Actuarial Math By: MacDonald R. Phillips March 20, 1996 E-mail: mphillia@osf1.gmu.edu phillipsm.ggd@gao.gov Life annuities and insurance differ from ordinary annuities in that they add a new factor, that of probability or contingency. The collection of a life annuity is contingent on a person being alive; the collection of life insurance is contingent on a person dying. These contingencies are dealt with in actuarial mathematics by life tables. Life tables usually contain tabulations by age of the basic functions q, l, d, and other derived functions. "q" is the proportion of people alive at the beginning of an age interval who die during the interval, i.e., "q" is the death rate for an age interval. "l" is the number of persons living at the beginning of an age interval. "d" is the number dying during an age interval. To construct life tables of the basic and derived functions, all that is needed is "q" and a stipulated interest rate. Two "q" tables are included with the programs: (1) CSO80 - Commissioners 1980 Standard Ordinary Mortality Table for Males and (2) IAM83 - 1983 Individual Annuity Table for Males. Their source is the 1992 Life Insurance Fact Book published by the American Council of Life Insurance. It is IMPORTANT to note that the "q" tables begin with age 0 (zero) if you want to use other mortality tables. All the programs are based on this fact. The IAM83 table begins at age 5, so for ages 0 to 4 I added zeros. The "q" tables, of course, are in a list format since the HP48G(X) processes lists so wonderfully. Installing Programs Simply copy the file INSUR from your PC to HP48G(X) in binary mode using any appropriate communications program. A directory name INSUR will be created with programs, data, and subdirectories. Basic TABLE Program In the INSUR directory is the basic TABLE program. This program takes 3 arguments and creates all the basic functions you will need to deal with simple problems of life probabilities, life annuities, and life insurance. The 3 arguments are: z: q y: radix x: percent interest rate "q" can be either the CSO80 or IAM83 tables (or other "q" table you may want to use); just enter the list onto the stack. The radix is the number of people living at age zero (or beginning age of the table). I normally use 10 million. The reason for the large number is that the number living is rounded to the nearest unit. If it gets below .5 before the end of the table, the rest of the numbers will be zero. The interest rate is entered as a percent, e.g., 6 for 6 percent. In the examples below, I used CSO80, a radix of 10,000,000, and an interest rate of 6 percent. After the 3 arguments are entered, simple press TABLE. The program calculates "l," "d," Dx, Nx, Cx, and Mx. The program takes about a minute to a minute and a half depending on the size of "q." "l" is the number living at the beginning of each age interval, in list form. "d" is the number dying during each age interval, in list form. Dx, Nx, Cx, and Mx are known as commutation functions; they greatly simplify calculations dealing with life annuities and insurance. If you want to know more about them, consult any introductory book on actuarial mathematics. Life Probabilities I have included five life probability programs in the PROBABIL directory. 1. X->L - The probability that a person age X will live one year. Enter X and press the key. 2. X->Q - The probability that a person age X will die during the year. Enter X and press the key. 3. XN->L - The probability that a person age X will live for N years. Enter X and then N and press the key. 4. XN->Q - The probability that a person age X will die in the next N years. Enter X and then N and press the key. 5. X->LE - The average remaining life of a person age X. Enter X and press the key. Examples using CSO80. 1. If X = 50, then L = .993290000756. 2. If X = 50, then Q = 6.70999924386E-3. 3. If X = 50 and N = 5, then L = .960400008119. 4. If X = 50 and N = 5, then Q = .03959991881. 5. If X = 50, then LE = 25.3560889959 years. Life Insurance Programs dealing with life insurance are in the LIFE directory. Press the CST key to bring up the custom menu. There are eight programs dealing with whole life insurance, term insurance, endowment insurance, and insurance reserves. Each program directly accesses the SOLVR menu. The variables associated with the programs are: 1. SP - single premium 2. AP - annual premium 3. INS - amount of insurance 4. TERM - years for term insurance 5. AGE - age at which insurance is purchased 6. N - number of premiums 7. RES - insurance reserve Only SP, AP, INS, and RES can be calculated, given the other variables. TERM, AGE, and N must be entered, if appropriate, and are used to get values from the commutation tables; they cannot be calculated. The programs are: 1. SPWL - Single Premium Whole Life Insurance 2. APWL - Annual Premium Whole Life Insurance 3. NPWL - N Premiums Whole Life Insurance 4. SPTL - Single Premium Term Life Insurance 5. NPTL - N Premiums Term Life Insurance 6. ENDW - Pure Endowment 7. ENIN - Endowment Insurance 8. PLRE - Policy Reserve for Annual and N Premium Life Insurance To use a program simply press the appropriate soft key. The equation is entered in EQ and the SOLVR is automatically started. Enter the known value and solve for the remaining value. NOTE: The commutation functions (DX, NX, CX, MX) appear in the SOLVR menu. DO NOT ENTER ANYTHING INTO THEM. Examples: Press APWL for annual premium whole life insurance. The equation is entered in EQ and the SOLVR started. If AGE = 25 and INS = 50,000.00, then AP = 278.18. (For this equation, either AP or INS may be calculated by pressing the left-shift key first and then the appropriate soft key.) Press CST again to get back to the custom menu. Press PLRE to calculate a policy reserve. If 25 annual premiums have been paid, what is the policy reserve? Enter 25 for N, the number of premiums already paid. The other values are already entered. Press left-shift RES to calculate the reserve: 9,919.11. Press CST. Press NPTL to set up the SOLVR for N payment term insurance. A person age 35 wants to buy a 10-year term policy for 100,000 and make only 5 annual payments for the policy. How much will the payments be? INS = 100,000 AGE = 35 TERM = 10 N = 5 AP = 473.51 a year for 5 years (calculated) If in the above example the person could afford a premium of 500 a year for 5 years, how much insurance could be purchased? Change AP to 500. INS = 105,593.60 (calculated) An pure endowment is, strictly speaking, a single payment annuity. A person agrees to pay a series of N premiums for the promise of a lump sum payment if the person is alive at the end of TERM years. If the person dies during the TERM, s/he receives nothing. Because of this, a pure endowment is rarely used. An endowment is most often combined with insurance to produce an endowment insurance policy. Example: A person age 25 purchases a 100,000 endowment to be paid at the end of 20 years. He make only 1 premium payment. How much would the pure endowment cost vs. endowment insurance? INS = 100,000 AGE = 25 TERM = 20 N = 1 AP (pure endowment) = 29,719.65 (Use ENDW program) AP (endowment insurance) = 32,171.93 (Use ENIN program) Life Annuities Life annuities are accessed in the ANN directory. Once in the directory press the CST key to bring up the custom menu. The programs are used in the same way as the ones in the insurance directory. The variables used in annuities calculations are: 1. SP - single premium 2. AP - annual premium 3. AN - annuity amount 4. TERM - years of a term annuity 5. N - number of premium payments 6. AGE - age at which annuity is purchased 7. K - number of years annuity payments are deferred As with the life insurance calculations, only SP, AP, and AN can be calculated. The other variables are used to get values from the commutation tables and therefore cannot be calculated. There are eight programs is the ANN directory. They are: 1. WA - whole-life annuity 2. WAD - whole-life annuity due 3. DWA - deferred whole-life annuity 4. DWAD - deferred whole-life annuity due 5. TA - term annuity 6. TAD - term annuity due 7. DTA - deferred term annuity 8. DTAD - deferred term annuity due Only deferred annuities can have annual premium payments N where N is <= K, the period of deferment. Deferred annuities can have a single premium by setting N = 1. Example 1: A person age 64 wants a life time annuity of 25,000 beginning at age 65. How much will cost? Use the WA program: AN = 25,000 AGE = 64 SP = 214,766.05 (calculated) If he purchased an annuity due at age 65, he would pay a single premium of 233,044.66 (calculated using the WAD program). Example 2: A person age 25 wants a life time annuity of 25,000 beginning at age 65. He will make 40 premium payments. What will be the amount of his annual premium? In this case it is easier to use the DWAD program because K, the period of deferment will be (65 - AGE) instead of (65 - AGE - 1) for the DWA program. AN = 25,000 AGE = 25 K = 40 N = 40 AP = 1,116.12 (calculated) Example 3: A person age 25 wants a temporary life annuity of 25,000 to begin at age 65 and last for 25 years. He will make 40 premium payments. What is the annual premium? Use the DTAD program. AN = 25,000 AGE = 25 TERM = 25 K = 40 N = 40 AP = 1108,08 (calculated) Other Matters There are many things one can do with the commutation tables in actuarial mathematics. I have set up only the basic equations. If you want to pursue matters further, get out some textbooks and practice, practice, practice. The calculations make with the programs in no way represent the premiums one would pay on any type of insurance or annuity. For one thing the premium calculations do NOT include loading, i.e., operating and overhead expenses of an insurance company. They are net premiums, not gross premiums. Disclaimer These programs are distributed in the hope that they will be useful to people studying finance or actuarial math. The are provided "as is" with no warranty of any kind as to fitness for any particular purpose. If you have any questions, suggestions for improvement, or find any bugs in the code, please send email.