Question

Expected return of stock Alpha is 8 percent and of stock Beta is 12 percent. The standard deviation of the stocks are 13 percent and 18 percent respectively. The correlation between these two stocks is 0.4. If the portfolio manager has decided to invest all the funds that he holdsin some proportion in these two assets. The expected return ofthe portfolio based on this proportion is 11.5%. What are the weightsin each of the stocks? What isthe standard deviationofthisportfolio?

Answer 1

Portfolio Ret= Weighted Avg Ret of securities in that portfolio.

Let y be the weight inStock Alpha.

Stock	Weight	Ret	WTd Ret
Stock Alpha	y	0.0800	0.08y
Stock Beta	1-y	0.1200	0.12 - 0.12y
Portfolio Ret Return			0.12 - 0.04y

Thus 0.12 - 0.04y = 0.115

0.04y = 0.12 - 0.115

= 0.005

y = 0.005 / 0.04

= 0.125

weight in Stock Alpha = 0.125 I.e 12.5%

Weight in STock Beta = 0.875 I.e 87.5%

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

A = Stock ALpha

B = Stock Beta

Particulars	Amount
Weight in A	0.1250
Weight in B	0.8750
SD of A	13.00%
SD of B	18.00%
r(A,B)	0.4

Portfolio SD = SQRT[((Wa*SDa)^2)+((Wb*SDb)^2)+2*(wa*SDa)*(Wb*SDb)*r(A,B)]
=SQRT[((0.125*0.13)^2)+((0.875*0.18)^2)+2*(0.125*0.13)*(0.875*0.18)*0.4]
=SQRT[((0.01625)^2)+((0.1575)^2)+2*(0.01625)*(0.1575)*0.4]
=SQRT[0.0271]
= 0.1647
= I.e 16.47 %

Portfolio SD is 16.47%