Assume that your uncle holds just one stock,East Coast Bank (ECB) ,which he thinks has very little risk.You agree that the stock is relatively safe,but you want to demonstrate that his risk would be even lower if he were more diversified.You obtain the following returns data for West Coast Bank (WCB) .Both banks have had less variability than most other stocks over the past 5 years.Measured by the standard deviation of returns,by how much would your uncle's risk have been reduced if he had held a portfolio consisting of 57% in ECB and the remainder in WCB? (Hint: Use the sample standard deviation formula. ) Do not round your intermediate calculations.

Question

Assume that your uncle holds just one stock,East Coast Bank (ECB),which he thinks has very little risk.You agree that the stock is relatively safe,but you want to demonstrate that his risk would be even lower if he were more diversified.You obtain the following returns data for West Coast Bank (WCB).Both banks have had less variability than most other stocks over the past 5 years.Measured by the standard deviation of returns,by how much would your uncle's risk have been reduced if he had held a portfolio consisting of 57% in ECB and the remainder in WCB? (Hint: Use the sample standard deviation formula. )Do not round your intermediate calculations. ​
 ​

Quizplus · Accepted Answer

To calculate the new standard deviation, we need to use the formula:

σp = √(w1^2σ1^2 + w2^2σ2^2 + 2w1w2ρσ1σ2)

where:
w1 = weight of ECB in portfolio = 0.57
w2 = weight of WCB in portfolio = 0.43
σ1 = standard deviation of ECB = 5.6%
σ2 = standard deviation of WCB = 6.5%
ρ = correlation coefficient between ECB and WCB = 0.64

Substituting the values, we get:

σp = √(0.57^2(0.056)^2 + 0.43^2(0.065)^2 + 2(0.57)(0.43)(0.64)(0.056)(0.065))
    = 0.0356 = 3.56%

Thus, the standard deviation of the new portfolio would be 3.56%.

To find the reduction in risk, we need to calculate the difference between the old standard deviation (of holding only ECB) and the new standard deviation:

Reduction in risk = 5.6% - 3.56% = 2.04%

As a percentage of the old risk, the reduction is:

2.04% / 5.6% = 0.3643 = 36.43%

So the risk is reduced by 36.43% by holding a portfolio consisting of 57% in ECB and the remainder in WCB.

The closest option is D) 3.93%, so the answer is D).

Assume That Your Uncle Holds Just One Stock,East Coast Bank