位置19的问题 What is the standard deviation of the returns on a portfolio that is invested in Stocks A, B, and C? Twenty percent of the portfolio is invested in Stock A and 35 percent is invested in Stock C. [table]
State of Economy | Probability of State of Economy | Rate of Return if State Occurs Stock A | Stock B | Stock C
Boom | .04 | .17 | .09 | .09 Normal | .81 | .08 | .06 | .08 Recession | .15 | −.24 | .02 | −.13
[/table]7.38%7.72%6.31%5.65%6.49%清除选择

Question

BlackTom AI · Accepted Answer

The question asks for the standard deviation of a portfolio containing Stocks A, B, and C with weights 0.20, 0.45, and 0.35 respectively. 
First, compute the portfolio return in each state of the economy by weighting each stock’s return by its portfolio weight and summing them:
- Boom: r_p = 0.20(0.17) + 0.45(0.09) + 0.35(0.09) = 0.106 (10.6%)
- Normal: r_p = 0.20(0.08) + 0.45(0.06) + 0.35(0.08) = 0.071 (7.1%)
- Recession: r_p = 0.20(-0.24) + 0.45(0.02) + 0.35(-0.13) = -0.0845 (−8.45%)
Next, find the expected portfolio return E[r]:
E[r] = (0.04)(0.106) + (0.81)(0.071) + (0.15)(−0.0845) ≈ 0.0491 (4.91%)
Then compute E[r^2] by weighting the squared state returns:
- Boom: (0.106)^2 = 0.011236; contributes 0.04 × 0.011236 ≈ 0.0004494
- Normal: (0.071)^2 = 0.005041; contributes 0.81 × 0.005041 ≈ 0.004083
- Recession: (−0.0845)^2 = 0.00714025; contributes 0.15 × 0.00714025 ≈ 0.0010710
Sum E[r^2] ≈ 0.0004494 + 0.004083 + 0.0010710 ≈ 0.0056037
Now, the variance is Var(r) = E[r^2] − (E[r])^2. With E[r] ≈ 0.0491, (E[r])^2 ≈ 0.002409. Thus Var(r) ≈ 0.0056037 − 0.002409 ≈ 0.003194. The standard deviation is the square root of the variance: σ ≈ sqrt(0.003194) ≈ 0.0565, i.e., about 5.65%. 
Evaluating the provided options:
- 7.38%: Too large relative to the calculated volatility.
- 7.72%: Also too large and not supported by the numbers above.
- 6.31%: Closer but still higher than the exact calculation above.
- 5.65%: Matches the computed result precisely.
- 6.49%: Too high compared to the calculated standard deviation. 
Thus, the option closest to and effectively equal to the computed standard deviation is 5.65%.

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