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花卷喵 · 2023年09月03日

factor-based vcv

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NO.PZ202105270100000501

问题如下:

Judith Bader is a senior analyst for a company that specializes in managing international developed and emerging markets equities. Next week, Bader must present proposed changes to client portfolios to the Investment Committee, and she is preparing a presentation to support the views underlying her recommendations.

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.

Bader then uses the Singer–Terhaar model and the final shrinkage-estimated VCV matrix to determine the equilibrium expected equity returns for all international asset classes by country. Three of the markets under consideration are located in Country A (developed market), Country B (emerging market), and Country C (emerging market). Bader projects that in relation to the global market, the equity market in Country A will remain highly integrated, the equity market in Country B will become more segmented, and the equity market in Country C will become more fully integrated.

Next, Bader applies the Grinold–Kroner model to estimate the expected equity returns for the various markets under consideration. For Country A, Bader assumes a very long-term corporate earnings growth rate of 4% per year (equal to the expected nominal GDP growth rate), a 2% rate of net share repurchases for Country A’s equities, and an expansion rate for P/E multiples of 0.5% per year.

In reviewing Countries B and C, Bader’s research assistant comments that emerging markets are especially risky owing to issues related to politics, competition, and accounting standards. As an example, Bader and her assistant discuss the risk implications of the following information related to Country B:

  • Experiencing declining per capita income
  • Expected to continue its persistent current account deficit below 2% of GDP
  • Transitioning to International Financial Reporting Standards, with full convergence scheduled to be completed within two years

Bader shifts her focus to currency expectations relative to clients’ base currency and summarizes her assumptions in Exhibit 1.


During a conversation about Exhibit 1, Bader and her research assistant discuss the composition of each country’s currency portfolio and the potential for triggering a crisis. Bader notes that some flows and holdings are more or less supportive of the currency, stating that investments in private equity make up the majority of Country A’s currency portfolio, investments in public equity make up the majority of Country B’s currency portfolio, and investments in public debt make up the majority of Country C’s currency portfolio.


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

选项:

A.Statement 1 B.Statement 2 C.Statement 3

解释:

C is correct.

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).

表述三是正确的。只要基于因素的风险现值模型中使用的所有因素都不是冗余的,而且没有任何资产回报完全由公共因素决定,就不会有任何投资组合错误地看起来是无风险的。因此,基于因子的VCV矩阵方法可能导致某些投资组合错误地看似无风险,如果任何资产收益完全由公共因子决定,或某些因子是多余的。


A是错误的,因为相对于样本VCV矩阵,收缩估计VCV矩阵会增加估计的效率,因为其均方误差(MSE)一般会小于(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是错误的,因为虽然所提出的方法是不可靠的,但原因不是样本VCV矩阵有偏差和不一致;相反,它是无偏的和一致的。相反,VCV矩阵的估计不可靠,因为观测的数量不满足至少10倍于资产数目的要求(例如,用十年的月度数据回测,只有120年的观察,但经验法则表明应该有至少200观察20资产类别)。


1.factor-based vcv和factor-model-based vcv是同一个东西吗?

2.框架图里面some portfolios that erroneously appear to be riskless是sample statistics(vcv)的缺点,题目里factor based vcv也有这个特点。那么这两者的关系是什么?

1 个答案

笛子_品职助教 · 2023年09月05日

嗨,从没放弃的小努力你好:


1.factor-based vcv和factor-model-based vcv是同一个东西吗?

都是factor的VCV,是同一个东西。


2.框架图里面some portfolios that erroneously appear to be riskless是sample statistics(vcv)的缺点,题目里factor based vcv也有这个特点。那么这两者的关系是什么?

同学注意。题目里的factor - based VCV 和我们教材上的factor - based VCV,有差别。

我们教材上的factor - based VCV,有一个前提,factor的回归模型是正确的,回归方程没有漏掉因子,也没有因子共线性的问题(虽然有几个因子,但这几个因子之间有相关性,都是受同一个因素影响)。在这个前提下,我们教材上的factor VCV,并不会有 erroneously appear to be riskless这个缺点。

知识点如下:教材里,riskless是sample的缺点,不是factor的缺点。


我们再看本题描述:

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.

声明3如果任何资产回报可以完全由共同因素决定,或者其中一些因素是多余的,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没有风险。

虽然也是factor - based VCV,但它的回归方程是错误的,某些因子之间有相关性,它们的背后有共同因素在影响。statement3的陈述“如果任何资产回报可以完全由共同因素决定”这句话,使得factor model的这个前提不存在了,因此本题factor VCV由于回归方程错误,也会有riskless的缺点。


这道题非常的特殊,同学也不必纠结。




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NO.PZ202105270100000501 问题如下 Whiof the following statements ma Bar’s researassistant is correct? A.Statement 1 B.Statement 2 C.Statement 3 C is correct. Statement 3 is correct. long none of the factors usein a factor-baseVmol are rennt annone of the asset returns are completely terminethe common factors, there will not any portfolios therroneously appeto riskless. Therefore, a factor-baseVmatrix approamresult in some portfolios therroneously appeto riskless if any asset returns ccompletely terminethe common factors or some of the factors are rennt.A is incorrebecause shrinkage estimation of Vmatrices will increase the efficienof the estimates versus the sample Vmatrix, because its mesquareerror (MSE) will in genersmaller ththe MSE of the (unbiase sample Vmatrix. Efficienin this context means a smaller MSE. B is incorrebecause, although the proposeapproais not reliable, the reason is not ththe sample Vmatrix is biaseaninconsistent; on the contrary, it is unbiaseanconsistent. Rather, the estimate of the Vmatrix is not reliable because the number of observations is not least 10 times the number of assets (i.e., with 10 years of monthly return tthere are only 120 observations, but the rule of thumb suggests there shoulleast 200 observations for 20 asset classes).表述三是正确的。只要基于因素的风险现值模型中使用的所有因素都不是冗余的,而且没有任何资产回报完全由公共因素决定,就不会有任何投资组合错误地看起来是无风险的。因此,基于因子的VCV矩阵方法可能导致某些投资组合错误地看似无风险,如果任何资产收益完全由公共因子决定,或某些因子是多余的。A是错误的,因为相对于样本VCV矩阵,收缩估计VCV矩阵会增加估计的效率,因为其均方误差(MSE)一般会小于(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是错误的,因为虽然所提出的方法是不可靠的,但原因不是样本VCV矩阵有偏差和不一致;相反,它是无偏的和一致的。相反,VCV矩阵的估计不可靠,因为观测的数量不满足至少10倍于资产数目的要求(例如,用十年的月度数据回测,只有120年的观察,但经验法则表明应该有至少200观察20资产类别)。 如果任何资产收益完全由公共因子决定,或某些因子是多余的。那么基于因子的VCV矩阵方法可能导致某些投资组合错误地被认为是无风险的。能解析一下这个中间推导过程吗1、为什么资产收益由完全由公共因子决定,那么基于因子的VCV矩阵方法可能导致某些投资组合错误地被认为是无风险的。2、为什么某些因子是多余的,那么基于因子的VCV矩阵方法可能导致某些投资组合错误地被认为是无风险的。

2024-08-08 18:43 1 · 回答

NO.PZ202105270100000501 问题如下 Whiof the following statements ma Bar’s researassistant is correct? A.Statement 1 B.Statement 2 C.Statement 3 C is correct. Statement 3 is correct. long none of the factors usein a factor-baseVmol are rennt annone of the asset returns are completely terminethe common factors, there will not any portfolios therroneously appeto riskless. Therefore, a factor-baseVmatrix approamresult in some portfolios therroneously appeto riskless if any asset returns ccompletely terminethe common factors or some of the factors are rennt.A is incorrebecause shrinkage estimation of Vmatrices will increase the efficienof the estimates versus the sample Vmatrix, because its mesquareerror (MSE) will in genersmaller ththe MSE of the (unbiase sample Vmatrix. Efficienin this context means a smaller MSE. B is incorrebecause, although the proposeapproais not reliable, the reason is not ththe sample Vmatrix is biaseaninconsistent; on the contrary, it is unbiaseanconsistent. Rather, the estimate of the Vmatrix is not reliable because the number of observations is not least 10 times the number of assets (i.e., with 10 years of monthly return tthere are only 120 observations, but the rule of thumb suggests there shoulleast 200 observations for 20 asset classes).表述三是正确的。只要基于因素的风险现值模型中使用的所有因素都不是冗余的,而且没有任何资产回报完全由公共因素决定,就不会有任何投资组合错误地看起来是无风险的。因此,基于因子的VCV矩阵方法可能导致某些投资组合错误地看似无风险,如果任何资产收益完全由公共因子决定,或某些因子是多余的。A是错误的,因为相对于样本VCV矩阵,收缩估计VCV矩阵会增加估计的效率,因为其均方误差(MSE)一般会小于(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是错误的,因为虽然所提出的方法是不可靠的,但原因不是样本VCV矩阵有偏差和不一致;相反,它是无偏的和一致的。相反,VCV矩阵的估计不可靠,因为观测的数量不满足至少10倍于资产数目的要求(例如,用十年的月度数据回测,只有120年的观察,但经验法则表明应该有至少200观察20资产类别)。 看到助教有回复“只要基于因素的风险现值模型中使用的所有因素都不是多余的,而且没有任何资产回报完全由共同因素决定,那么投资组合就不会被错误地认为是无风险的。这是咱们讲义P201第一段的原话。”课程视频里老师有讲解这句话吗?在哪个视频哪个位置?谢谢。

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NO.PZ202105270100000501 不仅仅是翻译原文,就是彻底一下statement3是想表达个什么意思

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