Business Mathematics: Insurance - Insurance Premiums

Suppose that you own a small business valued, say, at $100,000. If a fire or other disaster should occur, your business will be ruined and you will suffer financially. If someone would step in and write you a check for $100,000, you could rebuild your business and start over.
Of course, nobody will do this for you out of the goodness of their heart. But if there were some financial incentive, such a scenario might be possible.
This is the basic idea behind the concept of insurance.

Suppose, hypothetically, that you pay a $1,000 premium for the promise that you will receive $100,000 should a disaster occur. Since you pay relatively little for the assurance that you will be protected, you gladly agree to pay the money. Why would an insurance company agree to such an arrangement? The answer to this question is based on a little bit of common business sense and mathematical reasoning.

The likelihood of a disaster occurring to your business is small—but not impossible.

Insurance companies estimate this likelihood and adjust your premium, the amount of money you pay for protection, accordingly.

For example, suppose that it is estimated that 1 in every 200 businesses will have a $100,000 loss in a given year. If 200 insurance policies are sold for premiums of $1,000 each, the insurance company will collect 200 × $1,000 = $200,000. There will be enough money to pay the affected business $100,000, and the insurance company will have a hefty profit.

Of course, it is possible that two or more businesses will have disasters and the insurance company will have to pay each $100,000. However, if a large number of policies are sold, actuaries, using the mathematical theory of probability, together with data from past experience, can predict, with remarkable accuracy, the percentage that will result in catastrophe. The premium can then be adjusted accordingly.

There are many different types of insurance available to meet different requirements. A few examples are fire insurance, liability insurance, health insurance, accident insurance, theft insurance, and life insurance.

Some policies, such as homeowner’s and automobile policies, offer combinations of coverage tailored to meet specific needs.

Insurance Premiums
The amount of money you pay for insurance is called the premium. The amount of protection you purchase is called the coverage or face value of the policy. Premiums are usually stated as rates, that is, the amount of money you must pay per $100 or $1,000 unit of coverage.

In general
Premium = rate × number of units of coverage


Example:
The annual premium for a fire insurance policy is $0.58 per $100 of coverage.

To compute the premium for a $50,000 policy, we first compute the number of $100 units in $50,000:

Since the insurance rate is $0.58 per $100 unit, the premium is
0. 58 × 500 = $290

An insurance premium is generally paid in advance. If you purchase a fire insurance policy but fail to adhere to fire codes, or if you purchase an automobile policy but failed to list your traffic violations, the insurance company may cancel your policy before the end of its term. In this case you will be sent a refund proportional to the number of unused days in your policy.

The premium you used for protection before cancellation took effect is called the short premium.

Example
Suppose that the policy in Example 1 was effective March 1 and was canceled by the insurance company on May 15. The number of days the policy was in force is 76.

Therefore

If the insured cancels the policy before the end of its term, the calculation is just a bit different. As a penalty, you may be charged a higher rate per day than the annual rate. This rate varies with the number of days the policy is in force.


Example
Suppose that the short rate for policies canceled after 30 days in example 2 is 12% of the annual rate. To determine the short premium and subsequent refund in example 1, we multiply the annual premium by 12%:

Note that the short premium is somewhat larger than l/12th of the annual premium.

Solved Problems on Insurance Premiums:

10.1   The annual premium for a life insurance policy for a 25-year-old male who does not smoke is $23 per $1,000 of coverage. How much should a $50,000 policy cost?
 

Solution
Each unit of coverage is $1,000:


10.2   If liability insurance costs $41.20 per $10,000 of coverage, how much would the premium be for a policy whose face value is $250,000?
 

Solution


10.3   A $250,000 fire insurance policy written on June 15, whose annual premium is $672, was canceled by the insurance company at midnight June 30 because the sprinkler system was found to be inoperative. How much refund is due?
 

Solution
The policy was in force for 15 days. Therefore


10.4   Continental Insurance Company issued a $300,000 insurance policy to Ace Paper Company for protection against fire and theft. The annual premium was $11.50 per $1,000 of coverage. Their short rate for one month is 10% of the annual rate. How much refund is due to Ace

Paper if
(a)  Continental cancels the policy after one month (30 days)?
(b)  Ace cancels the policy after one month?
 

Solution
The number of $1,000 units of coverage = $300,000/$1,000 = 300 units. The premium paid = rate × number of units of coverage = $11.50 × 300 = $3,450.

(a) If Continental cancels the policy:

(b) If Ace cancels:


 

10.5  ABCO Tool and Die Company took out a $500,000 liability policy which had an annual premium of $3.40 per $1,000 of insurance. After 6 months they canceled their policy. If the insurance company’s short rate for 6 months is 55% of their annual rate, how much refund should ABCO receive?
 

Solution