1. Optimal Risky Portfolios

2. Review Mix one risky asset with the risk-free asset 1. E(r c ) = wE(r p ) + (1 - w)r f  c = w  p c= complete or combined portfolio W is the weight on the risky asset

3. Example Suppose as an investor, you want to invest $10000 There are 2 assets to pick from: S&P500 index and the risk-free T-bill For S&P500, the annual return from 1980- 2005 is around 10.5%, standard deviation is 15% annually For the risk-free, one-year Treasury security is about 3%

4. Possible Combinations for S&P500 and T-bill E(r) E(r p ) = 10.5% r f = 3% 15% 0 P F  cc E(r c ) = 8% C

5. New question Now introduce two risky assets, how can we find the optimal mix between the two to form a new portfolio p so that we can improve upon the reward-variability ratio (Sharpe ratio):

6. A new asset: real estate Average annual housing price increases from 1995-2004 around NYC is 7.5%, standard deviation is 8% Correlation coefficient between S&P500 and housing return is –0.3

7. Two-Security Portfolios p with Different Correlations  = 1 10.5% E(r) St. Dev 8%15%  =-.3  = -1 7.5%

8. 1 1 2   2 2 - Cov(r 1 r 2 ) W1W1 = + - 2Cov(r 1 r 2 ) W2W2 = (1 - W 1 ) 22 2 E(r 2 ) =.075=.08Sec 2 12 = -.3 E(r 1 ) =.105=.15Sec 1     2 Minimum-Variance Combination

9. W1W1 = (.08) 2 - (-.3)(.15)(.08) (.15) 2 + (.08) 2 - 2(-.3)(.15)(.08) W1W1 =.277 W2W2 = (1 -.277) =.723 Minimum-Variance Combination:  = -.03

10. r p =.277(.105) +.723(.075) =.08 p = [(.277) 2 (.15) 2 + (.723) 2 (.08) 2 + 2(.277)(.723)(-.3)(.15)(.08)] 1/2 p =.06     Minimum -Variance: Return and Risk with  = -.3

11. Optimal Risk Portfolio To achieve the optimal portfolio, we need to find the weights for both risky assets in the portfolio, the way to find them is to maximize the following: Where p stands for the optimal portfolio. The optimal portfolio weights are given:

12. Solution for the weights of the optimal risky portfolio with two risky assets (asset 1 and 2)

13. Example Risk-free rate as 3% Risky asset one has expected return as 10.5%, standard deviation as 15% The return-risk tradeoff for the above 2 assets: Reward-variability ratio is 0.5

14. Calculation

15. The new risky portfolio p By investing 33% in risky asset S&P500 and 67% in housing, the new risky portfolio has expected return as: 0.33*10.5%+0.67*7.5% = 8.5%, the standard deviation as  p = [w 1 2  1 2 + w 2 2  2 2 + 2W 1 W 2 Cov(r 1 r 2 )] 1/2 =6.1%

16. 2. the second step, combining the new risky portfolio and the risk-free asset, the return and risk tradeoff (CAL line) line becomes: