Earnings of a fund are estimated in terms of an interest rate. For example, if funds will earn 7% per year, then $100 put into a fund today will grow to $107 in one year and, with compounding of interest, to $140 in five years. Put the other way around, to have $140 in five years, fund $100 today at 7%. Starting from $140 and going back to $100 is called discounting for interest.
Example: Three coworkers age 65 and just retired contribute $100 apiece to a 70-th birthday fund. Only two survive to age 70. The two share $300 or $150 each. In effect $150 per person at age 70 has been discounted by survivorship to $100 at age 65.
A life table starts with some round number of specified individuals, for example, men at age twenty in a given occupation, and then shows how many remain at advancing ages.
The first life table in the modern sense was constructed by Edmund Halley at the close of the seventeenth century. Halley used the then new mathematical theory of probability and vital statistics from the city of Breslau.
In the days of the Roman Empire an official Domitius Ulpanius promulgated rules for setting life insurance premiums for trade guilds. Ulpianus based his rules on estimates of life expectancy rather than on an explicit life table. The "Ulpian" table below is implied by the numbers Ulpianus did use.
For modern life table see
Life Tables
In practice discounting for survivorship is based on experience with large groups of people. Survivorship under a pension plan involves turnover and disability as well as mortality. Estimates of survivorship are nowadays stated in terms of "life tables".
Comparative Life Tables
source Ulpianus Halley Karlsruhe MGA MIA
year: 230 1693 1864 1971 1983
age
20 1,000 1,000 1,000 1,000 1,000
25 897 948 956 997 997
30 825 888 916 994 994
35 749 819 874 989 990
40 643 744 829 983 985
45 643 664 777 973 976
50 643 579 718 955 962
55 412 488 647 924 939
60 263 405 559 878 907
65 139 321 454 810 863
70 51 237 337 708 797
75 19 147 216 566 697
80 7 69 112 401 559
85 3 0 40 231 390
The general actuarial rule is:
Step 1: Apply the general rule to a large group of lives.
Step 2: Divide both sides by l[65] and use the definition D[x] = l[x] v^(x).
Step 3: Assume MIA 1983 and 8% interest
Step 1. Find the value at age 65 of a life annuity of $50,000.
Step 1.1: Discount from older age x to age 65
Step 1.2: Divide both sides by l[65] and use the definition D[x] = l[x] v^(x).
Step 1.3: Sum for all years of life
Step 1.4: Use the definition N[x] = Sum[from x to omega] D[x].
Step 2. Find the value at age 65 of payments made at younger ages.
Step 2.1: The actuarial future value at age 65 of a payment made at younger age x is found from
Step 2.2: Use the definition of D[x] and solve for the future value.
Step 2.3: Sum for all working years.
Step 2.4: Use the definition of N[x] twice.
Step 3. Equate the value of the payments and the value of the pension.
Step 3.1: At age 65
Step 3.2: Solve for the payment.
Step 3.3: Assume MIA 1983 and 8% interest.
Summary: Starting at age 30, pay $2,193 a year up through age 64. Receive $50,000 a year starting at age 65 for as long as you live. This assumes no administrative costs and no extra benefits. For example, if you die before age 65, your estate receives nothing.
Step 1. Present value of premiums = (premium) l[30]
Step 2. Present value of total benefits paid = ($100,000) (l[30]-l[31]) (v)
Step 3. Equate and Solve.
Step 1. Present value of premiums = (premium) l[30]
Step 2. Present value of total benefits paid = ($100,000) (l[30]-l[31]) (ln(1.08)/(.08))
Step 3. Equate and Solve. premium = ($100,000) (.000850) (.962) = $81.80
Comment: When people die throughout the year, the insurance fund has less time to earn more money from the premiums paid at the start. That forces the premium higher.
Step 1. Present value of premiums = (premium) l[30]
Step 2. Present value of total benefits paid
Step 3. Equate and Solve.
Step 1. Present value of premiums
Step 2. Present value of total benefits paid
Step 3. Equate and Solve.
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