对外经济贸易大学投资学课件4

Asset allocation between one risky asset and one risk-free asset

Example: E(rA)=15%, (rA)=22%
E(rB)=7%, (rB)=0, so B is a risk-free asset For any portfolio consists of A and B E(rp)=yAE(rA)+(1-yA)E(rB) =15%+7%*yA (rp)=22%yA So, E(rp)=7%+8/22* (rp) The relation between expected return and its risk is linear
+ 2w1w3 Cov(r1,r3)
+ 2w2w3 Cov(r2,r3)
Table 7.2 Computation of Portfolio Variance From the Covariance Matrix
Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients
Mean-variance utility function

Investors only care about expected returns and risk, one type of utility function is :
U E (r ) 0.5 A
2
Mean-variance efficient frontier
Construct portfolios

The covariance and correlation coefficient measure the joint probability distribution of two stocks
Cov(rA , rB )
(r
t 1
N
A ,t
rA )( rB ,t rB ) N 1
E (rp ) wD E (rD ) wE E (rE )
w w 2w w Cov(rD , rE )
2 P 2 D
Two-Security Portfolio: Risk 2 2 2 D E E D E E

2 D
= Variance of Security D = Variance of Security E
Table 7.1 Descriptive Statistics for Two Mutual Funds
Three-Security Portfolio
E (rp ) w1E (r1 ) w2 E (r2 ) w3 E (r3 )
2p = w1212 + w2212 + w3232 + 2w1w2 Cov(r1,r2)
Why investors are risk-aversion?

Risk aversion implies diminishing marginal utility
We can also determine The certainty equivalent
Eg, suppose an investor has logarithmic utility function, an investment Can yield either 50000 or 150000 with equal possibility, what Is certainty equivalent amount for this investment?
2 P wD wDCov(rD , rD ) wE wE Cov(rE , rE ) 2wD wE Cov(rD , rE )
Covariance
Cov(rD,rE) = DEDE D,E = Correlation coefficient of returns
D = Standard deviation of returnsΒιβλιοθήκη Baidufor Security D E = Standard deviation of returns for Security E
Where xj is the portfolio weight of each stock j

Variance matrix
Stock A B C xA 2(A) Cov(A,B) Cov(A,C) xB Cov(A,B) 2(B) Cov(B,C) xC Cov(A,C) Cov(B,C) 2(C)
weight xA xB xC
rp rP rD rE
Two-Security Portfolio: Return wDr D wE r E

Portfolio Return
wD Bond Weight Bond Return Equity Return wE Equity Weight

CAL with Higher Borrowing Rate
E(r)
P ) S = 6/22 9% 7% ) S = 8/22

p = 22%
CAL with Higher Borrowing Rate
E(rp)=yAE(rA)+(1-yA)E(rB) =1.4*15%-0.4*9%=17.4% (rp)=22%*1.4=30.8% Reward-to-variability ratio is (17.4%9%)/30.8%=6/22
Lecture 4 Portfolio Theory
Learning Objectives



Calculate stock return from historical data Calculate mean, variance, covariance and correlation efficient of stock returns Find Markowitz mean-efficient portfolio for up to three stocks Chapters covered: Chapter 6,7 and 8


Rational investors are risk aversion, they prefer higher expected return given the risk level or lower risk given the expected return. If we graph expected return with risk, the portfolios that give highest expected return at each level of risk forms the efficient frontier
Risk aversion



Investors care about expected utility, not the expected value Investors’ expected utility is a concave function of risk level, it implies investors will not accept “fail game”, they demand additional reward for risks. The concept of risk aversion is one of the most important assumptions in investment theory
Asset allocation with leverage
Suppose the investment budget is $300000, and investor borrows an additional $120,000, investing a total of 420,000 in risky asset. E(rp)=yAE(rA)+(1-yA)E(rB) =1.4*15%-0.4*7%=18.2% (rp)=22%*1.4=30.8% Reward-to-variability ratio is (18.2%7%)/30.8%=8/22
Mean-variance indifference curve

A curve that connects all portfolio points with the same utility value in the meanvariance plane
utility
E(r)

Utility and Indifference Curves

Exp Ret 10 15 20 25
St Deviation U=E ( r ) - .5A2 20.0 0.02 25.5 0.02 30.0 0.02 33.9 0.02
Indifference Curves
Expected Return
Increasing Utility
Standard Deviation
Mean variance criterion

A mean-variance rational investor will always prefer portfolio A to portfolio B if:
E ( RA ) E ( RB )and
A B at least one inequality is strict
A, B
Cov(rA , rB ) (rA ) (rB )
The risk and expected return of a portfolio

A portfolio is a combination of securities
E (rp ) x j E (rj )
j 1 N
A portfolio consists of risk-free asset and risky asset

E(rp)
When there exists risk-free asset , the combination line is a straight line
borrowing xA=1 rf lending (rp) (rp)=xA (rA)

Individual investors’ indifference curve is tangent to mean-variance efficient frontier
Represent an investor’s willingness to trade-off return and risk. Example（A=4）

2 E
Cov(rD , rE )= Covariance of returns for
Security D and Security E
Two-Security Portfolio: Risk Continued

Another way to express variance of the portfolio:
Correlation Coefficients: Possible Values
Range of values for 1,2
+ 1.0 > > -1.0 If = 1.0, the securities would be perfectly positively correlated If = - 1.0, the securities would be perfectly negatively correlated
How does risk preference affect investor’s portfolio choice


Example: Consider historical data showing that average return on S&P 500 Index over the past 70 years is 8.5% above T bill with a standard deviation of 20%. Assume these values represents investors’ expectation about future and risk free rate is5%. 1. construct a combination line of two assets( T bill and S&P 500 index portfolio) 2 calculate the utility levels of each portfolio for an investor with risk averse level of A=3 3. recalculate the utility levels for an investor with A=5 What do you conclude?