Question

Explain optimal risky portfolio

Answer 1

Optimal Risky portfolio is the portfoli with Least risk.

I.e at the given weights, Portfolio will have lesser risk other than any combination weights of securities in portfolio.

Ex:

Particulars	Amount
SD of A	10%
SD of B	15%
r	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) ] ]

= [ [ (0.15)^2 ] - [ 0.1 * 0.15 * 0.5 ] ] / [ [ (0.1)^2 ] + [ ( 0.15 )^2 ] - [ 2 * 0.1 * 0.15 * 0.5 ] ]

= [ [ 0.0225 ] - [ 0.0075 ] ] / [ [ 0.01 ] + [ 0.0225 ] - [ 2 * 0.0075 ] ]

= [ 0.015 ] / [ 0.0175 ]

= 0.8571

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) ] ]

= [ [ (0.1)^2 ] - [ 0.1 * 0.15 * 0.5 ] ] / [ [ (0.1)^2 ] + [ ( 0.15 )^2 ] - [ 2 * 0.1 * 0.15 * 0.5 ] ]

= [ [ 0.01 ] - [ 0.0075 ] ] / [ [ 0.01 ] + [ 0.0225 ] - [ 2 * 0.0075 ] ]

= [ 0.0025 ] / [ 0.0175 ]

= 0.1429

If we invest in the specified weights portfolio will give Lesser risk.

Portfolio risk at Optimal portfolio:

Particulars	Amount
Weight in A	0.8571
Weight in B	0.1429
SD of A	10.00%
SD of B	15.00%
r(1,2)	0.5

Portfolio SD = SQRT[((Wa*SDa)^2)+((Wb*SDb)^2)+2*(wa*SDa)*(Wb*SDb)*r(1,2)]

=SQRT[((0.8571*0.1)^2)+((0.1429*0.15)^2)+2*(0.8571*0.1)*(0.1429*0.15)*0.5]

=SQRT[((0.08571)^2)+((0.021435)^2)+2*(0.08571)*(0.021435)*0.5]

=SQRT[0.0096]

9.82%

Portfolio SD at other combination of Weights:

For Ex:

Particulars	Amount
Weight in A	0.3000
Weight in B	0.7000
SD of A	10.00%
SD of B	15.00%
r(1,2)	0.5

Portfolio SD = SQRT[((Wa*SDa)^2)+((Wb*SDb)^2)+2*(wa*SDa)*(Wb*SDb)*r(1,2)]

=SQRT[((0.3*0.1)^2)+((0.7*0.15)^2)+2*(0.3*0.1)*(0.7*0.15)*0.5]

=SQRT[((0.03)^2)+((0.105)^2)+2*(0.03)*(0.105)*0.5]

=SQRT[0.0151]

12.28%

Pls do rate, if the answer is correct and comment, if any further assistance is required.