Consider the single-index model. The alpha of a stock is 3.75%. The return on the market index is 11%. The risk-free rate of return is 2.5%. During a sample period, the stock earns a return that exceeds the risk-free rate by 13%, and there is no firm-specific residual affecting the stocks return (ie e = 0). The β of the stock is单项选择题
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Question

Consider the single-index model. The alpha of a stock is 3.75%. The return on the market index is 11%. The risk-free rate of return is 2.5%. During a sample period, the stock earns a return that exceeds the risk-free rate by 13%, and there is no firm-specific residual affecting the stocks return (ie e = 0). The β of the stock is单项选择题

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BlackTom AI · Accepted Answer

The problem asks us to determine the stock's beta in a single-index model. We are given: alpha = 3.75%, market return R_m = 11%, risk-free rate R_f = 2.5%, and the stock’s realized return exceeded the risk-free by 13% with no residual e = 0.

First, recognize the single-index (CAPM-like) relation in excess terms: R_i - R_f = alpha_i + beta_i (R_m - R_f) when e_i = 0.

Compute the left-hand side: the stock’s excess return over the risk-free rate is 13%. That is R_i - R_f = 13%.

Compute the market excess return: R_m - R_f = 11% - 2.5% = 8.5%.

Set up the equation with the given alpha:
13% = 3.75% + beta * 8.5%

Isolate beta: beta = (13% - 3.75%) / 8.5% = 9.25% / 8.5% ≈ 1.088235...

Thus, the beta is approximately 1.088. This aligns with the idea that beta measures sensitivity of stock returns to market excess returns, adjusting for the given alpha and the realized excess return.

To check for potential pitfalls, ensure we used excess-return form rather than raw returns, and subtract the risk-free rate from both the stock and the market to isolate systematic risk, which is what beta captures. If one tried to use raw returns directly without subtracting R_f, the calculation would mix alpha and the risk-free baseline and yield an incorrect beta.