R4 练习: Probability Trees and Conditional Expectations

R4 练习: 概率树与条件期望

考纲范围

• Calculate expected values, variances, and standard deviations and demonstrate their application to investment problems.
• Formulate an investment problem as a probability tree and explain the use of conditional expectations in investment application.
• Calculate and interpret an updated probability in an investment setting using Bayes’ formula.

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Q1.

An analyst analyzes a stock and forecasts the return of this stock as follows:

Return	Probability
20%	20%
7%	50%
15%	30%

The expected return of the stock is closest to:

A. 14%.

B. 12%.

C. 13.87%.

查看答案与解析

答案：B

解析：期望收益率是各种可能收益率的概率加权平均。

计算过程： $E(R)=0.20\times 20\%+0.50\times 7\%+0.30\times 15\%$ $=4\%+3.5\%+4.5\%=12\%$

选项	判断	解析
A	✗	14%是简单算术平均 (20+7+15)/3 = 14%，未考虑概率权重
B	✓	概率加权期望收益 = 12%
C	✗	计算错误

关联：R4: Probability Trees and Conditional Expectations

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Q2.

The probability distribution for a stock’s return is as follows:

Probability	Return
0.3	20%
0.4	10%
0.3	5%

The standard deviation of return is closest to:

A. 5.50%.

B. 7.50%.

C. 5.94%.

查看答案与解析

答案：C

解析：先求期望值，再求方差，最后开方得标准差。

计算过程： $E(R)=0.3\times 20\%+0.4\times 10\%+0.3\times 5\%=6\%+4\%+1.5\%=11.5\%$

$Var(R)=0.3(20\%-11.5\%{)}^2+0.4(10\%-11.5\%{)}^2+0.3(5\%-11.5\%{)}^2$ $=0.3(8.5\%{)}^2+0.4(-1.5\%{)}^2+0.3(-6.5\%{)}^2$ $=0.3\times 72.25+0.4\times 2.25+0.3\times 42.25$ $=21.675+0.9+12.675=35.25$

$\sigma =\sqrt{35.25}=5.94\%$

选项	判断	解析
A	✗	计算不正确
B	✗	可能是方差的近似值
C	✓	标准差 = sqrt(35.25) = 5.94%

关联：R4: Probability Trees and Conditional Expectations

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Q3.

Which of the following is characteristic of the expected value?

A. It is a number greater than or equal to 0.

B. It is the probability-weighted average value of the possible results of the random variable.

C. It is a useful tool for estimating the risks of alternative investments.

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答案：B

解析：期望值（expected value）的定义和特征。

选项	判断	解析
A	✗	期望值可以是负数（如股票收益率的期望值可以为负）
B	✓	期望值正是随机变量所有可能结果的概率加权平均值
C	✗	方差和标准差用于估计风险，期望值用于估计收益

关联：R4: Probability Trees and Conditional Expectations

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Q4.

Which of the following statements is the least correct regarding variance?

A. The variance is a measure of the dispersion of results of the random variable around the expected value.

B. Variance is in the same units as the random variable.

C. Variance is a number greater than or equal to 0.

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答案：B

解析：方差的单位和特征。

选项	判断	解析
A	✓	正确，方差衡量随机变量结果围绕期望值的离散程度
B	✗	不正确。方差的单位是随机变量单位的平方。例如收益率的方差单位是%²，不同于收益率的单位%。标准差才与随机变量具有相同单位
C	✓	正确，方差 >= 0（平方和不可能为负）

关联：R4: Probability Trees and Conditional Expectations

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Q5.

The following information relates to three questions.

Based on a senior analyst’s research, there is a 45% chance that Company AXY will be acquired by Company BND next month. If being acquired, Company AXY’s stock price will go up by 35% with a probability of 0.6, or go up by 25% with a probability of 0.4. If the acquisition does not happen, its stock price will drop by 10% with a probability of 0.65, or drop by 20% with a probability of 0.35.

What is the expected rate of return, given that Company AXY is acquired by Company BND?

A. 31%.

B. -13.5%.

C. 6.53%.

查看答案与解析

答案：A

解析：这是一个条件期望（conditional expectation）问题。求在被收购条件下的期望收益率。

计算过程： $E(R|\text{acquired})=0.6\times 35\%+0.4\times 25\%=21\%+10\%=31\%$

选项	判断	解析
A	✓	条件期望 = 0.6×35% + 0.4×25% = 31%
B	✗	-13.5%是未被收购条件下的期望收益率
C	✗	6.53%可能是无条件期望收益率

关联：R4: Probability Trees and Conditional Expectations

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Q6.

(续上题) Based on previous question, what is the expected rate of return if one invests in Company AXY’s stock?

A. 11.5%.

B. 17.5%.

C. 6.53%.

查看答案与解析

答案：C

解析：无条件期望收益率 = 使用全概率公式计算。

计算过程： $E(R|\text{not acquired})=0.65\times (-10\%)+0.35\times (-20\%)=-6.5\%-7\%=-13.5\%$

$E(R)=P(\text{acquired})\times E(R|\text{acquired})+P(\text{not acquired})\times E(R|\text{not acquired})$ $=0.45\times 31\%+0.55\times (-13.5\%)$ $=13.95\%-7.425\%=6.525\%\approx 6.53\%$

选项	判断	解析
A	✗	计算错误
B	✗	计算错误
C	✓	全概率期望收益 = 0.45×31% + 0.55×(-13.5%) = 6.53%

关联：R4: Probability Trees and Conditional Expectations

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Q7.

(续上题) Based on previous questions, what is the variance of the return, given that Company AXY is acquired by Company BND?

A. 0.2275%.

B. 0.24%.

C. 6.23%.

查看答案与解析

答案：B

解析：在被收购条件下的方差。

计算过程： 已知 $E(R|\text{acquired})=31\%$

$Var(R|\text{acquired})=0.6\times (35\%-31\%{)}^2+0.4\times (25\%-31\%{)}^2$ $=0.6\times (4\%{)}^2+0.4\times (-6\%{)}^2$ $=0.6\times 16+0.4\times 36$ $=9.6+14.4=24$

即 $Var=0.0024=0.24\%$（以百分比的平方表示为24%²）

选项	判断	解析
A	✗	计算错误
B	✓	条件方差 = 0.6×(4%)² + 0.4×(-6%)² = 0.24%
C	✗	可能混淆了方差和标准差

关联：R4: Probability Trees and Conditional Expectations

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Q8.

A Pharmaceutical Company claims to discover a new approach to diagnose an infant’s disease. According to their research, the probability of the disease being detected when a baby has a disease is 99.9%. The probability of the disease being detected when the baby has no disease is 0.6%. The historical data shows that the probability of an infant getting the disease is 0.4%. What is the probability that an infant has the disease when it is diagnosed with the disease?

A. 40%.

B. 80%.

C. 60%.

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答案：A

解析：使用贝叶斯公式（Bayes’ theorem）更新概率。

计算过程：

• P(Disease) = 0.004
• P(No Disease) = 0.996
• P(Detected | Disease) = 0.999
• P(Detected | No Disease) = 0.006

$P(\text{Disease}|\text{Detected})=\frac{P(\text{Detected}|\text{Disease})\times P(\text{Disease})}{P(\text{Detected})}$

$P(\text{Detected})=P(\text{Detected}|\text{Disease})\times P(\text{Disease})+P(\text{Detected}|\text{No Disease})\times P(\text{No Disease})$ $=0.999\times 0.004+0.006\times 0.996=0.003996+0.005976=0.009972$

$P(\text{Disease}|\text{Detected})=\frac{0.003996}{0.009972}=0.4007\approx 40\%$

选项	判断	解析
A	✓	贝叶斯更新后的后验概率约为40%
B	✗	计算错误
C	✗	计算错误

关联：R4: Probability Trees and Conditional Expectations

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Q9.

An analyst has established the following prior probabilities regarding a company’s next quarter’s earnings per share (EPS).

	Prior probabilities
EPS below consensus	40%
EPS equal or exceed consensus	60%

Several days before releasing its earnings statement, the company announces a cut in its dividend. Given this information, the analyst revises his opinion regarding the likelihood that the company will have EPS below the consensus estimate. He estimates the likelihood the company will cut the dividend as reported below.

P(Cut div | EPS below consensus): 70% P(Cut div | EPS equal or exceed consensus): 20%

Given the information that the dividend is cut, the updated (posterior) probability that the company’s EPS is below the consensus is closest to:

A. 24%.

B. 70%.

C. 85%.

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答案：B

解析：使用贝叶斯公式更新先验概率。

计算过程：

• P(Below) = 0.40, P(Meet/Exceed) = 0.60
• P(Cut | Below) = 0.70, P(Cut | Meet/Exceed) = 0.20

$P(\text{Cut})=P(\text{Cut}|\text{Below})\times P(\text{Below})+P(\text{Cut}|\text{Meet})\times P(\text{Meet})$ $=0.70\times 0.40+0.20\times 0.60=0.28+0.12=0.40$

$P(\text{Below}|\text{Cut})=\frac{P(\text{Cut}|\text{Below})\times P(\text{Below})}{P(\text{Cut})}=\frac{0.28}{0.40}=0.70=70\%$

选项	判断	解析
A	✗	24%可能是某个联合概率
B	✓	后验概率 = 0.28/0.40 = 70%
C	✗	计算错误

关联：R4: Probability Trees and Conditional Expectations