Cash Refunds and Annuity-Certain Guarantees

Why the Contract Wording Matters

A pension may offer a member a life annuity with a cash refund, an annuity-certain guarantee, or a choice between them. Both options can provide value after the member’s death, but they do so through different payment streams. A cash refund is generally a lump sum determined by a refund formula; an annuity-certain guarantee continues scheduled installments for a specified period. Neither option has a fixed value independent of its terms.

The phrase “cash refund” is not enough to determine a value. A contract might return unused contributions, return a fixed purchase amount less payments already made, or apply interest to a balance. The formula below adopts one clearly specified hypothetical contract: the initial refundable amount is $C$, each pension installment is $B$, and the refund at death is $C$ less the installments already received, floored at zero. If a contract uses a different balance rule, its formula must replace this one.

Definition: Assume annual payments of $B$ are made in advance, at times $0,1,2,\ldots$, while the pensioner is alive. A cash refund contract with initial refundable amount $C$ pays a lump sum at death equal to $\max(C-mB,0)$, where $m$ is the number of installments received before death. An $n$-year annuity-certain guarantee pays the scheduled installments at times $0,1,\ldots,n-1$ whether or not the pensioner is alive; after that, installments are paid only while the pensioner is alive.

Use an annual effective interest rate $i$, with $v=1/(1+i)$. Let ${}_kp_x$ be the probability the pensioner aged $x$ survives $k$ years, and let $q_{x+k}=1-p_{x+k}$ be the probability of death in the following year, conditional on survival to age $x+k$. Thus the probability of death in year $t$, for $t=1,2,\ldots$, is ${}_{t-1}p_xq_{x+t-1}$. The examples use this annual discrete model and treat a death benefit for death in year $t$ as paid at time $t$.

Valuing the Two Payment Streams

The life annuity paid in advance has present value $B$ at time zero for each installment, weighted by the probability that it is payable. A payment at time $k$ is made only if the pensioner is alive then, so its expected present value is $Bv^k{}_kp_x$. Summing these terms gives the value of the life-contingent installments.

\operatorname{EPV}(\text{life annuity})=B\sum_{k=0}^{\infty}v^k{}_kp_x

For the cash refund, a death in year $t$ occurs after the installment at time $t-1$ has been paid and before an installment at time $t$. The member has therefore received $t$ installments. The refund is $\max(C-tB,0)$, paid at time $t$, and the probability of that death year is ${}_{t-1}p_xq_{x+t-1}$. The total value includes both the life annuity installments and the expected present value of these death benefits.

\operatorname{EPV}(\text{cash refund option}) = B\sum_{k=0}^{\infty}v^k{}_kp_x + \sum_{t=1}^{\infty}v^t{}_{t-1}p_xq_{x+t-1}\max(C-tB,0)

For an $n$-year annuity-certain guarantee, the first $n$ installments are payable with certainty. Later installments are payable only if the member survives to their payment dates. This gives a different expression:

\operatorname{EPV}(\text{\(n\)-year guarantee}) = B\left(\sum_{k=0}^{n-1}v^k+\sum_{k=n}^{\infty}v^k{}_kp_x\right)

The guarantee replaces the survival-weighted probabilities for the first $n$ installments with certainty. It does not add a separate lump-sum refund. Conversely, the cash-refund value adds a death benefit whose amount can reduce over time. Directly comparing the death benefit amounts, without discounting and weighting them by the probability of death at each time, does not compare the options’ values.

Why the Present-Value Formulas Hold

Each formula follows by valuing the possible payments separately and adding their expected present values. For any payment date $k$, let its amount be $Z_k$ and let it be payable only on an event $E_k$. Its expected present value is $v^kZ_kP(E_k)$. This uses the fact that the present value is the sum of discounted payments, and expectation is additive over that sum. For the refund, the events “death occurs in year $t$” are mutually exclusive, so each year’s refund contributes its own term.

List payment dates.
Record the annuity installments at times $0,1,2,\ldots$, and locate the refund payment at the end of the year of death.

Identify each payment condition.
A life installment at time $k$ requires survival to time $k$; a refund in year $t$ requires survival to year $t-1$ followed by death in year $t$.

Discount, weight, and sum.
Multiply each amount by its discount factor and payment probability, then add the resulting expected present values.

The guarantee formula is justified in the same way. For $k=0,\ldots,n-1$, the payment is made with probability one. At every later date $k$, it is made with probability ${}_kp_x$. The change in payment condition occurs at the guarantee boundary, which is why the two sums have different probability weights.

Worked Example: Comparing a Refund With a Three-Year Guarantee

A hypothetical pension pays $B=\$1{,}000$ annually in advance. Under the cash-refund option, $C=\$3{,}000$, and the refund is $C$ less installments received, floored at zero. Compare its value with a three-year annuity-certain guarantee for a pensioner whose annual probability of death is $0.20$ in every year. Assume $i=5\%$ effective annually, with independent annual survival probabilities under this constant mortality model.

Method. First value the life installments, then add the expected present value of the cash refunds. Separately, value the three guaranteed installments and the later survival-contingent installments. Here $v=1/1.05=20/21$, and the annual survival probability is $p=0.80$.

The life annuity factor is a geometric series with ratio $vp=(20/21)(0.80)=16/21$:

\sum_{k=0}^{\infty}(vp)^k = \frac{1}{1-16/21} = 4.2

Thus the life-contingent installment value is $1{,}000(4.2)=\$4{,}200.00$. At death in year 1, one installment has been received and the refund is $\$3{,}000-\$1{,}000=\$2{,}000$. At death in year 2, two installments have been received and the refund is $\$1{,}000$. At death in year 3 or later, the balance is zero. The probabilities of death in years 1 and 2 are $0.20$ and $(0.80)(0.20)=0.16$, respectively. Therefore:

\begin{aligned} \operatorname{EPV}(\text{refunds}) &=v(0.20)(2{,}000)+v^2(0.16)(1{,}000)\\ &=\frac{20}{21}(400)+\left(\frac{20}{21}\right)^2(160)\\ &=\$526.08 \end{aligned}

The total cash-refund option value is $\$4{,}200.00+\$526.08=\$4{,}726.08$.

For the three-year guarantee, the payments at times 0, 1, and 2 are certain. At times $k\geq 3$, a payment requires survival, so the value factor is:

\begin{aligned} 1+v+v^2+\sum_{k=3}^{\infty}(vp)^k &=1+\frac{20}{21}+\left(\frac{20}{21}\right)^2 +\frac{(16/21)^3}{1-16/21}\\ &=4.7170068\text{ (approximately)} \end{aligned}

The guarantee value is therefore $\$4{,}717.01$, rounded to the nearest cent. The cash-refund option is higher by $\$9.07$ under these assumptions. This small difference is specific to the chosen amounts, mortality, and interest rate; it does not establish a general ranking.

Common mistake. Applying survival probabilities to the first three guaranteed payments would incorrectly value them as life-contingent. The correction is to use probability one for payment dates 0, 1, and 2, and survival probabilities only from time 3 onward.

Worked Example: A Refund That Ends Earlier Than the Guarantee

A hypothetical plan pays $B=\$100$ annually in advance and has a cash-refund amount $C=\$200$, reduced by each installment already received. Compare this option with a three-year annuity-certain guarantee. Assume the pensioner has a constant annual death probability of $0.25$, and $i=0\%$. Find both expected present values.

Method. With zero interest, discount factors are one. The life-annuity factor is the sum of annual survival probabilities: $1+0.75+0.75^2+\cdots=1/(1-0.75)=4$. The life-contingent installment value is therefore $100(4)=\$400.00$.

At death in year 1, one installment has been received, so the refund is $\$200-\$100=\$100$, and the probability of that death is $0.25$. At death in year 2, two installments have been received, leaving no refund. Thus the expected present value of refunds is $(0.25)(\$100)=\$25.00$, and the cash-refund option is worth $\$425.00$.

The three-year guarantee pays three certain installments at times 0, 1, and 2. From time 3 onward, payments require survival. The value factor is:

1+1+1+\sum_{k=3}^{\infty}(0.75)^k = 3+\frac{0.75^3}{1-0.75} = 4.6875

The guarantee is worth $100(4.6875)=\$468.75$, which is $\$43.75$ more than the cash-refund option. The guarantee continues installments through its specified period; the cash refund here has no balance remaining after the second installment.

Common mistake. Treating $C$ itself as the amount paid on every death would overstate the cash-refund value. The contract specified a declining balance, so the year-1 refund is $C-B$, not $C$, and the year-2 refund is zero.

Worked Example: When a Refund and a Guarantee Have Equal Value

Consider a hypothetical pension with annual-in-advance installments $B=\$1{,}000$ and initial refundable amount $C=\$2{,}000$. Compare a two-year guarantee with the cash-refund option. Assume $i=10\%$, the probability of death in the first year is $0.50$, and anyone who survives the first year dies during the second year.

Method. The life annuity pays $\$1{,}000$ at time 0 with certainty and at time 1 with probability $0.50$. Since $v=1/1.10=10/11$, its value is:

1{,}000\left(1+\frac{10}{11}(0.50)\right) = \$1{,}454.55

A death in year 1 leaves a refund of $C-B=\$1{,}000$. Its probability is $0.50$, and it is paid at time 1. Anyone surviving to year 1 receives the second installment at time 1; after that, death in year 2 leaves no refund because two installments total $C$. Hence the expected present value of refunds is $v(0.50)(\$1{,}000)=\$454.55$. The cash-refund option is worth $\$1{,}909.09$.

Under the two-year guarantee, the installments at times 0 and 1 are both certain. Its value is:

1{,}000(1+v) = 1{,}000\left(1+\frac{10}{11}\right) = \$1{,}909.09

The values are equal. In this scenario, a member who dies in year 1 produces a refund at time 1 equal to the guaranteed installment that would otherwise be paid then; a member who survives to time 1 receives that installment directly. This equality depends on the specified contract, timing, and mortality assumptions.

Common mistake. A refund paid at death in year 1 is discounted one year, not zero years, because the model pays that benefit at time 1. Discounting it as if paid immediately would overstate its value.

Interpreting the Comparison

The formulas show why the two options need to be compared on a consistent basis. Use the same installment amount, payment timing, interest rate, and mortality assumptions for both. Then apply each contract’s actual payment conditions. If the cash-refund balance accrues interest, includes additional contributions, or uses a different deduction rule, calculate its death benefit using that rule rather than $\max(C-tB,0)$. Likewise, check whether a guarantee is measured from the retirement date and whether installments are paid in advance or in arrears.

A useful benchmark is the life-annuity value: the cash-refund option adds the expected present value of its death benefits to that benchmark. A guarantee instead removes the survival condition from the installments within the guaranteed period. The value difference is therefore driven by the size and timing of the refund balances relative to the guaranteed installments, along with mortality and discounting. A larger initial refund does not, by itself, prove that the cash-refund option is more valuable.

Key takeaway: Value each contractual payment by its amount, payment date, and probability of being paid. For a cash refund, count installments received before death to determine the balance; for an annuity-certain guarantee, treat each guaranteed installment as certain, even after the pensioner’s death.

Check Your Understanding

Use annual-in-advance payments and the stated payment timing. Answers are rounded to the nearest cent where money is requested.

A pension pays $\$500$ annually in advance and guarantees two years of payments. The annual death probability is $0.10$ in every year, and $i=0\%$. Find the value of the guarantee. (Answer: $\$5{,}050.00$)
A pension pays $B=\$400$ annually in advance with a cash-refund amount $C=\$1{,}200$. The balance is reduced by each installment received. The annual death probability is $0.25$ and $i=0\%$. Find the value of the cash-refund option. (Answer: $\$1{,}875.00$)
A pension pays $\$1{,}000$ annually in advance and has $C=\$2{,}000$. The annual effective interest rate is $10\%$; death occurs in year 1 with probability $0.50$, and all who survive year 1 die in year 2. Find and compare the values of the cash-refund option and a two-year guarantee. (Answer: Both are worth $\$1{,}909.09$)
A cash-refund contract has $B=\$600$, $C=\$1{,}800$, and $i=5\%$. The conditional death probabilities in years 1 and 2 are $0.10$ and $0.20$. Find only the expected present value of the refund benefits; no further refund is payable after year 2. (Answer: $\$212.24$)
An annuity pays $B=\$750$ annually in advance and guarantees two years. The interest rate is $4\%$, ${}_1p_x=0.90$, ${}_2p_x=0.72$, and ${}_3p_x=0$. Find the expected present value of the annuity. (Answer: $\$1{,}970.41$)

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