Topic 4: Portfolio Concepts Mean-Variance Analysis • Mean–variance portfolio theory is based on the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return. Assumptions of the Model 1. All investors are risk averse; they prefer less risk to more for the same level of expected return. 2. Expected returns for all assets are known. 3.The variances and covariances of all asset returns are known. 4. Investors need only know the expected returns, variances, and covariances of returns to determine optimal portfolios. 5. There are no transaction costs or taxes. Minimum Variance Frontier • An investor’s objective in using a mean– variance approach to portfolio selection is to choose an efficient portfolio. – An efficient portfolio is one offering the highest expected return for a given level of risk as measured by variance or standard deviation of return. – Portfolios that have the smallest variance for each given level of expected return are called minimum-variance portfolios. Portfolio Expected Return • The expected return for a portfolio is the weighted average of the expected returns of the securities in the portfolio. E(R P ) w1E(R1 ) w 2 E(R 2 ) w n E(R n ) Portfolio Variance • Although it might seem reasonable for the variance of a portfolio to be the weighted average of the variances of the securities in the portfolio, this is incorrect. • Portfolio variance consists of the variances of the individual securities, but must also consist of a factor that measures the interaction of each pair of securities. • Intuitively, if two risky securities are held in a portfolio, but Security A tends to do well when Security B does poorly, and vice versa, a portfolio of the two securities will have less risk. – we can account for the relationship between each pair of securities by using the covariance or the correlation. – Even though both assets are risky, a combination of the two will create a portfolio that is less risky than each of its components. • If we plot two assets in risk/expected return space we get : Negative Correlation Return A B Time Minimum Variance Frontier: Large Cap Stocks & Government Bonds Minimum Variance Frontier for Varied Correlations Portfolio Risk for a Two-Asset Case 2P w12 12 w 22 22 2w1w 21, 2 1 2 P w12 12 w 22 22 2w1w 21, 21 2 1 2 Portfolio Risk Three-Asset Case 2P w1212 w 22 22 w 32 32 2w1w 21, 21 2 2w1w 31,313 2w 2 w 32,3 2 3 P w w w 2w1w 21, 21 2 2w1w 31,313 2w 2 w 32,3 23 2 1 2 1 2 2 2 2 2 3 2 3 1 2 Portfolio Risk and Return n Asset Case n E(R P ) w jE(R j ) j1 n n 2P w i w jCov(R i , R j ) i 1 j1 n w j1 j 1 Example • Given the information in Table 11-1, find the expected return and variance for a portfolio consisting of 40% in large-cap stocks and 60% in government bonds. Example E ( R P ) w 1E ( R 1 ) w 2 E ( R 2 ) .40(15%) .60(5%) 9% 2P w 12 12 w 22 22 2 w 1w 21, 2 1 2 .4 2 152 .6 2 10 2 2(.4)(. 6)(0.5)(15)(10) 108.0 Example • We can find the expected return and variance for portfolios with different combinations of our two assets. Table 11-2 shows the different expected returns and risk for various portfolios. Example Capital Market Line E(R M R F ) E(R P ) R F P M E(R P ) the expected return of portfolio p lying on the CML R F the risk free rate E(R M ) the expected return on the market portfolio M the standard deviation of return on the market portfolio P the standard deviation of return on portfolio p Capital Asset Pricing Model • Assumptions of the CAPM – Investors need only know the expected returns, the variances, and the covariances of returns to determine which portfolios are optimal for them. – Investors have identical views about risky assets’ mean returns, variances of returns, and correlations. – Investors can buy and sell assets in any quantity without affecting price, and all assets are marketable (can be traded). – Investors can borrow and lend at the risk-free rate without limit, and they can sell short any asset in any quantity. – Investors pay no taxes on returns and pay no transaction costs on trades. Capital Asset Pricing Model E ( R i ) R F i [ E ( R M ) R F ] E(R i ) the expected return on asset i R F the risk free rate of return E(R M ) the expected return on the market portfolio Cov(R i , R M ) i Var (R M ) Mean Variance Portfolio Choice Rules • The Markowitz decision rule provides the principle by which a mean–variance investor facing the choice of putting all her money in Asset A or all her money in Asset B can sometimes reach a decision. – This investor prefers A to B if either • the mean return on A is equal to or larger than that on B, but A has a smaller standard deviation of return than B • the mean return on A is strictly larger than that on B, but A and B have the same standard deviation of return. Decision to Add an Investment to an Existing Portfolio • Adding the new asset to your portfolio is optimal if the following condition is met: E(R new ) R F E(R p ) R F Corr (R new , R p new p Market Model R i i i R M i R i the return on asset i R M the return on the market portfolio i average return on asset i unrelated to the market return i the sensitivit y of the return on asset i to the return on the market portfolio i an error term E ( R i ) i i E ( R M ) Var ( R i ) i2 2i Cov( R i R j ) i j 2 M Multifactor Models R i a i b i1F1 b i 2 F2 ... b iK FK i R i the return on asset i a i the expected return to asset i Fk the surprise in factor k, k 1, 2, . . ., K b ik the sensitivit y of the return on asset i to a surprise in factor k i an error term with zero mean that represents the portion of the portion of the return to asset i not explained by the factor model An Example R i a i b i1FINT b i 2 FGDP i R i the return on stock i a i the expected return to stock i FINT the surprise in interest rates b i1 the sensitivit y of the return to stock i to interest rate surprises FGDP the surprise in GDP growth b i1 the sensitivit y of the return to stock i to GDP growth surprises i an error term with zero mean that represents the portion of the portion of the return to asset i not explained by the factor model Arbitrage Pricing Theory (APT) and the Factor Model • APT relies on three assumptions: 1. A factor model describes asset returns. 2. There are many assets, so investors can form welldiversified portfolios that eliminate asset-specific risk. 3. No arbitrage opportunities exist among well-diversified portfolios. • Arbitrage is a risk-free operation that earns an expected positive net profit but requires no net investment of money. • An arbitrage opportunity is an opportunity to earn an expected positive net profit without risk and with no net investment of money. Arbitrage Pricing Theory E (R P ) R F 1 P ,1 ... 1 P ,K E (R P ) the expected return on portfolio P R F the risk free rate j the risk premium for factor j P , j the sensitivit y of the portfolio to factor j K the number of factors Analyzing Sources of Return • Multifactor models can help us understand in detail the sources of a manager’s returns relative to a benchmark. • The return on a portfolio, Rp, can be viewed as the sum of the benchmark’s return, RB, and the active return. K Active Return [( Portfolio sensitivit y) j (Benchmark sensitivit y) j ] (Factor return) j1 Asset selection • Active risk is the standard deviation of active returns. • Tracking error is the total return on a portfolio (gross of fees) minus the total return on a benchmark. • The information ratio (IR), is a tool for evaluating mean active returns per unit of active risk. j Analyzing Sources of Return • How can an analyst appraise the individual contributions of a manager’s active factor exposures to active risk squared? • We can usea a factor’s marginal contribution to active risk squared (FMCAR). • With K factors, the marginal contribution to active risk squared for a factor j, FMCARj is K FMCAR j b aj bia Cov(Fj , Fi ) i 1 Active risk squared Creating a Tracking Portfolio • In a risk-controlled active or enhanced index strategy, the portfolio manager may attempt to earn a small incremental return relative to the benchmark while controlling risk by matching the factor sensitivities of the portfolio to her benchmark. • A tracking portfolio is a portfolio having factor sensitivities that are matched to those of a benchmark or other portfolio.