Question

You are going to invest in Asset J and Asset S. Asset J has an expected return of 14.4 percent and a standard deviation of 55.4 percent. Asset S has an expected return of 11.4 percent and a standard deviation of 20.4 percent. The correlation between the two assets is .50. What are the standard deviation and expected return of the minimum variance portfolio? (Do not round intermediate calculations. Enter your answers 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) ] ]

A = Asset J

B = Asset S

Particulars	Amount
SD of A	55.40%
SD of B	20.40%
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) ] ]
= [ [ (0.204)^2 ] - [ 0.554 * 0.204 * 0.5 ] ] / [ [ (0.554)^2 ] + [ ( 0.204 )^2 ] - [ 2 * 0.554 * 0.204 * 0.5 ] ]
= [ [ 0.041616 ] - [ 0.056508 ] ] / [ [ 0.306916 ] + [ 0.041616 ] - [ 2 * 0.056508 ] ]
= [ -0.014892 ] / [ 0.235516 ]
= -0.0632

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) ] ]
= [ [ (0.554)^2 ] - [ 0.554 * 0.204 * 0.5 ] ] / [ [ (0.554)^2 ] + [ ( 0.204 )^2 ] - [ 2 * 0.554 * 0.204 * 0.5 ] ]
= [ [ 0.306916 ] - [ 0.056508 ] ] / [ [ 0.306916 ] + [ 0.041616 ] - [ 2 * 0.056508 ] ]
= [ 0.250408 ] / [ 0.235516 ]
= 1.0632
Expected Ret:

Portfolio Return is the weighted avg return of securities in that portfolio

Stock	Weight	Ret	WTd Ret
Stock J	(0.0632)	14.40%	-0.91%
Stock S	1.0632	11.40%	12.12%
Portfolio Ret Return			11.21%

Portfolio SD:

Particulars	Amount
Weight in A	-0.0632
Weight in B	1.0632
SD of A	55.40%
SD of B	20.40%
r(A,B)	0.5

Portfolio SD = SQRT[((Wa*SDa)^2)+((Wb*SDb)^2)+2*(wa*SDa)*(Wb*SDb)*r(A,B)]
=SQRT[((-0.0632*0.554)^2)+((1.0632*0.204)^2)+2*(-0.0632*0.554)*(1.0632*0.204)*0.5]
=SQRT[((-0.0350128)^2)+((0.2168928)^2)+2*(-0.0350128)*(0.2168928)*0.5]
=SQRT[0.0407]
= 0.2017
= I.e 20.17 %