How much should members of a pool put aside today to pay a specified benefit to those who experience a specified future event? The answer depends on what the funds set aside can earn and on how many members will share in the proceeds of the fund.
Earnings of a fund are estimated in terms of an interest rate. For example, if funds will earn 7% per year, then $100 funded 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%. Going from $140 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.
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 mortality are nowadays stated in terms of "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 25 893 948 956 997 997 30 821 888 916 994 994 35 747 819 874 989 990 40 635 744 829 983 985 45 635 664 777 973 976 50 635 579 718 955 962 55 353 488 647 924 939 60 202 405 559 878 907 65 81 321 454 810 863 70 27 237 337 708 797 75 9 147 216 566 697 80 3 69 112 401 559 85 1 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.
Tables