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蛋黄也酥酥 · 2021年05月10日

前面的解释都看明白了,就是不明白为什么要让ab和c的factor sensitivity相等

* 问题详情,请 查看题干

NO.PZ201710100100000102

问题如下:


2. The arbitrage opportunity identified by Zapata can be exploited with:

选项:

A.

Strategy 1: Buy $50,000 Fund A and $50,000 Fund B; sell short $100,000 Fund C.

B.

Strategy 2: Buy $60,000 Fund A and $40,000 Fund B; sell short $100,000 Fund C.

C.

Strategy 3: Sell short $60,000 of Fund A and $40,000 of Fund B; buy $100,000 Fund C

解释:

C is correct.

The expected return and factor sensitivities of a portfolio with a 60% weight in Fund A and a 40% weight in Fund B are calculated as weighted averages of the expected returns and factor sensitivities of Funds A and B: Expected return of Portfolio 60/40 = (0.60)(0.02) + (0.40)(0.04) = 0.028, or 2.8% Factor sensitivity of Portfolio 60/40 = (0.60)(0.5) + (0.40)(1.5) = 0.9

The factor sensitivity of Portfolio 60/40 is identical to that of Fund C; therefore, this strategy results in no factor risk relative to Portfolio C. However, Fund C’s expected return of 3.0% is higher than Portfolio 60/40’s expected return of 2.8%. This difference supports Strategy 3: buying Fund C and selling short Portfolio 60/40 to exploit the arbitrage opportunity.

考点:APT模型

解析:

根据题干,AB组合符合APT模型,而C不符合,因此存在套利空间。

首先求单因子的APT模型,公式写为:E(R)=Rf+βλ,代入AB组合的已知数:

Rf+0.5λ=0.02

Rf+1.5λ=0.04,

两个方程两个未知数,得Rf=1%,λ=2%。

根据E(R)=1%+β*2%,C组合在APT模型下的预期收益率为1%+0.9*2%=2.8%,而现在表格中给出的C组合的实际收益率为3%。所以C组合在市场上的实际收益率3%是高于APT模型的预期收益率,那么投资者可以通过long C组合的实际收益率,同时short APT模型下通过AB合成的C组合,来获得无风险收益率。

因此我们要找到AB组合的权重,使得合成后新组合的factor sensitivy=C组合的factor sensitivy,列出方程:

Wa+Wb=1

0.5Wa+1.5Wb=0.9

因此Wa=60%, Wb=40%

所以通过long1个C组合,short (60%的A组合+40%的B组合),可以获得套利机会。因此符合这样的头寸和投资比例的只有C选项。

前面的解释都看明白了,就是不明白为什么要让ab和c的factor sensitivity相等

1 个答案
已采纳答案

星星_品职助教 · 2021年05月10日

同学你好,

APT的套利原则是基于无风险的。

由于这里的风险因子(factor sensitivity)只有β,所以无风险的套利原则实际上就是使得β=0。

所以买入一个β=0.9的资产(long C),就必须同时卖出一个β=0.9的资产(short A&B的组合),这样才可以对冲掉β的影响,达到无风险的目的。

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