Question

A CEO is considering buying an insurance policy to cover possible losses incurred by marketing a new product. If the product is a complete failure, a loss of $450,000 would be incurred; if it is a moderately failure, a loss of $150,000 would be incurred; if it is only minor failure, a loss of $50,000 would be incurred. Insurance actuaries have determined that the probabilities that the product will be a failure or only moderately successful are 0.005, 0.056 and 0.153, respectively.

1. Assuming that the CEO is willing to ignore all other possible losses, what premium should the insurance company charge for a policy in order to have a profit of $5,000.
2. What is the variance of the profit, if there are 10 similar cases similar to this one?

Answer 1

Solution

Back-up Theory

If a discrete random variable, X, has probability function, p(x), x = x1, x2, …., xn, then

Mean (average) of X = E(X) = µ = Σ{x.p(x)} summed over all possible values of x…..…. (1)

E(X²) = Σ{x².p(x)} summed over all possible values of x…………………………………..(2)

Variance of X = Var(X) = σ² = E(X²) – {E(X)}²……………………………………………..(3)

If Y = X + a, E(Y) = a + E(X) and Var(Y) = Var(X) ………………………………………….. (3a)

If X₁, X₂, X₃, ……, X_n are iid (i.e., independently and identically distributed) with mean µ and variance σ², and S = (X₁ + X₂ + X₃+ …… + X_n) then

E(S) = E{∑[1,n](X_i)} = ∑[1,n](µ) = (nµ) …………………………………………………… (4)

Var(S) = Var{∑[1,n](X_i)} =Var{∑[1,n](X_i)} = {∑[1,n]Var(X_i)}

= {∑[1,n]σ² } = (nσ²) ………………………………………………………………………… (4a)

Now to work out the solution,

Part (a)

Let X represent the claim amount from the policy. Then,

X = 450000 with probability 0.005

    = 150000 with probability 0.056

    = 50000 with probability 0.153

Then, vide (1), expected claim amount = (450000 x 0.005) + (150000 x 0.056) + (50000 x 0.153)

= 18300.

Since the insurance company wants to make a profit of 5000 for a policy, the insurance premium for a policy should be: 18300 + 5000 = $23300 Answer

Part (b)

If Y = profit from a policy, then Y = premium – claim = 23300 – X

So, Var(Y) = Var(X) [vide (3a)]

Now, E(X²) = Σ{x².p(x)} [vide (2)]

= (450000² x 0.005) + (150000² x 0.056) + (50000² x 0.153)

= 2655000000

So, vide (3), Var(X) = 2655000000 - 5000²

= 263000000

If Xi = profit from a policy, then profit from 10 policies = (X₁ + X₂ + X₃+ …… + X₁₀), where Xi’s are iid. So, vide (4a),

Variance of profit from 10 policies = 10 x Var(y) = 10 x Var(X) = 2630000000 Answer

DONE