The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

the claims per policy to work out to be (just as the greater the number of coin flips, the

closer to 50% we expect the proportion of heads to be). Also, the company will no doubt

be selling many other policies as well, and so higher-than-predicted costs from one policy

type may be offset by lower-than-predicted costs from another. (Smaller insurers, who do

not sell enough policies to be confident of the law of large numbers’ protection, often make

financial arrangements to share risks with larger companies or with each other; these sorts

of arrangements are known as reinsurance.)

Example 13.1.1 An insurance company offers a policy that pays a fl at $5,000 if a

policyholder is a victim of identity theft. The company believes that on average one out

of every 287 policyholders will have a claim on their policies each year. Calculate the

pure premium for this policy for one year.

Pure premium 
 _______$5,000

287

$17.42

Note that the pure premium is not the actual premium that will be charged for the policy.

A large percentage, usually the largest percentage, of the premium paid for an insurance

policy may be the pure premium, but in addition the premium must cover administrative

expenses, sales commissions, and room for profit. While the pure premium is $17.42, the

actual premium for the policy will be higher.

Insurance Rates and Underwriting

The policy we considered in Example 13.1.1 was a very simple one: coverage was pro-

vided for a single type of event (identity theft) and for a specified dollar amount ($5,000).

Most insurance policies are far more complex. An automobile liability policy, for example,

might deal with claims ranging from a fatal accident resulting in a million dollar wrongful

death lawsuit down to comparatively minor repair costs resulting from a fender-bender.

To predict the average claims cost, the insurer must assess all the many different types

of events that could lead to a claim against the policy, considering both how often each

event will happen (the frequency of that type of claim) and the amount that could have to

be paid to settle such a claim (the claim’s severity). Obviously, this is far more complicated

than our initial example, where the single kind of claim covered had only one possible

benefit payment. Determining the pure premium for a policy covering many different types

of claims, each of which could result in a range of possible claim amounts, requires sig-

nificant analytical effort. Insurance companies employ financial professionals known as

actuaries to work this all out.

An insurer must also take into account the fact that, even though it is not possible to

say in advance which individual policyholders will have claims, it may be possible to say

in advance which policyholders are more likely than others to file a claim. In our disabil-

ity policy example, we quietly assumed that each policy was equally likely to produce a

claim. In reality, though, lion tamers are more likely than librarians to suffer a permanent

disability in the course of plying their trade. Reflecting this reality, an insurer will most

likely charge a higher premium to those who pose a greater risk of claims. In fact, in a

competitive market the insurer cannot afford not to do this; if Insurer A charges both the

same rate, while Insurer B charges different rates, lion tamers will buy their policies from

Insurer A and librarians will buy theirs from Insurer B. This will most likely lead to higher

than predicted claims at Insurer A, causing financial losses and possibly even jeopardizing

the insurer’s ability to survive and meet the benefit obligations it has promised.

Insurance companies group potential policyholders into different categories based on

their potential for filing claims. These groupings are called rating classes or underwriting

classes. Most readers are probably familiar with rating classes for automobile insurance.

The rates that you pay for auto insurance may be affected by your sex, your age, your mari-

tal status, how far you commute to work, the city where you live, your driving record, the

type of vehicle you drive, and other factors. In recent years, a number of insurers have even

taken to basing auto insurance rates on your credit rating (on the theory that people with

13.1 Property, Casualty, and Liability Insurance 527