R5 练习: Portfolio Mathematics

R5 练习: 组合数学

考纲范围

• Calculate and interpret the expected value, variance, standard deviation, covariances, and correlations of portfolio returns.
• Calculate and interpret the covariance and correlation of portfolio returns using a joint probability function for returns.
• Define shortfall risk, calculate the safety-first ratio, and identify an optimal portfolio using Roy’s safety-first criterion.

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Q1.

A fund manager is analyzing a portfolio of asset A and asset B:

Asset	Market Value	Book Value	Expected Return
Asset A	| | 8%
Asset B	| | 12%

The expected return of the portfolio is closest to:

A. 10.4%.

B. 9.6%.

C. 8.2%.

查看答案与解析

答案：A

解析：组合期望收益率是各资产期望收益率按市值权重的加权平均。注意使用市值而非账面价值。

计算过程： $w_A=\frac{200}{200+300}=40\%,w_B=\frac{300}{200+300}=60\%$ $E(R_p)=0.40\times 8\%+0.60\times 12\%=3.2\%+7.2\%=10.4\%$

选项	判断	解析
A	✓	按市值加权：40%×8% + 60%×12% = 10.4%
B	✗	9.6%可能是按账面价值加权的结果
C	✗	计算错误

关联：R5: Portfolio Mathematics

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Q2.

A fund consists of two assets A and B with the proportion of 35% and 65%, respectively. The standard deviation of asset A is 0.25 and the standard deviation of asset B is 0.18. Also knowing that the two assets’ correlation is 0.7, the standard deviation of the fund is closest to:

A. 0.1689.

B. 0.1889.

C. 0.2045.

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答案：B

解析：两资产组合的标准差公式。

计算过程： ${\sigma }_p^2=w_A^2{\sigma }_A^2+w_B^2{\sigma }_B^2+2w_A{w}_B{\sigma }_A{\sigma }_B{\rho }_{AB}$ $=(0.35{)}^2(0.25{)}^2+(0.65{)}^2(0.18{)}^2+2(0.35)(0.65)(0.25)(0.18)(0.7)$ $=0.1225\times 0.0625+0.4225\times 0.0324+2\times 0.35\times 0.65\times 0.25\times 0.18\times 0.7$ $=0.007656+0.013689+0.014355$ $=0.03570$

${\sigma }_p=\sqrt{0.03570}=0.1889$

选项	判断	解析
A	✗	可能忽略了协方差项或计算错误
B	✓	组合标准差 = sqrt(0.03570) = 0.1889
C	✗	可能使用了错误的相关系数

关联：R5: Portfolio Mathematics

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Q3.

To calculate the portfolio return variance for a given portfolio consisting of 7 stocks, how many unique covariance terms, excluding variances, are required to be determined?

A. 21.

B. 42.

C. 49.

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答案：A

解析：n个资产的组合方差需要的唯一协方差项数量。

计算过程： $\text{Unique covariance terms}=\frac{n(n-1)}{2}=\frac{7\times 6}{2}=21$

7×7矩阵共49个元素，减去7个对角线上的方差项 = 42个协方差项，但由于Cov(A,B) = Cov(B,A)，唯一协方差项为42/2 = 21。

选项	判断	解析
A	✓	C(7,2) = 7×6/2 = 21个唯一协方差项
B	✗	42是未考虑对称性时的协方差项数（含重复）
C	✗	49是整个协方差矩阵的总元素数（包含方差）

关联：R5: Portfolio Mathematics

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Q4.

An analyst produces the covariance matrix for the returns of portfolio X and portfolio Y as follows:

	Portfolio X	Portfolio Y
Portfolio X	289	99
Portfolio Y	99	144

The correlation of the returns between portfolio X and portfolio Y is closest to:

A. 0.485.

B. 0.568.

C. 0.324.

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答案：A

解析：从协方差矩阵中提取信息计算相关系数。

计算过程：

• $Var(X)=289$，所以 ${\sigma }_X=\sqrt{289}=17$
• $Var(Y)=144$，所以 ${\sigma }_Y=\sqrt{144}=12$
• $Cov(X,Y)=99$

${\rho }_{XY}=\frac{Cov(X,Y)}{{\sigma }_X\times {\sigma }_Y}=\frac{99}{17\times 12}=\frac{99}{204}=0.4853\approx 0.485$

选项	判断	解析
A	✓	ρ = 99/(17×12) = 0.485
B	✗	计算错误
C	✗	计算错误

关联：R5: Portfolio Mathematics

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Q5.

The following information relates to two questions.

Sophie, an analyst, collects the joint probability function of returns of asset X and asset Y as follows:

	Ry = 2.6%	Ry = 5.6%
Rx = 10%	0.2	0
Rx = 12%	0	0.8

The covariance of returns is closest to:

A. 0.0036%.

B. 0.0096%.

C. 0.0066%.

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答案：B

解析：利用联合概率函数计算协方差。

计算过程： $E(R_X)=0.2\times 10\%+0.8\times 12\%=2\%+9.6\%=11.6\%$ $E(R_Y)=0.2\times 2.6\%+0.8\times 5.6\%=0.52\%+4.48\%=5.0\%$

$Cov(X,Y)=\sum P_i\times (R_{Xi}-E(R_X))\times (R_{Yi}-E(R_Y))$ $=0.2\times (10\%-11.6\%)\times (2.6\%-5.0\%)+0+0+0.8\times (12\%-11.6\%)\times (5.6\%-5.0\%)$ $=0.2\times (-1.6\%)\times (-2.4\%)+0.8\times (0.4\%)\times (0.6\%)$ $=0.2\times 0.0384\%+0.8\times 0.0024\%$ $=0.00768\%+0.00192\%=0.0096\%$

选项	判断	解析
A	✗	计算错误
B	✓	Cov(X,Y) = 0.0096%
C	✗	计算错误

关联：R5: Portfolio Mathematics

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Q6.

(续上题) Based on previous question, the correlation of returns is closest to:

A. 0.

B. 1.

C. 0.5.

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答案：B

解析：利用已知协方差计算相关系数。

计算过程： $Var(R_X)=0.2\times (10\%-11.6\%{)}^2+0.8\times (12\%-11.6\%{)}^2$ $=0.2\times (-1.6\%{)}^2+0.8\times (0.4\%{)}^2$ $=0.2\times 0.0256\%+0.8\times 0.0016\%=0.00512\%+0.00128\%=0.0064\%$ ${\sigma }_X=\sqrt{0.0064\%}=0.8\%$

$Var(R_Y)=0.2\times (2.6\%-5.0\%{)}^2+0.8\times (5.6\%-5.0\%{)}^2$ $=0.2\times (-2.4\%{)}^2+0.8\times (0.6\%{)}^2$ $=0.2\times 0.0576\%+0.8\times 0.0036\%=0.01152\%+0.00288\%=0.0144\%$ ${\sigma }_Y=\sqrt{0.0144\%}=1.2\%$

$\rho =\frac{Cov(X,Y)}{{\sigma }_X\times {\sigma }_Y}=\frac{0.0096\%}{0.8\%\times 1.2\%}=\frac{0.0096\%}{0.0096\%}=1.0$

这表明X和Y完全正相关。从联合概率表可以看出，X=10%时Y必然=2.6%，X=12%时Y必然=5.6%，两者的组合没有交叉，呈完美线性关系。

选项	判断	解析
A	✗	0表示不相关
B	✓	完全正相关，ρ = 1
C	✗	0.5表示中度正相关

关联：R5: Portfolio Mathematics

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Q7.

Using Roy’s safety-first criterion with a shortfall level of 6%, a fund manager would prefer:

A. portfolio C with an expected return of 10% and a standard deviation of 15%.

B. portfolio B with an expected return of 8% and a standard deviation of 10%.

C. portfolio A with an expected return of 12% and a standard deviation of 16%.

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答案：C

解析：安全第一比率（SFRatio）越高越好，表示组合收益率低于最低可接受水平的概率越低。

计算过程： $SFRatio=\frac{E(R_p)-R_L}{{\sigma }_p}$

• Portfolio C: SFR = (10% - 6%) / 15% = 4/15 = 0.267
• Portfolio B: SFR = (8% - 6%) / 10% = 2/10 = 0.200
• Portfolio A: SFR = (12% - 6%) / 16% = 6/16 = 0.375

Portfolio A的SFRatio最高（0.375），是最优选择。

选项	判断	解析
A	✗	Portfolio C的SFR = 0.267，不是最高
B	✗	Portfolio B的SFR = 0.200，最低
C	✓	Portfolio A的SFR = 0.375，最高，shortfall risk最低

关联：R5: Portfolio Mathematics

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Q8.

Which of the following is the characteristic of the safety-first ratio (SFRatio)?

A. Substituting the risk-free rate, R_F, for the critical level R_L, the SFRatio becomes the Sharpe ratio.

B. Safety-first rules focus on the risk that portfolio return will fall below some maximum acceptable level over some time horizon.

C. The SFRatio is the lower the better.

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答案：A

解析：安全第一比率的特征和与夏普比率的关系。

选项	判断	解析
A	✓	正确。SFRatio = (E(Rp) - RL)/σp，当RL替换为Rf时，变为Sharpe Ratio = (E(Rp) - Rf)/σp
B	✗	应该是”minimum acceptable level”（最低可接受水平），不是”maximum acceptable level”
C	✗	SFRatio越高越好，代表收益低于阈值的概率越低

关联：R5: Portfolio Mathematics

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Q9.

A couple is investing to accumulate ample money to send their child to college. They would like to be able to take out \next year without eroding the initial capital of \The table below shows two available allocations.

	Allocation A	Allocation B
Expected annual return	12%	8%
Standard deviation of return	20%	5%

Based on the safety-first criterion, which allocation is better?

A. Allocation A.

B. Allocation B.

C. There is no difference between A and B.

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答案：B

解析：需要确定最低可接受收益率（shortfall level），然后计算SFRatio。

计算过程： 最低可接受收益率 = 需要提取的金额 / 初始资本 = / = 5.26%

$SF{R}_A=\frac{12\%-5.26\%}{20\%}=\frac{6.74\%}{20\%}=0.337$

$SF{R}_B=\frac{8\%-5.26\%}{5\%}=\frac{2.74\%}{5\%}=0.548$

Allocation B的SFRatio更高（0.548 > 0.337），因此更优。

选项	判断	解析
A	✗	Allocation A的SFR = 0.337，较低
B	✓	Allocation B的SFR = 0.548，较高，shortfall risk更低
C	✗	两者SFRatio不同

关联：R5: Portfolio Mathematics