# After how many years will the maximum payout be less than the premium, losing money by continuing with the policy in the scenario below?A machine is insured by paying an annual premium equal to 10%...

After how many years will the maximum payout be less than the premium, losing money by continuing with the policy in the scenario below?

A machine is insured by paying an annual premium equal to 10% of its value when new. This premium does nt change. The machine is depreciating at a constant percentage rate. If the machine is damaged or written off the maximum paid by the insurance company will be the value of the machine at that time.

After (2) two years, the machine is worth half its orginal value.

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The annual premium paid is 10% of the new value = P/10 , where P is the new value of the machine.

The value of the machine due to depreciation is 1/2 of new value for every two years. Thus the machine value at any year becomes sqrt(1/2) of its pevious year value. The new value of the machine is P . So after n years the value of the machine becomes {sqrt(1/2)} ^n * P= P/(2^(n/2)).

So the maximum pay out by the insurance company at any year = machine value at that time (or year) = P/(2^(n/2)).

So it is required to determine when the maximum pay out by the insurance company becomes less than the annual premum .

Therefore to determine n such that P/2^(n/2) < P/10.

Divide by P and cross multiply:

10 < 2^n/2.

log10 < (n/2) log2.

2 /log2 > n.

Or n > 2/log2 .

Clearly for n = 6.64 , 1/2(6.64/2) = 0.100226.. > 1/10.

For n = 6.65 , 1/2^(6-65/2) = 0.9979 < 1/10.

Therefore between the year 6 and 7 (or between 6.64 years and 6.65 years, or in the 8th month after 6 years) the maximum insurance becomes less than the annual premium. So making the annual payment of premium is a clear loss from the 7th year onwards.