Reserve Monotonicity Under Level-Premium Whole-Life Insurance

Why Whole-Life Reserves May Change With Duration

A level premium does not mean that a policy reserve must increase by the same amount each year—or even increase every year. The reserve compares the value of future benefits with the value of future premiums. As the insured ages, both values change, and their difference depends on the mortality pattern as well as on interest and the premium level.

This tutorial considers a hypothetical annual discrete whole-life policy with a fixed death benefit and level net premiums. The key exam skill is to separate two questions: what exact condition makes the reserve increase during one particular year, and what assumptions are sufficient to make reserves nondecreasing across durations?

Definition: Consider a whole-life policy issued at age $x$, with death benefit $B$ payable at the end of the year of death. Level annual net premiums $P$ are payable at the beginning of each year while the insured is alive. The reserve ${}_kV_x$ is measured immediately before the premium due at time $k$. Write $q_{x+k}$ for the probability of death during the next year and $p_{x+k}=1-q_{x+k}$. The effective annual interest rate is $i$, and $v=1/(1+i)$.

The net premium is determined by the equivalence principle on the issue basis: at issue, the expected present value of premiums equals the expected present value of benefits. There are no expenses, policyholder dividends, or other benefits in this model. Unless stated otherwise, assume the relevant annuity and insurance values are finite.

The One-Year Test for Reserve Growth

With the stated timing, the one-year reserve recursion is

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

The premium due at time $k$ is added to the reserve before interest is earned. At the end of the year, the benefit is paid if death occurs; if the insured survives, the policy continues with reserve ${}_{k+1}V_x$. Solving the recursion for the next reserve and subtracting the current one gives the exact local test.

{}_{k+1}V_x-{}_kV_x = \frac{i\,{}_kV_x+(1+i)P-q_{x+k}B}{p_{x+k}}, \qquad p_{x+k}>0.

Because $p_{x+k}>0$, the sign of the reserve change is the sign of the numerator. The reserve is nondecreasing over this year exactly when $i\,{}_kV_x+(1+i)P-q_{x+k}B\ge 0$, and it increases strictly exactly when that numerator is positive. Equality means the reserve is unchanged. If $p_{x+k}=0$, there is no surviving policy at the next duration, so this solved-for-next-reserve test does not apply.

Why the test holds: Divide the recursion by $p_{x+k}$ to obtain ${}_{k+1}V_x=((1+i)({}_kV_x+P)-q_{x+k}B)/p_{x+k}$. Subtracting ${}_kV_x$, and using $1-p_{x+k}=q_{x+k}$, leaves the numerator $i\,{}_kV_x+(1+i)P-q_{x+k}B$, divided by $p_{x+k}$. Since the denominator is positive, the numerator determines whether the reserve rises, stays level, or falls.

This criterion is exact for a given year, whether or not the premium is net. Its interpretation as a net-premium whole-life result requires the stated benefit and premium timing. For gross premiums or different expense timing, the recursion must include the corresponding expenses; the displayed numerator cannot be used unchanged.

Worked Example: Apply the Local Test to One Policy Year

A hypothetical whole-life policy has $B=\$5{,}000$, $i=4\%$, $q_{x+k}=0.08$, and ${}_kV_x=\$1{,}000$. The level annual premium is $P=\$300$. Use the one-year recursion to determine the next reserve and whether it increases. Assume $p_{x+k}>0$.

Method. Evaluate the local numerator first, then divide by the survival probability. This avoids treating a nonnegative numerator as evidence of a strictly positive increase.

The survival probability is $p_{x+k}=1-0.08=0.92$. The reserve-change numerator is

i\,{}_kV_x+(1+i)P-q_{x+k}B =0.04(1000)+1.04(300)-0.08(5000) =40+312-400=-48.

Therefore,

{}_{k+1}V_x-{}_kV_x=\frac{-48}{0.92} =-\frac{1200}{23}\approx-\$52.17.

The reserve decreases. Its next value is ${}_{k+1}V_x=1000-1200/23=21800/23\approx\$947.83$, rounded to the nearest cent. Check directly in the recursion: $(1000+300)(1.04)=\$1{,}352$, while $0.08(5000)+0.92(21800/23)=400+952=\$1{,}352$. The values agree.

A Sufficient Condition for Nondecreasing Net Reserves

For a whole-life policy, the net premium and prospective reserve give a useful structural explanation of reserve patterns. Let $\ddot{a}_{x+k}$ be the value at age $x+k$ of an annuity-due of 1 payable annually while alive, and let $A_{x+k}$ be the value of 1 payable at the end of the year of death. The usual identity for these discrete whole-life values is $A_{x+k}+d\ddot{a}_{x+k}=1$, where $d=i/(1+i)$.

Under the net premium equivalence principle, $P/B=A_x/\ddot{a}_x$. The prospective reserve at duration $k$, measured before the premium due then, is the value of future benefits less the value of future premiums:

{}_kV_x=B A_{x+k}-P\ddot{a}_{x+k}.

Using $A_{x+k}=1-d\ddot{a}_{x+k}$ and $P/B=A_x/\ddot{a}_x=1/\ddot{a}_x-d$, this becomes

{}_kV_x =B\left(1-\frac{\ddot{a}_{x+k}}{\ddot{a}_x}\right).

Theorem: For a fixed-benefit, annual discrete whole-life policy on a net level-premium basis, if the one-year death probabilities $q_{x+k}$ are nondecreasing with attained age, then the reserves are nondecreasing by duration. This is a sufficient condition, not a necessary one. Strict increase in a particular year requires a strict decrease in the corresponding future-life annuity value.

To justify the condition, write the annuity-due value as a sum of discounted survival probabilities:

\ddot{a}_{x+k} =\sum_{t=0}^{\infty}v^t\,{}_tp_{x+k}, \qquad {}_tp_{x+k}=\prod_{j=0}^{t-1}p_{x+k+j}, \qquad {}_0p_{x+k}=1.

If death probabilities are nondecreasing with age, then survival probabilities are nonincreasing with age. For each $t\ge 1$, every factor in ${}_tp_{x+k}$ is at least the corresponding factor in ${}_tp_{x+k+1}$. Thus ${}_tp_{x+k}\ge{}_tp_{x+k+1}$. Since the discount weights $v^t$ are positive, summing gives $\ddot{a}_{x+k}\ge\ddot{a}_{x+k+1}$. The prospective expression then yields

{}_{k+1}V_x-{}_kV_x =\frac{B}{\ddot{a}_x} \left(\ddot{a}_{x+k}-\ddot{a}_{x+k+1}\right)\ge 0.

The conclusion is nondecrease, not automatic strict increase. For example, if mortality is constant from one duration onward, the future-life annuity values at those durations are equal, so the corresponding reserves are equal. The local recursion test remains useful when mortality is not monotone, or when a particular year must be checked directly.

Worked Example: Increasing Mortality Gives Increasing Reserves

A hypothetical policy pays $B=\$2{,}980$ at the end of the year of death. The effective annual interest rate is $i=0\%$. Its annual death probabilities are $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.50$, and $q_{x+3}=1$. Premiums are level, payable annually in advance while alive, and are net premiums. Find $P$, ${}_1V_x$, and ${}_2V_x$, and check the changes with the recursion.

Method. First calculate the issue annuity-due value and benefit value to obtain the net premium. Then calculate each prospective reserve. The increasing mortality pattern should produce nondecreasing reserves, but the recursion provides an independent arithmetic check.

At issue, the successive survival probabilities for the premium annuity are $1$, $0.90$, $0.90(0.80)=0.72$, and $0.90(0.80)(0.50)=0.36$. Since $v=1$,

\ddot{a}_x=1+0.90+0.72+0.36=2.98.

Death is certain by the end of the fourth year, so the expected present value of the benefit is $A_x=1$. The equivalence principle gives $P=B A_x/\ddot{a}_x=2980/2.98=\$1{,}000$.

At age $x+1$, the annuity-due value is $1+0.80+0.80(0.50)=2.20$. The benefit value remains 1, because death is certain eventually and interest is zero. Therefore ${}_1V_x=2980-1000(2.20)=\$780$. At age $x+2$, the annuity-due value is $1+0.50=1.50$, so ${}_2V_x=2980-1000(1.50)=\$1{,}480$. The reserves rise from the issue reserve ${}_0V_x=0$ to $\$780$, then to $\$1{,}480$.

Check the first recursion: $(0+1000)(1+0)=0.10(2980)+0.90(780)=298+702=1000$. Check the second: $(780+1000)(1+0)=0.20(2980)+0.80(1480)=596+1184=1780$. Both identities hold.

Worked Example: A Level Reserve After Mortality Becomes Constant

A hypothetical whole-life policy has $B=\$4{,}830$ and $i=5\%$. The death probability is $q_x=0.10$ in the first year and $q_{x+k}=0.20$ for every $k\ge1$. Find the net level premium and the reserves at durations 1 and 2. Round money to the nearest cent only at the final step.

Method. Use the annuity-due recursion to value premiums. Once the mortality rate is constant, the future-life annuity value is the same at each of those later ages, so the prospective reserves at durations 1 and 2 should match.

Here $v=20/21$. At ages $x+1$ and later, the annuity-due value is geometric:

\ddot{a}_{x+1} =\frac{1}{1-v(0.80)} =\frac{1}{1-\frac{16}{21}} =\frac{21}{5}.

At issue, $\ddot{a}_x=1+v(0.90)\ddot{a}_{x+1}=1+(18/21)(21/5)=23/5$. Using $A_x=1-d\ddot{a}_x$, with $d=1/21$, gives $A_x=1-(1/21)(23/5)=82/105$. Thus

P=B\frac{A_x}{\ddot{a}_x} =4830\left(\frac{82}{105}\right)\left(\frac{5}{23}\right) =4830\left(\frac{82}{483}\right) =\$820.

At duration 1, $A_{x+1}=1-(1/21)(21/5)=4/5$, so ${}_1V_x=4830(4/5)-820(21/5)=3864-3444=\$420$. The same age-specific values apply at duration 2, giving ${}_2V_x=\$420$.

The recursion confirms both results. For the first year, $(0+820)(1.05)=861$, and $0.10(4830)+0.90(420)=483+378=861$. For the next year, $(420+820)(1.05)=1302$, and $0.20(4830)+0.80(420)=966+336=1302$. The reserve increases strictly in the first year and is unchanged in the second.

When the Pattern Can Fail

Nondecreasing mortality is sufficient, but it is not a universal description of mortality patterns. If mortality falls with age, a level-premium net reserve can decrease. Consider a hypothetical policy with $i=0$, $B=\$700$, $q_x=0.40$, and $q_{x+k}=0.10$ for every $k\ge1$. Death eventually occurs with probability 1, so $A_x=1$. At age $x+1$, $\ddot{a}_{x+1}=1/0.10=10$, while $\ddot{a}_x=1+0.60(10)=7$. The net premium is $P=700/7=\$100$, and ${}_1V_x=700-100(10)=-\$300$, below the issue reserve of zero. The recursion also checks: $P=0.40(700)+0.60(-300)=280-180=100$.

This example illustrates why a general claim that whole-life reserves always rise is unsafe. A lower future mortality rate can increase the value of future premiums, and the prospective reserve can fall. The one-year numerator gives the exact direction for the year being examined; the monotone-mortality theorem gives a convenient sufficient condition across durations.

Common Exam Traps

Confusing nondecrease with strict increase. A zero numerator gives ${}_{k+1}V_x={}_kV_x$, not a strict rise.
Reversing the sign test. The change is the numerator divided by $p_{x+k}$. When $p_{x+k}>0$, a negative numerator means the reserve decreases.
Mixing timing conventions. The recursion shown here includes the premium at time $k$ before interest is earned, with the reserve measured immediately before that premium. Moving the reserve measurement to after the premium changes the equation.
Calling a sufficient condition necessary. Nondecreasing mortality ensures nondecreasing net reserves here, but a reserve may still rise in a year even when mortality is not nondecreasing across all ages.
Applying the net-premium prospective formula to a different contract. Expenses, changing benefits, or a different premium pattern alter the relevant values and may alter the recursion.

Key takeaway: For a year with positive survival probability, test reserve growth using $i\,{}_kV_x+(1+i)P-q_{x+k}B$. For a fixed-benefit whole-life policy with level net premiums, nondecreasing mortality is a sufficient condition for reserves to be nondecreasing, with equality allowed.

Check Your Understanding

For each problem, use the stated assumptions and timing. Money answers are rounded to the nearest cent where needed.

A hypothetical whole-life policy has $i=0$, $B=\$4{,}000$, and $q_{x+k}=0.25$ at every duration. Premiums are level net premiums payable annually in advance. Find $P$ and ${}_1V_x$. (Answer: $P=\$1{,}000$, ${}_1V_x=\$0$.)
A hypothetical policy has $i=0$, $B=\$1{,}100$, $q_x=0.20$, and $q_{x+k}=0.30$ for every $k\ge1$. It has level net premiums. Find $P$ and ${}_1V_x$, and state whether the reserve is nondecreasing from issue through duration 2. (Answer: $P=\$300$, ${}_1V_x=\$100$; ${}_2V_x=\$100$, so the reserve is nondecreasing, with no increase from duration 1 to 2.)
In a policy year, $i=4\%$, $B=\$5{,}000$, $q_{x+k}=0.08$, $P=\$300$, and ${}_kV_x=\$1{,}000$. Find the reserve change and classify it. (Answer: ${}_{k+1}V_x-{}_kV_x=-\$52.17$, so the reserve decreases.)
A whole-life policy has fixed death benefit and level net premiums. Its death probabilities are nondecreasing with attained age. What can be concluded about its reserves, and does that imply a strict rise every year? (Answer: Reserves are nondecreasing by duration; no, equality can occur.)
For a policy year, $i=10\%$, $B=\$1{,}000$, $q_{x+k}=0.20$, $P=\$150$, and ${}_kV_x=\$200$. Assume $p_{x+k}>0$. Find the next reserve and classify the change. (Answer: The numerator is $0.10(200)+1.10(150)-0.20(1000)=-\$15$; the change is $-15/0.80=-\$18.75$, so ${}_{k+1}V_x=\$181.25$ and the reserve decreases.)

What You'll Learn