Mortality Sensitivity of One-Year Reserve Growth

Why the Fixed Quantity Matters

A one-year reserve recursion balances the funds available at the start of a policy year against the expected amount needed at year-end. Mortality affects this balance in two ways: it changes the probability of a death claim, and it changes the probability that a policy survives to require a next-year reserve. A comparison is therefore incomplete until it says which reserve quantities are fixed.

There are two different questions. If the next-year reserve is held fixed, a higher death probability changes the current reserve required to support it. If the current reserve and premium are held fixed, a higher death probability changes the next-year reserve required for surviving policies. The direction of change need not be the same in these two comparisons.

Model and timing: Consider a single annual policy year. The reserve $V_t$ is held immediately before the premium $P_t$ is paid at time $t$. The premium and reserve earn effective annual interest $i$. At the end of the year, a death during the year produces a benefit $B_{t+1}$; a policy that survives has reserve $V_{t+1}$ at time $t+1$. Let $q_t$ be the probability of death during the year and $p_t=1-q_t$ the probability of survival. Assume no expenses, lapses, or other cash flows.

The One-Year Recursion

At time $t$, the reserve plus premium is $V_t+P_t$. After earning interest, that amount is $(1+i)(V_t+P_t)$. At year-end, the required amount is $B_{t+1}$ if death occurred and $V_{t+1}$ if the policy survived. Taking the expectation over those two outcomes gives the recursion

(1+i)(V_t+P_t)=q_tB_{t+1}+p_tV_{t+1}.

The right side is an expected year-end outgo per policy in force at time $t$. It is not the amount paid on every policy: the death benefit is paid only with probability $q_t$, and the next-year reserve is needed only with probability $p_t$.

Equivalent forms: Solving the recursion for the current reserve gives $$ V_t=v\bigl(q_tB_{t+1}+p_tV_{t+1}\bigr)-P_t, $$ where $v=1/(1+i)$. Solving instead for the next-year reserve gives $$ V_{t+1}=\frac{(1+i)(V_t+P_t)-q_tB_{t+1}}{p_t}, \qquad p_t>0. $$

The second form requires $p_t>0$, because a reserve for surviving policies cannot be identified by dividing by a zero survival probability. Both forms use the same recursion; they answer different comparison questions because they solve for different quantities.

Why the Two Mortality Comparisons Differ

First hold $V_{t+1}$, $B_{t+1}$, $P_t$, and $i$ fixed. Replacing $p_t$ by $1-q_t$ in the current-reserve form gives

V_t=v\left[V_{t+1}+q_t(B_{t+1}-V_{t+1})\right]-P_t.

Thus the response of the current reserve to mortality depends on the difference between the death benefit and the surviving-policy reserve. When $B_{t+1}>V_{t+1}$, increasing $q_t$ raises the expected year-end amount and raises $V_t$. When $B_{t+1}<V_{t+1}$, increasing $q_t$ lowers that expected amount and lowers $V_t$. If the amounts are equal, mortality has no effect on $V_t$ in this comparison.

Now hold $V_t$, $P_t$, $B_{t+1}$, and $i$ fixed. Define the accumulated funds available at year-end by

$$ C_t=(1+i)(V_t+P_t). $$

The next-year reserve is $V_{t+1}=(C_t-q_tB_{t+1})/(1-q_t)$. Differentiating with respect to $q_t$, while holding $C_t$ and $B_{t+1}$ fixed, gives

\frac{dV_{t+1}}{dq_t} =\frac{-B_{t+1}(1-q_t)+C_t-q_tB_{t+1}}{(1-q_t)^2} =\frac{C_t-B_{t+1}}{p_t^2}.

The numerator simplifies because the two terms involving $q_tB_{t+1}$ cancel. If accumulated funds $C_t$ exceed the death benefit, higher mortality raises the required reserve per surviving policy. If $C_t$ is less than the benefit, higher mortality lowers that next-year reserve. The change in survival probability matters: the funds left after expected death claims must be allocated across the policies that survive.

Identify the target reserve.
Decide whether the comparison solves for $V_t$ or for $V_{t+1}$.

List what stays fixed.
For a current-reserve comparison, hold $V_{t+1}$ fixed. For a next-year-reserve comparison, hold $V_t$ and $P_t$ fixed, as well as the benefit and interest rate.

Use the matching form of the recursion.
Compare expected outgo for $V_t$; solve the survival-weighted balance for $V_{t+1}$.

Check the direction.
For fixed $V_{t+1}$, compare $B_{t+1}$ with $V_{t+1}$. For fixed $V_t$ and $P_t$, compare $B_{t+1}$ with $C_t$.

Worked Reserve Comparisons

Worked Example: Higher Mortality With the Next-Year Reserve Fixed

A hypothetical annual life policy has a death benefit of $B_{t+1}=\$100{,}000$, a next-year reserve of $V_{t+1}=\$60{,}000$, a premium of $P_t=\$2{,}000$, and an effective annual interest rate of $i=5\%$. Compare the current reserves when $q_t=0.05$ and when $q_t=0.10$, holding the benefit, premium, interest rate, and next-year reserve fixed.

Method. Use the current-reserve form of the recursion. The expected year-end amount at $q_t=0.05$ is

0.05(\$100{,}000)+0.95(\$60{,}000) =\$5{,}000+\$57{,}000 =\$62{,}000.

Discount that amount one year and subtract the premium paid at time $t$:

V_t=\frac{\$62{,}000}{1.05}-\$2{,}000 =\$59{,}047.6190-\$2{,}000 =\$57{,}047.62.

At $q_t=0.10$, the survival probability is $0.90$, so expected year-end outgo is

0.10(\$100{,}000)+0.90(\$60{,}000) =\$10{,}000+\$54{,}000 =\$64{,}000.

Therefore,

V_t=\frac{\$64{,}000}{1.05}-\$2{,}000 =\$60{,}952.3810-\$2{,}000 =\$58{,}952.38.

The current reserve increases by $\$58{,}952.38-\$57{,}047.62=\$1{,}904.76$. This agrees with the direction check: $B_{t+1}-V_{t+1}=\$40{,}000>0$, so higher mortality increases expected year-end outgo. Values are rounded to the nearest cent only at the final step.

Worked Example: Higher Mortality With the Current Reserve Fixed

A hypothetical policy has $V_t=\$5{,}000$, $P_t=\$1{,}000$, $i=5\%$, and a year-end death benefit $B_{t+1}=\$80{,}000$. Find the next-year reserve for $q_t=0.02$ and $q_t=0.04$, holding the current reserve, premium, benefit, and interest rate fixed.

Method. First calculate the accumulated funds:

C_t=(1.05)(\$5{,}000+\$1{,}000) =1.05(\$6{,}000) =\$6{,}300.

When $q_t=0.02$, the expected death-claim component is $0.02(\$80{,}000)=\$1{,}600$, and the survival probability is $0.98$. Hence

V_{t+1}=\frac{\$6{,}300-\$1{,}600}{0.98} =\frac{\$4{,}700}{0.98} =\$4{,}795.92.

When $q_t=0.04$, the expected death-claim component is $0.04(\$80{,}000)=\$3{,}200$, and the survival probability is $0.96$. Thus

V_{t+1}=\frac{\$6{,}300-\$3{,}200}{0.96} =\frac{\$3{,}100}{0.96} =\$3{,}229.17.

The next-year reserve decreases by $\$4{,}795.92-\$3{,}229.17=\$1{,}566.75$. Here $C_t-B_{t+1}=\$6{,}300-\$80{,}000<0$, which predicts a decrease as mortality rises. The calculation uses the same fixed current funds at both mortality rates. Values are rounded to the nearest cent only at the final step.

Worked Example: Higher Mortality Can Raise the Next-Year Reserve

A hypothetical policy has $V_t=\$2{,}000$, $P_t=\$1{,}000$, $i=5\%$, and $B_{t+1}=\$2{,}000$. Compare the next-year reserves at $q_t=0.10$ and $q_t=0.20$, with the current reserve, premium, benefit, and interest fixed.

Method. Accumulated funds are

C_t=1.05(\$2{,}000+\$1{,}000)=\$3{,}150.

At $q_t=0.10$, expected death claims are $0.10(\$2{,}000)=\$200$, leaving $\$3{,}150-\$200=\$2{,}950$ for the surviving policies. Therefore,

V_{t+1}=\frac{\$2{,}950}{0.90}=\$3{,}277.78.

At $q_t=0.20$, expected death claims are $0.20(\$2{,}000)=\$400$, leaving $\$2{,}750$ for survivors. Thus

V_{t+1}=\frac{\$2{,}750}{0.80}=\$3{,}437.50.

The next-year reserve rises by $\$159.72$. Although higher mortality increases the expected death-claim amount, it also leaves fewer surviving policies over which to allocate the remaining funds. Since $C_t-B_{t+1}=\$1{,}150>0$, the derivative formula predicts this increase. Values are rounded to the nearest cent only at the final step.

Common Exam Trap: Reversing What Is Fixed

A frequent error is to solve for the right reserve but then describe the wrong quantity as fixed. The two comparisons must be stated separately:

Comparison	Hold fixed	Changes	Direction depends on
Current reserve comparison	$V_{t+1}$, benefit, premium, and interest	$V_t$	$B_{t+1}-V_{t+1}$
Next-year reserve comparison	$V_t$, $P_t$, benefit, and interest	$V_{t+1}$	$C_t-B_{t+1}$, where $C_t=(1+i)(V_t+P_t)$

For example, in the second worked example, it would be wrong to say that $V_{t+1}$ is fixed while comparing next-year reserves at two mortality rates. It is the quantity being solved for. The correct calculation holds $V_t$ and $P_t$ fixed, so $C_t$ remains $\$6{,}300$, while $V_{t+1}$ changes.

Another error is to account for the expected death claim but omit the survival probability in the next-year-reserve calculation. Subtracting $q_tB_{t+1}$ from accumulated funds gives the amount left for survivors in expectation; it does not yet give the reserve per surviving policy. Dividing by $p_t$ is essential.

Key takeaway: Higher mortality does not have one universal effect on reserves. For a current-reserve comparison, hold $V_{t+1}$ fixed and compare the death benefit with that reserve. For a next-year-reserve comparison, hold $V_t$ and $P_t$ fixed and compare the death benefit with accumulated funds $C_t$. State the fixed quantities before judging the direction.

Check Your Understanding

For each problem, identify the reserve being solved for and use the stated fixed quantities. Money answers are rounded to the nearest cent.

A hypothetical policy has $B_{t+1}=\$50{,}000$, $V_{t+1}=\$30{,}000$, $P_t=\$500$, and $i=4\%$. How much does $V_t$ change when $q_t$ increases from $0.03$ to $0.05$, with the other listed quantities fixed? (Answer: $V_t$ increases by $\$384.62$.)
A hypothetical policy has $V_t=\$4{,}000$, $P_t=\$500$, $B_{t+1}=\$20{,}000$, and $i=4\%$. Find $V_{t+1}$ when $q_t=0.05$. (Answer: $\$3{,}873.68$.)
For a hypothetical policy, $V_t=\$2{,}000$, $P_t=0$, $B_{t+1}=\$4{,}000$, and $i=5\%$. With these quantities fixed, find $V_{t+1}$ at $q_t=0.10$ and $q_t=0.20$, and state the direction of change. (Answer: $\$1{,}888.89$ at $q_t=0.10$, $\$1{,}625.00$ at $q_t=0.20$; it decreases by $\$263.89$.)
In a current-reserve comparison, $B_{t+1}=\$42{,}000$ and $V_{t+1}=\$42{,}000$. If mortality changes while the premium and interest rate remain fixed, what is the effect on $V_t$? (Answer: No change in $V_t$ due to the mortality change.)
A hypothetical policy has $V_t=\$1{,}000$, $P_t=\$1{,}000$, $i=5\%$, and $B_{t+1}=\$2{,}000$. With $V_t$, $P_t$, the benefit, and interest held fixed, does increasing $q_t$ raise or lower $V_{t+1}$? (Answer: It raises $V_{t+1}$, since $C_t=\$2{,}100>B_{t+1}=\$2{,}000$.)

Comparison	Hold fixed	Changes	Direction depends on
Current reserve comparison	\(V_{t+1}\), benefit, premium, and interest	\(V_t\)	\(B_{t+1}-V_{t+1}\)
Next-year reserve comparison	\(V_t\), \(P_t\), benefit, and interest	\(V_{t+1}\)	\(C_t-B_{t+1}\), where \(C_t=(1+i)(V_t+P_t)\)

What You'll Learn