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超超 · 2025年01月12日

第二步之后能再解释一下吗

  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%

现在只看得懂第一步 后面的计算步骤完全是晕的

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1 个答案

源_品职助教 · 2025年01月13日

嗨，努力学习的PZer你好：


第一步是根据方差的公式，代入数字，倒推出残差项等于0.0070.


第二部是把数字打入到协方差的公式，计算出新的协方差的数值。

注意到因为题目中有说相关系数有变变成了0.25（Should be adjusted to incorporate a correlation coefficient of 0.25 ），所以这里要用0.25代入。



然后题目要求修正后的方差，就可以继续用第一步的公式代入求解。这里解析并没有列出公式，其实就是第一步的公式。

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

因为第一步算出了Var (ε)，第二部算出了新的 Cov (F1, F2)，所以现在就可以直接带入数字了，算的是0.3181。


最后题目要求标准差，再开个方就可以了。

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