September 1, 2011

Mathematics of Insurance


A recently engaged friend and I were discussing engagement rings, and she was shocked when I told her we did not insure our ring. “But doesn’t it have sentimental value to you?” She asked. Why yes, yes it does. But you can’t insure sentimental value, you can only insure monetary value. The insurance company won’t break out a search party should you lose your ring, or help in the police investigation if your ring gets stolen. They will write you a check to buy a new one.

Mathematically speaking, insurance doesn’t always make sense.

Insurance for possessions is similar extended warranties. Let’s consider an example from 2007. I purchased a Wii for $250. The store offered me a $10 extended warranty. Let’s say the store estimates the failure rate was 1 in a 100. Then for every 100 warranties sold, the store expects one customer’s Wii to break and to have to pay that customer the full price of a new wii, or $250. The store’s expected profit from selling 100 warranties is then $10×100 (the cost of the warranties) – $250×1 (the payout to one customer), or $750. This expectation is called the expected value. When the store sells tens of thousands of warranties, the statistical property called the Law of Large Numbers shows it is unlikely for a much larger percentage of wii’s to fail. Thus the store is not likely to lose money by paying out on the warranties. The store is a business, after all, not a charity and the goal of any business is revenue.

The expected value (EV) can be calculated for an individual warranty. This value represents the monetary worth of the warranty. The equation is: probability of failure x monetary value of failure + probability of no failure x monetary value of no failure
For the store:

EV(Store’s Value of the Warranty) = 1%x(-$250+$10) + 99%*($10) = -$2.40 + $9.90 = $7.50

In the first expression, the term -$250+$10 is the payout minus the revenue gained from the sale of the warranty. The monetary value of no failure, in the second expression, is simply the revenue from the sale of the warranty. Thus the store expects to earn $7.50 per warranty sold. Not coincidentally, it’s 1/100 of the expected value of selling 100 warranties.

We can also compute an individual consumer’s expected value of purchasing the warranty:

EV(Consumer’s Value of the Warranty) = 1%x($250-$10) + 99%*($10) = $2.40 + -$9.90 = -$7.50

The store’s expected gain is the customer’s expected loss. Each dollar the customer loses is a dollar the store gains.

We can also compare the expected value of the consumer not purchasing the warranty. In this case, the consumer does not pay the $10 fee, but is out of luck and must pay an additional $250 to replace the console, should it break. The consumer’s expected value of no warranty is:

EV(Consumer Value of No Warranty) = 1%x(-$250) + 99%*($0) = -$2.50

The expected value still negative, but the consumer’s expected value of not purchasing the warranty is less negative than the expected value of purchasing the warranty. Mathematically speaking, this means the consumer is expected to lose less money by not purchasing the warranty.

Engagement ring insurance works mostly the same way. A jewelry insurance company computes the probability that my ring will get lost, damaged or stolen. Many factors go into the calculation, including facts like the crime rate where I live, or whether or not I’ve ever reported a claim. The insurance company sets their rate accordingly, so their expected value is positive. In fact, they set their rate high enough that they can expect to be profitable and still pay their staffs wages, fixed costs for operating their business, and the women who file a claim on their rings. Again, they’re a business, not a charity. As before, the money the company is expected to make from me as a customer is equal to the money I am expected to lose.

This isn’t to say that insurance is never a good idea. In fact, it’s often a very good idea! Insurance and extended warranties are designed to provide protection for the worst case scenario, not the average case. Home, Auto, Health Insurance all have expensive worst case scenarios, beyond the financial capabilities of most people, which is why they are considered necessary. The worst case scenario when not purchasing the extended warranty is the Wii breaks, which would cost me $250 to replace. If you can afford the worst case (ie purchase a new Wii), you are usually better off not insuring. This is sometimes referred to as self insuring meaning you are setting money aside and relying on yourself to cover the financial burden should the worst case happen.

Should the worst case occur, and I lose my engagement ring, I will be okay financially speaking. Therefore, I expect to come out financially ahead by not insuring.

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