Home | Archives | Blog | Bonds | Credit & Debt | Forex | Futures | Insurance | Mutual Funds | Options | Real Estate | Stocks | Taxes | Other Investment Topics | New Money Articles
Insurance companies must determine what premium to charge that will cover losses, and be competitive with other insurance companies. To do this, insurance companies hire actuaries, who use statistics and the law of large numbers to determine expected losses and the probability of how much actual losses can deviate from expected losses.
Sometimes, the probability of an event can be determined a priori. Such is the case with the flip of a fair coin, or the roll of a fair die, because the possibilities are both limited and known. However, the probability of most insurable events cannot be known a priori, because there are too many factors that can influence the outcome, and the outcomes are highly variable. Thus, actuaries apply statistical analysis to past events in an attempt to determine the frequency of losses and the extent of those losses within a population, and how much they vary from year to year.
A probability distribution summarizes this data by plotting possible events against their probabilities. If there are only a limited number of possibilities, then the distribution is said to be discrete; otherwise, it is continuous. Plotting the roll of a die creates a discrete probability distribution, because there are only 6 possibilities. However, most insurable events have a continuous distribution because the outcomes have a continuous range of possibilities.
A probability distribution can be characterized by its central tendency and its dispersion. Central tendency is the expected value, or mean (common symbol: μ) of the distribution, and is equal to the sum of each possible event (X) times the probability (P) of that event. In the case for insurable losses, the mean is equal to the sum of the amount of each possible loss times the probability of that loss.
Equation for Mean: μ = Σ Xi Pi
| Amount of Loss (Xi) | Probability of Loss (Pi) | XiPi | ||
|---|---|---|---|---|
| $0 | x | .10 | = | $0 |
| $500 | x | .20 | = | $100 |
| $1,000 | x | .05 | = | $50 |
| μ | = | Σ Xi Pi | = | $150 |
| Amount of Loss (Xi) | Probability of Loss (Pi) | XiPi | ||
|---|---|---|---|---|
| $100 | x | .15 | = | $15 |
| $500 | x | .20 | = | $100 |
| $1,000 | x | .035 | = | $35 |
| μ | = | Σ Xi Pi | = | $150 |
However, the mean does not measure dispersion, which is a measure of how widely the individual events vary. A measure of dispersion is important because, in determining risk for insurance purposes, the greater the dispersion, the greater the variation in losses, and, thus, the greater the objective risk. Wide variations in dispersion can average to the same mean, as can be seen in the 2 tables above. Both have the same mean of $150, but the losses in the 1st table has a greater dispersion that varies from $0 to $100, and in the 2nd table, the losses vary from $15 to $100. Because the range is more limited in the 2nd example, it poses less risk—objective risk—and is, therefore, more predictable.
The statistical measure of dispersion is the variance (common symbol: σ2), which is equal to the square of the difference between the possible values and the mean.
Equation for Variance: σ2 = Σ Pi(Xi - μ)2
To make the units of central tendency and variance the same, the square root of the variance is used to represent dispersion, and is called the standard deviation (common symbol: σ).
For the 1st distribution, the variance and standard deviation are
σ2 = .1(0-150)2 + .2(500 - 150)2 + .05(1,000 - 150)2 = 2,250 + 24,500 + 36,125 = 62,875
σ = √62,875 = 250.75 (rounded).
For the 2nd distribution:
σ2 = .15(100-150)2 + .2(500 - 150)2 + .035(1,000 - 150)2 = 375 + 24,500 + 25,287.5 = 50,162.5
σ = √50,162.5 = 223.97 (rounded).
The greater the standard deviation for a loss event, such as fires, the greater the uncertainty of the event within a given time frame, and, therefore, the greater the potential for losses. However, the standard deviation can only be calculated from an observed population or a representative sample of the population. The law of large numbers is a useful tool because the standard deviation declines as the size of the population or sample increases, for the same reason that the number of heads in 1 million flips of a coin will probably be closer to the mean than in 10 flips of a coin.
The central limit theorem states that as the sample size (n) grows, the distribution becomes more like the normal distribution of the entire population, with the mean of the sample more nearly equal to the mean of the population, and the standard error (σs), which is the standard deviation of the sample, approaches the standard deviation of the population (σp).
Equation for Standard Error: σs = σp/√n
Thus, the difference between the standard error and the standard deviation of the population diminishes as the sample size, n, increases.
The normal distribution is represented by the bell-shaped curve, with 68% of the distribution lying within 1 standard deviation, 95% lying within 2 standard deviations, and 99.7% of the distribution lying within 3 standard deviations.
| As the size of the sample increases, dispersion decreases. The normal distribution curve for the sample size of 100,000 is narrower than the curve for 10,000, and they both have the same mean. Thus, losses are more predictable for the larger sample size. |
|
When an insurance company increases the size of its customer base, it increases its underwriting risk because the sample size is greater, and, therefore, there is a greater chance of loss. But the company also collects more premiums to finance those losses. In fact, premiums grow faster than the underwriting risk, because the underwriting risk is equal to the square root of n times the standard deviation for the population, and, thus, increases by the square root of the sample size, n, but the premiums grow by n.
Underwriting Risk = n x σs = n x σp/√n = √n x σp
Premiums Collected = n x Amount of Premium
Insurance companies expect losses—that’s their business, but, by increasing the customer base, actual losses more closely equal expected losses, thus, reducing objective risk, and allows insurance companies to charge a premium that covers losses and operating expenses, and that provides a profit, but no more.
Search this financial encyclopedia by using the search engine below.
 | Custom Search |