# How to calculate the present value of a defined benefit pension plan

This page explains how to calculate the present value of a defined benefit pension. A defined benefit pension (sometimes called an annuity) makes monthly benefit payments to a recipient upon retirement. Defined benefit pension plans are common among state and federal employees and many public school teachers. They are less common in private business, where “defined contribution” retirement plans such as 401k’s and 403b’s are more common.

The size of *monthly* benefits in a pension is determined by formulas that vary from employer to employer. The value of the monthly benefit is *not* determined by the amount of money that has been withheld by the employer, or pooled in an account, but by formulas based on such factors as age at retirement, years of service, and level of salary during the final years of employment. Your retirement plan administrator or online calculators for different states or employers can calculate what your future monthly benefit will be.

In some cases, people want to know how much their future, monthly, retirement benefits are worth *right now**, *while they are still working. This is often a question in cases of divorce. Spouses can agree to share the future monthly payments when the pension participant retires, or they can divide the value of the pension in the present. It is easy to divide a 401k or other retirement account in the present because the balance of the account represents the value of the account. It is much more difficult to figure out how much the promise of lifetime, monthly payments 20 years in the future is worth right now.

Figuring out how much a pension is worth in the present involves two basic steps and two formulas:

## Formula for present value of a pension or annuity

**Step One and Formula 1:**

First, one must calculate the value of the pension at the time when benefits begin, i.e., at the time of retirement. The question is, “How much would one have to pay to be guaranteed the monthly, lifetime benefits that the pension guarantees?” The answer depends on the size of the monthly payment, the age of the recipient, the likelihood that the recipient will survive each year, interest rates, and possible cost-of-living adjustments. Actuary Mark Altschuler, in his *Value of Pensions in Divorce* gives the following formula for this calculation:

PV = 1 * P65 + (1P65)/(1.06) + (2P65)/(1.06)^{2} + … + (45P65)/(1.06)^{45}

This example of the formula assumes a retirement age of 65 and a mortality table that ends at age 110, which explains why the last term in the formula accounts for the 45th year after retirement at age 65. This formula also assumes a single annual payment at the beginning of the year, so it does not consider mortality for the first year of benefits, i.e., that the recipient might die in the first year before receiving any benefits.

Altschuler’s rendering of the formula can be confusing for several reasons. First, in the formula he gives, “PV” does *not* represent the present value. “PV” in this formula is actually a *multiplier* (some might call it an “annuity factor”)–not a present value–that must be multiplied by the annual benefit to produce the present value. Second, the “PV” he uses on the left side of the equation represents a multiplier *at the moment of retirement* in this formula, which is not what most people think of as “present”. He then uses the SAME notation–“PV”–in Formula 2, below, that represents the present value at the moment of calculation (likely at the time of divorce, years before retirement). Finally, it is difficult to understand because the notation he uses does not correspond to general language meanings. A more comprehensible version, which substitutes English words for mathematical notation,and does not assume a specific retirement age, mortality table length, or interest rate, might look like this:

PV-on-Retirement-Date=

Annual-Benefit +

(Likelihood-of-surviving-1-year-after-retirement)(Annual-Benefit)/(1+30-year-treasury-rate) +

(Likelihood-of-surviving-2-years-after-retirement)(Annual-Benefit)/ (1+30-year-treasury-rate)^{2} +

(Likelihood-of-surviving-3-years-after-retirement)(Annual-Benefit)/ (1+30-year-treasury-rate)^{3 }+

(Likelihood-of-surviving-4-years-after-retirement)(Annual-Benefit)/ (1+30-year-treasury-rate)^{ 4} …………. +

(Likelihood-of-surviving-(last age in mortality table minus retirement age)(Annual-Benefit)/ (1+30-year-treasury-rate)^{last age in mortality table minus retirement age}

This formula only calculates the present value of the pension at the moment of retirement when one starts to collect benefits. To calculate the present value *before* retirement–which is typical in most divorce cases–one must take a second step, discounting the Present-Value-at-Retirement-Date back to the Present Value in the present, as described in Step Two.

**Step Two and Formula 2:**

Second, one must calculate the amount of money one would need right now to buy that annuity (lifetime monthly payment) in the future, when one retires. If one is 45 and plans to retire at age 65, one would be buying the annuity 20 years in the future. The present value of the pension at age 45 is *lower* than the cost to buy the annuity at age 65 for two reasons: one might not survive until age 65 (and therefore one would not collect any benefit) and money in the present can earn interest for 20 years and grow to the amount necessary to purchase the annuity at age 65. So the value of the pension at age 65 is “discounted” with 20 years of interest and the probability that the pension participant might not survive those 20 years. Altschuler gives this formula for this discounting:

PV={PV-on-Retirement-Date} x (15P50)/(1.06)^{15}

Translating the mathematical notation into language terms and generalizing the formula beyond the specific example he is using (a 50-year-old getting a present value with a projected retirement age of 65), results in the following formula:

PV={PV-on-Retirement-Date} x (Likelihood-of-surviving-until-retirement-age)/(1 + 30-year-treasury-rate)^{Number of years between present value determination date and retirement date}

## How to calculate the present value of a pension survivor benefit or survivor annuity

The present value of a pension depends fundamentally on a) long-term interest rates and b) mortality (how likely is a person to survive one more year).

Interest rates play the same role in survivor benefit calculations as they do in normal pension present value calculations. In each case, future payments are “discounted” by those interest rates, so that the *present* value of those payments is lower than the face value of those future payments.

Calculating a survivor benefit uses mortality in a different way than a regular pension present value calculation. In a regular pension present value calculation, the question is: “How likely is the benefit recipient to survive another year and collect benefits for that year?” If a person received $100/year and they are 90% likely to survive another year, that possible future annual payment is discounted to $90 for mortality (and then further discounted by interest rates).

For a survivor beneficiary to receive a benefit, TWO things have to happen: the pension participant must be dead, and the beneficiary has to survive another year. So the question for a pension survivor benefit is: “How likely is it for each year that a) the pension participant is dead AND b) the survivor beneficiary is alive?” If the pension participant has a 10% chance of being dead during a year, and the survivor beneficiary has 90% chance of surviving, the likelihood of receiving a survivor benefit is .10 multiplied by .90, which equals .09. If the survivor benefit for a particular pension is $100/year, that possible future payment is discounted to $9 for mortality (and then further discounted by interest rates).

In terms of the formulas in Steps One and Two, above, one can substitute “Survivor-beneficiary-likelihood-of-surviving” in each position where “Likelihood-of-surviving” appears. Each of these factors must then be multiplied by a corresponding “Pension-participant-likelihood-of-being-dead…”