1. 投資組合 Portfolio Theorem

2. 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

3. 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.

4. 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

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) (.15) = .0701
Var = (.8873)2(.12)2 + (.1127)2(.21)2 + 2(.8873)(.1127)(.01008) Var =
Std. Dev. =
Draw a graph of the investment set for a two asset portfolio. Show the efficient frontier.

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
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.

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

11. Expected Returns and Standard Deviations for Tangency Portfolio
E(R) = (.06) (.15) = .1677
var = ( )2(.12)2 + ( )2(.21)2 + 2( )( )(.01008) var =
std. dev. = .2428
reward-to-variability = ( ) / = .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.

12. 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
= in risk-free asset
(.6265) = in asset D
(.6265) = in asset E
Suppose you have $1000 to invest
Buy of T-bills
Short 123 of Asset D
Buy of Asset E

13. 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

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

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

16. 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
( )(10,000) = 3940 in T-bills
(.6060)(.4802)(10,000) = 2910 in Tasha’s Toys
(.6060)(.5198)(10,000) = 3150 in Cynthia’s

17. 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

18. 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

19. 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

20. 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.