Lecture 3 Foundations of Finance0Lecture 3: Portfolio Management-Characterizing the Return DistributionI. Reading.II. Examples.III. How to calculate expected return.IV. How to calculate the variance and standard deviation of return.V. How to calculate the covariance and correlation between two returns.VI. Regression.VII. Determining Expected Returns and Return Covariances for a Given Month at the Start ofthat Month.Lecture 3: Portfolio Management-A Risky and a Riskless Asset.I. Reading.II. Expected Portfolio Return: General FormulaIII. Standard Deviation of Portfolio Return: One Risky Asset and a Riskless Asset. IV. Graphical Depiction: Portfolio Expected Return and Standard Deviation.V. Investor Preferences.VI. Portfolio Management: One Risky Asset and a Riskless Asset.Lecture 3 Corrected Foundations of Finance1Lecture 3: Portfolio Management-Characterizing the Return DistributionI. Reading.A. BKM Appendix: A1:1007-1011, A2, A3.B. BKM Chapter 5: Section 5.2.II. Examples.A. Example 1: “Mock” DataState Probability (pr) Ford GM Reebok T-BillBOOM (BM) 50% 25% 22% 9% 5%RECOVERY (RC) 30% -3% 2% 9% 5%DEPRESSION (DP) 20% -10% -19% 3% 5%B. Example 2: Real Data1. Suppose an investor is interested in forming a portfolio at the start of May2006 from the following assets:a. ADM equity.b. IBM equity.c. Walgreen equity.d. S&P 500 index.e. small firms portfolio.f. value (high book-to-market)firms portfoliog. long-term government bond portfolio.h. U.S. one month t-bill.2. The investor also knows the dividend yield on the S&P 500 index at thestart of May (DP).3. The investor could use historical return data to approximate the jointprobability distribution:a. each month’s realization could be treated as a state.b. each state is equally likely; states that occur more frequently willoccur more frequently in the data.c. use 40 years of historical data from 1/65 to 12/04; so eachrealization has a 1 in 480 probability of occurring.d. here, abstract from the issue that this historical distribution is justan approximation of the true return distribution; instead, we willassume that this historical distribution is the true distribution.Lecture 3 Corrected Foundations of Finance2III. How to calculate expected return.A. Formula: If there are K possible states: s1,s2,...,,sK, E[RAsset] = Expected Return on the Asset = pr(s1) x RAsset(s1) +pr(s2) x RAsset (s2) + ... + pr(sK) x RAsset(sK)where RAsset (s) is the return on the Asset in state s; andpr(s) is the probability of state s.B. Example 1:1. Formula: There are 3 states. So,E[RAsset] = Expected Return on the Asset = pr(BM) x RAsset(BM) +pr(RC) x RAsset(RC) + pr(DP) x RAsset(DP)2. Calculations:E[RFord] = 0.50 x 25% + 0.30 x -3% + 0.20 x -10% = 9.6%. E[RGM] = 0.50 x 22% + 0.30 x 2% + 0.20 x -19% = 7.8%. E[RReebok] = 0.50 x 9% + 0.30 x 9% + 0.20 x 3% = 7.8%. E[RT-bill] = 0.50 x 5% + 0.30 x 5% + 0.20 x 5% = 5%. C. Example 2: There are 480 states.Asset E [R]ADM 1.52IBM 1.04WAG 1.78S&P 500 0.94Small Firms 1.25Value Firms 1.23Gov Bond 0.63DP (start) 3.29Lecture 3 Corrected Foundations of Finance3IV. How to calculate the variance and standard deviation of return.A. Formula: If there are K possible states: s1,s2,...,,sK, σ2[RAsset] = Variance of Return on the Asset = pr(s1) {RAsset(s1)-E[RAsset]}2 + pr(s2) {RAsset(s2)-E[RAsset}2 + ... + pr(sK) {RAsset(sK)-E[RAsset]}2 where RAsset (s) is the return on the Asset in state s; andpr(s) is the probability of state s.σ[RAsset ] = Standard Deviation of Return on the Asset = σ2[RAsset]B. Example 1:1. Formula: There are 3 states. So:σ2[RAsset] = Variance of Return on the Asset= pr(BM) {RAsset(BM)-E[RAsset]}2 + pr(RC) {RAsset(RC)-E[RAsset}2 + pr(DP) {RAsset(DP)-E[RAsset]}2 2. Calculations:σ2[RFord] = 0.50 x (25-9.6)(25-9.6) + 0.30 x (-3-9.6)(-3-9.6) + 0.20 x (-10-9.6)(-10-9.6) = 118.58 + 47.628 +76.832 = 243.04.σ[RFord] = = 15.5897%.243.04σ2[RGM] = 0.50 x (22-7.8)(22-7.8) + 0.30 x (2-7.8)(2-7.8) + 0.20 x (-19-7.8)(-19-7.8) = 100.82 + 10.092 + 143.648 = 254.56.σ[RGM] = = 15.9549%.254.56σ2[RReebok] = 0.50 x (9-7.8)(9-7.8) + 0.30 x (9-7.8)(9-7.8) + 0.20 x (3-7.8)(3-7.8) = 0.72 + 0.432 +4.608 = 5.76.σ[RReebok] = = 2.4%.5.76σ2[RT-bill] = 0.50 x (5-5)(5-5) + 0.30 x (5-5)(5-5) + 0.20 x (5-5)(5-5) = 0 +0 +0 = 0.σ[RT-bill] = = 0%.0Lecture 3 Corrected Foundations of Finance4C. Example 2:Asset σ [R]ADM 8.65IBM 7.27WAG 8.18S&P 500 4.38Small Firms 5.27Value Firms 5.67Gov Bond 2.34DP (start) 1.20Lecture 3 Corrected Foundations of Finance5V. How to calculate the covariance and correlation between two returns.A. Formula: If there are K possible states: s1,s2,...,,sK, σ[RA1,RA2] = Covariance of the Return on Asset 1with the Return on Asset 2= pr(s1) {RA1(s1)-E[RA1]} {RA2(s1)-E[RA2]} + pr(s2) {RA1(s2)-E[RA1]} {RA2(s2)-E[RA2]} + ... + pr(sK) {RA1(sK)-E[RA1]} {RA2(sK)-E[RA2]} where RA1 (s) is the return on Asset 1 in state s; RA2 (s) is the return on Asset 2 in state s; andpr(s) is the probability of state s.ρ[RA1,RA2] = Correlation of the Return on Asset 1with the Return on Asset 2= σ[RA1,RA2]σ[RA1] σ[RA2]B. Example 1:1. Formula: There are 3 states. So:σ[RA1,RA2] = Covariance of the Return on Asset 1with the Return on Asset 2= pr(BM) {RA1(BM)-E[RA1]} {RA2(BM)-E[RA2]} + pr(RC) {RA1(RC)-E[RA1]} {RA2(RC)-E[RA2]}+ pr(DP) {RA1(DP)-E[RA1]} {RA2(DP)-E[RA2]} 2. Calculations:σ[RFord,RGM] = 0.50 x (25-9.6)(22-7.8) + 0.30 x (-3-9.6)(2-7.8) + 0.20 x (-10-9.6)(-19-7.8) = 109.34 + 21.924 + 105.056 = 236.32.ρ[RFord,RGM] = 236.32/(15.5897x15.9549) = 0.95009.σ[RFord,RReebok] = 0.50 x (25-9.6)(9-7.8) + 0.30 x (-3-9.6)(9-7.8) + 0.20 x (-10-9.6)(3-7.8)= 9.24 - 4.536 + 18.816 = 23.52.ρ[RFord,RReebok] = 23.52/(15.5897x2.4) = 0.6286.σ[RGM,RReebok] = 0.50 x (22-7.8)(9-7.8) + 0.30 x (2-7.8)(9-7.8) + 0.20 x (-19-7.8)(3-7.8) = 8.52 - 2.088 + 128.64 = 32.16.ρ[RGM,RReebok] = 32.16/(15.9549x2.4) = 0.8398.σ[RFord,RT-bill] = 0.50 x (25-9.6)(5-5) + 0.30 x (-3-9.6)(5-5) + 0.20 x (-10-9.6)(5-5) = 48 - 48 = 0.Lecture 3 Corrected Foundations of Finance6C. Example 2:σ[Ri,Rj]i= σ[DP,Rj]j= ADM IBM WAG S&P500SmallFirmsValueFirmsGov BondDP(start)ADM 74.75 11.44 19.50 17.26 16.07 46.01 2.90 0.81IBM 11.44 52.79 16.74 18.69 29.40 15.06 0.63 0.12WAG 19.50 16.74 66.95 20.27 34.65 19.89 3.43 0.96S&P 500 17.26 18.69 20.27 19.20 19.38 18.23 2.30 0.46Small Firms 16.07 29.40 34.65 19.38 27.81 17.73 2.12 0.51Value Firms 46.01 15.06 19.89 18.23 17.73 32.12 2.60 0.64Gov Bond 2.90 0.63 3.43 2.30 2.12 2.60 5.48 0.23DP (start) 0.81