Suppose that the covariance between a stock and the market portfolio is 0.06 and that the standard deviation of the market portfolio is 0.30. Suppose also that the standard deviation of the stock’s returns is 0.55. Based on the CAPM, what is β of this stock?

(Please answer as a number and round to 2 decimal places. If the answer is 0.768, then in the box, write 0.77)

Question

Suppose that the covariance between a stock and the market portfolio is 0.06 and that the standard deviation of the market portfolio is 0.30. Suppose also that the standard deviation of the stock’s returns is 0.55. Based on the CAPM, what is β of this stock?

(Please answer as a number and round to 2 decimal places. If the answer is 0.768, then in the box, write 0.77)

BlackTom AI · Accepted Answer

Beta is defined as the covariance between the stock’s return and the market return divided by the variance of the market return: β = Cov(R_stock, R_market) / Var(R_market).

1) Relevant concepts:
- Covariance measures joint variability of stock and market returns.
- Variance of the market = (standard deviation of market)^2.
- In CAPM, β quantifies systematic risk: how much the stock moves with the market.

2) Step-by-step reasoning and calculations:
- Given Cov(R_stock, R_market) = 0.06.
- Given σ_market = 0.30, so Var(R_market) = σ_market^2 = 0.30^2.
- Compute variance: 0.30^2 = 0.09.
- Compute β: β = 0.06 / 0.09.

3) Intermediate arithmetic:
- 0.06 ÷ 0.09 = 0.666666... (repeating).

4) Rounding and interpretation:
- Rounded to two decimal places: 0.67.
- This β (≈0.67) means the stock has less systematic risk than the market: when the market moves 1% on average, the stock moves about 0.67% in the same direction.

(Alternate check) Using correlation form: ρ = Cov/(σ_stock·σ_market) = 0.06/(0.55·0.30) = 0.06/0.165 = 0.363636...; then β = ρ·(σ_stock/σ_market) = 0.363636·(0.55/0.30) = 0.363636·1.83333 = 0.666666..., consistent.

5) Why other possible mistakes are incorrect:
- Using σ_stock/σ_market (≈1.83) as β is wrong because β requires covariance with the market, not just relative volatility.
- Using correlation alone (≈0.36) is wrong because β scales correlation by volatility ratio.

Conclusion: The correct β equals 0.67, which matches the provided answer.

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