Consider a bank who has made a loan to a borrower, say Mr. Smith. Each month, Mr

Question

Consider a bank who has made a loan to a borrower, say Mr. Smith. Each month, Mr. Smith is supposed to pay $1,000 to the bank. Each month, however, Mr. Smith risks loosing his job and not be able to pay the bank. The probability of such event is 10%. The bank seeks to buy an insurance against this risk. Derive the price of an actuarially fair insurance contract that the bank can purchase, that is:

(a) First explain what it means for the contract to be actuarially fair.

(b) Second derive (and explain) the premium the bank needs to pay, say m in order to receive a compensation b if Mr Smith does not pay in a given month.

Explanation / Answer

a)

Insurance being actuarially fair, meaning that the premium is equal to expected claims: Premium = p*A where p is the expected probability of a claim, and A is the amount of the claim in event of an accident.

For insurance to be actuarially fair, the insurance company should have zero expected profits.When the probability of Mr.smith defaulting is 10% then the expected amount of loss for the bank is 10/100*$1000=$100.So if the bank looks for an actuarially fair insurance then the premium the bank has to pay the insurance company is $100.So the expected profit for insurance compnay is zero because it gets $100 as premium and the expected loses for bank is also $100 so effectively a zero expected profit.

b)

With probability p[that is when smith defaults on monthly payment] the insurance company must pay $x, while receiving lets say $rx in premiums. With probability (1-p)[that is when smith pays his monthly payment], they pay nothing but receive $rx in premiums. So their expected profit is:

p(rx - x) + (1-p)rx

But in actuarially fair insurance scheme expected profits =0
If this equals zero, we have: px(r-1) + (1-p)rx = 0

Dividing throughout by x, we get: pr - p + r - pr = 0

i.e. p = r.

So the premium $rx=$px=$0.1*x(as p is 10%)

premium(m) for a compensation of b in case of default is = p*b=0.1*b

equation is m=0.1b

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