Covariance Calculation of Company Returns K and L

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Question:

he shares of two companies, K and L have the following expected returns: Prob Return K Return L 0.3 20% 6% 0.5 15% 10% 0.2 5% 12%

The covariance of the returns of K and L is: Question 4Select one: a. -5.97 b. 0 c. 2.99 d. 5.97

Answer:

To calculate the covariance of the returns of companies K and L, we can use the following formula:

[ \text{Cov}(X, Y) = E[XY] - E[X]E[Y] ]

Where:

  • (E[XY]) is the expected value of the product of the returns of K and L.
  • (E[X]) is the expected return of K.
  • (E[Y]) is the expected return of L.

Step 1: Calculate the expected returns (E[X]) and (E[Y])

Expected Return of K ((E[X])): [ E[X] = (0.3 \times 20%) + (0.5 \times 15%) + (0.2 \times 5%) ] [ E[X] = (0.3 \times 0.20) + (0.5 \times 0.15) + (0.2 \times 0.05) ] [ E[X] = 0.06 + 0.075 + 0.01 = 0.145 \text{ or } 14.5% ]

Expected Return of L ((E[Y])): [ E[Y] = (0.3 \times 6%) + (0.5 \times 10%) + (0.2 \times 12%) ] [ E[Y] = (0.3 \times 0.06) + (0.5 \times 0.10) + (0.2 \times 0.12) ] [ E[Y] = 0.018 + 0.05 + 0.024 = 0.092 \text{ or } 9.2% ]

Step 2: Calculate (E[XY])

Now we calculate (E[XY]): [ E[XY] = (0.3 \times 20% \times 6%) + (0.5 \times 15% \times 10%) + (0.2 \times 5% \times 12%) ] [ E[XY] = (0.3 \times 0.20 \times 0.06) + (0.5 \times 0.15 \times 0.10) + (0.2 \times 0.05 \times 0.12) ] [ E[XY] = (0.3 \times 0.012) + (0.5 \times 0.015) + (0.2 \times 0.006) ] [ E[XY] = 0.0036 + 0.0075 + 0.0012 = 0.0123 ]

Step 3: Calculate Covariance

Now we can calculate the covariance: [ \text{Cov}(K, L) = E[XY] - E[X]E[Y] ] [ \text{Cov}(K, L) = 0.0123 - (0.145 \times 0.092) ] [ \text{Cov}(K, L) = 0.0123 - 0.01334 = -0.00104 ]

Conclusion

The covariance of the returns of K and L is approximately -0.00104, which does not match any of the provided options. However, if we consider the values in percentage terms, we can multiply by 100 to express it in a more conventional format:

[ \text{Cov}(K, L) \approx -0.104 ]

Since none of the options match this result, it seems there may be an error in the options provided. However, based on the calculations, the covariance is negative, indicating that the returns of K and L tend to move in opposite directions.

If we were to round or adjust based on the options, the closest would be option a. -5.97, but it is not an exact match. Please verify the options or the calculations.