1 / 16

Optimal Risky Portfolios: Balancing Return and Risk

Learn how to optimize a portfolio by balancing returns and risks of different assets to enhance the reward-variability ratio (Sharpe ratio) using the concepts of minimum-variance combinations and correlation coefficients. Understand the steps to calculate the weights for an optimal risky portfolio with two assets, considering their expected returns and standard deviations.

vberry
Download Presentation

Optimal Risky Portfolios: Balancing Return and Risk

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Optimal Risky Portfolios

  2. 1. E(rc) = wE(rp) + (1 - w)rf 2. c = w p c= complete or combined portfolio W is the weight on the risky asset Review • Mix one risky asset with the risk-free 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(rp) = 10.5% P E(rc) = 8% C rf = 3% F  0 c 15%

  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 withDifferent Correlations E(r) 10.5%  = -1  =- .3 7.5%  = -1  = 1 St. Dev 8% 15%

  8. Sec 1 E(r1) = .105 = .15  = -.3 12  Sec 2 E(r2) = .075 = .08 2 Minimum-Variance Combination 1 s22 - Cov(r1r2) = W1 s2 s2 - 2Cov(r1r2) + 2 1 W2 = (1 - W1)

  9. (.08)2 - (-.3)(.15)(.08) = W1 (.15)2 + (.08)2 - 2(-.3)(.15)(.08) W1 = .277 W2 = (1 - .277) = .723 Minimum-Variance Combination:  = -.03

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

  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= [w1212 + w2222 + 2W1W2 Cov(r1r2)]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:

More Related