Business Mathematics: Insurance - Life Insurance

Life insurance is designed to protect a person’s family or loved ones in the event of their death. The insured enters into a contract with an insurance company that promises to provide a certain amount of money upon his or her death. In return, the insured makes periodic payments, known as premiums, to the insurance company. The insured is the person receiving the coverage and the beneficiary is the person who will receive the benefits when the insured dies.

There are many different types of life insurance available, including the following:

1.   A term life insurance policy pays the face amount of the policy to the beneficiary if the insured dies during the term of the policy. The term may be 1, 5, 10 years, etc. Insurance is in effect only during the term of the policy; once the term ends, there is no protection. In contrast to other types of policies which “lock in” a fixed premium for the life of the policy, term life insurance premiums increase as the age of the insured increases. Term insurance is initially cheaper than other types of insurance for the same amount of protection and, therefore, can give the largest immediate coverage for the dollar. For this reason, it is useful for consumers who need large amounts of coverage for known periods of time, such as home buyers, parents of young children, or people with high debt obligations.

2.   A straight or whole life policy combines the protection of life insurance with a built-in cash value. As long as premiums are paid, the policy accumulates value on a tax-deferred basis and can be sold back to the company in return for a lump-sum payment or for an annuity, say, for retirement. While you are accumulating value, you are still protected for the full face value in the event of your death. Most straight life policies allow you to borrow against its cash value without surrendering the policy. Interest rates are usually less than bank rates for borrowing money. The premium for a straight life policy is constant for the life of the policy, but is higher than a term policy premium because it accumulates savings.
3.   A limited payment life insurance policy is similar to a straight life policy except that payments are made over a fixed period rather than for the rest of your life. However, because of the limited number of payments made, each premium is somewhat higher than a straight life premium for a comparable policy. Twenty-year terms are quite popular.
4.   An endowment policy is like a term policy with a cash value after a fixed number of years. During the payment period you make payments annually and are fully protected up to the face value of the policy. After the payment period, your life insurance ends and you receive the face value of the policy, either as a lump sum or as an annuity. During the payment period, however, premiums are high. An endowment policy is the most expensive policy you can buy. Twenty-year terms are common for this type of policy.
5.   A universal life insurance policy is similar to a straight life policy except that its premiums are tied to variable factors such as the prime interest rate, stock market indices, etc. If interest rates fall, for example, the policyholder may be forced to pay higher premiums or accept reduced benefits. Although somewhat cheaper than straight life, it is somewhat riskier.

For problems in this guide, we will use the hypothetical premium chart shown in the table below for nonsmoking males. Premiums for women are generally lower, and premiums for smokers are higher. These numbers are not meant to be representative, as rates vary between insurance companies, but only to illustrate the difference in premiums for different types of insurance.

Table: Annual Premiums per $1,000 for Nonsmoking Males


Solved Problems

10.14  Jim Brown is 35 years old and wants to purchase $50,000 of straight life insurance. What will be his annual premium?
 

Solution

His annual premium will be $562.50.

10.15  Bill Murphy is 25 years old. He needs $150,000 worth of life insurance. How much will his premium increase when he reaches age 45 if he purchases (a) term life insurance or (b) straight life insurance?
 

Solution
(a)  Per $1,000 the difference in premium for term life insurance between ages 25 and 45 is $4.65 – $1.93 = $2.72. The difference in annual premium for a $150,000 policy is
150 × $2.72 = $408.00
(b)  His premium will not increase. The premiums for straight life depend only on the age at time of purchase and are fixed for the term of the policy.

10.16  Abe Sanders, age 30, has budgeted $750 a year to purchase life insurance. How much coverage can he get if he purchases (a) a 5-year term, (b) straight life, (c) 20 payment life, or (d) 20-year endowment? Assume that insurance can only be purchased in $1,000 increments.
 

Solution
Since premium = rate × number of units, it follows that premium/rate = number of units.
(a)  750/2.05 = 365.85: He can purchase 365 × $1,000 = $365,000 worth of insurance.
(b)  750/8.95 = 83.8: He can buy 83 × $1000 = $83,000 worth of insurance.
(c)  750/12.25 = 61.22: He can afford 61 × $1,000 = $61,000 worth of insurance.
(d)  750/21.10 = 35.55: He can purchase 35 × $1,000 = $35,000 worth of insurance.

10.17  Scott Murray, age 45, and his son Joseph, age 20, both purchase life insurance policies. Scott buys a $125,000 20-year limited payment policy, and Joseph buys a $200,000 20-year endowment policy. How much more does Scott pay annually than his son?
 

Solution
Scott pays $22.45 per $1,000 of coverage:

Joseph pays $13.95 per $1,000 of coverage:

The difference in annual premium is $2,806.25 – $2,790.00 = $16.25. Note that Joseph gets more coverage and a more valuable policy for less money because of his younger age.