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Exam P

Deductible calculator: the expected insurance payment

Set the loss, the deductible, a limit and coinsurance. You get the expected payment per loss and per payment, with the working.

The loss is

Expected payment per loss

778.80

77.9% of losses lead to a payment.

Per payment
1,000.00
Mean loss
1,000.00
Kept by the policyholder
221.20

The working

  1. Payment for a loss XX: (X−250)\left(X - 250\right) when  , and 0 otherwise.
  2. For an exponential loss with mean θ=1,000\theta = 1{,}000: E[min⁡(X,m)]=θ(1−e−m/θ)E[\min(X, m)] = \theta\left(1 - e^{-m/\theta}\right).
  3. and  .
  4. Expected payment per loss =(1,000.00−221.20)=778.80= \left(1{,}000.00 - 221.20\right) = 778.80.
  5. A payment is made when  , with probability  .
  6. Expected payment per payment  .
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The terms

Deductible dThe first part of each loss, which the policyholder pays
Maximum covered loss uThe part of a loss above u is not covered
CoinsuranceThe share of the covered loss the insurer pays
Per lossAveraged over every loss, including those that pay nothing
Per paymentAveraged only over losses that lead to a payment

The exponential shortcut

An exponential loss is memoryless: given that it exceeds the deductible, the excess is exponential with the same mean. So the expected payment per payment is simply the mean loss, whatever the deductible.

Per loss, multiply by the chance the loss exceeds the deductible: θe^(−d/θ).

Try it on an exam question

Question 1

The loss from a theft is exponential with mean $1,000. The policy has a deductible of $250.

Calculate the expected payment by the insurer per loss.

Question 2

A loss from a burglary is uniformly distributed on [0,1,000][0, 1{,}000]. The policy pays the loss above a deductible of $200, up to a maximum payment of $500.

Calculate the expected payment.

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Questions people ask

How do you find the expected payment with a deductible?

The insurer pays the loss minus the deductible when the loss is larger, and nothing otherwise. Take the expected value of that payment: for an exponential loss with mean θ and deductible d it is θe^(−d/θ).

What is the difference between per loss and per payment?

Per loss averages over every loss, including those below the deductible that pay nothing. Per payment averages only over losses that lead to a payment, so it is larger: divide the per-loss figure by the probability that the loss exceeds the deductible.

Why can you not just subtract the deductible from the mean loss?

Because a loss below the deductible pays zero, not a negative amount. Subtracting the deductible from the mean treats those small losses as negative payments and gives an answer that is too low.

How does coinsurance work?

With coinsurance the insurer pays a fixed share, such as 80%, of the covered loss after the deductible. Multiply the expected covered amount by that share.

What does inflation do to a policy with a deductible?

If losses rise and the deductible stays the same, more losses exceed it and each exceeds it by more. The expected payment rises by more than the rate of inflation.