Refunds Versus Deferred Vested Pensions

Why the Comparison Requires a Common Valuation Basis

When a member withdraws from a pension plan, the member may be entitled to an immediate refund of contributions, a deferred vested pension, or—if the plan permits a choice—an election between them. Comparing the refund amount directly with the annual pension amount is not meaningful: one is a lump sum and the other is a stream of future payments. The amounts must first be valued at the same date using the same interest and survival assumptions.

Eligibility is a separate question from value. A plan might require a minimum service period for vesting. If that requirement has not been met at withdrawal, a deferred vested pension is not an available option, regardless of how attractive its hypothetical value might be. The comparison below is conditional on the member satisfying the plan’s vesting and other eligibility conditions.

Definition: At the withdrawal date, let $R$ be the refund payable immediately. Let $B$ be the annual pension payable in advance, beginning $n$ years later, provided the member is alive on each payment date. The break-even pension is the value of $B$ for which the pension’s expected present value at withdrawal equals $R$. This comparison excludes taxes, expenses, guarantees, and survivor benefits unless they are explicitly included in the benefit description and valuation assumptions.

A deferred pension valuation must reflect both discounting and the chance that the member survives to receive each payment. The examples use annual effective interest $i$, annual survival probability $p$, and $v=1/(1+i)$. For a constant annual survival probability, the probability of surviving $k$ years is $p^k$. These assumptions are deliberately stated rather than inferred: a pension with payments at the start of each year has a different value from one paying at year-end.

Present Value of a Deferred Pension

Take the withdrawal date as time 0. If the first annual payment is made at time $n$, then the first payment is made only if the member survives $n$ years. Later payments are made at times $n+1,n+2,\ldots$, each contingent on survival to its payment date. Under a constant annual survival probability $p$, the expected present value of an unlimited annual pension is

\operatorname{EPV}(B) = B\sum_{j=0}^{\infty}v^{n+j}p^{n+j} = B\sum_{j=0}^{\infty}q^{n+j} = \frac{Bq^n}{1-q}, \qquad q=pv=\frac{p}{1+i},

This formula requires $q<1$, which holds when the discount and survival assumptions make the geometric series converge. It also assumes level annual payments, a constant annual survival probability, and no payment after death. If the pension pays for only $m$ annual payments, the corresponding value is

\operatorname{EPV}(B) = B\sum_{j=0}^{m-1}q^{n+j} = \frac{Bq^n(1-q^m)}{1-q}.

For either payment pattern, the break-even pension is found by setting the expected present value equal to the immediate refund. The finite-term version uses the number of payments $m$; it must not be replaced by a lifetime formula unless payments really continue for life.

B^*_{\text{lifetime}}=\frac{R(1-q)}{q^n}, \qquad B^*_{\text{finite}}=\frac{R(1-q)}{q^n(1-q^m)}.

Derivation: At time $n+j$, the pension payment $B$ has probability $p^{n+j}$ of being paid and discount factor $v^{n+j}$. Its expected present value is therefore $Bv^{n+j}p^{n+j}=Bq^{n+j}$. Summing these values over the permitted payment dates gives a geometric series. For a lifetime pension, the sum is $Bq^n/(1-q)$; for $m$ payments, it is $Bq^n(1-q^m)/(1-q)$. Equating that value to $R$ and solving for $B$ gives the break-even formulas.

Worked Comparisons

Worked Example: Break-Even Lifetime Pension After a Two-Year Deferral

A hypothetical plan member has completed the vesting requirement and is offered either an immediate refund of $5,000 or a level lifetime pension paid annually in advance, with the first payment two years from the withdrawal date. Assume an effective annual interest rate of $i=0.04$, a constant annual survival probability of $p=0.96$, and no pension payment after death. Find the break-even annual pension.

Method. Value the pension at the withdrawal date. The first payment is at time 2, so its survival and discount factors are both applied for two years.

First calculate the one-year combined survival-discount factor:

v=\frac{1}{1.04}=\frac{25}{26}, \qquad q=pv=0.96\left(\frac{25}{26}\right) =\frac{24}{26}=\frac{12}{13}.

For a lifetime pension deferred two years, the present-value factor per dollar of annual pension is

\frac{q^2}{1-q} =\frac{(12/13)^2}{1/13} =\frac{144}{13} =11.076923\ldots.

Set the pension value equal to the $5,000 refund and solve for $B$:

B^*=\frac{5{,}000}{144/13} =5{,}000\left(\frac{13}{144}\right) =451.388888\ldots.

Rounded to the nearest cent, the break-even pension is $451.39 per year. At this amount, its expected present value is $451.388888\ldots(144/13)=$5{,}000$. A larger annual pension has greater expected present value under these assumptions; a smaller one has less.

Common mistake avoided: Omitting the two-year deferral would use $B/(1-q)$ rather than $Bq^2/(1-q)$. The omitted $q^2$ overstates the value of each pension dollar at withdrawal and understates the break-even annual pension.

Worked Example: A Pension Limited to Three Payments

A hypothetical vested member may take an immediate refund of $1,952 or receive a pension of $B$ dollars annually in advance for exactly three payments, beginning one year after withdrawal. Assume $i=0.10$, a constant annual survival probability of $p=0.88$, and that each payment is made only if the member is alive on its payment date. Calculate the break-even annual pension.

Method. Use the finite-payment formula, since there are exactly three possible payments, at times 1, 2, and 3.

The combined annual factor is

q=pv=\frac{0.88}{1.10}=0.8.

The present-value factor per dollar of annual pension is the sum of the three payment factors:

$$ q+q^2+q^3 =0.8+0.64+0.512 =1.952. $$

Thus the pension’s expected present value is $1.952B$. Equating it to the refund gives

1.952B=1{,}952, \qquad B=\frac{1{,}952}{1.952}=1{,}000.

The break-even pension is $1,000 per year. As a check, the three expected present values are $1{,}000(0.8)=$800$, $1{,}000(0.64)=$640$, and $1{,}000(0.512)=$512$. Their sum is $1,952.

Common mistake avoided: Treating a three-payment pension as a lifetime annuity adds payments after time 3 that the plan does not provide. Conversely, using only the first payment omits the two later contingent payments.

Worked Example: Apply Eligibility Before Comparing Values

A hypothetical plan requires five completed years of service for vesting. At withdrawal, a member has 6.25 years of service and qualifies for a deferred pension. The member may instead take a $4,800 immediate refund. The pension is $350 annually in advance for life, beginning two years after withdrawal. Assume $i=0.04$, a constant annual survival probability of $p=0.98$, and no payments after death. Which option has the higher expected present value at withdrawal?

Method. First confirm eligibility; then calculate the deferred pension value on the specified assumptions and compare it with the immediate refund.

The member has more than the required five completed years, so the deferred pension is available. The combined annual factor is

q=\frac{0.98}{1.04} =\frac{49}{52}, \qquad 1-q=\frac{3}{52}.

The lifetime pension factor for a two-year deferral is

\frac{q^2}{1-q} =\frac{(49/52)^2}{3/52} =\frac{2{,}401}{156} =15.3910256\ldots.

The expected present value of the $350 annual pension is therefore

350\left(\frac{2{,}401}{156}\right) =\frac{840{,}350}{156} =5{,}386.858974\ldots.

Rounded to the nearest cent, the pension’s expected present value is $5,386.86, which exceeds the $4,800 refund by $586.86. On this valuation basis, the pension has the greater expected present value.

Common mistake avoided: Eligibility is not established by comparing values. If the member had fewer than five completed years, the deferred pension would not be an available choice, even if this calculation suggested a high hypothetical value.

Interpreting the Result and Avoiding Exam Traps

The break-even amount is conditional on the stated assumptions and benefit design; it is not a universal conversion between a refund and a pension. A change in interest, survival, deferral, or payment duration changes the value. In particular, increasing the deferral generally reduces the value of each dollar of pension at withdrawal, so a larger annual pension is needed to match the same immediate refund. Lower survival probabilities also reduce the expected value of payments contingent on being alive.

The comparison is made at the withdrawal date and conditional on reaching an eligible withdrawal with the choice available. It does not, by itself, include the probability that a current active member withdraws, dies, retires, or meets the vesting condition in the future. When a valuation is required before the exit event, the relevant event probabilities and benefit cash flows must also be included, as in multiple-decrement valuation.

A frequent timing error is to treat an annual pension as payable at the end of a year when the problem says it is payable in advance, or to count the first payment at time 0 when it begins after a deferral. Write down the payment dates before calculating. For example, a pension deferred two years and payable in advance has its first payment at time 2, not time 0 or time 3. Apply survival and discounting to each payment date, not just to the first payment or to the refund.

Check eligibility.
Apply the plan’s vesting, service, age, and election conditions at the relevant withdrawal date.

Set the valuation date and payment dates.
Use the withdrawal date as the common present-value date, and specify whether the first pension payment is immediate, deferred, or contingent on survival to a future date.

Value every contingent payment.
For each date, multiply the amount by its survival probability and discount factor, then sum the expected present values.

Compare like with like.
Compare the pension’s expected present value with the immediate refund. Use the break-even formula only when its lifetime or finite-payment assumptions match the benefit.

Key takeaway: A refund-versus-pension comparison is valid only after eligibility is confirmed and both alternatives are valued at the same date. Deferred pension value depends on payment timing, survival to each payment, and discounting; the break-even annual amount follows by equating that value to the immediate refund.

Check Your Understanding

For each problem, use the stated assumptions and compare values at the withdrawal date. Round dollar amounts to the nearest cent where needed.

A member is eligible for a lifetime pension of $B$ dollars annually in advance, beginning one year after withdrawal, or an immediate $6,000 refund. Assume $i=0.05$ and annual survival probability $p=0.95$, constant for life. Find the break-even annual pension. (Answer: $631.58 per year.)
A vested member can take an immediate refund of $1,889.28 or receive four annual payments in advance beginning two years after withdrawal, with no payments after the fourth payment. Assume $i=0.125$ and annual survival probability $p=0.90$, constant. Find the break-even annual payment. (Answer: $1,000.00 per year.)
A member is eligible for a lifetime pension of $300 annually in advance, beginning two years after withdrawal, or a $3,500 immediate refund. Assume $i=0.04$ and annual survival probability $p=0.96$, constant for life. Which option has the higher expected present value, and by how much? (Answer: The refund is higher by $176.92.)
A hypothetical plan requires five completed years of service for vesting. A member withdraws with 4.9 years of service and is entitled to an immediate contribution refund. Can the member elect a deferred vested pension under the stated rule? (Answer: No; the member has not completed five years, so the deferred vested pension is not available.)
A member may take a $1,000 immediate refund or receive a $600 annual pension in advance for exactly two payments, beginning one year after withdrawal. Assume $i=0.05$ and annual survival probability $p=0.95$. Which alternative has the higher expected present value, and by how much? (Answer: The pension is worth $1,034.01, so it is higher by $34.01.)

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