portfolio is given by σ 2 p= E(R p ER p ) 2 = E[w 1 (R 1 μ 1 ) + w 2 (R 2 μ 2 )] 2 = w 2 1 σ 2 1+ w 2 2σ 2 2+ 2w 1 w 2 ρσ 1 σ 2 (6) For the moment, we are assuming the investor is not concerned about expected return when choosing w i. To minimise portfolio risk, σ 2 p : σ 2 p/ w 1 = 2w 1 σ 2 1 2(1 w 1 )σ 2 2+ 2(1 2w 1 )ρσ 1 σ 2 = 0 (7) Solving (7) for w 1, we have 2
w 1 = (σ 2 2 ρσ 1 σ 2 )/(σ 2 1+ σ 2 2 2ρσ 1 σ 2 )= (σ 2 2 σ 12 )/(σ 2 1+ σ 2 2 2σ 12 ) (8) Note that, in general, the minimum variance portfolio has a positive expected return although w i was chosen independently of a desired expected return. For the special case where ρ = 1, we obtain σ 2 p= 0. Thus, all risk can be diversified when the two-asset returns are perfectly negatively correlated. An individual asset may be highly risky taken in isolation (i.e. its own variance of returns is high) but if it has a negative covariance with assets already held in the portfolio, then investors will be willing to add it to their existing portfolio even if its expected return is relatively low since such an asset tends to reduce overall portfolio risk σ 2 p. This basic intuitive notion lies behind the explanation of determination of equilibrium asset returns in the CAPM. 3
Principle of Insurance Generalising the above, from equation (6) for n assets, we have σ For the special case where asset returns are totally uncorrelated (i.e. all n assets have ρ ij = 0), the portfolio variance can be reduced to σ 2 p= (w 2 1σ 2 1+ w 2 2σ 2 2+ +w 2 nσ 2 n) (10) Simplifying further, if all the variances are equal (σ 2 i= σ 2 ) and all the assets are held in equal proportions (1/n), we have σ 2 p= (1/n 2 )nσ 2 = σ 2 /n (11) Hence, as n, the variance of the portfolio approaches 0. Thus, if uncorrelated risks are pooled, much of the portfolio risk is diversified away. 4
The total risk attached to each individual security is: in part due to factors that affect all securities (e.g. interest rates, growth of the economy, etc.), and this is known as market risk and in part, is due to events that are specific to the individual firm (e.g. luck in gaining new orders, an unexpected fall in input costs, etc.) this is specific idiosyncratic) risk. It is this specific risk that can be completely eliminated when we hold many shares in a diversified portfolio. Essentially, the good luck of several firms is broadly cancelled out by some other firms who are currently experiencing bad luck on average, these specific risks cancel out if you hold many shares. 5
Intuitively, one is inclined to suggest that such specific risk should not be reflected in the average return on any stock you hold (provided you hold all stocks in a diversified portfolio).this intuition carries through to the CAPM. 6
Portfolio Expected Return and Variance Clearly, individuals are interested in both expected portfolio return μ p ER p and the risk of the portfolio σ p. The question we now ask is how μ p and σ p vary, relative to each other, as the investor alters the proportion of her own wealth held in each of the risky assets. Take the two-asset case. Remember that μ 1, μ 2, σ 1, σ 2 and σ 12 (or ρ) are fixed and known. As we alter w 1 (and w 2 = 1 w 1 ), equation (5) and equation (6) allow us to calculate the combinations of (μ p, σ p ) that ensue for each of the values of w 1 (and w 2 ) that we have arbitrarily chosen. A numerical example is is plotted in Figure 2. 7
In general, as ρ approaches 1, the (μ p, σ p ) locus moves closer to the vertical axis as shown in Figure 3, indicating that a greater reduction in portfolio risk is possible for any given expected return. 8
(Compare portfolios A and B corresponding to ρ = 0.5 and ρ = 0.5, respectively.) For ρ = 1, the curve hits the vertical axis, indicating there are values for wi which reduce risk to zero. 9
For ρ = 1, the risk return locus is a straight line between the (μ i, σ i ) points for each individual security. In the above example, we have arbitrarily chosen a specific set of w i values, and there is no maximisation problem involved. Also, in the real world, there is only one value of ρ (at any point in time) and hence only one risk return locus corresponding to different values of w i. This risk return combination is part of the feasible set or opportunity set available to every investor, and for two assets, it gives a graph known as the risk return frontier. 10
More Than Two Securities Holding more securities reduces portfolio risk for any given level of expected return. Intuitively, more choice gives you unambiguously better outcomes in terms of (μ p, σ p ) combinations. The slope of the efficient frontier is a measure of how the agent can trade off expected return against risk by altering the proportions w i held in the three assets. The dashed portion of the curve below the minimum variance point Z (for the frontier with ρ = 0.5 in Figure 3) indicates mean-variance inefficient portfolios. An investor would never choose portfolio-c rather than B because C has a lower expected return but the same level of risk, as portfolio-b. Portfolio-B is said to dominate portfolio-c on the mean-variance criterion. 11
Efficient Frontier Consider the case of N assets. When we vary the proportions w i (i =1, 2,..., N) to form portfolios. We can also form portfolios consisting of the same number of assets but in different proportions. Every possible portfolio is given by crosses marked in Figure 4. 12
If we now apply the mean-variance dominance criterion, then all of the points in the interior of the portfolio opportunity set (e.g. P 1, P 2 in Figure 4) are dominated by those on the curve AB. How does the investor calculate the w i values that make up the efficient frontier? Building the efficient frontier involves the following steps. 1. The investor chooses optimal proportions w i, which satisfy the budget constraint w i= 1 and minimise σ p for any given level of expected return on the portfolio μ p. 2. She repeats this procedure and calculates the minimum value of σ p for each level of expected return μ p and hence maps out the (μ p, σ p ) points that constitute the efficient frontier. There is only one efficient frontier for a given set of μ i, σ i, ρ ij. 3. Each point on the efficient frontier corresponds to a different set of optimal proportions w 1,w 2,w 3,... in which the stocks are held. 13
Points 1 3 constitute the first decision the investor makes in applying the separation theorem the second part of the decision process still misses. Note that only the upper portion of the curve, that is AULB, yields the set of efficient portfolios, and this is the efficient frontier. The general solution to the above problem could (and usually does) involve some w i being negative as well as positive. A positive w i indicates stocks that have been purchased (i.e. stocks held long ) Negative w i represent stocks held short, that is, stocks that are owned by someone else (e.g. a broker) that the investor borrows and then sells in the market. She therefore has a negative proportion held in these stocks (i.e. she must return the shares to the broker at some point in the future). She uses the proceeds from these short sales to augment her holding of other stocks. 14
Borrowing and Lending: Transformation Line We now allow our agent to borrow or lend at the risk-free rate of interest, r. Because r is fixed over the holding period, its variance and covariance (with the set of n risky assets) are both zero. Thus, we can allow the agent to invest more than her total wealth in the risky assets by borrowing the additional funds at the risk-free rate. In this case, she is said to hold a levered portfolio. The transformation line is a relationship between expected return and risk on a portfolio that consists of (i) a riskless asset and (ii) a portfolio of risky assets. The transformation line holds for any portfolio consisting of these two assets, and it turns out that the relationship between expected return and risk is linear. 15
Assume the individual has somehow already chosen a particular combination of proportions (i.e. the w i ) of q risky assets (stocks) with actual return R, expected return μ R and variance σ 2 R. Note that the w i are not optimal proportions but can take any values (subject to w i = 1). If she invests a proportion x of her own wealth in the risk-free asset, then she invests (1 x) in the risky bundle. Denote the actual return and expected return on this new portfolio as RN and μ N respectively. R N = xr + (1 x)r (15) μ N = xr + (1 x)μ R (16) where (R, μ R ) is the (actual, expected) return on the risky bundle of her portfolio held in stocks. 16
For x < 0, the agent borrows money at the risk-free rate r to invest in the risky portfolio (i.e. a levered position). Since r is known and fixed over the holding period, the standard deviation of this new portfolio depends only on the standard deviation of the risky portfolio of stocks σ R. From (15) and (16), we have σ 2 N = E(R N μ N ) 2 = (1 x) 2 E(R μ R ) 2 (17) σ N = (1 x)σ R (18) where σ R is the standard deviation of the return on the set of risky assets. Equations (16) and (18) are both definitional, but it is useful to rearrange them into a single equation in terms of mean and standard deviation (μ N, σ N ) of the new portfolio. Rearranging (18) (1 x) = σ N /σ R (19) and substituting for x and (1 x) from (19) in (16) gives the identity 17
µ N r µ R R σ N (20) For any portfolio consisting of two assets, one of which is a risky asset (portfolio) and the other is a risk-free asset, the relationship between the expected return on this new portfolio μ N and its standard error σ N is linear. Equation (20) is, of course, an identity; there is no behaviour involved. We can see from (20) that when all wealth is held in the set of risky assets, x = 0 and hence σ N = σ R, and this is designated the 100% equity portfolio (point X, Figure 5). When all wealth is invested in the risk-free asset, x = 1 and μ N = r (since σ N /σ R = 0). Now consider Figure 5; at points like Z, the individual holds a levered portfolio (i.e. he borrows some funds at a rate r and also uses all his own wealth to invest in equities). 18
19
Optimal Tangent Portfolio At each point on a given transformation line, the agent holds the risky assets in the same fixed proportions w i. Suppose point X (Figure 5) represents a combination of w i = 20%, 25% and 55% in the three risky securities of firms, alpha, beta and gamma. Then points Q, L and Z also represent the same proportions of the risky assets. The only quantity that varies along the transformation line is the proportion held in the one risky bundle of assets relative to that held in the risk-free asset. The investor can borrow or lend and be anywhere along the transformation line rz. Where he ends up along rz depends on his preferences for risk versus return. 20
Although an investor can attain any point along rz, any investor (regardless of his preferences) would prefer to be on the transformation line rz (see Figure 6). This is because at any point on rz, the investor has a greater expected return for any given level of risk compared to points on rz. 21
Point M represents a bundle of stocks held in certain fixed proportions. As M is on the efficient frontier, the proportions w i held in risky assets are optimal (i.e. the w i referred to earlier). An investor can be anywhere along rz, but M is always a fixed bundle of stocks held by all investors. Hence, point M is known as the market portfolio, and rz is known as the capital market line (CML). The CML is therefore that transformation line which is tangential to the efficient frontier. Investor preferences only determine where along the CML each individual investor ends up. For example, an investor with little or no risk aversion would end up at a point like K where she borrows money (at r) to augment her own wealth and then 22
invests all of these funds in the bundle of securities represented by M (but she still holds all her risky stocks in the fixed proportions wi). Although each investor holds the same proportions in the risky assets along the CML, the dollar amount in each risky asset differs. Note that all levered portfolios have to be north-east of point M, such as point K Separation principle 1. He uses his knowledge of expected returns, variances and covariances to calculate the set of stocks represented by the efficient frontier. He then determines point M as the point of tangency of the straight line from r to the efficient frontier. 2. The investor now determines how he will combine the market portfolio of risky assets with the riskless asset. This decision does depend on his subjective risk return preferences. 23
At a point to the left of M, the individual investor is reasonably riskaverse, the reverse is also true. The CML, rz which is tangential at M, the market portfolio, must have the form given by (20):. µ N r µ σ N (21) Slope of CML = (μ m r)/σ m = slope of the indifference curve (22) The slope of the CML is often referred to as the market price of risk. The slope of the indifference curve is referred to as the marginal rate of substitution (MRS), since it is the rate at which the individual will trade-off more return for more risk. All investor s portfolios lie on the CML, and, therefore, they all face the same market price of risk. 24
Hence, in equilibrium, all individuals have the same trade-off between risk and return. λ m = (μ m r)/σ 2 m (23) then λ m is also frequently referred to as the market price of risk. Capital Asset Pricing Model The mean-variance model in which agents choose optimal asset proportions also yields a model of equilibrium expected returns known as the CAPM (providing we assume homogeneous expectations). We now undertake an experiment whereby we move from M (which contains all assets in fixed proportions) and create an artificial portfolio by investing some of the funds at present in the assets represented by M, in any risky security i. This artificial portfolio (call it p) consists of two risky portfolios with proportions x i in asset-i and (1 x i ) in the portfolio at M. 25
This portfolio p has expected return μ p and standard deviation σ p : μ p = x i μ i + (1 x i )μ m (26a) σ p = [x 2 i σ 2 i + (1 x i ) 2 σ 2 m+ 2x i (1 x i )σ im ] 1/2 (26b) The portfolio p lies along the curve AMB and is tangent at M. Note also that at M there is no borrowing or lending. 26
As we alter x i and move along MA, we are shorting security-i and investing more than 100% of the funds in portfolio M. The key element in this derivation is to note that at point M, the curves LMY and AMB coincide and since M is the market portfolio, x i = 0. To find the slope of the efficient frontier at M, we require (27) 2 2 1 2 4 (28) At x i = 0 (point M), we know σ p = σ m and hence (30) 27
But at M, the slope of the efficient frontier (equation (30)) equals the slope of the CML (equation (25)) µ r σ From (31), we obtain the CAPM relationship µ r 32 When borrowing and lending in the risk-free asset is allowed, then in order for asset i to be willingly held, it must command an expected or required return in the market given by ER i = r + β i (ER m r) where β i = cov(r i,r m )/ var(r m ) (34) There is one further rearrangement of (34) we wish to consider. 28
Substituting for (ER m r) from (23) in (34) gives, ER i = r + λ m cov(r i,r m ) (35) Beta and Systematic Risk If we define the extra return on asset i over and above the risk-free rate as a risk premium, ER i = r + rp i (36) then the CAPM gives the following expressions for the risk premium rp i = β i (ER m r) = λ m cov(r i,r m ) (37) The CAPM predicts that only the covariance of returns between asset i and the market portfolio influence the cross-section of excess returns, across assets. 29
No additional variables such as the dividend price ratio, the size of the firm or the earnings price ratio should influence the cross-section of expected excess returns. All changes in the risk of asset i is encapsulated in changes in cov(r i,r m ). Strictly, this covariance is a conditional covariance the agent at each point in time forms her best view of the value for the covariance/beta. Security market line The linear relationship between the cross-section of average returns ER i and the asset s beta β i is known as the security market line (SML): ER i = r + β i (ER m r) If the CAPM is correct, then all securities should lie on the SML. According to the CAPM/SML, the average excess monthly return on each asset ER i r should be proportional to that asset s beta. 30
31