Continuous Whole-Life Insurance Reserves

Continuous Whole-Life Reserves

A continuous whole-life policy pays a fixed benefit when death occurs and receives premiums continuously while the insured is alive. Its reserve at a duration is the conditional value, for an insurer with a living policyholder, of future benefits less future premiums. With a constant force of mortality, a constant force of interest, and level continuous premiums, the calculation has a useful feature: the reserve does not depend on the duration already elapsed.

The result depends on the cash-flow assumptions. In particular, a continuous premium stream is not the same as annual premiums, and a force of interest is not generally the same as an effective annual interest rate. The formula below applies to the stated continuous model, not automatically to policies with expenses, lapses, changing benefits, or premiums that cease at a fixed duration.

Definition: Consider a hypothetical whole-life policy issued to a life whose future lifetime $T$ has an exponential distribution with constant force of mortality $\mu>0$, so that ${}_tp_x=e^{-\mu t}$. Interest accumulates at constant force $\delta\geq 0$. The insurer pays a benefit $B$ immediately at death and receives premiums continuously at rate $P$ per year while the insured is alive. The prospective reserve $V_t$, conditional on the insured being alive at duration $t$, is the expected present value at time $t$ of future benefits less the expected present value of future premiums.

The model has no maturity benefit: the insured eventually dies, and the only benefit is $B$ at death. Premiums stop at death. There are no expenses, policyholder lapses, or other cash flows. Because $T$ is exponential, the remaining lifetime conditional on survival to duration $t$ has the same distribution as the original lifetime. Thus, at any duration, the future cash flows have the same conditional distribution as they had at issue.

Prospective Reserve Formula

At a reserve time $t$, let $s$ denote time measured forward from $t$. The probability of surviving a further $s$ years is $e^{-\mu s}$. The discounted expected benefit can be calculated by integrating over the possible times of death: the probability density of death at $s$ is $\mu e^{-\mu s}$, and the discount factor is $e^{-\delta s}$. The expected present value of the future continuous premium stream uses the probability of being alive at each future time.

\begin{aligned} \operatorname{EPV}_t(\text{benefit}) &= \int_0^\infty B e^{-\delta s}\mu e^{-\mu s}\,ds = \frac{\mu B}{\delta+\mu},\\ \operatorname{EPV}_t(\text{premiums}) &= \int_0^\infty P e^{-\delta s}e^{-\mu s}\,ds = \frac{P}{\delta+\mu},\\ V_t &= \frac{\mu B-P}{\delta+\mu}. \end{aligned}

These integrals converge because $\delta+\mu>0$. In the premium integral, $P\,ds$ is the premium received over a short interval of length $ds$, conditional on being alive at its start; the survival probability weights the chance that this payment is received. In the benefit integral, death must occur in that interval, which is why the density $\mu e^{-\mu s}$, rather than the survival probability alone, appears.

Theorem: Under the stated assumptions, the continuous whole-life reserve is constant over policy duration and is $V_t=(\mu B-P)/(\delta+\mu)$ for every $t\geq 0$, conditional on the insured being alive. The continuous net premium rate is $P=\mu B$, which gives $V_t=0$ at every duration.

The net premium result follows by equating the expected present values at issue: $\mu B/(\delta+\mu)=P/(\delta+\mu)$, so $P=\mu B$. The interest force cancels from this equation. It still affects a reserve when the actual premium rate differs from the net premium rate, through the denominator $\delta+\mu$.

Why the Reserve Satisfies Thiele’s Equation

The prospective calculation gives the reserve directly. A second check comes from the single-state Thiele equation. Over a short interval, the reserve earns interest at rate $\delta V_t$ and receives premium income at rate $P$ while the insured is alive. Death occurs at rate $\mu$. If death occurs, the insurer pays $B$ and releases the reserve $V_t$; the net outgo at death is therefore $B-V_t$. The resulting reserve equation is

\frac{dV_t}{dt} =\delta V_t+P-\mu(B-V_t) =(\delta+\mu)V_t+P-\mu B.

To check the equation, substitute the prospective reserve. Its right-hand side becomes $(\delta+\mu)(\mu B-P)/(\delta+\mu)+P-\mu B=0$. Hence $dV_t/dt=0$, as required for a reserve constant over time. This also verifies the mortality sign: the mortality term is $-\mu(B-V_t)$, because death produces a benefit outgo but releases the reserve.

Specify the cash flows.
Identify the death benefit, when it is paid, the premium rate while alive, and any cash flows excluded from the model.

Use consistent force assumptions.
Write the survival probability as $e^{-\mu s}$ and the discount factor as $e^{-\delta s}$.

Calculate the two expected present values.
Use the death-time density for the benefit and the survival probability for the continuous premiums.

Subtract premiums from benefits and check.
Apply $V_t=\operatorname{EPV}_t(\text{benefit})-\operatorname{EPV}_t(\text{premiums})$, then substitute the result into Thiele’s equation.

Worked Applications

Worked Example: Calculating the Reserve at Any Duration

A hypothetical policy has a death benefit of $B=\$120{,}000$, payable immediately at death. The future lifetime is exponential with force of mortality $\mu=0.0125$ per year. Interest is at force $\delta=0.0375$ per year, and the policyholder pays a continuous premium at rate $P=\$900$ per year while alive. Calculate the reserve at duration 12 years, conditional on survival, and check it using Thiele’s equation.

Method. Use the prospective formula. The exponential assumption means that, conditional on survival to year 12, the remaining lifetime has the same exponential distribution.

First, $\delta+\mu=0.0375+0.0125=0.0500$. The benefit numerator is $\mu B=0.0125(\$120{,}000)=\$1{,}500$. Therefore,

V_{12}=\frac{\$1{,}500-\$900}{0.0500} =\frac{\$600}{0.0500} =\$12{,}000.

The separate expected present values are $\$1{,}500/0.0500=\$30{,}000$ for the benefit and $\$900/0.0500=\$18{,}000$ for premiums. Their difference is $\$30{,}000-\$18{,}000=\$12{,}000$, which agrees with the reserve formula.

For the Thiele check, $(\delta+\mu)V_{12}+P-\mu B=0.0500(\$12{,}000)+\$900-\$1{,}500=\$600+\$900-\$1{,}500=\$0$. The reserve is thus constant, including at duration 12. The common mistake avoided here is to treat the 12 years already survived as changing the remaining lifetime distribution; that would be incorrect under constant force of mortality.

Worked Example: Finding the Premium From a Target Reserve

A hypothetical whole-life policy pays $\$150{,}000$ immediately at death. The future lifetime is exponential with $\mu=0.008$ per year, and the force of interest is $\delta=0.042$ per year. At duration 8, conditional on survival, the prospective reserve is $\$10{,}000$. Find the level continuous premium rate $P$.

Method. Rearrange the prospective reserve formula to solve for the premium rate. The memoryless property permits use of the same formula at duration 8 as at issue.

The denominator is $0.008+0.042=0.050$, and the benefit term is $\mu B=0.008(\$150{,}000)=\$1{,}200$. Substitute the target reserve:

\$10{,}000=\frac{\$1{,}200-P}{0.050}.

Multiplying both sides by $0.050$ gives $\$500=\$1{,}200-P$, so $P=\$1{,}200-\$500=\$700$ per year. Checking, $(\$1{,}200-\$700)/0.050=\$500/0.050=\$10{,}000$. This premium is below the net premium rate of $\$1{,}200$, so a positive reserve is consistent with the insurer’s future expected benefit outgo exceeding future expected premium income.

The common mistake avoided is to assume that a reserve is automatically the full expected present value of benefits. It is the expected present value of benefits less the expected present value of future premiums.

Worked Example: Converting an Effective Rate to a Force

A hypothetical policy pays $\$100{,}000$ immediately at death and receives continuous premiums of $\$800$ per year while the insured is alive. Assume an exponential future lifetime with $\mu=0.010$ per year. The effective annual interest rate is $i=0.05$. Calculate the reserve, in dollars to the cent.

Method. Convert the effective annual rate to the equivalent force of interest before applying the continuous-time formula. Since $1+i=e^\delta$,

\delta=\ln(1.05)=0.048790164\ldots, \qquad \delta+\mu=0.048790164\ldots+0.010=0.058790164\ldots.

The benefit term is $\mu B=0.010(\$100{,}000)=\$1{,}000$, so the numerator is $\$1{,}000-\$800=\$200$. Thus

V_t=\frac{\$200}{0.058790164\ldots} =\$3{,}401.93.

The reserve is $\$3{,}401.93$ at any duration, rounded to two decimal places. As a check, multiplying $\$3{,}401.93$ by $0.058790164\ldots$ gives approximately $\$200$, the numerator in the reserve formula.

The common mistake is to use $0.05$ directly as the force of interest. That gives the incorrect value $\$200/(0.05+0.010)=\$3{,}333.33$. The correction is to use $\delta=\ln(1+i)$ for a continuous-time discounting calculation; the effective annual rate and the force are different quantities.

Interpretation and Exam Traps

A reserve may be positive, zero, or negative under this mathematical definition. If $P<\mu B$, then the future expected present value of benefits exceeds that of premiums and the reserve is positive. If $P=\mu B$, the reserve is zero. If $P>\mu B$, it is negative because the future premiums have greater expected present value than the future benefit under the assumptions. A negative value is not, by itself, an algebra error; whether such a premium structure is appropriate is a separate product or valuation question.

The constancy of $V_t$ is specific to this model. It relies on constant mortality and interest forces, a fixed death benefit, and a premium stream that continues at the same rate for as long as the insured is alive. For a non-exponential lifetime, survival to a later duration generally changes the conditional distribution of remaining lifetime. Likewise, a benefit that changes with duration or a premium that stops at a specified term changes the future cash flows. The formula should not be transferred to those settings without revisiting the model.

Use the death density for the benefit. The probability of death at time $s$ is weighted by $\mu e^{-\mu s}$, not by $e^{-\mu s}$ alone.
Use survival for premiums. Continuous premiums arrive only while the insured is alive, so the premium integral includes $e^{-\mu s}$.
Keep the timing and units consistent. Here $B$ is paid at death, $P$ is a per-year continuous rate, and both $\mu$ and $\delta$ are per year.
Check the mortality sign in Thiele’s equation. The death term is $-\mu(B-V_t)$, the net outgo after allowing for reserve released at death.

Key takeaway: For continuous whole-life insurance with constant forces and a level continuous premium, the prospective reserve is $V_t=(\mu B-P)/(\delta+\mu)$ at every duration. The expected present value calculation and Thiele’s equation provide independent checks of the same result.

Check Your Understanding

For each problem, use the assumptions stated and check that the units and cash-flow timing match the continuous whole-life model.

A hypothetical policy pays $\$100{,}000$ immediately at death, has $\mu=0.010$ and $\delta=0.040$ per year, and charges $P=\$700$ per year continuously while alive. Find the reserve at duration 6, conditional on survival. (Answer: $\$6{,}000$)
A hypothetical policy has benefit $\$120{,}000$, constant mortality force $0.0125$, and a level continuous premium set at the net rate. Find that premium rate and the reserve at duration 10. (Answer: premium rate $\$1{,}500$ per year; reserve $\$0$)
A policy pays $\$200{,}000$ at death, receives continuous premiums of $\$1{,}000$ per year, and has $\mu=0.008$ and $\delta=0.032$ per year. Find the reserve at any duration and verify that the Thiele-equation derivative is zero. (Answer: reserve $\$15{,}000$; derivative $0$)
A hypothetical contract has $B=\$100{,}000$, $\mu=0.015$, and $\delta=0.035$. If its reserve is $\$12{,}000$ at duration 4, find the continuous premium rate and state whether it is above or below the net premium rate. (Answer: $P=\$900$ per year, below the net rate of $\$1{,}500$ per year)
A policy pays $\$50{,}000$ at death, charges $P=\$800$ per year continuously while alive, and has $\mu=0.020$. The effective annual interest rate is $i=0.06$. Convert the interest rate to a force and calculate the reserve to the nearest cent. (Answer: $\delta=\ln(1.06)=0.0582689\ldots$; reserve $\$2{,}555.29$)

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