1 / 17

Optimal Risky Portfolios

Optimal Risky Portfolios. Objective. Objective - finding the optimal risky portfolio Start with two risky assets find portfolios with lowest possible risk, given a set of assets find the tangency portfolio based on the set of risky assets and the risk-free asset

nau
Télécharger la présentation

Optimal Risky Portfolios

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. Objective • Objective - finding the optimal risky portfolio • Start with two risky assets • find portfolios with lowest possible risk, given a set of assets • find the tangency portfolio based on the set of risky assets and the risk-free asset • Generalize to more than two risky assets

  3. Weighted sums of random variables: Applied in portfolio theory • E(rp)=w1E(r1)+w2E(r2), • VAR(rp)= w1212+ w2222+2 w1w212= w1212+ w2222+2 w1w2121

  4. Example: Assuming the correlation between East Co. and West Co. is 0.6, and portfolio weights for East Co. and West Co. are 0.25 and 0.75 respectively.. E(rp)=0.25E(rE)+0.75E(rW)=0.25*11.44%+0.75*9.52%=0.10 VAR(rp)= 0.252E2+ 0.752W2+2 0.25*0.75EWEW =0.252*9.0181%+ 0.752*0.7851%+2 0.25*0.75*0.6*30.0301%*8.8606% =7.1158% rp=10.585%

  5. Minimum-Variance Weights for Two Asset Portfolio

  6. Example - Problem Statement Given the following information, what is the expected return and standard deviation for the minimum-variance portfolio?

  7. Example - Minimum-Variance Portfolio Weights

  8. Example - Portfolio Return and Standard Deviation • E(R) = .8873(.06) + .1127(.15) = .0701 • Var = (.8873)2(.12)2 + (.1127)2(.21)2 + 2(.8873)(.1127)(.01008)Var = .01391 • Std. Dev. = .11795

  9. Investment Set • Efficient Frontier • The portfolio that provides the highest return for a given level of risk • For a risky portfolio this is the upper portion of the curve beginning at the minimum-variance portfolio • For a combination of a risk-free asset and a risky portfolio it is the tangency line emanating from the risk-free return

  10. Findng the Tangency Portfolio From previous example, assume risk-free rate is 4%.

  11. Another Example - Problem Statement Suppose the risk-free raet of return is 5% and you have two risky assets, Cynthia Enterprises and Tasha’s Toys. They have the following expected returns: E(RCE) = .15 and E(RTT) = .25. Covariance Matrix

  12. Example - Minimum-Variance Risk and Return • E(R) = .8048(.15) + .1952(.25) = .1695 • Var = (.8048)2(.0169) + (.1952)2(.0441) + 2(.8048)(.1952)(.00819) = .0152 • Std. Dev. = .1233 • Reward-to-variability = (.1695 - .05)/.1233 = .9694

  13. Capital Allocation Decision • Suppose you have $10,000 to invest and you want to earn an expected return of 17%. How much should you invest in each asset? • .17 = .1980y + (1-y)(.05)y = .6060 • Investment • (1-.6060)(10,000) = 3940 in T-bills • (.6060)(.4802)(10,000) = 2910 in Tasha’s Toys • (.6060)(.5198)(10,000) = 3150 in Cynthia’s

  14. Markowitz Portfolio Selection Model • Generalize the two asset case to many risky assets and a risk-free security • Identify the risk-return combinations • Identify the optimal risky portfolio - steepest CAL • Choose the complete portfolio based on preferences

  15. Introduce Risk-free Asset • Vary CAL until the tangency portfolio is found • This is the optimal portfolio for ALL investors regardless of risk preferences for this given set of assets. • Investor imposed constraints prevent everyone from holding the same risky portfolio • The more risk averse the investor, the more they will hold of the risk-free asset

  16. Separation Property • Investment decision is separated into two parts • technical determination of optimal risky portfolio • decision on investment between risky portfolio and risk-free asset based on personal preference • This property makes professional money management very efficient.

More Related