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沪上小王子 · 2024年01月20日

此题的另类解法思考

  1. CFA III
  2. Capital Market Expectations

NO.PZ2022122601000064

问题如下：

The SCI risk premium, equal to the SCI return minus the risk-free rate, denoted as SCIRP, is used as the dependent variable in a two-factor regression in which the independent variables are index returns minus the risk-free rate for the consumer credit industry (CCIRP) and the telecommunications industry (TELIRP). The regression results are in Exhibit 2.

Although volatility information is available from the SCI data and correspondingly for the SCIRP, Li’s team wants to determine the statistical relationship between the SCIRP and both the CCIRP and the TELIRP because forecasting the CCIRP and TELIRP is much less difficult than forecasting the SCIRP. After some discussion, the team believes that the volatility measure for the SCIRP data based on the volatility of CCIRP and TELIRP through the regression should be adjusted to incorporate a correlation coefficient of 0.25 between the CCIRP and TELIRP. Although the two index risk premiums were uncorrelated in the past and within the regression, Li’s team believes the two technologies will become more correlated in the future.

Based on the correlation that Li's team believes to exist between the CCIRP and TELIRP, the new volatility for the SCIRP is closest to:

选项：

A.

31.8%

B.56.4% C.49.1%

解释：

Correct Answer: B

Begin with: Var (M) = Var (F1)× (b1)2 + Var (F2) × (b2)2 + 2 × b1 × b2 × Cov (F1, F2) +Var (ε).

Find the variance of the error term using values from Exhibit 2:

0.2704 = 0.0784 × (1.020)2+ 0.1024 × (1.045)2 + 2 × 1.020 × 1.045 × 0 +Var (ε),Var (ε) = 0.0770.

The adjustment is stated as being a correlation of 0.25.

Change the correlation into a covariance:

Cov(F1,F2) = Corr(F1,F2) × Std Dev (F1) × Std Dev (F2)= 0.25 × (0.0784)^0.5 × (0.1024)^0.5 = 0.0224

The volatility of SCI after adjusting for the correlation is0.3181^0.5=56.4%

中文解析：

Var (M) = Var (F1)× (b1)2 +Var (F2) × (b2)2 + 2 × b1 × b2 × Cov (F1, F2) +Var (ε)。

使用表2中的值找到误差项的方差:

0.2704 = 0.0784××0.1024(1.020)2 +(1.045)2 + 2×1.020×1.045×0 + Var(ε),Var(ε)= 0.0770。

调整的相关系数为0.25。

将相关性转化为协方差:

Cov(F1,F2) = Corr(F1,F2) × Std Dev (F1) × Std Dev (F2)= 0.25 × (0.0784)^0.5 × (0.1024)^0.5 = 0.0224

经相关系数调整后的上证综指波动率为0.3181^0.5=56.4%

  1. 不需要计算残差项的方差，直接将题中给出的数据相乘计算出在ρ=0.25情况下，需要加到原有SCIRP方差上的新方差，具体计算如下：

2x1.02x1.045x0.25x(0.0784x0.1024)^0.5=0.047752

新方差=0.2704+0.047752=0.3181

标准差=0.3181^0.5=0.564

请问老师这种思考路径是否可以？


另外，题目中给出了三个index的mean值，并且老师讲解过程中还提到了R=α+∑biFi+残差，能否用mean值或“R=α+∑biFi+残差”去计算该题，如果可以，应该如何计算？

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2 个答案

笛子_品职助教 · 2024年01月22日

嗨，爱思考的PZer你好：


老师，我的第二个问题也请帮助回答一下，谢谢。 题目中给出了三个index的mean值，并且老师讲解过程中还提到了R=α+∑biFi+残差，能否用mean值或“R=α+∑biFi+残差”去计算该题，如果可以，应该如何计算？


Hello，亲爱的同学~

这道题还是围绕公式：

Var (M) = Var (F1)× (b1)2 +Var (F2) × (b2)2 + 2 × b1 × b2 × Cov (F1, F2) +Var (ε)。来进行计算。

这个公式里，Var (M)是方差，Var (ε)是残差的方差（注意不是残差本身）。


老师看了一下，本题似乎，确实不能用mean值，或者R=α+∑biFi+残差 来解题。

mean值与R=α+∑biFi+残差，比较适合于做收益分析的场景。而本题是对风险（方差）进行分析。


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笛子_品职助教 · 2024年01月22日

嗨，从没放弃的小努力你好：


请问老师这种思考路径是否可以？

Hello，亲爱的同学~

这种思考方式是可以的。

这里是相关系数改变，从0变到0.25。

因此，用0.25去替代0就可以。

----------------------------------------------
就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

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