Deriving the Whole-Life Reserve Identity

Introduction

A prospective reserve is the present value of future benefits less the present value of future premiums, evaluated at a specified time and under a specified information condition. For a whole-life policy with annual level premiums, deriving this value requires careful attention to whether the valuation is just before or just after a premium is due.

This tutorial derives the reserve identity for a discrete whole-life policy and shows how the equivalence premium makes the reserve at issue zero. It also connects the identity to the annual reserve recursion. The methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Model and Timing

Assume a life aged $x$ at issue is covered for life. The insurer pays a benefit $B$ at the end of the year of death. The policyholder pays a level annual premium $P$ at the beginning of each year while alive, including at issue. Premiums stop at death. There are no expenses, lapses, or other cash flows, and the effective annual interest rate $i$ is constant, with $v=1/(1+i)$.

Write ${}_kp_x$ for the probability the life survives $k$ years from age $x$, and ${}_kq_x$ for the probability of death within $k$ years. In particular, $p_x={}_1p_x$ and $q_x={}_1q_x=1-p_x$. The whole-life insurance factor $A_x$ is the expected present value of a unit benefit paid at the end of the year of death. The whole-life annuity-due factor $\ddot{a}_x$ is the expected present value of unit payments at the beginnings of years while the life is alive, starting immediately.

Definition: For this tutorial, ${}_kV_x$ is the prospective reserve at time $k$, conditional on survival to that time, immediately before the premium due at time $k$. It includes that premium among the future premiums. For a unit benefit, the benefit value is $A_{x+k}$, and the future premium value is $P\ddot{a}_{x+k}$.

This timing convention matters. At time $k$, the benefit is valued using mortality from age $x+k$, conditional on the life being alive then. The annuity-due factor includes a payment immediately at time $k$, because the valuation is just before that premium is paid.

For a unit benefit, the expected present value of future benefits at duration $k$ is $A_{x+k}$. The expected present value of premiums still due, including the current premium at time $k$, is $P\ddot{a}_{x+k}$. Subtracting the latter from the former gives the prospective reserve. A benefit $B$ scales the insurance value by $B$.

Theorem: Under the stated model, the reserve immediately before the premium due at time $k$, conditional on survival to that time, is $$ {}_kV_x = B A_{x+k} - P\ddot{a}_{x+k}. $$ For a unit benefit, set $B=1$. The formula does not describe a reserve immediately after the time-$k$ premium has been paid; that valuation excludes the premium just received from future premiums.

The equivalence principle sets the premium so that the expected present value of premiums at issue equals the expected present value of benefits at issue. Thus, for a benefit $B$,

P=\frac{B A_x}{\ddot{a}_x}.

Substituting this premium into the prospective formula at issue proves the cancellation:

{}_0V_x = B A_x-P\ddot{a}_x = B A_x-\frac{B A_x}{\ddot{a}_x}\ddot{a}_x =0.

The cancellation depends on matching the premium and benefit values under the same assumptions and timing convention. In particular, $\ddot{a}_x$ includes the premium payable immediately at issue. Using an annuity-immediate factor instead would omit that payment and would not give a zero reserve at issue under this definition.

Why the Reserve Recursion Agrees

The same reserve can be checked one year at a time. Consider time $k$, immediately before the premium, and assume a unit benefit. The current premium reduces the net liability by $P$. Over the next year, death occurs with probability $q_{x+k}$, in which case the insurer pays 1 at year-end. With probability $p_{x+k}$, the life survives and the reserve at time $k+1$ is ${}_{k+1}V_x$, measured before the next premium. Discounting these outcomes one year gives

{}_kV_x=-P+v\left(q_{x+k}+p_{x+k}{}_{k+1}V_x\right).

Multiplying by $1+i$ and rearranging produces the annual reserve recursion:

\left({}_kV_x+P\right)(1+i) =q_{x+k}+p_{x+k}{}_{k+1}V_x,

or, when $p_{x+k}>0$,

{}_{k+1}V_x = \frac{\left({}_kV_x+P\right)(1+i)-q_{x+k}}{p_{x+k}}.

The prospective identity satisfies this recursion because the insurance and annuity factors themselves obey one-year decompositions:

A_{x+k}=v\left(q_{x+k}+p_{x+k}A_{x+k+1}\right), \qquad \ddot{a}_{x+k}=1+v p_{x+k}\ddot{a}_{x+k+1}.

The insurance decomposition separates death in the coming year from survival to the next age. The annuity-due decomposition counts the payment now, then values the remaining annuity only if the life survives. Substituting both decompositions into $A_{x+k}-P\ddot{a}_{x+k}$ yields the same recursion. The recursion is not a substitute for the timing assumptions: changing the valuation from before to after a premium changes which premium is included.

Identify the valuation time.
State whether the reserve is immediately before or after the premium due at that time.

Condition on survival.
At duration $k$, use insurance and annuity values from age $x+k$.

Subtract future premium value.
For a before-premium reserve, include the premium due now in the annuity-due value.

Check issue and recursion.
Verify the equivalence cancellation at issue and, when useful, the one-year reserve recursion.

Worked Applications

Worked Example: Calculating Reserves from an Age-Specific Mortality Model

A hypothetical whole-life policy pays 1 at the end of the year of death. The effective annual interest rate is $25\%$. The mortality assumptions are $q_x=0.5$, $q_{x+1}=0.5$, and $q_{x+2}=1$, so death occurs by the end of the third policy year. Premiums are level and payable annually in advance while the life is alive. Calculate the net premium and the reserves at durations 0, 1, and 2, each immediately before its due premium.

Method. Calculate the issue insurance and annuity-due values, set the premium by equivalence, then apply the prospective formula at each duration. The final-year mortality assumption $q_{x+2}=1$ means $p_{x+2}=0$, so no annuity payment is due at time 3.

The discount factor is $v=1/1.25=0.8$. The probabilities of surviving one and two years are ${}_1p_x=0.5$ and ${}_2p_x=0.5(0.5)=0.25$. Therefore,

$$ A_x=(0.5)(0.8)+(0.5)(0.5)(0.8^2)+(0.5)(0.5)(1)(0.8^3) =0.4+0.16+0.128=0.688. $$

The annuity-due value is

\ddot{a}_x=1+(0.5)(0.8)+(0.25)(0.8^2) =1+0.4+0.16=1.56.

Thus the equivalence premium is

P=\frac{0.688}{1.56}=\frac{86}{195}\approx 0.441026.

At issue, ${}_0V_x=0.688-(86/195)(1.56)=0$, as required. At duration 1, the conditional insurance value is

A_{x+1}=(0.5)(0.8)+(0.5)(1)(0.8^2)=0.4+0.32=0.72, \qquad \ddot{a}_{x+1}=1+(0.5)(0.8)=1.4.

Hence

{}_1V_x=0.72-\frac{86}{195}(1.4) =\frac{702}{975}-\frac{602}{975} =\frac{4}{39}\approx 0.102564.

At duration 2, the life is certain to die at the end of the next year. Thus $A_{x+2}=0.8$ and $\ddot{a}_{x+2}=1$, giving

{}_2V_x=0.8-\frac{86}{195} =\frac{14}{39}\approx 0.358974.

As a check, the recursion from time 0 gives $(0+P)(1.25)=0.551282$, while $q_x+p_x{}_1V_x=0.5+0.5(4/39)=0.551282$. The recursion from time 1 gives $({}_1V_x+P)(1.25)=0.679487$, equal to $0.5+0.5(14/39)=0.679487$. The calculations agree. A common mistake is to use the age-$x$ factors again at duration 1; the reserve is conditional on survival and must use the mortality values from age $x+1$.

Worked Example: Scaling the Identity to a Dollar Benefit

A hypothetical whole-life policy pays $50,000 at the end of the year of death. The effective annual interest rate is $25\%$, and the mortality assumptions are $q_x=0.2$, $q_{x+1}=0.5$, and $q_{x+2}=1$. Find the annual net premium and the reserve at duration 1 immediately before the premium due then.

Method. First calculate factors per dollar of benefit, then multiply by $50,000. This avoids treating a unit-benefit factor as though it were already a dollar amount.

Again $v=0.8$. Survival probabilities from issue are $0.8$ for one year and $0.8(0.5)=0.4$ for two years. The issue insurance value per dollar is

$$ A_x=(0.2)(0.8)+(0.8)(0.5)(0.8^2)+(0.4)(1)(0.8^3) =0.16+0.256+0.2048=0.6208. $$

The annuity-due value is

\ddot{a}_x=1+(0.8)(0.8)+(0.4)(0.8^2) =1+0.64+0.256=1.896.

The premium per dollar of benefit is $0.6208/1.896=388/1185$. The annual premium is therefore

P=50{,}000\left(\frac{388}{1185}\right) =\$16{,}371.31.

At age $x+1$, the conditional insurance value per dollar is $0.4+0.32=0.72$, and the annuity-due value is $1+0.4=1.4$. Thus the reserve per dollar of benefit is

0.72-\frac{388}{1185}(1.4) =\frac{62}{237}.

The dollar reserve is $$50{,}000(62/237)=\$13{,}080.17$. As a recursion check, the reserve at issue is zero, so the reserve at duration 1 per dollar is

\frac{(0+388/1185)(1.25)-0.2}{0.8} =\frac{62}{237}.

The prospective calculation and the recursion agree. The common error avoided here is multiplying the benefit by the dollar benefit twice: $A_{x+1}$ is a value per dollar, so the whole expression must be scaled by $50,000 once.

Worked Example: Before-Premium and After-Premium Reserves

A hypothetical whole-life policy has a benefit of $100,000, an effective annual interest rate of $25\%$, and mortality assumptions $q_x=0.4$ and $q_{x+1}=1$. Find the level annual net premium and compare the reserves at duration 1 immediately before and immediately after the premium due at that time.

Method. Use the annuity-due value for the before-premium reserve. For the after-premium value, exclude the premium already paid and value only premiums payable later.

Here $v=0.8$. From age $x$, the insurance value per dollar is

A_x=(0.4)(0.8)+(0.6)(1)(0.8^2)=0.32+0.384=0.704, \qquad \ddot{a}_x=1+(0.6)(0.8)=1.48.

The premium per dollar of benefit is $0.704/1.48=88/185$. Thus

P=100{,}000\left(\frac{88}{185}\right)=\$47{,}567.57.

At duration 1, death is certain at the end of the next year, so $A_{x+1}=0.8$ and $\ddot{a}_{x+1}=1$. Immediately before the premium,

{}_1V_x=100{,}000(0.8)-47{,}567.57 =\$32{,}432.43.

Immediately after that premium is paid, it is no longer a future premium. Since $q_{x+1}=1$, there is no chance of a premium one year later, and the value of premiums still outstanding is zero. The after-premium reserve is therefore the benefit value, $$100{,}000(0.8)=\$80{,}000$. The increase of $47,567.57 is exactly the premium just received.

A common wrong calculation labels $32,432.43 as the after-premium reserve. That amount subtracts the current premium and is correct before it is paid. After payment, the current premium must not be counted as a future outflow from the insurer.

Common Timing and Equivalence Traps

Omitting the premium at issue. With premiums payable immediately, the issue premium stream is valued by $\ddot{a}_x$, not by an annuity-immediate factor. The equivalence cancellation is $BA_x-P\ddot{a}_x=0$.
Using the wrong age after conditioning. At duration $k$, conditional on survival, use $A_{x+k}$ and $\ddot{a}_{x+k}$, not the issue-age values.
Mixing before- and after-premium values. A before-premium reserve includes the premium due now among future premiums. An after-premium reserve does not.
Ignoring the benefit amount. $A_{x+k}$ is the value for a unit benefit; multiply it by $B$ when the policy benefit is $B$.
Applying equivalence to inconsistent assumptions. The premium and reserve must use the same interest, mortality, benefit timing, and premium timing assumptions for the issue cancellation to hold.

Key takeaway: For a whole-life policy valued immediately before each annual premium, ${}_kV_x=BA_{x+k}-P\ddot{a}_{x+k}$. Under the equivalence principle, $P=BA_x/\ddot{a}_x$, so the issue reserve is zero. At later durations, conditioning on survival changes the age used in the factors.

Check Your Understanding

For each problem, use the stated timing convention and check that the premium and reserve refer to the same benefit amount.

A hypothetical unit-benefit whole-life policy has $i=10\%$, $q_x=0.2$, and $q_{x+1}=1$. Premiums are annual in advance. Find the equivalence premium and the duration-1 reserve immediately before the premium. (Answer: $P=102/209\approx0.4880$; ${}_1V_x=88/209\approx0.4211$.)
A hypothetical whole-life policy pays $20,000 at the end of the year of death. Let $i=20\%$, $q_x=0.3$, and $q_{x+1}=1$. Find the level annual net premium and the duration-1 reserve immediately before the premium. (Answer: $P=\$9{,}298.25$; ${}_1V_x=\$7{,}368.42$.)
At a given duration, under a hypothetical mortality model and the stated interest assumption, suppose the unit-benefit factor is $A_{x+k}=0.63$, the annuity-due factor is $\ddot{a}_{x+k}=1.4$, and the level premium per unit of benefit is $P=0.30$. Find the before-premium reserve per unit of benefit. (Answer: $0.21$.)
A policy has benefit $80,000. At a duration, its conditional factors are $A_{x+k}=0.75$ and $\ddot{a}_{x+k}=1.5$, and its annual premium is $30,000. Find the reserve immediately before the premium. (Answer: $$15{,}000$.)
For a hypothetical unit-benefit policy, $i=25\%$, $q_x=0.4$, and $q_{x+1}=1$. Calculate $A_x$, $\ddot{a}_x$, and the equivalence premium. Then find the reserve at duration 1 immediately before the premium. (Answer: $A_x=0.704$, $\ddot{a}_x=1.48$, $P=88/185\approx0.4757$, and ${}_1V_x=12/37\approx0.3243$.)

Deriving the Whole-Life Reserve Identity

What You'll Learn

Introduction

Model and Timing