prob & stat question

Question

A stock broker wishes to put together a portfolio from two types of stocks available A and B. The return on stock A is XA, and the return on stock B is XB. Stock A is less risky, but has less potential reward than stock B; in fact, it is known that

• E[XA] = 10 and Var[XA] = 4

• E[XB] = 12 and Var[XB] = 100

• XA is independent of XB (all amounts in thousands). The broker will make a portfolio with a combination of the two stocks, and the combined return will be Y = αXA + (1 − α)XB where 0 ≤ α ≤ 1 is a constant representing the proportion of money that is invested in stock A.

(a) Find E[Y ] (in terms of α)

(b) ind Var[Y ] (in terms of α)

(c)Find the value of α which minimizes Var[Y ].

Answer 1

(a) E[Y] = E[XA] + (1 − α)E[XB] = 10α + 12(1 - α) = 12 - 2α

 

We know Var(Y) = E[Y^2] –  (E[Y])^2)

or

Var(Y) = (α^2 x Var[XA] ^2) + ((1 - α)^2 x Var[XB]^2) + (2 x (αVar[XA])(1 - α)Var[XB]q(XA,XB)) where q(1,2) is the correlation between the 2 stocks. Note: ^ denotes to the power of or an exponent.

 

q(XA,XB) = cov(XA,XB)/sd(XA)sd(XB) where cov(XA,XB) is the covariance of the stocks and sd are standard deviations(square root of variance for the two stocks given above.) The problem states the stocks are independent so q(XA,XB) = 0 and thus it is crossed out in the formula above. Note: I can't turn italic font off by toggling the italics icon or Ctrl + I.

 

(b) The formula above becomes:  Var(Y) = (α^2 x Var[XA] ^2) + ((1 - α)^2 x Var[XB]^2) = 16α^2 + 100(1 - α)^2 = 116α^2 - 200α + 100

 

(c) Differentiate Var(Y) wrt α and set equal to 0 to get minimum variance in terms of weight (α).

 

d(Var(Y))/d(α) = 232α - 200 = 0 so α = 200/232 = 0.86