The basic statistics of risk-pooling

Insurance exploits the statistical properties of pooled independent risks.  Let's assume the typical driver has a 1% chance of having a car accident in any given year, and each driver's risk of an accident is completely independent of any other driver's risk.

Excel's RAND() function yields a random number uniformly distributed between 0 and 1.  The formula =INT(RAND()+0.01) represents a random driver with a 99% likelihood of yielding a zero (no accident), and a 1% probability of yielding a one (accident).  To create a Monte Carlo simulation of an insurance pool of 1,000 drivers, I copied the formula =INT(RAND()+0.01) down a column of 1,000 cells to represent the drivers.  The sum of this column represents the number of accident claims from the pool.  The number of claims changes each time the worksheet is recalculated, and varies randomly around an expected value of 10.

So I ran 1,000 recalculations of this worksheet, using an Excel macro to record the column sums from these cells, representing the number of accidents that occurred in each trial of the insurance pool.  I then used the COUNTIF function to obtain the frequencies of these accident counts.

I created an Monte Carlo experiment that explains these procedures in detail.  A typical result is shown here:

Obviously we can't predict which individual drivers will have accidents, but this experiment demonstrates that the number of accidents in the whole pool of drivers is normally-distributed.  So this simulation predicts that 1,000 drivers will have mean of 10 accidents per year with a standard deviation of about 3 accidents, i.e., +/-30% of the mean.  We can expect between 7 and 13 accidents in this pool of drivers about 68% of the time, and between 4 and 16 accidents 95% of the time.  The probability that this pool will have more than 16 accidents in a year is only about 2.5%.

Suppose the average accident loss is $50,000.  To insure this pool of drivers with 97.5% certainty that the premiums they pay will cover your loss payouts, you would set the annual premium to cover 16 accidents = $800,000 in payouts, or $800 per driver.  With a typical loss payout of $500,000, you would net an average $300,000 per year.  You would build up a cash reserve for the rare year with more than 16 accidents.

A second experiment demonstrates how a larger insurance pool yields tighter confidence intervals around the mean.  Here is the distribution of 10,000 trials from a Monte Carlo simulation of a larger insurance pool with 10,000 drivers:

These examples illustrate the central limit theorem.  The larger insurance pool yields a tighter normal distribution with a predicted mean of 100 accidents per year and a standard deviation of 10 accidents, or only +/-10% of the mean.  So we can expect between 90 and 110 accidents in this pool of drivers about 68% of the time, and between 80 and 120 accidents 95% of the time.  The probability that this pool will have more than 120 accidents in a year is only about 2.5%.

To insure this larger pool of drivers with 97.5% certainty that the premiums they pay will cover your loss payouts, you would set the premium to cover 120 accidents, or $6,000,000 in payouts.  The annual premium per driver in this larger pool would be only $600.  So the central limit theorem implies that insurance pools have natural scale economies.

Failures of insurance markets

Insurance markets may be vulnerable to various failures.  First, insurance creates moral hazard: when people can transfer their risk costs to an insurer, they become less risk-averse.  For example, having auto insurance reduces your own financial risks from a car accident, so you don't have to drive quite as carefully as you would if an accident would immediately bankrupt you.  If your homeowner's insurance covers your liability for somebody who falls down your front steps, your incentive to clear the ice off your steps is weaker.  Obviously your insurance company doesn't want you to be careless or reckless, so your policy specifies some deductible amount that you are still responsible for, such as the first $500 of damages from a car accident, before the insurer pays. One way to lower the cost of your car insurance premiums is to increase your deductible.

Insurers have incomplete information about the risks they are insuring.  You might not tell your life insurance company that you are a smoker because it will increase your premiums.  You wouldn't want your health insurer to find out that your cancer might have been a pre-existing condition.  The costs of this incomplete information get factored into everyone's insurance premiums.

Customers don't necessarily understand their own risks very well.  For example, a large majority of drivers think their driving skills are "above average," and many drivers would opt out of car insurance if they could.  That's one reason why states require car insurance.

Incomplete information contributes to adverse selection which may cause an insurance pool to fail.  Suppose an insurer cannot differentiate high-risk and low-risk policy-holders very efficiently, so it charges everybody the same premium.  The lowest-risk customers are overcharged, so they tend to exit the insurance pool, which increases average risk-level in the remaining pool.  So the insurer has to increase the premiums, which drives more low-risk customers out of the pool, etc.  Eventually the insurer is left with just the worst risks, and if those customers aren't willing to pay exorbitant premiums, the insurance pool fails entirely.

Auto insurers assign drivers to different risk pools based upon their accident and traffic violation histories, so that higher-risk drivers do pay higher premiums.  With mandatory car insurance, adverse selection is not a significant problem.  (Unfortunately some people do drive illegally without insurance, so your auto policy generally covers you if you have an accident caused by an uninsured motorist.)

There is a serious adverse selection problem in US health insurance, however.  About 50 million Americans don't have insurance from employers and don't qualify for Medicare (over 65) or Medicaid (low-income children, disabled people, etc.).  A disproportionate share of this uninsured population are young adults with low health risks, for whom health insurance is simply overpriced.  A lot of younger Americans are taking an economically rational gamble on staying healthy.  Why should a physically fit, clean-living 26-year-old with a low income subsidize the health care of an obese 60-year-old smoker with a high income? Charging higher car insurance premiums for drivers with bad driving records seems perfectly fair, but charging higher health insurance premiums for sick people doesn't.

Risk-pooling in credit markets

Banks manage loan default risks exactly the way insurance companies cover accidents risks.  Mortgage banks pool their home mortgages into portfolios, and credit-card banks pool the credit card debt of their customers into portfolios.  The larger the portfolio, the tighter the confidence interval on the expected default rate (assuming each borrower's risk of default remains independent of other defaults).  The expected aggregate default rate is factored into interest rates that customers are charged. 

The stream of interest payments earned by a large portfolio of mortgage debt or credit card debt is very predictable.  So having loaned out their money for customers' mortgages, credit-card purchases or whatever, banks typically raise more cash by re-selling these loan portfolios to investors as bonds.

Now unlike a home mortgage or a car loan, your credit card debt is unsecured, meaning there is no specified asset of yours that the bank can seize if you don't pay.  So credit cards carry higher interest charges to cover higher default risks.  Just as your car insurance is based on your driving history, your credit card's interest rate and credit limit are based on your credit history, as reported by credit-reporting bureaus (Equifax, Experian and Transunion).  Being late on your credit card payments is like getting points on your license for speeding: it can be expensive!