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yuqijeffery · 2022年07月23日

请问为什么是50% 50%?

  1. CFA II
  2. Portfolio

NO.PZ2021101401000003

问题如下：

Assume that the following one-factor model describes the expected return for portfolios:

E(Rp)=0.10+0.12βp,1E{(R_p)}=0.10+0.12\beta_{p,1}

Also assume that all investors agree on the expected returns and factor sensitivity of the three highly diversified Portfolios A, B, and C given in the following table:

Assuming the one-factor model is correct and based on the data provided for Portfolios A, B, and C, determine if an arbitrage opportunity exists and explain how it might be exploited.

选项：

解释：

According to the one-factor model for expected returns, the portfolio should have these expected returns if they are correctly priced in terms of their risk:

Portfolio A

E(RA) = 0.10 + 0.12βA,1 = 0.10 + (0.12)(0.80) = 0.10 + 0.10 = 0.20

Portfolio B

E(RB) = 0.10 + 0.12βB,1 = 0.10 + (0.12)(1.00) = 0.10 + 0.12 = 0.22

Portfolio C

E(RC) = 0.10 + 0.12βC,1 = 0.10 + (0.12)(1.20) = 0.10 + 0.14 = 0.24

In the table below, the column for expected return shows that Portfolios A and C are correctly priced but Portfolio B offers too little expected return for its risk, 0.15 or 15%. By shorting Portfolio B (selling an overvalued portfolio) and using the proceeds to buy a portfolio 50% invested in A and 50% invested in C with a sensitivity of 1 that matches the sensitivity of B, for each monetary unit shorted (say each euro), an arbitrage profit of €0.22 -€0.15 = €0.07 is earned.

请问为什么是50% 50%?

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星星_品职助教 · 2022年07月23日

同学你好，

套利的原则之一是无风险，所以需要将A和C组合出一个等于B的β值，即A和C组合完后要得到β=1.

由此，可列出如下方程组：

Wa+Wc=1,

Wa×0.8+Wc×1.2=1

解得到Wa=50%，Wc=50%

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