第七讲作业-最优风险资产组合

第七讲作业-最优风险资产组合 下面的数据可用于第1至第8题：

一位养老基金经理正在考虑三种共同基金。第一种是股票基金，第二种是长期政府债券与公司债券基金，第三种是回报率为8%的以短期国库券为内容的货币市场基金。这些风险基金的概率分布如下：

名称期望收益率(%) 标准差(%) 股票基金(S) 20 30 债券基金(B) 12 15

基金回报率之间的相关系数为0.10。

1. 两种风险基金的最小方差资产组合的投资比例是多少？这种资产组合回报率的期望值与标准差各是多少？

2. 制表并画出这两种风险基金的投资机会集合，股票基金的投资比率从0%到100%，按照20%的幅度增长。

3. 从无风险回报率到机会集合曲线画一条切线，你的图表表现出来的最优资产组合的期望收益与标准差各是多少？

4. 计算出最优风险资产组合下每种资产的比率以及期望收益与标准差。

5. 最优资本配置线下的最优酬报与波动性比率是多少？

6. 投资者对他的资产组合的期望收益率要求为14%，并且在最佳可行方案上是有效率的。

a. 投资者资产组合的标准差是多少？

b. 投资在短期国库券上的比率以及在其他两种风险基金上的投资比率是多少？

7. 如果投资者只用两种风险基金进行投资并且要求14%的收益率，那么投资者资产组合中的投资比率是怎样安排的？把现在的标准差与第6题中的相比，投资者会得出什么结论？

8. 假设投资者面对同样的机会集合，但是不能够借款。投资者希望只由股票与债券构成期望收益率为4%的资产组合。合适的投资比率

是多少？由此的标准差是多少？如果投资者被允许以无风险收益率借款，那么投资者的标准差可以降低多少？

9.Risk and Return for Security Portfolios

Suppose you have $100 to invest in two assets, A and B. A and B are the only assets available. A is a risky asset and B is a risk-free asset. The expected return on A is 5%, and B earns a risk-free rate of 3%. The standard deviations of returns on A and B are 10% and 0%, respectively. The covariance between the returns on the two assets is 0.

(a)If you invest $30 in A and $70 in B, what is the expected return on your portfolio?

(b)What is the standard deviation of return for the portfolio in (a)?

(c)If you want to invest $130 in A, how much do you have to short sell B? Assume

you can fully use the proceeds from the short sale, and ignore margin and collateral requirements.

(d)Calculate the expected return and the standard deviation of return for the portfolio in (c).

(e)If you short sell $x of B and close your position a year later, how much do you pay when you buy back B? Ignore margin and collateral requirements.

10.Optimal Allocation

There are only two assets: a risk-free asset which earns 7%, and a risky asset which earns an expected return of 15% and has a standard deviation of 22%.

(a) What is the optimal allocation, y*, invested in the risky portfolio if the risk aversion parameter A = 4?

(b) What are the expected return E(rC) and the standard deviation σC of the complete portfolio, and the utility UC when

holding the complete portfolio?

(c) The borrowing rate is 9% (the lending rate is still 7%). Does this affect the investor with risk aversion A = 4? For another investor whose risk aversion A is 1.1, what is his optimal allocation y*?

11．Your utility function is U = E(r) –? Aσ2, and you are risk averse. (a) Draw the shape of two utility indifference curves with utility scores 5% and 10%. Label the axes. (The curves do not have to be exact. Simply draw the shape.)

(b) IBM has an expected return of 10% and standard deviation of 20%. Is it possible that IBM lies on the indifference curve with utility score = 5%? Is it possible that IBM lies on the indifference curve with utility score = 10%? Briefly explain.

(c) HSBC has an expected return of 8% and standard deviation of 15%. The Risk-free rate is 4%. Using IBM, HSBC, and the risk-free asset, you form a portfolio that gives you the highest utility score. What additional information do you need in order to find this portfolio? Explain how you can find this portfolio if you are given such information.

12. Consider two stocks A and B with the following characteristics:

Both stocks have the same standard deviation of 30%. The expected returns on A and B are E(rA) and E(rB), respectively.

The correlation between A and B is -1.

(a) In a portfolio P you invest w in Stock A and 1 –w in Stock B, at what w can you achieve the minimum portfolio variance σP2?

(b) What is the standard deviation of your portfolio in part (a)? Show your calculations.

(c)What should the risk-free rate be (in terms of E(rA) and E(rB))? Briefly

explain

(d)Can E(rA) and E(rB) be different in equilibrium? Briefly explain 13．Your utility function is U = E(r) –? Aσ2, where A > 0.

(a) Stock Q has an expected return of 15% and a standard deviation of 15%, while Stock R has an expected return of 12% and a standard deviation of 17%. Could Q and R be on the same utility indifference curve? Why or why not? (Do not use your answer in part (b) to answer this part.)

(b) The risk-free rate is 6%. What value of A makes you indifferent between Stock Q and the risk-free asset?

(c) Use the value of A in part (b). What is the maximum utility you can get from a portfolio that consists of Stock Q and the risk-free asset?

14.You are managing a portfolio P, with an expected return of 13% and a standard deviation of 20%. The risk-free rate is 3% and the borrowing rate is 5%.

(a) Your client C wants to invest 60% in P, and 40% in the risk-free asset. Find the expected return and standard deviation of your client C’s complete portfolio.

(b) What is the Sharpe ratio of your portfolio P? What is the Sharpe ratio of your client C’s complete portfolio?

(c) Another client D wants to invest in P and have a standard deviation of 28%. What is the expected return of his complete portfolio?

(d) Draw the Capital Allocation Line (CAL). Show in the diagram where you would find Portfolio P and your clients’ portfolios (Both C’s and D’s). Label the axes.

15、选择题

1)Adding additional risky assets will generally move the efficient frontier to the_____ and to the _______.

A) up, right B) up, left C) down, right D) down, left

2)The term efficient frontier refers to the set of portfolios that ______.

A) Yield the greatest return for a given level of risk B) Involve the least risk for a given level of return C) Both a and b above

D) None of the above answers are correct

3)Asset A has an expected return of 15% and a reward-to-variability ratio of .4. Asset B has an expected return of 20% and a reward-to-variability ratio of .3. A risk-averse investor would prefer a portfolio using the risk-free asset and _______.

A) asset A；B) asset B；C) no risky asset；D) can’t tell from the data given

4)Diversification is most effective when security returns are __________.

A) high B) negatively correlated C) positively correlated D) uncorrelated

5)The standard deviation of return on investment A is .10 while the standard deviation of

return on investment B is .05. If the covariance of returns on A and B is .0030, the correlation coefficient between the returns on A and B is _____.

A) .12 B) .36 C) .60 D) .77

6)Consider two perfectly negatively correlated risky securities, A and B. Security A has an expected rate of return of 16% and a standard deviation of return of 20%. B has an expected rate of return 10% and a standard deviation of return of 30%. The weight of security B in the global minimum variance is _____.

A) 10% B) 20% C) 40% D) 60%

7)Which of the following portfolios cannot lie on the efficient

frontier?

I 8% 10% J 16% 20% K 15% 25% L 25% 38%

A) Portfolio I B) Portfolio J C) Portfolio K D) Portfolio L 8)The ______ average ignores compounding.

A) geometric B) arithmetic C) both a and b above D) none of the above

9)The geometric average of 10%, 20% and 25% is __________. A) 15% B) 18.2% C) 18.3% D) 23%

10)Consider a treasury bill with a rate of return of 5% and the following risky securities: Security A: E(r) = .15; s 2 = .0400

Security B: E(r) = .10; s 2 = .0225 Security C: E(r) = .12; s 2 = .1000 Security D: E(r) = .13; s 2 = .0625

The investor must develop a complete portfolio by combining the risk-free asset with one of the securities mentioned above. The security the investor would choose as part of his complete portfolio would be __________.

A) security A ；B) security B ； C) security C ； D) security D 11)An investor can design a risky portfolio based on two stocks, A and B. The standard deviation of return on stock A is 20% while the standard deviation on stock B is 15%.The correlation coefficient between the return on A and B is 0%. The expected return onstock A is 20% while on stock B it is 10%. The proportion of the minimum variance portfolio that would be invested in stock B is ________.

A) 6% B) 50% C) 64% D) 100%

12)In the mean-standard deviation graph, the line that connects the risk-free rate and the optimal risky portfolio, P, is called __________.

A) the capital allocation line

B) the indifference curve C) the investor’s utility line D) the security market line

16.（马克威茨的资产组合理论）市场中有三种可投资的资产，其收益率分别为321r r r ，，。方差协

方差矩阵和期望收益率为 =210121012V =0.80.80.4r

求最小方差的投资组合，该投资组合的预期收益是多少？ 17.金融市场有无风险资产和两个风险资产，已知 r f =5%;E (r ?1)=10%;E (r ?2)=15% var (r ?1)=0.04;var (r ?2)=0.09;corr (r ?1,r ?2)=0.5

a. 对资产组合计算(w 0,w 1,w 2)=(0.4,0.3,0.3)；计算E(r ?p )和 Var(r ?p )

b. 对于风险厌恶系数为2的投资者，计算他的最优资产组合。