Lecture 9-10 Foundations of Finance0Lecture 9-10: Portfolio Management - N Risky Assets and a Riskless AssetI. Reading.II. Standard Deviation of Portfolio Return: N Risky Assets.III. Effect of Diversification. IV. Opportunity Set: N Risky Assets.V. Portfolio Choice: N Risky Assets and a Riskless AssetLecture 9-10 Foundations of Finance1Lecture 9-10: Portfolio Management - N Risky Assets and a Riskless AssetI. Reading.A. BKM, Chapter 8, Sections 8.4 and 8.5 and Appendix 8.A.II. Standard Deviation of Portfolio Return: N Risky Assets.1. Formula. σ2[Rp(t)] 'jNi'1jNj'1ωi,pωj,pσ[Ri(t), Rj(t)]whereσ[Ri(t), Rj(t)] is the covariance of asset i’s return and asset j’sreturn in period t;ωi,p is the weight of asset i in the portfolio p;σ2[Rp(t)] is the variance of return on portfolio p in period t.2. The formula says that σ2[Rp(t)] is equal to the sum of the elements in thefollowing N x N matrix.1 2 ... j ... N-1 N1 ω1,p ω1,p σ[R1, R1]ω1,p ω2,p σ[R1, R2]ω1,p ωj,p σ[R1, Rj]ω1,p ωN-1,p σ[R1, RN-1]ω1,p ωN,p σ[R1, RN]2 ω2,p ω1,p σ[R2, R1]ω2,p ω2,p σ[R2, R2]ω2,p ωj,p σ[R2, Rj]ω2,p ωN-1,p σ[R2, RN-1]ω2,p ωN,p σ[R2, RN]...i... ωi,p ω1,p σ[Ri, R1]ωi,p ω2,p σ[Ri, R2]ωi,p ωj,p σ[Ri, Rj]ωi,p ωN-1,p σ[Ri, RN-1]ωi,p ωN,p σ[Ri, RN]N-1 ωN-1,p ω1,p σ[RN-1, R1]ωN-1,p ω2,p σ[RN-1, R2]ωN-1,p ωj,p σ[RN-1, Rj]ωN-1,p ωN-1,p σ[RN-1,RN-1]ωN-1,p ωN,p σ[RN-1, RN]N ωN,p ω1,p σ[RN, R1]ωN,p ω2,p σ[RN, R2]ωN,p ωj,p σ[RN, Rj]ωN,p ωN-1,p σ[RN, RN-1]ωN,p ωN,p σ[RN, RN]a. Notice that there are N2 terms.b. The diagonal elements are the variance terms since σ2[Ri(t)] =σ[Ri(t), Ri(t)]; so there are N variance terms and (N-1)N covarianceterms.Lecture 9-10 Foundations of Finance2c. Notice that this formula specializes to the formula used above forthe two asset case:σ2[Rp(t)] ' ω21,pσ[R1(t)]2% ω22,pσ[R2(t)]2% 2 ω1,pω2,pσ[R1(t), R2(t)]121 ω1,p ω1,p σ[R1, R1]ω1,p ω2,p σ[R1, R2]2 ω2,p ω1,p σ[R2, R1]ω2,p ω2,p σ[R2, R2]Lecture 9-10 Foundations of Finance3III. Effect of Diversification. A. Consider an equal weighted portfolio (So ωi,p= 1/N for all i.). For example, whenN=2, an equal weighted portfolio has 50% in each asset.B. Suppose all assets have the same E[R] = R' and σ[R] = σ and have returns whichare uncorrelated. Then, for the equal weighted portfolio:1. N=2:E[Rp(t)] = ½ E[R1(t)] + ½ E[R2(t)] = R'.σ[Rp(t)]2 = (½)2 σ[R1(t)]2 + (½)2 σ[R2(t)]2 = ½ σ2.2. N=3:E[Rp(t)] = a E[R1(t)] + a E[R2(t)] + a E[R3(t)] = R'.σ[Rp(t)]2 = (a) 2 σ[R1(t)]2 + (a) 2 σ[R2(t)]2 + (a) 2 σ[R3(t)]2 = a σ2 .3. Arbitrary N: E[Rp(t)] = R'. σ[Rp(t)]2 = σ2 /N.4. As N increases:a. the variance of the portfolio declines to zero.b. the portfolio’s expected return is unaffected. 5. This is known as the effect of diversification.Lecture 9-10 Foundations of Finance4C. Suppose all assets have the same σ[R] = σ and have returns which are correlated. 1. Formulas for expected return and standard deviation of return for the equalweighted portfolio can be written:E[Rp(t)] ' average expected returnσ2[Rp(t)] ' σ2[1N1 % (1 &1N) average correlation ]whereaverage expected return =1NjNi'1E[Ri(t)]average correlation = 1N(N&1)jNi'1jNj'1, j…iρ[Ri(t), Rj(t)].2. As N increases:a. Expected portfolio return is unaffected.b. Variance of portfolio return:(1) Expressed as a fraction of firm variance, portfolio varianceconverges to the average pairwise correlation betweenassets.3. Shows the benefit of diversification depends on the correlation betweenthe assets.4. Can see that assets with low correlation maximize the diversificationbenefits.Lecture 9-10 Foundations of Finance5D. Suppose assets have non-zero covariances and differing expected returns andstandard deviations.1. Formulas for expected portfolio return and standard deviation can bewritten:E[Rp(t)] ' average expected returnσ2[Rp(t)] '1Naverage variance % (1 &1N) average covariancewhereaverage expected return =1NjNi'1E[Ri(t)]average variance =1NjNi'1σ[Ri(t)]2average covariance = 1N(N&1)jNi'1jNj'1, j…iσ[Ri(t), Rj(t)].2. As N increases:a. Expected portfolio return is unaffected.b. Variance of portfolio return:(1) First term (the unique/ firm specific/ diversifiable/unsystematic risk) goes to zero.(2) Second term (the market/ systematic/ undiversifiable risk)remains.(a) When the assets are uncorrelated (the case above inIII.B), this second term is zero.Lecture 9-10 Foundations of Finance6IV. Opportunity Set: N Risky Assets.A. Set of Possible Portfolios.1. No longer a curve as in the two asset case.2. Instead, a set of curves.B. Minimum Variance Frontier.1. Since individuals are risk averse, can restrict attention to the set ofportfolios with the lowest variance for a given expected return. 2. This curve is known as the minimum variance frontier (MVF) for the riskyassets.3. Every other possible portfolio is dominated by a portfolio on the MVF(lower variance of return for the same expected return).4. Example 2 (cont): Ignoring DP. The basic shape of the MVF is the sameas that for the MVF for three of funds in this example (S&P 500, smallfirm fund and govt bond fund) which is graphed below.5. Further, risk averse individuals would never hold a portfolio on thenegative sloped portion of the MVF; so can restrict attention to thepositive sloped portion. This portion is known as the efficient frontier.Lecture 9-10 Foundations of Finance7C. Adding risky assets.1. Adding risky assets to the opportunity set always causes the minimumvariance frontier to shift to the left in {σ[R],E[R]} space. Why?a. For any given E[R], the portfolio on the MVF for the subset ofrisky assets is still feasible using the larger set of risky assets.b. Further, there may be another portfolio which can be formed fromthe larger set and which has the same E[R] but an even lower σ[R].2. Example 2 (cont): Ignoring DP. MVF for the S&P 500, the small firmfund, the value firm fund and the govt bond fund is to the left of the MVFfor the S&P 500, the small firm fund and the govt bond fund excluding thevalue firm fund . This happens even though the value firm fund has an{σ[R],E[R]) denoted by W which lies to the right of the MVF for the 3funds excluding the value firm fund.Lecture 9-10 Foundations of Finance83. Example 2 (cont): Ignoring DP. MVF for all 6 stock assets (including the3 stock funds or portfolios: S&P 500 fund, small firm fund, and value firmfund) is to the left of the MVF for the 3 individual