投資組合 Portfolio Theorem

課程內容 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

Minimum-Variance Portfolios The minimum-variance is the combination of a specific set of assets that leads to the least variation in returns. Portfolios of assets that aren’t perfectly correlated offer better risk-return opportunities than the individual securities The weighted average of standard deviations is greater than the ortfolio standard deviation.

Finding the minimum-variance portfolio Start with the formula for portfolio variance For a two asset portfolio: The weights are wi in asset i and (1-wi) in asset j Take a first derivative with respect to wi and set equal to 0 to find the minimum-variance point For portfolios with more than two assets: you would have to take multiple partial derivates and solve a set of equations to find the weights for the minimum-variance portoflio

Minimum-Variance Weights for Two Asset Portfolio

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

Example - Minimum-Variance Portfolio Weights

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 Draw a graph of the investment set for a two asset portfolio. Show the efficient frontier.

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 Point out the various efficient frontiers (with or without risk-free asset) on graph. Want to choose that portfolio that will provide the highest reward-to-variability.

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

Expected Returns and Standard Deviations for Tangency Portfolio E(R) = -.19626(.06) + 1.19626(.15) = .1677 var = (-.19626)2(.12)2 + (1.19626)2(.21)2 + 2(-.19626)(1.19626)(.01008) var = .05893 std. dev. = .2428 reward-to-variability = (.1677-.04) / .2428 = .5259 If you choose any combination of the two assets other than the one given above, the reward-to-variability ratio will be smaller. The tangency portfolio is the optimal risky portfolio that we used in Chapter 6. The optimal allocation between the risky asset and the tangency portfolio depends on investor preferences. Go back to graph.

Capital Allocation Suppose you want an expected return of 12%. What percent of your investment should you invest in the risk-free asset and what percent should you invest in each of the risky assets? .12 = .1677y +(1-y)(.04) y = .6265 Investment 1-.6265 = .3735 in risk-free asset -.19626(.6265) = -.1230 in asset D 1.19626(.6265) = .7495 in asset E Suppose you have $1000 to invest Buy 373.50 of T-bills Short 123 of Asset D Buy 749.50 of Asset E

Another Example - Problem Statement Suppose the risk-free rate 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

Example - Minimum-Variance Risk and Return Std. Dev. = .1233 Reward-to-variability = (.1695 - .05)/.1233 = .9694

Example - Tangency Portfolio E(R)=.5198(.15) + .4802(.25) = .1980 Var = (.5198)2(.0169) + (.4802)2(.0441) + 2(.5198)(.4802)(.00819) Var = .0188 Std Dev = .1372 Reward-to-Variability = (.1980-.05) / .1372 = 1.0789

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

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 Draw a graph for multi assets Point out: minimum variance frontier - lowest possible portfolio variance for a given portfolio return global minimum-variance portfolio efficient frontier individual assets and inefficient portfolios

Identify risk-return combinations Obtain estimates of expected returns and covariances (historical data or probability distributions) Use estimates to find minimum-variance frontier and discard everything that has a return less than the minimum-variance portfolio In practice, you don’t have to try every combination of asset weights. Choose a series of weights at relatively small intervals and then graph return vs std. dev. Around the tangency portfolio, you might decrease your intervals to get a more precise estimate. This provides the efficient frontier if no restrictions or constraints. If there are restraints on short-selling you would need to adjust your portfolio returns accordingly. discussion - social consciousness

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

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.