Question at position 9 (5 marks, difficulty level: Easy/Medium) A portfolio consists of two stocks, Microsoft and Apple. The investor holds $40,000 in Microsoft and $60,000 in Apple. Microsoft has a standard deviation of 22%, Apple has a standard deviation of 30%, and the correlation between the two stocks is 0.45. Calculate the standard deviation of this two-stock portfolio. (Hint: Calculate the portfolio weights for each stock first.)22.24%24.20%23.32%25.30%

Question

Question at position 9 (5 marks, difficulty level: Easy/Medium) A portfolio consists of two stocks, Microsoft and Apple. The investor holds $40,000 in Microsoft and $60,000 in Apple. Microsoft has a standard deviation of 22%, Apple has a standard deviation of 30%, and the correlation between the two stocks is 0.45. Calculate the standard deviation of this two-stock portfolio. (Hint: Calculate the portfolio weights for each stock first.)22.24%24.20%23.32%25.30%

BlackTom AI · Accepted Answer

Let’s start by identifying the given data and converting the dollar amounts into portfolio weights. The investor holds $40,000 in Microsoft and $60,000 in Apple, for a total of $100,000. So the weight of Microsoft, w1, is 0.4 and the weight of Apple, w2, is 0.6. The standard deviations are s1 = 0.22 and s2 = 0.30, and the correlation between the stocks is ρ = 0.45.

Now, calculate the portfolio variance using the two-asset formula: Var(P) = w1^2 * s1^2 + w2^2 * s2^2 + 2 * w1 * w2 * s1 * s2 * ρ.

Option by option analysis:

- Option 22.24%: If the variance were 0.2224 (i.e., 22.24%), the standard deviation would be sqrt(0.2224) ≈ 0.472, which is far larger than typical two-asset portfolios with the given weights and volatilities. This implies either a miscalculation of weights or a misapplication of the formula, making this value unlikely given the inputs.

- Option 24.20%: This would correspond to a variance of about 0.0520. While closer to the computed range than 22.24%, it still does not match the precise result from applying the formula with w1 = 0.4, w2 = 0.6, s1 = 0.22, s2 = 0.30, and ρ = 0.45. It suggests a small arithmetic discrepancy but not the exact outcome.

- Option 23.32%: This option translates to a variance of 0.0544. Let’s check by plugging into the formula:
  - w1^2 * s1^2 = (0.4)^2 * (0.22)^2 = 0.16 * 0.0484 = 0.007744
  - w2^2 * s2^2 = (0.6)^2 * (0.30)^2 = 0.36 * 0.09 = 0.0324
  - 2 * w1 * w2 * s1 * s2 * ρ = 2 * 0.4 * 0.6 * 0.22 * 0.30 * 0.45 = 0.014256
  - Sum = 0.007744 + 0.0324 + 0.014256 = 0.0544
  - Std dev = sqrt(0.0544) = 0.2332 = 23.32%
  This aligns perfectly with the standard two-asset portfolio calculation using the given inputs.

- Option 25.30%: A standard deviation of 0.2530 would correspond to a variance of about 0.0640. This is noticeably higher than the variance computed from the data and suggests either overestimating the cross-term or misusing the inputs. Given the exact computation above, this value is not supported by the formula with w1 = 0.4, w2 = 0.6, s1 = 0.22, s2 = 0.30, and ρ = 0.45.

Overall, the only option that exactly matches the result from the portfolio variance formula with the provided weights, volatilities, and correlation is 23.32%. The others either overstate the risk (leading to unrealistically high standard deviations) or do not align with the precise arithmetic of the variance components.

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类似问题

Part 2 of 7: What is the standard deviation of a Complete Portfolio that invests 40% in Portfolio A and 60% in the risk-free rate? Round your answer to four decimals. DO NOT use percentage format. E.g. 10.51% should be entered as 0.1051

Part 4 of 4 If the investor allocates $8,000 to SPY and $2,000 to BND, but the correlation coefficient is now -0.54, what is the standard deviation of the portfolio return? Round your answer to four decimals. DO NOT use percentage format. E.g. 10.51% should be entered as 0.1051

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