Question

You invest $100 in a risky asset with an expected rate of return of 0.25 and a standard deviation of 0.20 and a T-bill with a rate of return of 0.05.

If the portfolio standard deviation is 0.12, what is the expected return of this portfolio?

A.

14.0%

B.

12.0%

C.

13.0%

D.

17.0%

E.

Cannot be determined

Answer 1

Solution:
	Answer is D. 17.0%
Working Notes:
	Expected return of the portfolio
	We get Expected return of the portfolio by getting Weighted average return of individual assets. Portfolio consist of T bill and risky assets
	Expected return of the portfolio= Weighted average return of individual assets
	Let W be weight of risky asset in the portfolio
	Then weight of risk free assets in the portfolio =1-W
	r = return of risky assets = 0.25
	rf = return of T bill = 0.05
	Expected return of the portfolio= W x r + (1-W) x rf
	As we no details of weights of risky assets and T bill in the portfolio , we will use the given standard deviation of the portfolio to gets weights of each assets in the portfolio.
	Standard deviation of the portfolio	0.12
	there is T bill in the portfolio as it is also known as risk free means standard deviation must be zero .
	We calculate by first calculating variance of the portfolio having two risky assets formula. Then we get standard deviation of the portfolio as square root of variance
	Variance of two risky assets portfolio we get
	(S.d P)^2 = W^2 x (S.d r)^2 + Wrf^2 x (s.d rf)^2 + 2 x W x Wrf x S.d r x Sd. rf x Rr,rf
	Where
	S.d P = Standard deviation of the portfolio and (S.d P)^2 is the variance of the portfolio
	Wr = weight of risky assets in the portfolio = W
	Wrf = weight of T bill in the portfolio =(1-W)
	S.d r = Standard deviations of risky assets = 0.20
	S.d rf = Standard deviations of T bill = 0   as it is risk free
	Rr,rf = Correlation coefficient of risky asset & T bill
	Variance of two risky assets portfolio we get
	(S.d P)^2 = W^2 x (S.d r)^2 + Wrf^2 x (s.d rf)^2 + 2 x W x Wrf x S.d r x Sd. rf x Rr,rf
	As above formula give standard deviation of two risky assets but in our case there is only one risky assets the above formula get reduced to small formula as below
	(S.d P)^2 = W^2 x (S.d r)^2 + Wrf^2 x (s.d rf)^2 + 2 x W x Wrf x S.d r x Sd. rf x Rr,rf
	(S.d P)^2 = W^2 x (S.d r)^2 + Wrf^2 x (0)^2 + 2 x W x Wrf x S.d r x 0 x Rr,rf
	(S.d P)^2 = W^2 x (S.d r)^2
	S.d P = Standard deviation of the portfolio =0.12
	Wr = weight of risky assets in the portfolio = W
	S.d r = Standard deviations of risky assets = 0.20
	(S.d P)^2 = W^2 x (S.d r)^2
	(0.12)^2 = W^2 x (0.20)^2
	W^2 = ((0.12)^2 )/(0.20)^2
	W^2 = 0.360
	W = 0.360^(1/2)
	W = 0.60
	W is the weight of risky assets in the portfolio which is 0.60 computed above
	Hence weight of T bill in the portfolio becomes = 1-W =1-0.60 =0.40
Now	r = return of risky assets = 0.25
	rf = return of T bill = 0.05
	Expected return of the portfolio= W x r + (1-W) x rf
	Expected return of the portfolio= 0.60 x 0.25 + (1-0.60) x 0.05
	Expected return of the portfolio=0.17
	Expected return of the portfolio=   17.0%
Please feel free to ask if anything about above solution in comment section of the question.