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eva · 2024年03月27日

beta如何计算？看不明白

NO.PZ2022070603000008

问题如下：

An investment advisor is analyzing the range of potential expected returns of a new fund designed to replicate the directional moves of the China Shanghai Composite Stock Market Index (SHANGHAI) but with twice the volatility of the index. SHANGHAI has an expected annual return of 7.6% and a volatility of 14.0%, and the risk free rate is 3.0% per year. Assuming the correlation between the fund’s returns and that of the index is 1.0, what is the expected return of the fund using the CAPM?

选项：

A.

12.2%

B.

19.0%

C.

22.1%

D.

24.6%

解释：

中文解析：

A正确。如果CAPM成立，那么Ri = Rf + βi * (Rm – Rf).

βi决定了基金的收益率随着指数收益率的变化而变化的程度。

Ri = Rf + βi * (Rm – Rf) = 0.03 + 2.0*(0.076 – 0.03)= 0.1220 = 12.2%.

-------------------------------------------------------------------------------------------------------------------

A is correct. If the CAPM holds, then Ri = Rf + βi * (Rm – Rf).

Beta (βi), which determines how much the return of the fund fluctuates in relation to the index return is expressed as follows:

${\beta_{i}=\frac{\operatorname{Cov}\left(\mathrm{R}_{i}, \mathrm{R}_{m}\right)}{\sigma_{m}^{2}}=\frac{\operatorname{Corr}\left(\mathrm{R}_{i}, \mathrm{R}_{m}\right) * \sigma_{i} \sigma_{m}}{\sigma_{m}^{2}}=\frac{\operatorname{Corr}\left(\mathrm{R}_{i}, \mathrm{R}_{m}\right) * \sigma_{i}}{\sigma_{m}}}$

Where i and m denote the new fund and the index, respectively, and Ri = expected return on the fund, Rm = expected return on the index, Rf = risk-free rate, σi = volatility of the fund, σm = volatility of the index, Cov(Ri,Rm) = covariance between the fund and the index returns, and Corr(Ri,Rm) = correlation between the fund and the index returns.

If the new fund has twice the volatility of the index, then σi = 2σi = 2σm, and given that Corr(Ri,Rm) = 1.0, the beta of the new fund then becomes:

$\beta_{i}=\frac{\operatorname{Corr}\left(\mathrm{R}_{i}, \mathrm{R}_{m}\right) * 2 \sigma_{m}}{\sigma_{m}}=1.0 * 2.0=2.0$

Therefore, using CAPM, Ri = Rf + βi * (Rm – Rf) = 0.03 + 2.0*(0.076 – 0.03)

= 0.1220 = 12.2%.

beta如何计算？看不明白，可以写过程么？

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pzqa27 · 2024年03月28日

嗨，从没放弃的小努力你好：

原理在数学那门课里有涉及，就是代入下图中我框起来的公式即可，由于the China Shanghai Composite Stock Market Index (SHANGHAI) with twice the volatility of the index，所以分子分母刚好可以抵消一部分。

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加油吧，让我们一起遇见更好的自己！

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