CHAPTER VII POLICY OPTIONS. LOADING AND SURPLUS 54. Options.—In a standard life insurance policy there is a table of options giving cash or surrender value, paid-up insurance, and extended insurance at the end of each policy year for a period of years. If the insured desires to cease paying premiums at the end of a policy year, he may surrender his policy and receive there- for the amount in cash stated in the table or a paid-up policy for the amount stated in the table. Or he may retain his policy and remain fully insured for the time stated in the table. If, after the expiration of this time, he desires to retain his insurance, he must either reinsure at his then attained age or furnish unpaid premiums with interest. 55. Surrender Value.—The cash or surrender value is the value of the policy at the time of surrender. Thus, in Sec. 44, we found the value of a whole-life $1000 policy, issued to a person of age twenty, to be $72.79 at the end of the tenth year. This is the cash or surrender value of the policy at the end of the tenth policy year. It is usual to publish only even dollars in the option table. Thus the surrender value of the above policy at the end of the tenth year would appear in the table as $72. Loans made to a policyholder by the insurance company are based upon the surrender value of the policy, the policy being security against the loan. The surrender value of a limited-payment policy at the end of the premium-paying period must equal the net single premium for an insurance for the face value of the policy chargeable to a per- son whose age is equal to the age of the policyholder at that time. If preliminary term valuation is used, surrender values are lowered. 74 §56] POLICY OPTIONS. LOADING AND SURPLUS 75 56. Paid-up Insurance.—If, at the end of any policy year, the holder should desire to cease paying premiums, he can surrender his policy and receive therefor a paid-up policy whose face value, or amount of insurance, depends upon the surrender value of his policy at that time. Thus, for the policy considered in the pre- ceding section, the value at the end of the tenth year is $72.79. At this time the insured is thirty years of age and an insurance of $1 would cost A 30 = M^ = 0.337 ^'30 net single premium. Hence the insured can buy as many dollars of insurance as 0.337 is contained in $72.79, or approximately of $216. A policy for $216 can then be issued to him upon surrender his original policy and no further premiums will be charged to him. In general, if F represents the amount of insurance, then nVx'F represents the surrender value of the policy at the end of the nth policy year, and »7, F/A,+n represents the amount of paid-up insurance that can replace the policy at the end of the nth policy year. In practice, only even dollars of paid-up insurance is published in the option table. In case the policy is a term insurance policy for (years, one must divide nVi F by the net single premium for an insurance of $1 for the unexpired term; namely ( — n years, instead of the net single premium for the rest of life. If preliminary term methods of valuations are used, the face value of a paid-up policy is lessened thereby. 57. Extension of Insurance.—The length of time an insurance company can carry an insurance policy, for the full amount of insurance, beyond the date a premium falls due, without the pay- ment of further premiums, depends upon the surrender value of the policy at that date. For example, the value of the policy considered in the preceding sections is $72.79 at the end of the tenth policy year. The question is: For how long a time will this amountof money keepa person of age thirty insured for $1000? To answer this question, we must solve the equation 1000 X lAao = 72.79 76
MATHEMATICS OF LIFE INSURANCE
[§57
for t. But
I .A.
(Mso — Mso+i) DM
and hence M'sM-f = MM - 0.07279 X Dso = 8043.234. This value of Myo+i lies between Mw and Af<i, and by interpola- tion we find 30 + ( = 40.2 approximately. Thus the value $72.79 is sufficient to purchase a term policy of $1000 for 10.2 years, or about 10 years and 2 months. In general, we must solve the equation |iA,+, = „?, (1) for(. Making use of former results, we have the following table for a policy of $1000 issued to a person twenty years of age, annual premiums throughout life:
Preliminary term methods of valuation shorten the time an insurance can be carried without further premiums. Exercises 1. Make a table like the one in this section for a $1000, ten-payment life policy to a person of age thirty-five. 2. How would you determine the surrender values, paid-up insur- ance, and extension of time for an endowment policy? Illustrate by the policy in Exercise 1, Sec. 61.
§58] POLICY OPTIONS. LOADING AND SURPLUS 77 58. Loading.—There are various methods for loading a net premium to care for expense of management, or overhead expenses. A very common method is to increase the net premium by a certain percentage of itself and add thereto a fixed amount per $1000 of insurance. If Pz represents the loaded, or office, premium per dollar of insurance, we have the equation P/ = (1 + ^ + ^ (1) For example, if A = 20 per cent and c = $2, the office premium for an ordinary whole-life insurance of $1000 to a person of age twenty is 1000 X Px' = 1.20 X 13.48 + 2 = $18.18. Another method is to make the loading a percentage of the total cost of insurance covering the premium-paying period plus a fixed amount per $1000 of insurance. This method distributes the overhead expenses of an insurance company more equitably than the former method, but requires more computation for its determination. In general, one may say that the loading must be sufficient to meet overhead expenses; otherwise the company is in danger of becoming insolvent. On the other hand, overloaded premiums compare unfavorably with other companies. 59. Surplus.—There must always be a surplus of funds, since no insurance company can do business upon an exact theoretical basis. In all mutual companies this surplus belongs to the policy- holders, since they have furnished the funds. In stock companies, the surplus belongs to the stockholders. The following are sources of surplus: 1. If actual mortality is less than that indicated by the table in use, the actual cost of assurance is less than that computed from the table. The difference becomes a salvage and must be returned to the policyholders in all mutual companies. The mortality table in use should actually overstate mortality; that is, there should be, in general, fewer deaths each year than are expected from the table. The American Experience Table overstates mortality by something like 40 per cent. 78 MATHEMATICS OF LIFE INSURANCE (§59 2. If the company earns a greater rate of interest than that upon which its premiums are computed, the increased earnings must be returned to the policyholders in all mutual companies. An insurance company should always compute its premiums upon a lower rate of interest than that which it can actually earn upon its investments. 3. If the overhead expenses of a company are less than the loading of the premiums, the difference is a salvage returned to the policyholders. An insurance company should load its pre- miums for more than enough to cover the expected expenses of management. 4. There are always some forfeitures, either from inability to pay premiums as they fall due or from careless neglect. Some surplus arises from forfeitures, especially when full net reserves are not returned to the policyholder. For exclusively stock insurance companies, the problem of disposing of the surplus is very simple, since it is divided among the stockholders in the form of dividends upon their stock. For mutual insurance companies, the problem of distributing the surplus equitably is one of considerable complexity. To illustrate, suppose the insurance company has issued exactly 1000 ordinary whole-life $1000 policies to persons of age twenty. The loaded premium for each of these policies is $18.18 and hence the fund contributed by the policyholders is $18,180. We will now suppose the effective rate of interest earned by the company on its investments is 4 per cent, instead of 3% per cent upon which it has based its premiums. The amount of the fund at the end of the year is then $18,907.20. Suppose the mortality experienced in this group of 1000 persons is three persons for the first year, instead of the number expected from the American Experience Table (approximately eight). The death claims due at the end of the first year then amount to $3000, leaving a fund of $15,907.20. Suppose the overhead expense apportioned to each policy is $3.76 (=80 per cent of the load $4.70). Overhead expense then amounts to $3760, leaving a fund of $12,149.20. This fund divided among the 997 living policyholders gives $12.19 as each person's share. §59] POLICY OPTIONS. LOADING AND SURPLUS 79 If we assume net reserves, as computed by former methods, are held by the company, each policy must have a reserve of $6.19 at the end of the first year, leaving a surplus, or dividend of $6. At the beginning of the second year, each of the 997 living policyholders contributes his premium $18.18 and the reserve $6.19, or $24.37, making a total fund of $24,296.89, which at 4 per cent interest amounts to $25,268.77 at the end of the year. With the same death claims (= $3000) and the same apportion- ment of overhead expense (= $3.76 X 997 = $3748.72), we have a fund of $18,520.05 at the end of the year to be divided among the 994 living policyholders, or $18.63 for each. The net reserve is $12.60, leaving a dividend of $6.03. The process outlined for the first two years may be extended as far as desired, or may be formulated as follows: ^ ^ (P,' +n-iy.)^+n-l(l + i') - d'^n-l - El'^n-l _ ^ ^ 1'is+n where A indicates the dividend at the end of the nth year on a policy of 1, i' the effective interest rate, Px' office premium, 1',+n-i the number of living policyholders at the beginning of the nth year, d',+n-i the number of deaths during the year, E each policy's share of overhead expense and ;',+„ the number of living policy- holders at the end of the year. This method for determining surplus is known as the contribu- tion plan and is due to David Parks Fackler. Other methods have been devised which, if aiming at equitable distribution of surplus, are modifications of the contribution plan. Whatever method is adopted, it must be said that the determina- tion of the amount of surplus is a matter for the actuarial depart- ment of the insurance company, subject, of course, to inspection by properly constituted legal authorities to insure protection of policyholders. A few years ago it was the general practice of insurance com- panies to withhold surplus for a period of years and then return it to the insured either as cash or increased insurance if the insured were then living. But, in case of the death of the insured during this period, the surplus was not returned to the beneficiary. In case of limited-payment policies, the first distribution of surplus was ordinarily at the end of the premium-paying period. 80 MATHEMATICS OF LIFE INSURANCE (§5 Today, surplus is returned when it falls due at the end of eac year as outlined above. The dividend may be paid in cash o used to reduce the annual premium or applied to purchase addi tional insurance at the option of the insured. The symbol V in Eq. (1) may be either the net reserve or tb reserve determined by preliminary term methods of valuation according to the usage of the insurance company. Exercise Discuss surplus as in this section, using for illustrations the sam< policy and full preliminary term reserves. Would surplus be increasec or diminished by this change in valuation?