Inferring Mortality from Adjacent Reserves

Why Adjacent Reserves Contain Mortality Information

A reserve recursion balances the funds available at the start of a policy year against the expected amounts needed at its end. The balance depends on both the interest earned and the chance of each outcome: death or survival. If the interest rate, premium, death benefit, and two consecutive reserves are known, the recursion can often be rearranged to infer the one-year probability of death.

The result is only meaningful when the cash-flow timing and reserve definitions match the recursion. In particular, the premium must be assigned to the correct point in the year, the death benefit must be the amount actually paid on death, and the next reserve must be the amount held for a surviving policyholder under the stated timing convention.

Definition: Let ${}_kV_x$ be the reserve at time $k$, immediately before a premium due at that time. Suppose a premium $P$ is paid at that time, the effective annual interest rate is $i$, a death benefit $B$ is paid at the end of the year of death, and ${}_{k+1}V_x$ is the reserve at time $k+1$ for a survivor, immediately before the next premium. The one-year death and survival probabilities are ${}_1q_{x+k}$ and ${}_1p_{x+k}=1-{}_1q_{x+k}$, respectively.

The Mortality Inference Formula

Under the assumptions in the definition, the reserve recursion is

({}_kV_x+P)(1+i) = {}_1q_{x+k}B + {}_1p_{x+k}{}_{{k+1}}V_x .

The left side is the value at year-end of the reserve plus the premium collected at the start of the year. The right side is the probability-weighted year-end outgo: the death benefit if death occurs, or the next reserve if the policyholder survives. Since ${}_1p_{x+k}=1-{}_1q_{x+k}$, this equation can be rearranged to isolate the unknown death probability.

{}_1q_{x+k} = \frac{({}_kV_x+P)(1+i)-{}_{k+1}V_x} {B-{}_{k+1}V_x}, \qquad B\ne{}_{k+1}V_x.

The denominator is the difference between the death outgo and the survivor reserve. It appears because increasing the death probability shifts probability weight away from the survivor reserve and toward the death benefit. The next reserve must therefore be subtracted from the death benefit; dividing only by $B$ would ignore the outgo on survival.

Condition: The formula requires $B\ne{}_{k+1}V_x$. A valid inferred probability must also satisfy $0\le{}_1q_{x+k}\le1$. If the calculation falls outside that range, the supplied figures are inconsistent with the stated recursion and assumptions; do not force the result into the probability range.

Why the Rearrangement Works

Let $q={}_1q_{x+k}$ and $V_{\text{next}}={}_{k+1}V_x$ for this derivation. Substituting the survival probability $1-q$ into the recursion gives

({}_kV_x+P)(1+i) = qB+(1-q)V_{\text{next}} = V_{\text{next}}+q(B-V_{\text{next}}).

Subtract $V_{\text{next}}$ from both sides. If $B-V_{\text{next}}\ne0$, divide by that difference to obtain

q= \frac{({}_kV_x+P)(1+i)-V_{\text{next}}} {B-V_{\text{next}}}.

This also explains the feasibility check. For a death benefit larger than the next reserve, the expected year-end amount must lie between those two amounts. In that case,

V_{\text{next}} \le ({}_kV_x+P)(1+i) \le B

is equivalent to $0\le q\le1$. If the death benefit is less than the survivor reserve, the endpoints reverse, but the same requirement that the inferred probability lie in $[0,1]$ still applies.

A Reliable Calculation Sequence

Fix the timing.
Confirm when the reserve is measured, when the premium is paid, when interest is credited, and when each claim or next reserve is valued.

Write the recursion before rearranging.
Place the death benefit on the death branch and the next reserve on the survival branch, each weighted by its probability.

Calculate the accumulated funds.
Evaluate $({}_kV_x+P)(1+i)$, keeping the premium and reserve on the same timing basis.

Solve and check.
Use the mortality formula, check the probability range, and substitute the result back into the original recursion.

Worked Applications

Worked Example: Inferring Mortality in a Final Term Year

A hypothetical term policy is in its final year. The reserve immediately before the annual premium is ${}_kV_x=\$680$, the premium is $P=\$120$, the effective annual interest rate is $i=5\%$, and the end-of-year death benefit is $B=\$4{,}200$. A survivor receives no maturity payment, so ${}_{k+1}V_x=\$0$. Find the one-year death probability. Assume the premium is paid at the start of the year and the death benefit at the end of the year.

Method. Apply the annual recursion with the stated before-premium reserve convention. First, accumulate the reserve and premium:

({}_kV_x+P)(1+i) = (\$680+\$120)(1.05) = \$800(1.05) = \$840.

The survivor branch has value zero. Thus the recursion is $\$840=q(\$4{,}200)+(1-q)(\$0)$, giving

{}_1q_{x+k} = \frac{\$840-\$0}{\$4{,}200-\$0} = 0.20.

The inferred probability is $20\%$, which is between zero and one. Checking the recursion, a $20\%$ chance of a \$4,200 claim and an $80\%$ chance of zero outgo give $0.20(\$4{,}200)+0.80(\$0)=\$840$, equal to the accumulated funds. The mistake to avoid is treating the zero survivor reserve as a maturity benefit; it is simply the reserve for a survivor after coverage expires.

Worked Example: Including the Survivor Reserve in the Denominator

For a hypothetical policy year, ${}_kV_x=\$1{,}800$, $P=\$200$, $i=4\%$, $B=\$12{,}000$, and ${}_{k+1}V_x=\$2{,}000$. The premium is paid at the start of the year, the death benefit is paid at year-end, and the reserves are measured immediately before premiums. Infer the one-year death probability.

Method. Accumulate the starting reserve and premium:

(\$1{,}800+\$200)(1.04) = \$2{,}000(1.04) = \$2{,}080.

The death and survival amounts differ by $\$12{,}000-\$2{,}000=\$10{,}000$. Therefore,

{}_1q_{x+k} = \frac{\$2{,}080-\$2{,}000}{\$12{,}000-\$2{,}000} = \frac{\$80}{\$10{,}000} = 0.008.

The inferred death probability is $0.8\%$, and the survival probability is $0.992$. Substitution checks the result: $0.008(\$12{,}000)+0.992(\$2{,}000)=\$96+\$1{,}984=\$2{,}080$.

Common mistake and correction. Dividing the \$80 excess by the death benefit alone gives $\$80/\$12{,}000=0.0067$ approximately. That is wrong because it treats survival as if its year-end outgo were zero. The correct excess over the survivor amount is $B-{}_{k+1}V_x=\$10{,}000$, so the correct probability is $0.008$.

Worked Example: Checking a High Inferred Probability

A hypothetical benefit has a death amount $B=\$50{,}000$. The reserve before the premium is ${}_kV_x=\$40{,}000$, the premium is $P=\$8{,}000$, the effective annual interest rate is $i=3\%$, and the next reserve for a survivor is ${}_{k+1}V_x=\$48{,}000$. All reserves are measured immediately before premiums. Find the one-year death probability and verify the result.

Method. The accumulated amount is

(\$40{,}000+\$8{,}000)(1.03) = \$48{,}000(1.03) = \$49{,}440.

The gap between the death benefit and the survivor reserve is $\$50{,}000-\$48{,}000=\$2{,}000$. The excess of accumulated funds over the survivor reserve is $\$49{,}440-\$48{,}000=\$1{,}440$. Hence

{}_1q_{x+k} = \frac{\$1{,}440}{\$2{,}000} = 0.72.

The result is a valid probability, though it is high. The survival probability is $1-0.72=0.28$. To check, the probability-weighted year-end amount is

0.72(\$50{,}000)+0.28(\$48{,}000) = \$36{,}000+\$13{,}440 = \$49{,}440.

This matches the accumulated funds. A useful exam check is to compare the accumulated amount with the two possible year-end amounts: it lies between \$48,000 and \$50,000, consistent with a probability between zero and one.

Timing, Identifiability, and Exam Traps

The formula above assumes a reserve immediately before a premium. If instead the starting reserve is stated immediately after the premium, call it $W_k$. There is then no premium to add again in the recursion:

W_k(1+i) = {}_1q_{x+k}B + {}_1p_{x+k}{}_{k+1}V_x, \qquad {}_1q_{x+k} = \frac{W_k(1+i)-{}_{k+1}V_x} {B-{}_{k+1}V_x}.

Here the next reserve is still defined immediately before the next premium. Adding $P$ to $W_k$ would count the premium twice. If a benefit is paid at a different time, or if the premium is payable at a different time, the cash-flow equation must be adjusted before inferring mortality; the displayed formulas do not automatically apply.

There are cases where the data cannot identify mortality. If $B={}_{k+1}V_x$, the two year-end branches have the same amount, so changing the probability does not change the expected outgo. If the accumulated funds also equal that common amount, every probability satisfies the recursion; if they do not, no probability satisfies it. More generally, an inferred value outside $[0,1]$ signals inconsistent inputs, a timing mismatch, or a recursion that does not describe the contract as stated.

Using the wrong denominator. The relevant difference is $B-{}_{k+1}V_x$, not $B$ alone.
Mixing reserve timings. A before-premium reserve requires adding the premium; an after-premium reserve does not.
Using the face amount without checking the claim. $B$ must be the benefit actually payable on death under the specified contract, including any adjustments stated in the problem.
Skipping the probability check. A numerical answer is not a valid mortality probability unless it lies between zero and one.

Inferring mortality from reserves is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The essential discipline is to write the cash-flow recursion with consistent timing, solve for the probability, and verify the result in the original equation.

Check Your Understanding

For each problem, use the timing and contract assumptions stated. Check whether the result is a valid probability.

A hypothetical policy has ${}_kV_x=\$500$, $P=\$100$, $i=4\%$, $B=\$3{,}000$, and ${}_{k+1}V_x=\$500$. The reserve is before the premium; the death benefit is paid at year-end. Find ${}_1q_{x+k}$. (Answer: $0.0496$, or $4.96\%$.)
A reserve of $W_k=\$2{,}100$ is measured immediately after the premium. With $i=5\%$, $B=\$12{,}000$, and next before-premium survivor reserve ${}_{k+1}V_x=\$2{,}000$, infer the death probability. (Answer: $0.0205$, or $2.05\%$.)
For a before-premium reserve of ${}_kV_x=\$9{,}600$, a premium $P=\$400$, $i=3\%$, $B=\$20{,}000$, and ${}_{k+1}V_x=\$10{,}000$, find and verify the death probability. (Answer: $0.03$, or $3\%$.)
Suppose ${}_kV_x=\$1{,}000$, $P=\$50$, $i=4\%$, $B=\$5{,}000$, and ${}_{k+1}V_x=\$1{,}100$, with before-premium reserves. What does the recursion imply? (Answer: ${}_1q_{x+k}=-8/3{,}900\approx-0.0021$; the data are inconsistent with the stated assumptions because this is not a valid probability.)
With ${}_kV_x=\$6{,}000$, $P=\$400$, $i=5\%$, $B=\$9{,}000$, and ${}_{k+1}V_x=\$9{,}000$, can the recursion identify a death probability? (Answer: No. The two outcome amounts are equal, but accumulated funds are $\$6{,}720$, not $\$9{,}000$, so no probability satisfies the recursion.)

What You'll Learn