Tuesday, August 23, 2011

The Deal With Insurance


     Cat did not understand. "They pay him gold and silver, but he only gives them writing. Are they stupid?"
     "A few, mayhaps. Most are simply cautious. Some think to cozen him. He is not a man easily cozened, however."
     "But what is he selling them?"
     "He is writing each a binder. If their ships are lost in a storm or taken by pirates, he promises to pay them the value of the vessel and all its contents."
     "Is it some kind of wager?"
     "Of a sort. A wager every captain hopes to lose."

George R.R. Martin
A Dance with Dragons

What is he selling them, Cat asks? Well, insurance, of course.

Albert Einstein was quoted as saying that insurance is one of the most important innovations of modern history. Insurance makes it possible for us to live in this chaotic world, amid all the natural and man-made volatility and unpredictability. Insurance protects people from natural disasters, families from a burning house or the death of the breadwinner, individuals from car accidents or unforeseen hospital expenses, and even ship captains from storms or pirate attacks.

In the passage above, Martin correctly describes insurance as a wager or bet for both the individual who wants to be insured--the captain--and the person or organization (e.g., an insurance company) that sells insurance--the insurer. To understand how, let's take a look at the situation below.

Let's say a captain in Martin's story, let's call him Daario, owns a ship with precious cargo and is set to sail across the Narrow Sea and back for six months. The Narrow Sea is known for its violent waters and bloodthirsty pirates; while Daario does not consider himself to be a coward, he nonetheless recognizes that the threat of losing his ship and its precious cargo is real. A nameless man--the insurer--offers to pay Daario the value of his ship and its cargo--around one hundred thousand gold dragons--if it gets lost in a storm or taken by pirates within a six month window, at the low, low price of fifty gold dragons--the insurance premium. Say, Daario estimates that the probability of his ship getting lost in a storm or taken by pirates is one in one thousand, should he buy the offered insurance or not?

In other words, Daario faces the following choices:

To not buy insurance, which has the following consequences:
  • a 1 in 1,000 (0.1%) chance that he will lose 100,000 gold dragons
  • a 999 in 1,000 (99.9%) chance that nothing will happen
To buy insurance, in which case Daario will lose 50 gold dragons for certain for the insurance premium

If you were in Daario's shoes, what would you do? For many of us, 50 gold dragons (or Philippine pesos or US dollars, if you prefer) would be a small price to pay to prevent a possible monumental loss--be it your ship, your house, your husband's earning power, and so on--no matter how remote that possibility is. We would not hesitate to pay a certain amount just to avoid risk.

How about the insurer? Things don't look quite so good from his end, do they? Surely, in the 1 in 1,000 chance that Daario loses his ship, the insurer will be wiped out since the 50 gold dragons he receives from the captain is far from enough to cover his obligation. This is where the true beauty of insurance comes to light, when we look at things from the other side of the fence. Most definitely, Daario won't be the only captain who would be willing to buy insurance from the nameless man. When individuals or organizations with independent or low-correlated risk (like say, how a person dying from cancer does not increase the likelihood that another person will die from the same disease; or how a fire in a slum area in Quezon City is unlikely to result in another fire in Makati) buy insurance, risk is pooled and the total risk for the whole group is reduced. Therefore, as more ship captains buy insurance, the insurance provider is able to collect enough premiums to cover possible losses at any given time. This is how insurance companies make money. 

That's the end of it. In many instances, insurance is the simplest and cheapest way to manage risk, and we all should have it in one form or another. However, if you're interested in the math of how insurance is able to pool and reduce risk, read on. 


(WARNING: Elementary statistics involved!) Let's say n captains buy insurance from the nameless insurer and that the probability that a captain will lose his ship is p. If the possibility that a captain will lose his ship is independent of other similar occurrences, then the probability that x out of n ships will be lost, P(x), is said to follow a binomial distribution; the mean percentage or proportion of ships that will be lost is always equal to p, but the standard deviation of the proportion, a measure of how much the proportion can move away from the mean, is the square root of p*(1 - p)/n. With these formulas, we see that as more captains buy insurance from the insurer--as n increases--the mean proportion of lost ships will be constant but the standard deviation of the proportion--a practical proxy for risk--decreases.

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