Pension Promises and Insurer Capital

Why Pension Promises Need Careful Valuation

A pension promise is defined not only by its annual amount but also by who receives each payment, when payments stop, and what happens after the pensioner dies. A pension guaranteed for a minimum period can continue to be paid after the pensioner’s death. A survivor pension, by contrast, is payable only when the pensioner has died and the eligible survivor is still alive. These conditions affect both the expected liability and its sensitivity to mortality.

For solvency purposes, a best-estimate liability for pension benefits is the expected present value of the future benefit cash flows under the stated assumptions. Capital is then assessed against adverse changes in the insurer’s financial position under the applicable framework. The examples here use simplified, explicitly stated stress scenarios to illustrate that relationship; they are not a complete regulatory capital formula.

Definition: A guaranteed period is a stated minimum payment duration for which the benefit remains payable even if the pensioner dies. A survivor benefit is a benefit payable to an eligible survivor after the pensioner’s death, usually only while that survivor remains alive. A pension benefit liability is the present value of expected future benefit payments under specified benefit, mortality, and discount assumptions.

Set the Cash-Flow Conditions Before Calculating

A valuation needs a precise payment rule. In the examples below, payments are annual and made at the end of each year. The effective annual discount rate is $i$, so a payment at the end of year $t$ is discounted by $v^t$, where $v=1/(1+i)$. Mortality probabilities are annual. A payment to the pensioner is made only if the pensioner is alive at that payment date, unless the payment falls within a guaranteed period. A survivor payment is made only if the pensioner has died by that date and the survivor is alive then.

Where two lives are involved, independence must not be silently assumed. If the model specifies independent lifetimes, the probability that the pensioner has died by year $t$ and the spouse is alive at year $t$ is the product of those probabilities. If dependence is relevant, the joint probability must instead come from an appropriate joint-life model.

A simple discrete-time liability formula makes the benefit conditions visible:

\text{Liability}=\sum_{t=1}^{n}v^t E[B_t],

where $B_t$ is the total benefit payable at the end of year $t$, and $n$ is the projection horizon. This expression values only the benefits included in $B_t$. Unless stated otherwise, the examples exclude expenses, premiums, taxes, and risk margins.

Worked Example: Value of a Four-Year-Certain Pension

A hypothetical pension promise pays $12,000 at each year-end for four years with certainty, even if the pensioner dies during that period. The annual effective discount rate is 3%. Calculate the present value of this guaranteed component.

Method. Since all four payments are certain, mortality does not reduce the probability of any payment. Set $i=0.03$, so $v=1/1.03=0.9708737864$. The present value is:

12{,}000(v+v^2+v^3+v^4).

The discount factors are $v=0.9708737864$, $v^2=0.9425959091$, $v^3=0.9151416593$, and $v^4=0.8884870479$. Their sum is $3.7170984027$, giving $12{,}000(3.7170984027)=44{,}605.1808$. Rounded to the nearest cent, the guaranteed component is $44,605.18.

Check. The discounted value is below the undiscounted total of $4(12{,}000)=$48{,}000$, as it should be at a positive interest rate. This is only the four-year-certain component; any payments after year four require separate survival assumptions.

Common mistake avoided. Multiplying a guaranteed payment by the pensioner’s survival probability would understate this component. The guarantee means the payment remains due during the stated period even if the pensioner dies.

Valuing Pensioner and Survivor Payments

For a pension paid only while its original recipient is alive, payment probability generally decreases with duration. A survivor pension adds another contingent stream. Under an annual mortality model, let $p$ be the pensioner’s probability of surviving one year and $r$ the spouse’s probability of surviving one year. Assume these probabilities are constant by year and the two lifetimes are independent. Then the probability the pensioner is alive at the end of year $t$ is $p^t$. The probability the pensioner has died by then while the spouse is alive is $(1-p^t)r^t$.

If the annual pension while the pensioner is alive is $P$, and the survivor pension is a fraction $c$ of $P$, the year-$t$ expected payment is $Pp^t+cP(1-p^t)r^t$. This applies when the survivor pension begins after the pensioner’s death, with no guaranteed-period provision or other transition rule affecting that year’s payment.

Theorem: Under the annual payment timing, constant annual survival probabilities, and independence assumptions just stated, the present value through year $n$ is $$ P\sum_{t=1}^{n}v^t p^t+cP\sum_{t=1}^{n}v^t(1-p^t)r^t. $$ The first sum values pensioner payments; the second values survivor payments. The formula does not apply unchanged if the benefit has a guarantee period, a different survivor eligibility rule, or dependent lifetimes.

The formula follows by valuing each possible payment using its probability. Let $I_t$ equal 1 if a pensioner payment is due at year $t$, and 0 otherwise; define $J_t$ similarly for the survivor payment. The year-$t$ benefit is $B_t=PI_t+cPJ_t$. By linearity of expectation, $E[B_t]=P P(I_t=1)+cP P(J_t=1)$. The two probabilities are $p^t$ and $(1-p^t)r^t$, respectively. Multiplying the expected benefit by $v^t$ and summing over payment dates gives the stated present value. The two payment events are mutually exclusive under this benefit design, but the expectation calculation does not require treating their probabilities as independent.

Worked Example: Pension with a Contingent Spouse Benefit

A hypothetical contract pays $1,000 annually to a pensioner while alive and $500 annually to the spouse after the pensioner has died, while the spouse is alive. Payments are made at the ends of years 1, 2, and 3 only. Assume independent lifetimes, annual survival probabilities $p=0.80$ for the pensioner and $r=0.90$ for the spouse, and an effective annual interest rate of 5%. Calculate the liability for these three years.

Method. Here $P=$1{,}000$, $c=0.5$, $v=1/1.05=0.9523809524$, and $v^2=0.9070294785$, $v^3=0.8638375986$. For each payment date, calculate both eligibility probabilities before discounting.

Year $t$	Pensioner alive $p^t$	Pensioner dead, spouse alive $(1-p^t)r^t$	Discounted pensioner probability	Discounted survivor probability
1	0.800000	0.200000 × 0.900000 = 0.180000	0.800000 × 0.9523809524 = 0.761905	0.180000 × 0.9523809524 = 0.171429
2	0.640000	0.360000 × 0.810000 = 0.291600	0.640000 × 0.9070294785 = 0.580499	0.291600 × 0.9070294785 = 0.264490
3	0.512000	0.488000 × 0.729000 = 0.355752	0.512000 × 0.8638375986 = 0.442285	0.355752 × 0.8638375986 = 0.307312

The discounted pensioner probabilities sum to $0.761904762+0.580498866+0.442284850=1.784688478$. The discounted survivor probabilities sum to $0.171428571+0.264489796+0.307311953=0.743230320$. Thus the liability is $1{,}000(1.784688478)+500(0.743230320)=$2{,}156.303638$, or $2,156.30 to the nearest cent.

Check. The survivor amount is half the pensioner amount, and the survivor discounted probability sum is less than one, so the survivor component is $500(0.743230320)=$371.62$, less than $500. The total also exceeds the pensioner-only value of $1,784.69, as adding a nonnegative survivor benefit should.

Common mistake avoided. The survivor benefit is not payable whenever the pensioner is dead: the spouse must also be alive at the payment date. Omitting the factor $r^t$ overstates the liability.

From Pension Liability to a Solvency Stress

A pension promise can affect solvency through the size of its expected liability and through the change in that liability under adverse assumptions. For example, a longevity stress increases the probability that pensioner payments continue. It may also reduce the probability that a survivor pension has started by a given date. The net effect depends on the pension and survivor amounts, the payment rules, and the mortality model; it should be calculated rather than presumed.

For a simplified one-factor illustration, suppose assets are held fixed during a specified stress. Define the stress loss as the increase in liability:

\text{Stress loss}=\max(0,\text{stressed liability}-\text{base liability}).

This is an illustrative calculation, not a universal definition of a solvency capital requirement. Actual frameworks can prescribe different stresses, horizons, aggregation rules, asset responses, and adjustments. A complete capital assessment must follow the applicable basis and cannot be inferred from this one liability difference alone.

Worked Example: Longevity Stress on Pension and Survivor Benefits

Use the three-year contract and base assumptions from the previous example: $1,000 a year to the pensioner while alive, $500 a year to the spouse after the pensioner’s death while the spouse is alive, independent lifetimes, spouse annual survival probability $r=0.90$, and interest $i=5\%$. Now apply a hypothetical longevity stress that raises the pensioner’s annual survival probability from $0.80$ to $0.85$, leaving all other assumptions unchanged. Calculate the stressed liability and the increase over the base liability.

Method. Recalculate the payment probabilities; do not simply increase the base liability by a percentage. The discount factors remain $0.9523809524$, $0.9070294785$, and $0.8638375986$.

For the pensioner, the survival probabilities in years 1 to 3 are $0.85$, $0.85^2=0.7225$, and $0.85^3=0.614125$. The discounted values of those probabilities are $0.85(0.9523809524)=0.809523810$, $0.7225(0.9070294785)=0.655328798$, and $0.614125(0.8638375986)=0.530504$ (approximately). Their sum is approximately $1.995357$.

For the spouse benefit, the probabilities before discounting are:

Year 1: $(1-0.85)(0.90)=0.135000$.
Year 2: $(1-0.7225)(0.90^2)=0.2775(0.81)=0.224775$.
Year 3: $(1-0.614125)(0.90^3)=0.385875(0.729)=0.281302875$.

After discounting, the values are $0.135000(0.9523809524)=0.128571429$, $0.224775(0.9070294785)=0.203877551$, and $0.281302875(0.8638375986)\approx0.243000$. The survivor discounted probability sum is approximately $0.575449$.

Therefore the stressed liability is approximately $1{,}000(1.995357)+500(0.575449)=$2{,}283.08$. The base liability was $2,156.30, so the stress loss is $2{,}283.08-2{,}156.30=\mathbf{$126.78}$, to the nearest cent. On a fixed-asset basis, this stress reduces net assets by the same $126.78.

Check. The stressed pensioner survival probabilities are higher, while the probability of the pensioner having died by each date is lower. Consequently, the pensioner component rises and the survivor component falls. The net liability still rises in this example because the increase in pensioner payments is larger than the decrease in survivor payments.

Common mistake avoided. It is incorrect to conclude that a longevity stress must increase every part of a pension liability. The pensioner and survivor streams respond in opposite directions here; the total effect depends on their amounts and the stated terms.

Duration, Guarantees, and Common Traps

The number and timing of payments matter as much as the annual amount. A certain payment duration creates cash flows that do not depend on mortality during the guarantee. A life-contingent continuation beyond that duration depends on the appropriate survival probabilities. In a contract that pays a guarantee and then continues for life, the payment rule must also specify whether the guarantee is a minimum number of payments or a separate benefit. The two designs can produce different cash flows.

A reliable valuation process keeps benefit terms, probabilities, and discounting separate until the final sum:

Specify the benefit.
Record the amount, payment dates, guarantee duration, survivor fraction, and conditions for stopping or transferring payments.

Identify who qualifies at each date.
For each payment, state whether it is certain, dependent on pensioner survival, or dependent on pensioner death and survivor survival.

Calculate expected cash flows.
Use the mortality assumptions and any necessary joint-life probabilities. Do not assume independence unless the model states it.

Discount and compare scenarios.
Apply the stated discount basis, calculate the base liability, and then repeat consistently under each specified stress.

A frequent exam error is to discount a benefit for the right number of years but use the wrong probability of payment. Another is to treat a survivor pension as payable immediately on the pensioner’s death without checking whether the contract pays only at scheduled dates. A third is to call one stress difference the full capital requirement when the problem has not supplied the broader capital framework. State the model and scope of the calculation explicitly.

Key takeaway: Pension liability and solvency effects follow from the exact payment conditions. Value guaranteed payments without inappropriate mortality reductions, value survivor benefits only when all eligibility conditions are met, and calculate stress effects on each cash-flow stream before interpreting the total.

Check Your Understanding

Use end-of-year payments and the assumptions stated in each question.

A hypothetical pension pays $2,000 with certainty at the ends of years 2, 3, and 4. At an effective annual interest rate of 4%, what is the present value? (Answer: $5,336.71)
A pension pays $1,000 at each of the next two year-ends only if the pensioner is alive. Annual survival probability is $0.90$, and the effective annual interest rate is 0%. What is the liability? (Answer: $1,710.00)
For two years, a contract pays $800 to a living pensioner, or $400 to the spouse if the pensioner has died and the spouse is alive. Assume independent lives, annual survival probabilities $0.80$ and $0.90$, respectively, and 0% interest. What is the two-year liability? (Answer: $1,340.64)
For the three-year pension and survivor contract in the longevity-stress example, what is the increase in liability when pensioner annual survival rises from 0.80 to 0.85, with the other assumptions unchanged? (Answer: $126.78)

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