Question

Consider two stocks, Stock D, with an expected return of 16 percent and a standard deviation of 31 percent, and Stock I, an international company, with an expected return of 9 percent and a standard deviation of 19 percent. The correlation between the two stocks is −.17. What are the expected return and standard deviation of the minimum variance portfolio? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Answer 1

Minimum Variance Portfolio:

A minimum variance portfolio is a collection of securities that combine to minimize the price volatility of the overall portfolio. with the given weights to securities/ Assets in portfolio,portfolio risk will be minimal.

Weight in A = [ [ (SD of B)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
Weight in B = [ [ (SD of A)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]

Assume

A = Stock D

B = Stock I

Particulars	Amount
SD of A	31%
SD of B	19%
r(A,B)	-0.1700

Weight in A = [ [ (SD of B)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
= [ [ (0.19)^2 ] - [ 0.31 * 0.19 * -0.17 ] ] / [ [ (0.31)^2 ] + [ ( 0.19 )^2 ] - [ 2 * 0.31 * 0.19 * -0.17 ] ]
= [ [ 0.0361 ] - [ -0.010013 ] ] / [ [ 0.0961 ] + [ 0.0361 ] - [ 2 * -0.010013 ] ]
= [ 0.046113 ] / [ 0.152226 ]
= 0.3029

Weight in B = [ [ (SD of A)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
= [ [ (0.31)^2 ] - [ 0.31 * 0.19 * -0.17 ] ] / [ [ (0.31)^2 ] + [ ( 0.19 )^2 ] - [ 2 * 0.31 * 0.19 * -0.17 ] ]
= [ [ 0.0961 ] - [ -0.010013 ] ] / [ [ 0.0961 ] + [ 0.0361 ] - [ 2 * -0.010013 ] ]
= [ 0.106113 ] / [ 0.152226 ]
= 0.6971

Expected Ret:

Stock	Weight	Ret	WTd Ret
Stock D	0.3029	16.00%	4.85%
Stock I	0.6971	9.00%	6.27%
Portfolio Ret Return			11.12%

Expected Ret from Portfolio is 11.12%

Expected SD:

It is nothing but volataility of Portfolio. It is calculated based on three factors. They are
a. weights of Individual assets in portfolio
b. Volatality of individual assets in portfolio
c. Correlation betwen individual assets in portfolio.
If correlation = +1, portfolio SD is weighted avg of individual Asset's SD in portfolio. We can't reduce the SD through diversification.
If Correlation = -1, we casn reduce the SD to Sero, by investing at propoer weights.
If correlation > -1 but <1, We can reduce the SD, n=but it will not become Zero.

Wa = Weight of A
Wb = Weigh of B
SDa = SD of A
SDb = SD of B

Particulars	Amount
Weight in A	0.3029
Weight in B	0.6971
SD of A	31.00%
SD of B	19.00%
r(A,B)	-0.17

Portfolio SD = SQRT[((Wa*SDa)^2)+((Wb*SDb)^2)+2*(wa*SDa)*(Wb*SDb)*r(A,B)]
=SQRT[((0.3029*0.31)^2)+((0.6971*0.19)^2)+2*(0.3029*0.31)*(0.6971*0.19)*-0.17]
=SQRT[((0.093899)^2)+((0.132449)^2)+2*(0.093899)*(0.132449)*-0.17]
=SQRT[0.0221]
= 0.1488
= I.e 14.88 %

Expected SD from Portfolio is 14.88%