# ## 16.13 Calculation of Present Value of a Defined Benefit Pension

The fundamental concept in calculating a present value of future payments is the concept of discounting. In order to understand discounting, first consider the concept of compounding. Assume one invests \$1,000 in a five-year CD, with a compound interest rate of 6 percent. The future value of the CD at maturity is given by:

(1) Future Value = \$1,000 x (1.06)5 = \$1,338

Therefore, the present value today of \$1,338 in five years, discounting at 6 percent interest is given by:

(2) Present Value (PV) = \$1,338 x 1/(1.06)5 = \$1,000

The fact that the present value is \$1,000 is not surprising. Thus, discounting without mortality is simply the inverse process of compounding. Instead of a lump sum, consider discounting a five-year annuity of compounding.

Instead of a lump sum, consider discounting a five-year annuity of \$1 per year. This is called a five-year certain only annuity. Instead of discounting the single lump sum of \$1,338 for five years, each \$1 payment is discounted separately, then added up. In essence, there are a series of \$1 lump sums. The present value of the first dollar payment is \$1, assuming payment at the beginning of the year. This is because the first dollar is received now, so there is no discounting. The second dollar payment has one year of discounting at 6 percent interest. The third dollar payment has two years of discounting, and the last payment has four years of discounting. After discounting each installment, the present values are then added up. Mathematically, this is expressed as follows:

(3) PV = 1 + 1/1.06 + 1/(1.06)2 + 1/(1.06)3 + 1/(1.06)4
= 1 + .94 + .89 + .84 + .79 = \$4.46

This is a commonsense result. The present value of a lump sum payment of \$5 four years from now, discounted at 6 percent interest is \$3,9605. However, in the annuity, only the last payment is discounted four years. The average amount of discounting is two years, with two payments discounted less than two years, and two payments discounted more than two years. As an approximation, imagine that the five payments of \$1 is a single \$5 payment two years from now. Then the present value would be:

(4) PV = \$5 x 1/(1.06)2 = \$4.45

This result of \$4.45 is a reasonable approximation to the correct result of \$4.46. Please note that this approximation is not used in real actuarial calculations but merely used here as an illustration to provide some commonsense, intuitive feel for the calculations.

Instead of the five-year annuity used above, consider a more realistic \$1,000 per month annuity, payable over the annuitant's lifetime, for a male age 65, using the technique of life expectancy. Since the life expectancy of a male age 65 is 17 years, the annuity now is for 17 years, not five. This is called a 17 year certain only annuity. In actuality, the annuity is payable over the annuitant's lifetime. This is called a single life annuity. The annuity does not go 17 years and then stop. An accurate calculation, performed by an actuary, will account for the probability of mortality in each and every year from now (age 65) until age 110 (where the mortality table ends). However, life expectancy is a starting point. First, consider the \$1 per year annuity, now for 17 years. The present value of this annuity is given by:

(5) PV = 1 + 1/(1.06) + 1/(1.06)2 + ... + 1/(1.06)16 = 10.8147

However, the payments are not \$1 per year, but \$12,000 per year. Thus, the present value (using life expectancy) is as follows:

(6) PV = \$1,000 x 12 x 10.8147 = \$129,776

As mentioned previously, life expectancy is not a method used by actuaries. The reason is that the most likely outcome for a male age 65 is to live 17 years. It means that with 100 percent probability, the individual will live 17 years, then die. But the individual may die tomorrow, or live to be 110. Since the probability of anyone living past the age of 110 is minute, the table ends at age 110. The only way to account for each possible outcome is to use probability of mortality, for each and every year from 65 until 110. Thus, the \$1 per year annuity now goes on for 45 years, until age 110. However, each year's payment is discounted with interest and mortality. Thus, using life expectancy, the lust year's payment is not discounted. While there is no discounting for interest, there is discounting for mortality, since the individual may die within the first year. Therefore, the \$1 is multiplied by the probability of surviving the first year. The second year's payment of \$1 is multiplied by the probability of a male age 65 living one more year, and is discounted with one year's interest (at 6 percent). The third year's payment is multiplied by the probability of living two additional years, and discounted with two years' interest. This goes on until age 110. The \$1 payment in that year is multiplied by the probability of the individual living to collect it, and discounted back to the present with 45 years' interest. The present values are then added up, as in the life expectancy example. The procedure just described is given mathematically as follows:

(7) PV = 1 * P65 + (1P65)/(1.06) + (2P65)/(1.06)2 + ... + (45P65)/(1.06)45
= 9.9165

This is the present value of a \$1 per year annuity. But the annuity is actually \$12,000 per year, and so the present value is given as:

(8) PV = \$1,000 x 12 x 9.9165 = \$118,998

Note that there is a significant difference between the life expectancy result and the correct result using probability of mortality. Life expectancy is a simplistic method used by non-actuaries, such as accountants, in valuing pensions, and is not accurate.

Now consider the most common case, in which the individual has not yet retired. Assume that the annuity of \$1,000 per month is payable at age 65, the normal retirement age of the pension plan, but the individual is age 50. First, consider the life expectancy case. At age 50, the life expectancy is 29 years, as opposed to 17 years at age 65. Thus, in this computation, the pension only goes for 14 years, not 17-that is, from age 65 to age 79. Thus, the present value at age 65 is \$115,133, not \$129,776, using life expectancy, since the annuity is now for only 14 years, not 17. The present value at age 65 for an individual age 50, versus the present value at age 65 for an individual age 65, is a confusing concept necessitated by the use of life expectancy, which is a mathematically unsophisticated technique. This cumbersome step is not needed using probability of mortality. The next step is to discount this lump sum of \$115,133 at age 65 back 15 years, at 6 percent interest. Note that equation (9) has the same form as the calculation in equation (1):

(9) PV = \$115,133 x 1/(1.06)15 = \$48,041

The \$115,133 is not actually a lump sum, but it is mathematically equivalent to a lump sum. Of course, as noted previously, this result is not accurate anyway, but it is the correct result, given the use of life expectancy.

The last step is to calculate the present value for a male age 50, payable at age 65, using probability of mortality. Remember that the present value at age 65 is \$118,898. As in the life expectancy case, this is discounted with 15 years interest, at 6 percent. However, in addition, it is necessary to multiply by the probability of the individual actually living from age 50 until age 65. Thus, the present value is given as follows:

(10) PV = \$118,898 x 1/(1.06)15 x 15P50

Following equation (8), the value at age 65 is given by:

\$118,898 = \$1,000 x 12 x 9.9165

Substituting this expression for the \$118,898 gives the following expression for present value:

(11) PV = {\$1,000 x 12 x 9.9165} x (15P50)/(1.06)15

Collecting terms and "plugging in" the probability of living from age 50 to age 65, the present value at age 50 is given by:

(12) PV = \$1,000 x 12 x {9.9165 x .8896/(1.06)15}
= \$1,000 x 12 x 3.6810
= \$44,172

Note that at age 65, the use of life expectancy overestimated the value of the pension. At age 50, the use of life expectancy underestimated the value of the pension. Again, this is one more flaw with the use of life expectancy.

The last term, 3.6810, is the present value for a male age 50, for a \$1 per year annuity, payable at age 65. This is called the annuity factor: Now look at Exhibit 16-5, showing this result as a pension valuation report. Within the following chapters on federal, military, and Fortune 500 company pensions there will be several such sample reports.

### EXHIBIT 16-5 Sample Pension Valuation Report

 DATE: March 5, 1998 PREPARED FOR: Robert D. Feder, Attorney At Law NAME OF EMPLOYEE: Paul Aspen EMPLOYER: Doe Ind., Inc. PLAN: Pension Plan (Defined Benefit) Birth Date: Entry Date: Marriage Date: Cut-off Date: Valuation Date: 03/05/48 01/01/68 01/01/68 03/05/98 03/05/98 Retirement Age: 65 Retirement Date: 04/01/13 Status: Active Sex: Male Age: 50
 1) Accrued monthly pension at cut-off date \$1,000 2) Annuity factor 3.681 3) Present value (12 x item 1 x item 2) 44,172 4) Length of Plan service while married 30.16667 5) Length of Plan service to cut-off date 30.16667 6) Coverture fraction (item 4 divided by item 5) 1.00000 7) Marital present value (item 3 x item 6) 44,172 8) Marital employee contributions/account 0 9) Marital present value as of valuation date     (greater of item 7 or item 8) \$44,172 Marital portion contingent on jurisdictional cut-off date. Pension form: Life Annuity Calculations in accordance with generally accepted actuarial standards Interest rate: 6.00%          Mortality: GAM-83

Reprinted with permission. © 2002, Aspen Publishers, Inc., from Valuing Specific Assets in Divorce, edited by Robert D. Feder.

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