Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 3.0%. The probability distributions of the risky funds are:

	Expected Return	Standard Deviation
Stock fund (S)	12%	41%
Bond fund (B)	5%	30%

The correlation between the fund returns is 0.0667.

What is the expected return and standard deviation for the minimum-variance portfolio of the two risky funds? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Expected Return ?

Standard Deviation ?

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) ] ]

Particulars	Amount
SD of A	41%
SD of B	30.0%
r(A,B)	0.0667

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) ] ]
= [ [ (0.3)^2 ] - [ 0.41 * 0.3 * 0.0667 ] ] / [ [ (0.41)^2 ] + [ ( 0.3 )^2 ] - [ 2 * 0.41 * 0.3 * 0.0667 ] ]
= [ [ 0.09 ] - [ 0.0082041 ] ] / [ [ 0.1681 ] + [ 0.09 ] - [ 2 * 0.0082041 ] ]
= [ 0.0817959 ] / [ 0.2416918 ]
= 0.3384

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) ] ]
= [ [ (0.41)^2 ] - [ 0.41 * 0.3 * 0.0667 ] ] / [ [ (0.41)^2 ] + [ ( 0.3 )^2 ] - [ 2 * 0.41 * 0.3 * 0.0667 ] ]
= [ [ 0.1681 ] - [ 0.0082041 ] ] / [ [ 0.1681 ] + [ 0.09 ] - [ 2 * 0.0082041 ] ]
= [ 0.1598959 ] / [ 0.2416918 ]
= 0.6616
A = Stock Fund

B = Bond Fund

Expected Ret:

Expected Ret = Weighted avg ret of securities in that portfolio.

Stock	Weight	Ret	WTd Ret
Stock Fund	0.3384	12.00%	4.06%
Bond Fund	0.6616	5.00%	3.31%
Portfolio Ret Return			7.37%

Expected Ret from Portfolio is 7.37%

Portfolio 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	0.3384
Weight in B	0.6616
SD of A	41.00%
SD of B	30.00%
r(A,B)	0.0667

Portfolio SD = SQRT[((Wa*SDa)^2)+((Wb*SDb)^2)+2*(wa*SDa)*(Wb*SDb)*r(A,B)]
=SQRT[((0.3384*0.41)^2)+((0.6616*0.3)^2)+2*(0.3384*0.41)*(0.6616*0.3)*0.0667]
=SQRT[((0.138744)^2)+((0.19848)^2)+2*(0.138744)*(0.19848)*0.0667]
=SQRT[0.0623]
= 0.2496
= I.e 24.96 %

Portfolio SD is 24.96%