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Diversification Class 10 Financial Management, 15.414MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Today Diversification • Portfolio risk and diversification • Optimal portfolios Reading • Brealey and Myers, Chapters 7 and 8.1MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Example Fidelity Magellan, a large U.S. stock mutual fund, is considering an investment in Biogen. Biogen has been successful in the past, but the payoffs from its current R&D program are quite uncertain. How should Magellan’s portfolio managers evaluate the risks of investing in Biogen? Magellan can also invest Microsoft. Which stock is riskier, Microsoft or Biogen? 3MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Biogen stock price, 1988 – 2001 120 100 80 60 40 20 0 Jul-88 Jul-90 Jul-92 Jul-94 Jul-96 Jul-98 Jul-00 Average stock return = 3.22% monthly Std deviation = 14.31% monthly 4MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Fidelity Magellan 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% Average returnFidelity Magellan (Over past 10 years) 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Std dev 5MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Example Exxon is bidding for a new oil field in Canada. Exxon’s scientist estimate that there is a 40% chance the field contains 200 million barrels of extractable oil and a 60% chance it contains 400 million barrels. The price of oil is $30 and Exxon would have to spend $10 / barrel to extract the oil. The project would last 8 years. What are the risks associated with this project? How should each affect the required return? 6MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Plan Portfolio mean and variance Two stocks* Many stocks* How much does a stock contribute to the portfolio’s risk? How much does a stock contribute to the portfolio’s return? What is the best portfolio? * Same analysis applies to portfolios of projects 7MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Portfolios Two stocks, A and B You hold a portfolio of A and B. The fraction of the portfolio invested in A is wA and the fraction invested in B is wB. Portfolio return = RP = wA RA + wB RB What is the portfolio’s expected return and variance? Portfolio E[RP] = wA E[RA] + wB E[RB] var(RP) = )w)w)w BABAB 2 BA 2 A ++ R , Rcov( w 2 Rvar( Rvar( 8MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Example 1 Over the past 50 years, Motorola has had an average monthly return of 1.75% and a std. dev. of 9.73%. GM has had an average return of 1.08% and a std. dev. of 6.23%. Their correlation is 0.37. How would a portfolio of the two stocks perform? E[RP] = wGM 1.08 + wMot 1.75 var(RP) = wGM2 6.232 + wMot2 9.732 + 2 wMot wGM (0.37×6.23×9.73) wMot wGM E[RP] var(RP) stdev(RP) 0 1 1.08 38.8 6.23 0.25 0.75 1.25 36.2 6.01 0.50 0.50 1.42 44.6 6.68 0.75 0.25 1.58 64.1 8.00 1 0 1.75 94.6 9.73 1.25 -0.25 1.92 136.3 11.67 9MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 GM and Motorola Motorola GM 0.5% 0.9% 1.3% 1.7% 2.1% Mean75% GM / 25% Mot 50% GM / 50% Mot 25% GM / 75% Mot -25% GM / 125% Mot 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% Std dev 10MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Example 1, cont. Suppose the correlation between GM and Motorola changes. What if it equals –1.0? 0.0? 1.0? E[RP] = wGM 1.08 + wMot 1.75 var(RP) = wGM2 6.232 + wMot2 9.732 + 2 wMot wGM (corr×6.23×9.73) Std dev of portfolio wMot wGM E[RP] corr = -1 corr = 0 corr = 1 0 1 1.08% 6.23% 6.23% 6.23% 0.25 0.75 1.25 2.24 5.27 7.10 0.50 0.50 1.42 1.75 5.78 7.98 0.75 0.25 1.58 5.74 7.46 8.85 1 0 1.75 9.73 9.73 9.73 11MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 GM and Motorola: Hypothetical correlations Mean 2.2% 1.8% 1.4% 1.0% 0.6% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% cor=-1 cor=-.5 cor=0 cor=1 Std dev 12MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Example 2 In 1980, you were thinking about investing in GD. Over the subse-quent 10 years, GD had an average monthly return of 0.00% and a std dev of 9.96%. Motorola had an average return of 1.28% and a std dev of 9.33%. Their correlation is 0.28. How would a portfolio of the two stocks perform? E[RP] = wGD 0.00 + wMot 1.28 var(RP) = wGD2 9.962 + wMot2 9.332 + 2 wMot wGD (0.28×9.96×9.33) wMot wGD E[RP] var(RP) stdev(RP) 0 1 0.00 99.20 9.96 0.25 0.75 0.32 71.00 8.43 0.50 0.50 0.64 59.57 7.72 0.75 0.25 0.96 64.92 8.06 1 0 1.28 87.05 9.33 13MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 GD and Motorola Motorola GD -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 4.0% 6.0% 8.0% 12.0% 14.0% 16.0% Std dev Mean 10.0% 14MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Example 3 You are trying to decide how to allocate your retirement savings between Treasury bills and the stock market. The Tbill rate is 0.12% monthly. You expect the stock market to have a monthly return of 0.75% with a standard deviation of 4.25%. E[RP] = wTbill 0.12 + wStk 0.75 wvar(RP) = wTbill2 0.02 + wStk2 4.252 + 2 wTbill wstk (0.0×0.0×4.25) Stk2 4.252 wStk wTbill E[RP] var(RP) stdev(RP) 0 1 0.12 0.00 0.00 0.33 0.67 0.33 1.97 1.40 0.67 0.33 0.54 8.11 2.85 1 0 0.75 18.06 4.25 15MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Stocks and Tbills 1.0% 0.8% 0.6% Mean 0.4% 0.2% 0.0% 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% Std dev Stock market 50/50 portfolio Tbill 16MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Many assets Many stocks, R1, R2, …, RN You hold a portfolio of stocks 1, …, N. The fraction of your wealth invested in stock 1 is w1, invested in stock 2 is w2, etc. Portfolio return = RP = w1 R1 + w2 R2 + … + wN RN = ∑ R w i i i Portfolio mean and variance E[RP] = ]ii i∑ (weighted average) var(RP) = )) jiji jiii 2 i ∑ ∑∑ ≠+ E[R w R , (Rcov w w Rvar( w 17MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Many assets Variance = sum of the matrix ))) ))) ))) N 2 NN2N2N1N1 N2N22 2 22121 N1N121211 2 1 L MOMMM L L L var(R w R , cov(R w w R , cov(R w w N Stk R , cov(R w w var(R w R , cov(R w w 2 Stk R , cov(R w w R , cov(R w w var(R w 1 Stk N Stk 2 Stk 1 Stk The matrix contains N2 terms N are variances N(N-1) are covariances In a diversified portfolio, covariances are more important than variances. A stock’s variance is less important than its covariance with other stocks. 18MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 10 Fact 1: Diversification Suppose you hold an equal-weighted portfolio of many stocks (inves-ting the same amount in every stock). What is the variance of your portfolio?


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MIT 15 414 - Diversification

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