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 standard deviations are: Question 3Select one: a. K 27.25 L 4.96 b. K 14.5 L 9.2 c. K 6.8 L 3.4 d. K 5.22 L 2.23
To calculate the standard deviation of the returns for companies K and L, we first need to find the expected return for each company, and then use that to calculate the variance and standard deviation.
The expected return ( E(R) ) is calculated as follows:
[ E(R) = \sum (P_i \times R_i) ]
Where ( P_i ) is the probability and ( R_i ) is the return.
For Company K:
[ E(R_K) = (0.3 \times 20%) + (0.5 \times 15%) + (0.2 \times 5%) ] [ E(R_K) = (0.3 \times 0.20) + (0.5 \times 0.15) + (0.2 \times 0.05) ] [ E(R_K) = 0.06 + 0.075 + 0.01 = 0.145 \text{ or } 14.5% ]
For Company L:
[ E(R_L) = (0.3 \times 6%) + (0.5 \times 10%) + (0.2 \times 12%) ] [ E(R_L) = (0.3 \times 0.06) + (0.5 \times 0.10) + (0.2 \times 0.12) ] [ E(R_L) = 0.018 + 0.05 + 0.024 = 0.092 \text{ or } 9.2% ]
The variance ( \sigma^2 ) is calculated as follows:
[ \sigma^2 = \sum (P_i \times (R_i - E(R))^2) ]
For Company K:
[ \sigma^2_K = (0.3 \times (20% - 14.5%)^2) + (0.5 \times (15% - 14.5%)^2) + (0.2 \times (5% - 14.5%)^2) ] [ = (0.3 \times (0.055)^2) + (0.5 \times (0.005)^2) + (0.2 \times (-0.095)^2 ] [ = (0.3 \times 0.0003025) + (0.5 \times 0.000025) + (0.2 \times 0.009025) ] [ = 0.00009075 + 0.0000125 + 0.001805 = 0.00190825 ]
For Company L:
[ \sigma^2_L = (0.3 \times (6% - 9.2%)^2) + (0.5 \times (10% - 9.2%)^2) + (0.2 \times (12% - 9.2%)^2) ] [ = (0.3 \times (-0.032)^2) + (0.5 \times (0.008)^2) + (0.2 \times (0.028)^2) ] [ = (0.3 \times 0.001024) + (0.5 \times 0.000064) + (0.2 \times 0.000784) ] [ = 0.0003072 + 0.000032 + 0.0001568 = 0.000496 ]
The standard deviation ( \sigma ) is the square root of the variance:
[ \sigma_K = \sqrt{0.00190825} \approx 0.0437 \text{ or } 4.37% ] [ \sigma_L = \sqrt{0.000496} \approx 0.0223 \text{ or } 2.23% ]
The standard deviations are approximately:
Thus, the correct answer is:
d. K 5.22 L 2.23 (Note: The value for K seems to be slightly off; it should be around 4.37% based on the