Question

Here are returns and standard deviations for four investments. Standard Return (%) Deviation(%) Treasury bills Slock P Stock Q Stock R 4.0 9.5 13.5 21.5 12 25 24 Calculate the standard deviations of the following portfolios a. 50% in Treasury bills, 50% in stock P (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) Standard deviation b 50% each in and R, assuming the shares have: (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places) Standard Deviation Perfect positive correlation Perfect negative correlation No correlation

Answer 1

Part a:
The formula for standard deviation is given by:
Portfolio standard deviation = wiσi + wjσj
wi = the portfolio weight of the asset i
wj = the portfolio weight of the asset j
σi = the standard deviation of returns on asset i
σj = the standard deviation of returns on asset j
We are given that, in this investment 50% weightage is given to treasury bill and 50% to stock P.
Standard deviation of the treasury bill is zero and standard deviation of stock P is 12%

Portfolio standard deviation = 50%*0%+50%*12%=50/100*12/100=6/100=.06=6.00%

Part b:
As per question stock Q and R are given 50% weightage each.
Standard deviation of Q is 25%=.25 and standard deviation of R is 24%=.24

So, (weight of Q)*(Standard deviation of Q)=.5*.25=.125

And, (weight of R)*(Standard deviation of R)=.5*.24=.12

Explanation:

In a two asset portfolio Var(RiRj) = wi^2σi^2 + wj^2σj^2 + 2wiwjCov(Ri,Rj)

wi = the portfolio weight of the asset i
wj = the portfolio weight of the asset j
σi = the standard deviation of returns on asset i
σj = the standard deviation of returns on asset j
Var(RiRj) = Variance of the two asset portfolio returns.
Cov(Ri,Rj) = the covariance between the returns on the two assets
This covariance can be further simplified as Cov(Ri,Rj)=σ(i)σ(j)*corr(Ri,Rj)

corr(Ri,Rj) = the correlation between the returns on asset i and j
σi = the standard deviation of returns on asset i
σj = the standard deviation of returns on asset j

On simplifying variance equations by substituting the value of Cov(Ri,Rj) with σ(i)σ(j)*corr(Ri,Rj), we get
Var(RiRj) = wi^2σi^2 + wj^2σj^2 + 2wiwjσ(i)σ(j)*corr(Ri,Rj)

Perfect positive correlation (when corr(Ri,Rj)=1): In this case the portfolio standard deviation is the weighted average of the standard deviation on individual assets.
Var(RiRj) = wi^2σi^2 + wj^2σj^2 + 2wiwjσ(i)σ(j)= (wiσi + wjσj)^2
Portfolio standard deviation = wiσi + wjσj, as standard deviation is the square root of variance
Portfolio standard deviation=(weight of Q)*(Standard deviation of Q) + (weight of R)*(Standard deviation of R)
=.125+.12
=.245
=24.50% (rounded to 2 decimal places)

Perfect negative correlation (when corr(Ri,Rj)=-1): In this case, portfolio standrad deviation is the difference (non-negative value) caused by the standard deviation of returns on individual assets weighted by their respective shares in the portfolio.
When the correlation coefficient between asset returns is negative unity, it is possible to combine them in a manner that will eliminate all the risk.
Var(RiRj) = wi^2σi^2 + wj^2σj^2 - 2wiwjσ(i)σ(j)=(wiσi - wjσj)^2
Portfolio standard deviation= wiσi - wjσj, as standard deviation is the square root of variance
Portfolio standard deviation=(weight of Q)*(Standard deviation of Q) - (weight of R)*(Standard deviation of R)
=.125-.12
=.005
=.5%
Zero correlation (when corr(Ri,Rj)=0): When the returns on two assets are uncorrelated, their correlation is zero and hence covariance term becomes zero. In this case, portfolio variance is the sum of the square of the standard deviation of each asset weighted by its proportion in the portfolio and standard deviation id the square root of variance.
Var(RiRj) = wi^2σi^2 + wj^2σj^2
Portfolio standard deviation = [wi^2σi^2 + wj^2σj^2]^(1/2) , as standard deviation is the square root of variance.
Portfolio standard deviation=[(weight of Q)^2*(Standard deviation of Q)^2 + (weight of R)^2*(Standard deviation of R)^2]^1/2
=[.5^2*.25^2+.5^2*.24^2]^1/2
=[.25*.0625+.25*.0576]^1/2
=[.015625+.0144]^1/2
=.030025^1/2
=.54795=5.4795% or 5.48%(rounded to 2 decimal places)