位置23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]6.52%9.64%7.39%10.89%12.72%清除选择

Question

BlackTom AI · Accepted Answer

We start by clearly restating what the question asks and the given data, so we can work through the calculation step by step.

- Portfolio weights: wA = 0.25, wB = 0.50, wC = 0.25.
- States of the economy with probabilities and asset returns:
  • Boom: p = 0.07, A = 0.28, B = 0.14, C = 0.11
  • Good: p = 0.55, A = 0.19, B = 0.12, C = 0.09
  • Poor: p = 0.36, A = −0.21, B = 0.07, C = 0.06
  • Bust: p = 0.02, A = −0.65, B = 0.03, C = −0.03

- Compute the portfolio return in each state by summing the weighted returns of A, B, and C:
  • Boom: R_p(Boom) = 0.25(0.28) + 0.50(0.14) + 0.25(0.11) = 0.07 + 0.07 + 0.0275 = 0.1675
  • Good: R_p(Good) = 0.25(0.19) + 0.50(0.12) + 0.25(0.09) = 0.0475 + 0.06 + 0.0225 = 0.13
  • Poor: R_p(Poor) = 0.25(−0.21) + 0.50(0.07) + 0.25(0.06) = −0.0525 + 0.035 + 0.015 = −0.0025
  • Bust: R_p(Bust) = 0.25(−0.65) + 0.50(0.03) + 0.25(−0.03) = −0.1625 + 0.015 − 0.0075 = −0.155

- Next, determine the expected portfolio return: E[R] = Σ p_s * R_p(s) = 0.07(0.1675) + 0.55(0.13) + 0.36(−0.0025) + 0.02(−0.155).
  • 0.07 × 0.1675 ≈ 0.011725
  • 0.55 × 0.13 = 0.0715
  • 0.36 × (−0.0025) = −0.0009
  • 0.02 × (−0.155) = −0.0031
  • Sum ≈ 0.011725 + 0.0715 − 0.0009 − 0.0031 ≈ 0.079225
  • So E[R] ≈ 0.0792 (about 7.92%).

- Now compute the portfolio variance using the law of total variance: Var(R) = Σ p_s [R_p(s) − E[R]]^2.
  • For Boom: d1 = 0.1675 − 0.079225 ≈ 0.088275; d1^2 ≈ 0.00778
  • For Good: d2 = 0.13 − 0.079225 ≈ 0.050775; d2^2 ≈ 0.00258
  • For Poor: d3 = −0.0025 − 0.079225 ≈ −0.081725; d3^2 ≈ 0.00668
  • For Bust: d4 = −0.155 − 0.079225 ≈ −0.234225; d4^2 ≈ 0.05485
  • Multiply by probabilities and sum:
    − Boom contribution: 0.07 × 0.00778 ≈ 0.00054
    − Good contribution: 0.55 × 0.00258 ≈ 0.00142
    − Poor contribution: 0.36 × 0.00668 ≈ 0.00240
    − Bust contribution: 0.02 × 0.05485 ≈ 0.00110
  • Var(R) ≈ 0.00054 + 0.00142 + 0.00240 + 0.00110 ≈ 0.00546

- Finally, the standard deviation is the square root of the variance: SD(R) = sqrt(0.00546) ≈ 0.0739, or about 7.39%.

Note: All intermediate steps are based on standard portfolio theory, where the state-by-state returns are combined with the state probabilities to yield the distribution of portfolio returns, from which the mean and variance (and hence the standard deviation) are derived. The arithmetic confirms that the closest option to the computed value is 7.39%.

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