Question

Consider a coin toss experiment and the following assets. A gives £200 if the first is heads, £50 for tails. B gives £200 if the second is heads and £50 for tails. C is half of A plus half of B. A and B are independent. Show that the expected value of each asset isthe same. Show that C reduces risk and explain why this is so. [Hint: You need to calculate E(A), E(B), and E(C). Also calculate the standard deviation (or risk) of returns from each asset]

Answer 1

Let P(.) indicates the probability of the event.

Since the tossed coin is assumed to be unbiased in nature, hence for both A and B, the event of giving 200 and 50 are equi-likely i.e.  

P(200) = P(50) = 0.5

So, Expected amount that A needs to pay E(A) = 0.5*200 + 0.5*50 = 125

Similalrly Expected amount that B needs to pay E(B) = 0.5*50 + 0.5*200 = 125

Hence E(A) = E(B)

Since C = 0.5(A+B) , therefore E(C) = 0.5 * (125 +125) = 125

Now, E(A²) = 0.5*200² + 0.5*50² = E(B²) = 20,000 +1250 = 21250

So E(C²) = 0.5 * (E(A²) + E(B²)) = 21250

Since, risk is measured as the standard deviation of the random variable, hence ,

Var(A) = Var(B) = E(A²) - (E(A))² = 21250 - (125)² = 5625

So, Standard deviation of A & B = Risk of A & B = 75

Now, Since C = 0.5(A+B) => Var (C) = (0.5)² * ( Var(A) + Var(B)+ 2*Covariance(A,B))

Since, its given that A and B are independent, therefore,

Var (C) = (0.5)² * ( Var(A) + Var(B) ) = 0.25 * (5625 + 5625) = 2812.5

Therefore risk of C = 53.03

hence risk of C is lower than risk of A and risk of B.