如下

问题如下：

Bader begins by analyzing portfolio risk. She decides to forecast a variance–covariance matrix (VCV) for 20 asset classes, using 10 years of monthly returns and incorporating both the sample statistics and the factor-model methods. To mitigate the impact of estimation error, Bader is considering combining the results of the two methods in an alternative target VCV matrix, using shrinkage estimation.

Bader asks her research assistant to comment on the two approaches and the benefits of applying shrinkage estimation. The assistant makes the following statements:

Statement 1

Shrinkage estimation of VCV matrices will decrease the efficiency of the estimates versus the sample VCV matrix.

Statement 2

Your proposed approach for estimating the VCV matrix will not be reliable because a sample VCV matrix is biased and inconsistent.

Statement 3

A factor-based VCV matrix approach may result in some portfolios that erroneously appear to be riskless if any asset returns can be completely determined by the common factors or some of the factors are redundant.

Which of the following statements made by Bader’s research assistant is correct?

选项：

Statement 1

Statement 2

Statement 3

解释：

Correct Answer: C

Statement 3 is correct. As long as none of the factors used in a factor-based VCV model are redundant and none of the asset returns are completely determined by the common factors, there will not be any portfolios that erroneously appear to be riskless. Therefore, a factor-based VCV matrix approach may result in some portfolios that erroneously appear to be riskless if any asset returns can be completely determined by the common factors or some of the factors are redundant.

A is incorrect because shrinkage estimation of VCV matrices will increase the efficiency of the estimates versus the sample VCV matrix, because its mean squared error (MSE) will in general be smaller than the MSE of the (unbiased) sample VCV matrix. Efficiency in this context means a smaller MSE.

B is incorrect because, although the proposed approach is not reliable, the reason is not that the sample VCV matrix is biased and inconsistent; on the contrary, it is unbiased and consistent. Rather, the estimate of the VCV matrix is not reliable because the number of observations is not at least 10 times the number of assets (i.e., with 10 years of monthly return data, there are only 120 observations, but the rule of thumb suggests there should be at least 200 observations for 20 asset classes).

中文解析：

表述3是正确的。只要在基于因素的VCV模型中使用的因素没有一个是多余的，没有一个资产回报完全由共同因素决定，就不会有任何投资组合错误地看起来是无风险的。因此，如果任何资产收益完全由共同因素决定，或者某些因素是冗余的，那么基于因素的VCV矩阵方法可能会导致一些投资组合错误地看起来是无风险的。

A是不正确的，因为相对于样本VCV矩阵，VCV矩阵的收缩估计会增加估计的效率，因为它的均方误差(MSE)通常会小于(无偏)样本VCV矩阵的MSE。在这种情况下，效率意味着较小的MSE。

B是不正确的，因为虽然提出的方法不可靠，但原因不是样本VCV矩阵有偏差和不一致;相反，它是公正和一致的。相反，VCV矩阵的估计是不可靠的，因为观测值的数量至少不是资产数量的10倍(即，对于10年的月度回报数据，只有120个观测值，但经验法则表明，20个资产类别至少应该有200个观测值)。