Question

Stock A has an expected return of 17% and a standard deviation of 33%. Stock B has an expected return of 13% and a standard deviation of 17%. The risk-free rate is 2.2% and the correlation between Stock A and Stock B is 0.5. Build the optimal risky portfolio of Stock A and Stock B. What is the standard deviation of this portfolio?

Answer 1

Optimal Risky Portfolio or 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) ] ]

Particulars	Amount
SD of A	33.0%
SD of B	17.0%
r(A,B)	0.5000

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) ] ]
= [ [ (0.17)^2 ] - [ 0.33 * 0.17 * 0.5 ] ] / [ [ (0.33)^2 ] + [ ( 0.17 )^2 ] - [ 2 * 0.33 * 0.17 * 0.5 ] ]
= [ [ 0.0289 ] - [ 0.02805 ] ] / [ [ 0.1089 ] + [ 0.0289 ] - [ 2 * 0.02805 ] ]
= [ 0.000850000000000004 ] / [ 0.0817 ]
= 0.010404

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) ] ]
= [ [ (0.33)^2 ] - [ 0.33 * 0.17 * 0.5 ] ] / [ [ (0.33)^2 ] + [ ( 0.17 )^2 ] - [ 2 * 0.33 * 0.17 * 0.5 ] ]
= [ [ 0.1089 ] - [ 0.02805 ] ] / [ [ 0.1089 ] + [ 0.0289 ] - [ 2 * 0.02805 ] ]
= [ 0.08085 ] / [ 0.0817 ]
= 0.989596

POrtfolio SD at Optimal Risky Portfolio:

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	0.0104
Weight in B	0.9896
SD of A	33.00%
SD of B	17.00%
r(A,B)	0.5

Portfolio SD = SQRT[((Wa*SDa)^2)+((Wb*SDb)^2)+2*(wa*SDa)*(Wb*SDb)*r(A,B)]
=SQRT[((0.0104*0.33)^2)+((0.9896*0.17)^2)+2*(0.0104*0.33)*(0.9896*0.17)*0.5]
=SQRT[((0.003432)^2)+((0.168232)^2)+2*(0.003432)*(0.168232)*0.5]
=SQRT[0.0289]
= 0.17
= I.e 17 %