位置3的问题 Your portfolio is invested 25 percent each in Stocks A and C, and 50 percent in Stock B. What is the standard deviation of your portfolio given the following information? [table]
State of Economy | Probability of State of Economy | Rate of Return if State Occurs Stock A | Stock B | Stock C
Boom | .07 | .28 | .14 | .11 Good | .55 | .19 | .12 | .09 Poor | .36 | −.21 | .07 | .06 Bust | .02 | −.65 | .03 | −.03
[/table]10.89%6.52%9.64%7.39%12.72%清除选择

Question

BlackTom AI · Accepted Answer

We start by laying out the portfolio: weights are A = 0.25, B = 0.50, C = 0.25. For each state of the economy, we compute the portfolio return as the weighted sum of the individual stock returns.
Boom state: R_p = 0.25*(0.28) + 0.50*(0.14) + 0.25*(0.11) = 0.07 + 0.07 + 0.0275 = 0.1675 (16.75%).
Good state: R_p = 0.25*(0.19) + 0.50*(0.12) + 0.25*(0.09) = 0.0475 + 0.06 + 0.0225 = 0.13 (13.0%).
Poor state: R_p = 0.25*(-0.21) + 0.50*(0.07) + 0.25*(0.06) = -0.0525 + 0.035 + 0.015 = -0.0025 (-0.25%).
Bust state: R_p = 0.25*(-0.65) + 0.50*(0.03) + 0.25*(-0.03) = -0.1625 + 0.015 - 0.0075 = -0.155 (-15.50%).
Next, determine the expected portfolio return using state probabilities p:
Boom 0.07, Good 0.55, Poor 0.36, Bust 0.02.
E[R_p] = 0.07*(0.1675) + 0.55*(0.13) + 0.36*(-0.0025) + 0.02*(-0.155)
         = 0.011725 + 0.0715 - 0.0009 - 0.0031 ≈ 0.079225 (7.9225%).
Now compute the variance by summing p*(R_p - E[R_p])^2 for all states:
- Boom: d = 0.1675 - 0.079225 = 0.088275; d^2 ≈ 0.0077925; contribution ≈ 0.07 * 0.0077925 ≈ 0.0005455.
- Good: d = 0.13 - 0.079225 = 0.050775; d^2 ≈ 0.0025781; contribution ≈ 0.55 * 0.0025781 ≈ 0.0014180.
- Poor: d = -0.0025 - 0.079225 = -0.081725; d^2 ≈ 0.0066790; contribution ≈ 0.36 * 0.0066790 ≈ 0.0024044.
- Bust: d = -0.155 - 0.079225 = -0.234225; d^2 ≈ 0.0548613; contribution ≈ 0.02 * 0.0548613 ≈ 0.0010972.
Total variance ≈ 0.0005455 + 0.0014180 + 0.0024044 + 0.0010972 ≈ 0.0054651. The standard deviation is sqrt(0.0054651) ≈ 0.0739, i.e., about 7.39%.
Now appraising the answer choices:
- 10.89%: This is too large; it would require a larger dispersion or a different weighting of state outcomes. The calculated spread is smaller.
- 6.52%: This underestimates the variability; several state outcomes contribute nontrivially to risk, pushing the std above this value.
- 9.64%: While closer, this still overshoots relative to the precise computed dispersion given the state-by-state differences and probabilities.
- 7.39%: This matches the computed standard deviation from the probability-weighted deviations above.
- 12.72%: This is clearly too high for this portfolio given the numbers, implying far larger dispersion than observed.
Thus, the option that aligns with the calculation is 7.39%.

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