Diversification. Class 10 Financial Management,

Transcription

1 Diversification Class 10 Financial Management,

2 Today Diversification Portfolio risk and diversification Optimal portfolios Reading Brealey and Myers, Chapters 7 and 8.1

3 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? 3

4 Biogen stock price, Average stock return = 3.% monthly Std deviation = 14.31% monthly Jul-88 Jul-90 Jul-9 Jul-94 Jul-96 Jul-98 Jul-00 4

5 Fidelity Magellan 1.6% 1.4% 1.% Average return 1.0% 0.8% 0.6% 0.4% Fidelity Magellan (Over past 10 years) 0.% 0.0% 0.0%.0% 4.0% 6.0% 8.0% 10.0% Std dev 5

6 Example Exxon is bidding for a new oil field in Canada. Exxon s scientist estimate that there is a 40% chance the field contains 00 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? 6

7 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 7

8 Two stocks, A and B Portfolios You hold a portfolio of A and B. The fraction of the portfolio invested in A is w A and the fraction invested in B is w B. Portfolio return = R P = w A R A + w B R B What is the portfolio s expected return and variance? Portfolio E[R P ] = w A E[R A ] + w B E[R B ] var(r P ) = w var( R ) + w var( R ) w w cov( R, R ) A A B B + A B A B 8

9 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 6.3%. Their correlation is How would a portfolio of the two stocks perform? E[R P ] = w GM w Mot 1.75 var(r P ) = w GM w Mot w Mot w GM ( ) w Mot w GM E[R P ] var(r P ) stdev(r P )

10 GM and Motorola.1% -5% GM / 15% Mot 1.7% Motorola Mean 1.3% 0.9% 5% GM / 75% Mot 50% GM / 50% Mot 75% GM / 5% Mot GM 0.5%.0% 4.0% 6.0% 8.0% 10.0% 1.0% 14.0% Std dev 10

11 Example 1, cont. Suppose the correlation between GM and Motorola changes. What if it equals 1.0? 0.0? 1.0? E[R P ] = w GM w Mot 1.75 var(r P ) = w GM w Mot w Mot w GM (corr ) Std dev of portfolio w Mot w GM E[R P ] corr = -1 corr = 0 corr = % 6.3% 6.3% 6.3%

12 .% GM and Motorola: Hypothetical correlations 1.8% Mean 1.4% cor=-1 cor=-.5 cor=0 cor=1 1.0% 0.6% 0.0%.0% 4.0% 6.0% 8.0% 10.0% 1.0% 14.0% 16.0% Std dev 1

13 Example In 1980, you were thinking about investing in GD. Over the subsequent 10 years, GD had an average monthly return of 0.00% and a std dev of 9.96%. Motorola had an average return of 1.8% and a std dev of 9.33%. Their correlation is 0.8. How would a portfolio of the two stocks perform? E[R P ] = w GD w Mot 1.8 var(r P ) = w GD w Mot w Mot w GD ( ) w Mot w GD E[R P ] var(r P ) stdev(r P )

14 GD and Motorola.0% 1.5% 1.0% Motorola Mean 0.5% 0.0% 4.0% 6.0% 8.0% 10.0% 1.0% 14.0% 16.0% GD -0.5% Std dev -1.0% 14

15 Example 3 You are trying to decide how to allocate your retirement savings between Treasury bills and the stock market. The Tbill rate is 0.1% monthly. You expect the stock market to have a monthly return of 0.75% with a standard deviation of 4.5%. E[R P ] = w Tbill w Stk 0.75 var(r P ) = w Tbill w Stk w Tbill w stk ( ) w Stk 4.5 w Stk w Tbill E[R P ] var(r P ) stdev(r P )

16 Stocks and Tbills 1.0% 0.8% Stock market Mean0.6% 0.4% 50/50 portfolio 0.% 0.0% Tbill 0.0% 1.0%.0% 3.0% 4.0% 5.0% 6.0% Std dev 16

17 Many assets Many stocks, R 1, R,, R N You hold a portfolio of stocks 1,, N. The fraction of your wealth invested in stock 1 is w 1, invested in stock is w, etc. Portfolio return = R P = w 1 R 1 + w R + + w N R N = w R i i i Portfolio mean and variance E[R P ] = w E[R ] i i i (weighted average) var(r P ) = w var( R + w w cov (R, R ) i i i) i j i j i j 17

18 Many assets Variance = sum of the matrix Stk 1 Stk L Stk N Stk 1 Stk M 1 w w w 1 var(r 1 ) cov(r, R M 1 ) w w 1 w cov(r, R var(r M 1 ) ) L L O w w w 1 w N N cov(r, R cov(r M 1, R N N ) ) Stk N w w 1 N cov(r, R 1 N ) w w N cov(r, R N ) L w N var(r N ) The matrix contains N 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. 18

19 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? Portfolio of N assets, w i = 1/N 1 N -1 var(r P ) = Avg. variance + Avg. c ovariance N N For a diversified portfolio, variance is determined by the average covariance among stocks. An investor should care only about common variation in returns ( systematic risk). Stock-specific risk gets diversified away. 19

20 Example The average stock has a monthly standard deviation of 10% and the average correlation between stocks is If you invest the same amount in each stock, what is variance of the portfolio? What if the correlation is 0.0? 1.0? cov(r i, R j ) = correlation stdev(r i ) stdev(r j ) = = var(r P ) = N if N is large N N stdev(r P ) = 6.3% 0

21 Diversification 1% 10% If correlation = 1.0 Std dev of portfolio 8% 6% 4% % If correlation = 0.4 If correlation = 0.0 0% Number of stocks 1

22 Fact : Efficient portfolios With many assets, any portfolio inside a bullet-shaped region is feasible. The minimum-variance boundary is the set of portfolios that minimize risk for a given level of expected returns.* The efficient frontier is the top half of the minimum-variance boundary. * On a graph, the minimum-variance boundary is an hyperbola.

23 Example You can invest in any combination of GM, IBM, and Motorola. Given the following information, what portfolio would you choose? Variance / covariance Stock Mean Std dev GM IBM Motorola GM IBM Motorola E[R P ] = (w GM 1.08) + (w IBM 1.3) + (w Mot 1.75) var(r P ) = (w GM 6.3 ) + (w IBM 6.34 ) + (w Mot 9.73 ) + ( w GM w IBM 16.13) + ( w GM w Mot.43) + ( w IBM w Mot 3.99) 3

24 Feasible portfolios.1% 1.8% Efficient frontier Motorola Mean 1.5% 1.% Minimum-variance portfolio IBM GM 0.9% 0.6% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0% 11.0% 1.0% Std dev 4

25 Tangency portfolio Fact 3 If there is also a riskless asset (Tbills), all investors should hold exactly the same stock portfolio! All efficient portfolios are combinations of the riskless asset and a unique portfolio of stocks, called the tangengy portfolio.* * Harry Markowitz, Nobel Laureate 5

26 Combinations of risky and riskless assets.4% 1.8% Motorola Mean 1.% IBM GM P 0.6% Riskfree asset 0.0% 0.0%.0% 4.0% 6.0% 8.0% 10.0% 1.0% 14.0% 16.0% Std dev 6

27 .4% Optimal portfolios with a riskfree asset Mean 1.8% 1.% Tangency portfolio IBM GM Motorola 0.6% Riskfree asset 0.0% 0.0%.0% 4.0% 6.0% 8.0% 10.0% 1.0% 14.0% 16.0% Std dev 7

28 Fact 3, cont. With a riskless asset, the optimal portfolio maximizes the slope of the line. The tangency portfolio has the maximum Sharpe ratio of any portfolio, where the Sharpe ratio is defined as Sharpe ratio = E[R ] σ P r f P Put differently, the tangency portfolio has the best risk-return trade-off of any portfolio. Aside Alpha is a measure of a mutual fund s risk-adjusted performance. A mutual fund should hold the tangency portfolio if it wants to maximize its alpha. 8

29 Summary Diversification reduces risk. The standard deviation of a portfolio is always less than the average standard deviation of the individual stocks in the portfolio. In diversified portfolios, covariances among stocks are more important than individual variances. Only systematic risk matters. Investors should try to hold portfolios on the efficient frontier. These portfolios maximize expected return for a given level of risk. With a riskless asset, all investors should hold the tangency portfolio. This portfolio maximizes the trade-off between risk and expected return. 9