Answer:
a. Standard deviation of the portfolio = 7.00%
b(i) Standard deviation of the portfolio = 30.00%
b(ii) Standard deviation of the portfolio = 4.00%
b(iii) Standard deviation of the portfolio = 21.40%
Explanation:
Note: This question is not complete. The complete question is therefore provided before answering the question as follows:
Here are returns and standard deviations for four investments.
Return (%) Standard Deviation (%)
Treasury bills 4.5 0
Stock P 8.0 14
Stock Q 17.0 34
Stock R 21.5 26
Calculate the standard deviations of the following portfolios.
a. 50% in Treasury bills, 50% in stock P. (Enter your answer as a percent rounded to 2 decimal places.)
b. 50% each in Q and R, assuming the shares have:
i. perfect positive correlation
ii. perfect negative correlation
iii. no correlation
(Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.)
The explanation to the answer is now provided as follows:
a. Calculate the standard deviations of 50% in Treasury bills, 50% in stock P. (Enter your answer as a percent rounded to 2 decimal places.)
Since there is no correlation between Treasury bills and stocks, it therefore implies that the correlation coefficient between the Treasury bills and stock P is zero.
The standard deviation between the Treasury bills and stock P can be calculated by first estimating the variance of their returns using the following formula:
Portfolio return variance = (WT^2 * SDT^2) + (WP^2 * SDP^2) + (2 * WT * SDT * WP * SDP * CFtp) ......................... (1)
Where;
WT = Weight of Stock Treasury bills = 50%
WP = Weight of Stock P = 50%
SDT = Standard deviation of Treasury bills = 0
SDP = Standard deviation of stock P = 14%
CFtp = The correlation coefficient between Treasury bills and stock P = 0.45
Substituting all the values into equation (1), we have:
Portfolio return variance = (50%^2 * 0^2) + (50%^2 * 14%^2) + (2 * 50% * 0 * 50% * 14% * 0) = 0.49%
Standard deviation of the portfolio = (Portfolio return variance)^(1/2) = (0.49%)^(1/2) = (0.49)^0.5 = 7.00%
b. 50% each in Q and R
To calculated the standard deviation 50% each in Q and R, we first estimate the variance using the following formula:
Portfolio return variance = (WQ^2 * SDQ^2) + (WR^2 * SDR^2) + (2 * WQ * SDQ * WR * SDR * CFqr) ......................... (2)
Where;
WQ = Weight of Stock Q = 50%
WR = Weight of Stock R = 50%
SDQ = Standard deviation of stock Q = 34%
SDR = Standard deviation of stock R = 26%
b(i). assuming the shares have perfect positive correlation
This implies that:
CFqr = The correlation coefficient between stocks Q and = 1
Substituting all the values into equation (2), we have:
Portfolio return variance = (50%^2 * 34%^2) + (50%^2 * 26%^2) + (2 * 50% * 34% * 50% * 26% * 1) = 9.00%
Standard deviation of the portfolio = (Portfolio return variance)^(1/2) = (9.00%)^(1/2) = (9.00%)^0.5 = 30.00%
b(ii). assuming the shares have perfect negative correlation
This implies that:
CFqr = The correlation coefficient between stocks Q and = -1
Substituting all the values into equation (2), we have:
Portfolio return variance = (50%^2 * 34%^2) + (50%^2 * 26%^2) + (2 * 50% * 34% * 50% * 26% * (-1)) = 0.16%
Standard deviation of the portfolio = (Portfolio return variance)^(1/2) = (0.16%)^(1/2) = (0.16%)^0.5 = 4.00%
b(iii). assuming the shares have no correlation
This implies that:
CFqr = The correlation coefficient between stocks Q and = 0
Substituting all the values into equation (2), we have:
Portfolio return variance = (50%^2 * 34%^2) + (50%^2 * 26%^2) + (2 * 50% * 34% * 50% * 26% * 0) = 4.58%
Standard deviation of the portfolio = (Portfolio return variance)^(1/2) = (4.58%)^(1/2) = (4.58%)^0.5 = 21.40%
