Just PENSIONS

Introduction

The actuarial present value process is the product of three components — the contingency, interest, and the benefit. To an actuary, a contingency is an event to which a probability or percentage of occupance can be assigned. If an event has a 20% probability of occupance and if the event happens a benefit of $100 is made then expected value of the event is 20% x $100 = $20. This differs from the concept some persons with a legal background have of a contingent event — if an event has a probability of less than 50% of occurring it is assumed that it will not occur and if the event has a probability of 50% or more of occurring is assumed the event will occur. This flawed logic would value an event with a probability of just 20% as an event that would not happen.

The probability of occupance times the benefit to be paid gives the expected value of the benefit at the time of occupance. If the time of occupance is at some point in the future this value has to be discounted back to a present value. This is called discounting for interest. The higher the interest rate used for discounting back to present value the lower the present value.

Stephenson [1984] attempts to provide an explanation of the actuarial valuation process.

Three Components - Contingency, Interest and Benefits

1. The Three Components Separately

   1. Contingency/Actuarial Component

      Being alive to qualify for a benefit is the contingency necessary to trigger an annuity payment. The probability of being alive to trigger a benefit payment is calculated from a mortality table suitable for the purpose. This probability of being alive to trigger payment is shown in Chart 1. The middle line shows the probability that a male participant, age 65, will be alive at any time in the future. The right-most line shows the probability that a female alternate payee, age 55, will be alive at any time in the future. The left-most line shows the probability that a male, now age 65, and a female, now age 55, will both be alive at anytime in the future.

      

   2. Interest/Economic Component

      The value today of a benefit or benefits payable in the future are discounted for interest to obtain the present value of the benefit(s).

      

   3. Benefit Component

      The benefit is determined from a formula defined in the plan. Generally, it is assumed that any automatic post-retirem cost-of-living will be paid.

2. The Components Together - The Actuarial Present Value

   The actuarial present value process discounts each benefit for interest and the probability that the payment of the benefit will be made.Chart 3 shows the results of combining the three components. The area under the dashed line represents the actuarial present value of a benefit of $1,000 per month payable to a male age 65 without cost-of-living. The area under the dotted line represents the actuarial present value of a benefit of $1,000 per month payable to a female now age 55. The area under the solid line represents the actuarial present value payable to a male age 65 while the Alternate Payee, age 55, is alive.

Mechanics of the Actuarial Valuation Process — FIDO

To illustrate the mechanics of the actuarial valuation process, we’ll walk through a K-9 example providing pension benefits to Fido, the dog. Suppose we want to provide Fido a pension of $100 at the beginning of each year that Fido is alive. Assume Fido is age 15 and there is one chance is six of Fido’s dying during any year. If Fido is alive at age 25, the pension plan will stop. The first step is to develop a cash flow analysis that is independent of Fido’s survival. The cash flow analysis is given in Columns 1 through 6 of Table 1. The last entry in Column 6 of $724.69 is present value of pension benefits Fido will recieve if he does not die. The next step is to reflect the actuarial, or contingent, nature of payments; that is, Fido must survive (be alive) to recieve the payments. Several methods are available to take this into account:

Calculating Life Expectancy

The data on Fido can be used to illustrate how life expectancy is calculated. If we remove the decimal point in Column 7 of Table 1 and assume we have 100,000 dogs alive at age 15 and 83,333 alive at age 16, then each of the 83,333 dogs who survive to age 16 have lived one year; in other words, 83,333 years of life have been lived between age 15 and age 16 of the 83,333 dogs who ended the year. If we assume each of the 16,667 dogs who died between age 15 and age 16, on the average, lived one-half year, then these dogs who died lived a total of 8,333 years during their last year of life. This process, repeated for each year of age, is summarized in Table 2. The total of Column 6 (443,403) represents the total years of life lived after age 15, until age 25, by the original 100,000 dogs at age 15; thus, each dog, on the average, lived 4.434 years after age 15 or until age 19.434.