Question

Shares of company A are sold at $10 per share. Shares of company B are sold at $50 per share. According to a market analyst, 1 share of each company can either gain $1, with a probability of 0.5 or lose $1, with probability 0.5, independently of the other company. Which of the following portfolios has the lowest risk:

a) 100 shares of A
b) 50 shares of A + 10 shares of B
c) 40 shares of A + 12 shares of B

--- For E(X) for both A and B I get:
PA(1) = PB(1) = .5, PB(-1)=0
EA(X)= EB(X) =(1)(.5) + (-1)(.5) = 0.

To obtain variance:
EA(X^2)= EB(X^2)= (1^2)(.5) + (-1^2)(.5) = (1)(.5) + (1)(.5) = 1.

VarA(X) = VarB(X) = EA(x^2) - EA(X)^2 = 1 - 0 = 1

a) 100 shares * $10 * X/10 (<-you gain X for every 10 spent) = 100X = A

E(A) = E(100x) = 100E(x) = 100 * 0 = 0
Var(A) = 100^2*Var(X) = 10,000 * 1 = 10,000

b) 50 shares * 10 * X/10 + 10 shares * 50 + Y/50 (<-you gain Y for every 50 spent) ) = 50X + 10Y

E(X) = E(50x) + E(10Y) = 50*0 + 10*0 = 0
Var(X) = 50^2 * Var(X) + 10^2 * Var(Y) = 50^2 * 1 + 10^2 * 1 =2500 + 100 = 2,600

c) 40 shares * 10 * X/10 + 12 shares * 50 + Y/50 (<-you gain Y for every 50 spent) ) = 40X + 12Y
E(X) = E(40x) + E(12Y) = 40*0 + 12*0 = 0
Var(X) = 40^2 * Var(X) + 12^2 * Var(Y) = 40^2 * 1 + 12^2 * 1 =1600 + 144 = 1,744

Based on the solutions on the book I know the answer is c, but I am pretty certain my approach is wrong. Can you please tell me where my mistake is and how to fix it? The fact the E(X)=0 really throws me off.