Reversionary Fractions and Survivor Pension Amounts

What a Reversionary Fraction Specifies

A pension option may pay a retired member an annual pension for as long as the member lives, then pay an eligible spouse a stated proportion of that pension if the member dies first. That proportion is the reversionary fraction. For example, a fraction of $0.60$ means that the spouse’s annual pension, if it becomes payable, is 60% of the member’s pension—not 60% of the pension’s present value.

The fraction determines the amount of the survivor benefit. The expected cost also depends on the probability that the member dies while the spouse is alive, the timing and duration of payments, and interest. Keep those two tasks separate: first identify the pension amount; then value the contingent payment stream.

Definition: Let the member’s annual pension be $B$, and let the reversionary fraction be $r$, where $0\leq r\leq 1$. If the member dies before the eligible spouse and the spouse is alive when a survivor payment is due, the spouse’s annual pension is $rB$. The fraction applies to the stated member pension. Unless the contract says otherwise, it does not create a payment while the member is alive, and no survivor pension is payable after the spouse’s death.

The valuation below uses a simple annual-payment model to make the timing explicit. Payments are made at times $0,1,2,\ldots$. The member receives $B$ at a payment date if alive then. At a date $k\geq1$, the spouse receives $rB$ if the member has died by that date and the spouse is alive then. Thus the member’s payment at date $k$ and a survivor payment at that same date are mutually exclusive.

Assume independent lifetimes with constant annual survival probabilities $p_x$ for the member and $p_y$ for the spouse. Survival for $k$ years is $p_x^k$ and $p_y^k$. Deaths are represented by annual intervals, and the survivor pension starts at the first scheduled payment date at which the member is no longer alive. The effective annual interest rate is $i$, with $v=1/(1+i)$. These assumptions are a hypothetical valuation model, not a description of any particular pension plan.

Present Value of the Member and Survivor Payments

At date $k$, the probability that the member is alive is $p_x^k$. The probability that the member is dead and the spouse is alive is $(1-p_x^k)p_y^k$, using independence. Discounting each possible payment and summing gives the expected present value.

\operatorname{EPV}(r) = B\sum_{k=0}^{\infty}v^k p_x^k + rB\sum_{k=1}^{\infty}v^k(1-p_x^k)p_y^k.

The first sum includes the date-0 member payment. The second starts at date 1 because a survivor payment cannot be due at retirement at time 0 under this setup. Since the sums are geometric when $vp_x<1$, $vp_y<1$, and $vp_xp_y<1$, they can be written as

\operatorname{EPV}(r) = B\left[ \frac{1}{1-vp_x} + r\left\{ \frac{vp_y}{1-vp_y} - \frac{vp_xp_y}{1-vp_xp_y} \right\} \right].

The first term is the present value per dollar of the member’s pension. The expression in braces is the present value per dollar of survivor pension, before multiplying by $rB$. In particular, the survivor value is not the present value of a spouse’s pension from retirement regardless of when the member dies: it is restricted to dates when the member is dead and the spouse is alive.

Why the survivor factor has this form: Independence gives $P(\text{member dead and spouse alive at }k)=(1-p_x^k)p_y^k$. Expand this as $p_y^k-(p_xp_y)^k$. After discounting, summing over $k\geq1$ gives $\sum (vp_y)^k-\sum (vp_xp_y)^k$, which equals the difference of geometric-series sums in the formula. The subtraction removes dates when the member is still alive but the spouse is alive.

This derivation also explains why the expected present value is linear in $r$. Under a fixed interest and mortality basis, write it as $C_0+rC_1$, where $C_0$ is the value of the member pension and $C_1$ is the value of a full survivor pension of amount $B$. Increasing the fraction by $\Delta r$ therefore increases the expected present value by $\Delta r\,C_1$.

Worked Valuations

Worked Example: Converting a Fraction into a Survivor Pension and Its Value

A hypothetical plan pays a retired member an annual pension of $\$12{,}000$. The spouse’s reversionary fraction is $0.60$. Use an effective annual interest rate of 5%, independent lifetimes, and constant annual survival probabilities $p_x=0.90$ for the member and $p_y=0.92$ for the spouse. Payments follow the annual schedule defined above. Find the annual survivor pension and the expected present value of the option.

Method. First calculate the spouse’s annual amount. Then calculate the member and survivor factors on the stated basis, and multiply each by its applicable pension amount.

The spouse’s pension, if it becomes payable, is

rB=0.60(12{,}000)=\$7{,}200.

The discount factor is $v=1/1.05=20/21$. Therefore $vp_x=(20/21)(0.90)=6/7$, and the member factor is

\frac{1}{1-vp_x}=\frac{1}{1-6/7}=7.

For the survivor factor, $vp_y=(20/21)(0.92)=92/105$, so its geometric sum is $(92/105)/(13/105)=92/13$. Also, $vp_xp_y=(20/21)(0.90)(0.92)=138/175$, so the second geometric sum is $(138/175)/(37/175)=138/37$. The survivor factor is consequently

\frac{92}{13}-\frac{138}{37} = \frac{1610}{481} = 3.347193347\ldots.

The member pension’s value is $\$12{,}000(7)=\$84{,}000$. The survivor component is $\$12{,}000(0.60)(3.347193347\ldots)=\$24{,}099.7921\ldots$. Thus the total expected present value is $\$84{,}000+\$24{,}099.7921\ldots=\$108{,}099.79$, rounded to the nearest cent.

Check. The total exceeds the value of the member-only pension because the option adds a nonnegative survivor benefit. The common mistake avoided here is applying the fraction to the present value of the member pension; it applies to the annual pension amount, while the survivor factor values the contingent payments.

Worked Example: Comparing Alternative Reversionary Fractions

A hypothetical plan offers a member an annual pension of $\$18{,}000$. Compare the expected present values for survivor fractions $0.25$, $0.50$, and $0.75$. Use an effective annual interest rate of 5%, independent lifetimes, constant annual survival probabilities $p_x=0.90$ and $p_y=0.92$, and the same annual payment timing as above.

Method. The mortality and interest assumptions are unchanged, so the member factor is 7 and the survivor factor is $1610/481=3.347193347\ldots$. The member-only value is $\$18{,}000(7)=\$126{,}000$. The cost of a full survivor pension, before applying the fraction, is

\$18{,}000\left(\frac{1610}{481}\right) = \$60{,}249.480249\ldots.

For $r=0.25$, the survivor component is $0.25(\$60{,}249.480249\ldots)=\$15{,}062.370062\ldots$, giving a total of $\$141{,}062.37$. For $r=0.50$, it is $\$30{,}124.740124\ldots$, giving $\$156{,}124.74$. For $r=0.75$, it is $\$45{,}187.110187\ldots$, giving $\$171{,}187.11$. All dollar amounts are rounded to the nearest cent.

Check. Raising the fraction by $0.25$ raises the expected present value by the same amount each time: $0.25(\$60{,}249.480249\ldots)=\$15{,}062.37$. This equal increment confirms the linear relationship between the fraction and expected cost.

Worked Example: Finding the Largest Feasible Fraction

A hypothetical plan has a $\$10{,}000$ annual member pension and a maximum permitted expected present value of $\$100{,}000$. Use an effective annual interest rate of 5%, independent lifetimes, constant annual survival probabilities $p_x=0.90$ and $p_y=0.92$, and the stated annual payment timing. Find the largest reversionary fraction that can be selected in increments of $0.0001$ without exceeding the cost limit.

Method. The member-only value is $\$10{,}000(7)=\$70{,}000$. The cost of a full survivor pension is $\$10{,}000(1610/481)=\$33{,}471.933472\ldots$. Thus the budget leaves $\$100{,}000-\$70{,}000=\$30{,}000$ for the survivor component. The continuous fraction limit is

r_{\max} = \frac{30{,}000}{33{,}471.933472\ldots} = 0.8968944099\ldots.

Because the fraction must be selected in increments of $0.0001$, the largest feasible choice is $0.8968$, not $0.8969$. At $r=0.8968$, the total expected present value is approximately $\$99{,}996.84$. At $r=0.8969$, it is approximately $\$100{,}000.19$, which exceeds the limit. The answer is therefore $0.8968$. Rounding the continuous limit to the nearest four decimals would give a fraction that breaches the constraint.

Check. The chosen fraction is below the continuous threshold and satisfies the cost limit; the next available fraction is above the threshold and does not. This is the relevant check when the question asks for the largest permissible fraction on a specified grid.

Assumptions, Interpretation, and Common Traps

The formula is specific to its payment and mortality assumptions. If the spouse’s lifetime is dependent on the member’s, the joint probability $P(\text{member dead by }k,\text{ spouse alive at }k)$ must replace the product $(1-p_x^k)p_y^k$. If payments are monthly, continuous, or made at different points in the year, the payment times and discounting must be changed accordingly. A stated benefit payable immediately on death also requires a different timing model. Do not use the annual formula without checking these conditions.

The member’s own pension continues while the member is alive in this model. A reversionary fraction changes the contingent spouse pension; it does not reduce the member’s pension during the member’s lifetime. A plan may price or define options differently, so use the contract terms if they specify a reduction to the member’s pension.

Applying the fraction to the wrong amount. The spouse’s annual pension is $rB$, not $r$ times the member’s expected present value.
Paying the spouse while both are alive. The survivor stream requires the member to have died and the spouse to be alive at the payment date.
Ignoring whether the spouse survives. A member’s death alone does not produce a payment at dates when the spouse is no longer alive.
Rounding a constrained answer upward. If a rounded fraction must remain within a budget, check the selected value and the next permitted increment. Ordinary rounding to nearest may not preserve feasibility.

For example, if a separate hypothetical valuation has a member-only value of $\$40{,}828.4024$ and a full-fraction survivor cost of $\$13{,}280.3254$, a $\$46{,}000$ limit gives the continuous threshold

\frac{46{,}000-40{,}828.4024}{13{,}280.3254} = 0.389418\ldots.

To four decimal places this is $0.3894$. Indeed, $0.3894$ gives an expected present value of about $\$45{,}999.76$, while $0.3895$ gives about $\$46{,}001.09$. The first is within the limit and the next increment is not. This arithmetic check illustrates why the rounded answer should be tested against the original constraint.

Key takeaway: A reversionary fraction scales the spouse’s pension after the member’s death. Under a fixed valuation basis, the total expected present value equals the member-only value plus the fraction times the full survivor-pension value. State the payment timing and joint-life assumptions, then verify any constrained fraction at the requested precision.

Check Your Understanding

Use the annual payment timing and independent, constant annual survival assumptions stated in each question. Give monetary answers to the nearest cent where needed.

A member’s annual pension is $\$30{,}000$ and the reversionary fraction is $0.40$. What annual pension is payable to the spouse after the member dies, if the spouse is then eligible and alive? (Answer: $\$12{,}000$ per year.)
For $i=5\%$, $p_x=0.90$, and $p_y=0.92$, a member has an annual pension of $\$8{,}000$ and the fraction is $0.30$. Calculate the total expected present value. (Answer: $\$64{,}033.26$.)
On the same interest and mortality basis, a $\$15{,}000$ annual pension changes from fraction $0.20$ to $0.40$. By how much does expected present value increase? (Answer: $\$10{,}041.58$.)
On the same basis, a $\$5{,}000$ annual pension has an expected-present-value limit of $\$40{,}000$. Fractions are selectable only in increments of $0.0001$. What is the largest feasible fraction? (Answer: $0.2987$.)
Under independent lifetimes, at payment date $k$ the member’s survival probability is $0.70$ and the spouse’s survival probability is $0.80$. What probability should multiply the discounted survivor pension at that date? Explain why multiplying by the member’s death probability alone is insufficient. (Answer: $0.30(0.80)=0.2400$; the spouse must also be alive to receive the payment.)

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