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	$200	$300	8%
Asset B	$300	$200	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。账面价值（book value）反映历史成本，与当前组合配置无关，因此不作为权重。

计算过程：

$w_A=\frac{\$200}{\$200+\$300}=40\%,w_B=\frac{\$300}{\$500}=60\%$

$E(R_p)=0.40\times 8\%+0.60\times 12\%=3.2\%+7.2\%=10.4\%$

选项	判断	解析
A	✓	市值加权：40%×8% + 60%×12% = 10.4%
B	✗	9.6% 是按账面价值加权（书面权重 wA=300/500=60%, wB=40% → 0.6×8% + 0.4×12% = 9.6%）——干扰项
C	✗	8.2% 计算错误

⚠️ 易错点：组合权重永远使用当前市值（market value），不是账面价值或购买成本。题目同时给出 MV 和 BV 就是在测试这个概念。

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 $50,000 next year without eroding the initial capital of $950,000. 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 = $50,000 / $950,000 = 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