R83 练习: Portfolio Risk and Return Part I

R83 练习: 组合风险与收益（上）

考纲范围

• Describe characteristics of the major asset classes that investors consider in forming portfolios. 描述：主要投资资产类别的特征
• Explain risk aversion and its implications for portfolio selection. 说明：风险厌恶及其对投资组合选择的影响
• Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. 说明：最佳投资组合的选择（根据投资者的效用（或风险厌恶）和资本配置线）
• Calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data. 计算/解释：资产回报的平均值，方差和协方差（或相关性）
• Calculate and interpret portfolio standard deviation. 计算/解释：投资组合标准差
• Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated. 描述：投资不完全相关的资产对投资组合风险的影响
• Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio. 描述/解释：风险资产的最小方差边界和有效边界，以及全球最小方差投资组合

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Q1.

An examination of the returns and standard deviation of returns across major investable classes provides evidence for the concept of a tradeoff between risk and return. It shows that the relationship between risk and return is:

A. positive.

B. irrelevant.

C. negative.

查看答案与解析

答案：A

解析：历史数据表明，风险（以收益率标准差衡量）与回报之间存在正向关系。高风险资产类别（如小盘股）通常提供更高的平均回报，而低风险资产类别（如国债）则提供较低的回报。这一正向关系是现代投资组合理论的基石之一。

选项	判断	解析
A	✓	风险与收益呈正相关：更高的风险对应更高的预期回报，这是资本市场的核心规律
B	✗	风险与收益并非无关；大量实证数据支持两者之间的正向关系
C	✗	负相关意味着高风险对应低回报，这与历史数据和经济逻辑不符

关联：R83: Portfolio Risk and Return I

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Q2.

Assessing investments based on expected return and variance of returns is an oversimplification due to the fact that returns do not follow a normal distribution. Distributions are usually skewed, with kurtosis than a normal distribution.

A. positive, higher.

B. negative, lower.

C. negative, higher.

查看答案与解析

答案：C

解析：实际投资回报的分布通常不是正态分布。实证研究表明，回报率分布通常呈现负偏（negative skewness）和高峰度（higher kurtosis/leptokurtic）的特征。负偏意味着左尾更长，即极端负回报出现的概率大于正态分布所预测的；高峰度（也称为”肥尾”）意味着极端事件（无论正负）发生的概率都高于正态分布的预测。

选项	判断	解析
A	✗	正偏不符合实际；实际回报分布通常为负偏
B	✗	虽然负偏正确，但峰度应高于正态分布，而非更低
C	✓	负偏（左尾更长）+ 高峰度（肥尾），这是实际回报分布的典型特征

关联：R83: Portfolio Risk and Return I

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Q3.

In terms of trading costs, liquidity is the factor least likely to have impact on:

A. brokerage commissions.

B. bid-ask spreads.

C. trading price.

查看答案与解析

答案：A

解析：流动性影响交易成本的方式主要通过买卖价差（bid-ask spread）和交易价格（price impact）。高流动性资产的买卖价差较小，且大额交易对价格的影响较小。然而，经纪佣金（brokerage commissions）通常由经纪商固定设定，主要取决于交易金额或合约规模，与资产的流动性关系最小。

选项	判断	解析
A	✓	经纪佣金通常是固定费率或按交易金额计算，与流动性关系最小
B	✗	流动性直接影响买卖价差：流动性越低，价差越大
C	✗	流动性影响交易的价格冲击（price impact）：低流动性资产的大额交易会显著影响价格

关联：R83: Portfolio Risk and Return I

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Q4.

Using the utility formula $U=E(R)-0.5A{\sigma }^2$, if the measure for risk aversion (A) has a value of 4, which of the following funds will a risk-averse investor choose?

Fund	Expected Return	Standard Deviation
Fund 1	10%	5%
Fund 2	12%	10%
Fund 3	24%	30%

A. Fund 1.

B. Fund 2.

C. Fund 3

查看答案与解析

答案：B

解析：使用效用函数 $U=E(R)-0.5A{\sigma }^2$ 计算每只基金的效用值。

计算过程：

$U_1=0.10-0.5\times 4\times (0.05{)}^2=0.10-0.005=0.095$

$U_2=0.12-0.5\times 4\times (0.10{)}^2=0.12-0.02=0.10$

$U_3=0.24-0.5\times 4\times (0.30{)}^2=0.24-0.18=0.06$

Fund 2 的效用值最高（0.10），因此风险厌恶型投资者会选择 Fund 2。

选项	判断	解析
A	✗	Fund 1 效用 = 0.095，低于 Fund 2
B	✓	Fund 2 效用 = 0.10，三者中最高
C	✗	Fund 3 效用 = 0.06，虽然预期回报最高，但风险过大导致效用最低

关联：R83: Portfolio Risk and Return I

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Q5.

Which of the following statements relating to utility theory is most likely correct?

A. A risk-free asset will generate the same utility for all individuals.

B. A risk-free asset’s utility is dependent on the investor’s risk attitude.

C. A risk-free asset has a variance of nonzero.

查看答案与解析

答案：A

解析：对于无风险资产，${\sigma }^2=0$，因此效用公式简化为 $U=E(R)-0.5A\times 0=E(R)$。无论投资者的风险厌恶系数 A 为多少，无风险资产的效用都等于其预期收益率（即无风险利率），对所有人相同。

选项	判断	解析
A	✓	无风险资产方差为零，效用 = E(R) = Rf，与风险态度无关，对所有人相同
B	✗	由于无风险资产方差为零，风险态度（A值）不影响其效用
C	✗	无风险资产的方差为零，这正是”无风险”的定义

关联：R83: Portfolio Risk and Return I

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Q6.

There are two comments about risk-averse investors:

Comment 1: Risk-averse investors will minimize the risk for a certain level of return.

Comment 2: Risk-averse investors do not concern about the return for a certain level of risk.

Are the two comments correct?

A. Yes

B. No, only Comment 1 is correct.

C. No, only Comment 2 is correct.

查看答案与解析

答案：B

解析：风险厌恶型投资者有两个核心行为特征：(1) 在给定回报水平下，最小化风险；(2) 在给定风险水平下，最大化回报。Comment 1 正确描述了第一个特征。Comment 2 错误，因为风险厌恶投资者在给定风险下仍然追求最高回报，他们并非不关心回报。

选项	判断	解析
A	✗	Comment 2 是错误的
B	✓	只有 Comment 1 正确：风险厌恶投资者在给定回报下最小化风险
C	✗	Comment 2 错误：风险厌恶投资者同样关心在给定风险下最大化回报

关联：R83: Portfolio Risk and Return I

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Q7.

Which of the following statements about the investor’s optimal portfolio is most accurate?

A. It is the tangent point of the investor’s indifference curve and the efficient frontier if we don’t consider the risk-free asset.

B. It provides the investor with the highest expected return among all investable portfolios.

C. All investors share the same optimal portfolio.

查看答案与解析

答案：A

解析：在不考虑无风险资产的情况下，投资者的最优组合是其无差异曲线与有效前沿的切点。该点代表投资者在其风险偏好约束下能获得的最高效用。不同投资者的无差异曲线形状不同（取决于风险厌恶程度），因此最优组合也不同。

选项	判断	解析
A	✓	无差异曲线与有效前沿的切点即为不考虑无风险资产时的最优组合
B	✗	最优组合并非回报最高的组合，而是效用最高的组合（综合考虑风险和回报）
C	✗	不同风险偏好的投资者有不同的最优组合

关联：R83: Portfolio Risk and Return I

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Q8.

An investor wants to achieve a higher return than that of the optimal risky portfolio. Which portfolio is most appropriate?

A. A lending portfolio

B. A borrowing portfolio

C. The optimal investor portfolio

查看答案与解析

答案：B

解析：最优风险组合位于资本配置线（CAL）与有效前沿的切点。要获得比最优风险组合更高的预期回报，投资者需要借入资金（borrowing portfolio），即以无风险利率借款并将借入资金和自有资金全部投入最优风险组合。这使投资者在CAL上位于最优风险组合的右上方。

选项	判断	解析
A	✗	贷出组合（lending）是将部分资金投入无风险资产，回报低于最优风险组合
B	✓	借入组合通过杠杆放大回报，可获得高于最优风险组合的预期回报
C	✗	最优投资者组合取决于个人风险偏好，不一定高于最优风险组合的回报

关联：R83: Portfolio Risk and Return I

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Q9.

With regard to the modern portfolio theory, the optimal portfolio for an investor:

A. is a combination of the optimal risky portfolio and the market portfolio.

B. best matches the investor’s risk preference.

C. lies above the optimal capital allocation line.

查看答案与解析

答案：B

解析：根据现代投资组合理论，投资者的最优组合是资本配置线（CAL）上最佳匹配其风险偏好的点，即CAL与投资者无差异曲线的切点。该组合最好地平衡了投资者对风险和回报的偏好。

选项	判断	解析
A	✗	最优投资者组合是无风险资产与最优风险组合的组合，而非最优风险组合与市场组合的组合
B	✓	最优组合位于CAL上，最佳匹配投资者的风险偏好
C	✗	CAL上方的点是不可达到的；最优组合位于CAL上

关联：R83: Portfolio Risk and Return I

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Q10.

Stock A had the following returns over the past 5 years:

Year	1	2	3	4	5
Return	4%	5.5%	0.5%	-1.5%	0%

The arithmetic mean return over the 5 years is closest to:

A. 0.5%.

B. 1.7%.

C. 8.5%.

查看答案与解析

答案：B

解析：算术平均收益率是所有期间收益率的简单平均。

计算过程：

$\bar{R}=\frac{4\%+5.5\%+0.5\%+(-1.5\%)+0\%}{5}=\frac{8.5\%}{5}=1.7\%$

选项	判断	解析
A	✗	0.5% 不是正确的平均值
B	✓	8.5% / 5 = 1.7%，为正确的算术平均值
C	✗	8.5% 是总和而非平均值

关联：R83: Portfolio Risk and Return I

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Q11.

The following information relates to the below 2 questions. Consider the monthly returns on hedge funds A and B as shown in the following table:

Month	January	February	March	April	May
Fund A’s Return	4%	2%	-2%	-5%	3%
Fund B’s Return	-6%	3%	-1%	-2%	5%

The sample variance of Fund A is closest to:

A. 0.14%.

B. 0.11%.

C. 3.78%.

查看答案与解析

答案：A

解析：首先计算均值，再计算样本方差。

计算过程：

${\bar{R}}_A=\frac{4\%+2\%+(-2\%)+(-5\%)+3\%}{5}=\frac{2\%}{5}=0.4\%$

$s^2=\frac{\sum (R_i-\bar{R}{)}^2}{n-1}=\frac{(3.6{)}^2+(1.6{)}^2+(-2.4{)}^2+(-5.4{)}^2+(2.6{)}^2}{4}$

$=\frac{12.96+2.56+5.76+29.16+6.76}{4}=\frac{57.20}{4}=14.30$

注意：上述计算单位为 %^2，转换为小数形式：$s^2=0.001430\approx 0.14\%$

使用计算器：Sx = 0.037815，$s^2=(0.037815{)}^2=0.001430\approx 0.14\%$

选项	判断	解析
A	✓	样本方差约为 0.14%（即 0.001430）
B	✗	0.11% 是总体方差而非样本方差（除以 n 而非 n-1）
C	✗	3.78% 是样本标准差（Sx = 3.78%），不是方差

关联：R83: Portfolio Risk and Return I

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Q12.

Based on the previous question, the sample covariance between the two funds is closest to:

A. 0.000235.

B. 0.000188.

C. 0.1437.

查看答案与解析

答案：A

解析：样本协方差可以通过相关系数和两只基金的标准差来计算。

计算过程：

使用计算器得出：

• $S_x=0.037815$（Fund A 样本标准差）
• $S_y=0.043243$（Fund B 样本标准差）
• $r=0.143707$（样本相关系数）

$Cov(A,B)=r\times S_x\times S_y=0.143707\times 0.037815\times 0.043243=0.000235$

选项	判断	解析
A	✓	正确计算的样本协方差 = 0.000235
B	✗	0.000188 可能是用总体标准差计算的结果
C	✗	0.1437 是相关系数，不是协方差

关联：R83: Portfolio Risk and Return I

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Q13.

An analyst gathers the following information about two securities:

Security	Expected Return	Standard Deviation
Security A	10%	6%
Security B	12%	10%

If the covariance between the two securities is 0.002, the correlation coefficient is closest to:

A. 0.33.

B. 0.45.

C. -0.45.

查看答案与解析

答案：A

解析：相关系数等于协方差除以两个标准差的乘积。

计算过程：

${\rho }_{A,B}=\frac{Cov(A,B)}{{\sigma }_A\times {\sigma }_B}=\frac{0.002}{0.06\times 0.10}=\frac{0.002}{0.006}=0.33$

选项	判断	解析
A	✓	0.002 / (0.06 x 0.10) = 0.33
B	✗	计算错误
C	✗	协方差为正，相关系数应为正值

关联：R83: Portfolio Risk and Return I

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Q14.

A portfolio includes two risky assets. Holding other things constant, if the correlation coefficient between the two risky assets increases, what is most likely to happen to the expected return and standard deviation of the portfolio?

A. Both the expected return and the expected standard deviation will decrease.

B. Both the expected return and the expected standard deviation will increase.

C. The expected return will remain the same, and the expected standard deviation other will increase.

查看答案与解析

答案：C

解析：组合预期收益率 $E(R_p)=w_{1}E(R_1)+w_{2}E(R_2)$，仅取决于权重和各资产预期收益，与相关系数无关。组合方差 ${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\rho }_{1,2}{\sigma }_1{\sigma }_2$，当相关系数增大时，组合方差和标准差增大。

选项	判断	解析
A	✗	预期收益不变（与相关系数无关），标准差增加而非减少
B	✗	预期收益不变，不会增加
C	✓	预期收益不变，标准差随相关系数增大而增大（分散化效果减弱）

关联：R83: Portfolio Risk and Return I

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Q15.

According to capital market theory, the variance of a portfolio is most likely affected by:

A. the returns and the standard deviations of assets.

B. the standard deviations of assets and the correlation between assets.

C. the covariance between assets and the returns on assets.

查看答案与解析

答案：B

解析：两资产组合的方差公式为 ${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\rho }_{1,2}{\sigma }_1{\sigma }_2$。从公式可以看出，组合方差由各资产的标准差（方差）和资产间的相关系数（协方差）决定，与资产的预期收益率无关。

选项	判断	解析
A	✗	组合方差与资产收益率无关，收益率只影响组合的预期收益
B	✓	组合方差由各资产的标准差和资产间的相关系数决定
C	✗	虽然协方差影响组合方差，但收益率不影响

关联：R83: Portfolio Risk and Return I

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Q16.

Which of the following statements regarding the correlation between the assets in the portfolio is most accurate?

A. A zero-variance portfolio can only be constructed if the correlation coefficient is -1.

B. Diversification can mitigate risk when the correlation is +1.

C. The higher the correlation coefficient, the greater the potential benefits from diversification.

查看答案与解析

答案：A

解析：当两资产完全负相关（$\rho =-1$）时，可以通过适当配置权重构建零方差组合。当 $\rho =+1$ 时，组合标准差等于各资产标准差的加权平均，没有分散化效果。相关系数越低，分散化效果越大。

选项	判断	解析
A	✓	只有当 $\rho =-1$ 时，才能通过适当权重使组合方差为零
B	✗	当 $\rho =+1$ 时，没有分散化效果，组合风险是各资产风险的加权平均
C	✗	相关系数越低（而非越高），分散化效果越大

关联：R83: Portfolio Risk and Return I

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Q17.

An investor wants to reduce the overall risk of his current portfolio by adding a new asset. Which of the following correlations between the new asset and the existing portfolio best serves the goal?

A. 0

B. -0.8

C. 0.5

查看答案与解析

答案：B

解析：要最大程度降低组合风险，应选择与现有组合相关系数最低（最负）的资产。相关系数越负，新资产与现有组合的反向运动程度越大，分散化效果越好。在三个选项中，-0.8 是最低的相关系数。

选项	判断	解析
A	✗	相关系数为 0 有一定分散化效果，但不如 -0.8
B	✓	-0.8 是最低的相关系数，提供最大的风险降低效果
C	✗	正相关会使分散化效果较弱

关联：R83: Portfolio Risk and Return I

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Q18.

A portfolio contains two risky assets and its risk is equal to the weighted average of the standard deviations of the two asset returns. The correlation between the two assets is closest to:

A. 0.

B. 1.

C. -1.

查看答案与解析

答案：B

解析：当 $\rho =1$（完全正相关）时，组合标准差公式简化为：

计算过程：

${\sigma }_p=\sqrt{w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2(1){\sigma }_1{\sigma }_2}=\sqrt{(w_1{\sigma }_1+w_2{\sigma }_2{)}^2}=w_1{\sigma }_1+w_2{\sigma }_2$

这正好等于标准差的加权平均值。只有在完全正相关时，组合风险才等于个别资产风险的加权平均。

选项	判断	解析
A	✗	当 $\rho =0$ 时，组合风险低于加权平均
B	✓	当 $\rho =1$ 时，${\sigma }_p=w_1{\sigma }_1+w_2{\sigma }_2$，等于加权平均
C	✗	当 $\rho =-1$ 时，组合风险远低于加权平均

关联：R83: Portfolio Risk and Return I

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Q19.

Cedric, CFA candidate, is preparing for the exam and he makes the following statements about Markowitz efficient frontier.

Statement I: Portfolios on the efficient frontier provide the highest return given a certain level of risk.

Statement II: The efficient frontier, which is the part of the minimum-variance frontier, lies below the global minimum-variance portfolio.

Statement III: The shape of the efficient frontier is affected by adding or removing the risk-free asset.

Which of the statements is most accurate?

A. Statement I

B. Statement II

C. Statement III

查看答案与解析

答案：A

解析：有效前沿（efficient frontier）是最小方差边界（minimum-variance frontier）的上半部分，位于全球最小方差组合（global minimum-variance portfolio）之上（而非之下）。有效前沿上的组合在给定风险水平下提供最高的预期回报。有效前沿的形状由风险资产决定，不受无风险资产的影响。

选项	判断	解析
A	✓	有效前沿上的组合在给定风险下提供最高回报
B	✗	有效前沿位于全球最小方差组合之上，而非之下
C	✗	有效前沿的形状由风险资产决定，添加无风险资产不改变有效前沿的形状

关联：R83: Portfolio Risk and Return I

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Q20.

Cedric, CFA, makes three statements about the global minimum-variance portfolio as follows:

Statement I: It is the leftmost point on the minimum-variance frontier.

Statement II: It is the optimal risky portfolio for investors.

Statement III: It has the lowest standard deviation relative to other portfolios on the minimum-variance frontier.

How many statements above are correct?

A. One

B. Two

C. Three

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答案：B

解析：全球最小方差组合（GMVP）是最小方差边界上风险最低的点（最左端的点）。Statement I 正确（最左端点）。Statement II 错误（GMVP 不一定是最优风险组合；最优风险组合是CAL与有效前沿的切点）。Statement III 正确（GMVP 在最小方差边界上具有最低的标准差）。因此两个陈述正确。

选项	判断	解析
A	✗	有两个陈述正确（I 和 III）
B	✓	Statement I 和 III 正确；Statement II 错误
C	✗	Statement II 错误：GMVP 不等于最优风险组合

关联：R83: Portfolio Risk and Return I

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Q21.

Which of the following statements about the Markowitz efficient frontier is most accurate?

A. The portfolios which don’t lie on the efficient frontier are unattainable.

B. The efficient frontier represents the set of risky portfolios on the minimum-variance frontier below the global minimum-variance portfolio.

C. The portfolios on the efficient frontier have the lowest variance for a given expected return level.

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答案：C

解析：有效前沿上的组合在给定预期收益水平下具有最低的方差（或在给定风险水平下具有最高的预期收益）。不在有效前沿上的组合可能是可达到的，但不是最优的。有效前沿是最小方差边界上位于全球最小方差组合之上的部分。

选项	判断	解析
A	✗	不在有效前沿上的组合仍然可达到（attainable），但效率较低
B	✗	有效前沿位于全球最小方差组合之上，而非之下
C	✓	有效前沿上的组合在给定预期收益下具有最低方差

关联：R83: Portfolio Risk and Return I