Closed-Form Reserves Under Constant Forces

Introduction

When the forces and cash-flow rates in a single-state reserve model are constant, Thiele’s differential equation can be solved directly rather than evaluated through repeated numerical steps. The main challenge is to keep the cash-flow signs and the direction of time consistent: a reserve at duration $t$ is determined by the time still remaining, not just by the time already elapsed.

This tutorial derives and applies the closed-form reserve for a hypothetical term contract with continuous premiums and expenses and a death benefit paid immediately on death. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and Conventions

Consider a policy that remains in the insured state while the policyholder is alive, up to a fixed term $n$. Let $V_t$ be the reserve at duration $t$, for $0\le t\le n$. Assume the force of interest is the constant $\delta$, and the force of mortality is the constant $\mu$. Thus, conditional on being alive at duration $t$, the probability of surviving a further $s$ years is $e^{-\mu s}$, and the density of death at $s$ years from $t$ is $\mu e^{-\mu s}$.

While the policy is in force, premiums are received continuously at rate $p$ dollars per year and expenses are paid continuously at rate $e$ dollars per year. A death benefit $B$ dollars is paid immediately on death. Assume $p,e,B,\delta,\mu$ are constant for the remaining term, there are no other benefits or cash flows, and no maturity benefit is payable. The reserve at the term’s end is therefore $V_n=0$.

Definition: The net continuous cash-flow rate while the policy is in force is $c=p-e$. Under these assumptions the single-state Thiele equation is $$ \frac{dV_t}{dt}=\delta V_t+c-\mu(B-V_t) =(\delta+\mu)V_t+c-\mu B, \qquad V_n=0. $$

The term $\delta V_t$ is interest earned on the reserve. The net rate $c$ is positive when premium income exceeds expenses. On death, the insurer pays $B$ and no longer needs to hold the reserve $V_t$; the expected death-related outgo per unit time is therefore $\mu(B-V_t)$. These signs give a useful check before solving: a larger death benefit increases the reserve, while greater net premium income tends to reduce it.

Closed-Form Reserve

Write $k=\delta+\mu$. For nonnegative interest and mortality forces with $k>0$, the equation becomes $V'_t-kV_t=c-\mu B$. The terminal condition is imposed at $n$, so the equation must be solved backward to find $V_t$ at an earlier duration.

Theorem: Under the stated constant-force and constant-cash-flow assumptions, with no maturity benefit and $V_n=0$, the reserve at duration $t$ is $$ V_t=\frac{\mu B-c}{\delta+\mu} \left(1-e^{-(\delta+\mu)(n-t)}\right), \qquad 0\le t\le n. $$

The expression uses the remaining duration $n-t$. The factor $\delta+\mu$, not just $\delta$, appears because death both triggers the benefit and releases the reserve. If $\mu B=c$, the constant net premium income exactly balances the mortality-weighted benefit in the differential equation, and the formula gives $V_t=0$ at every duration. If $c>\mu B$, the formula can produce a negative reserve under this valuation model; that is an algebraic consequence of the assumed cash flows, not a reason to change the signs in the equation.

Why the Formula Holds

The closed form follows by an integrating factor. This derivation also makes the backward terminal condition explicit.

Collect the reserve terms.
Set $k=\delta+\mu$ and rewrite the equation as $V'_t-kV_t=c-\mu B$.

Apply the integrating factor.
Multiply by $e^{-kt}$. The left side becomes a product derivative: $$ \frac{d}{dt}\left(e^{-kt}V_t\right)=e^{-kt}(c-\mu B). $$

Integrate from $t$ to $n$.
Using $V_n=0$, $$ -e^{-kt}V_t=(c-\mu B)\int_t^n e^{-ks}\,ds =(c-\mu B)\frac{e^{-kt}-e^{-kn}}{k}. $$

Isolate the reserve.
Multiply by $-e^{kt}$, then simplify: $$ V_t=\frac{\mu B-c}{k}\left(1-e^{-k(n-t)}\right). $$

Two direct checks confirm the result. First, at $t=n$, the exponential is $1$, so $V_n=0$. Second, differentiating gives $V'_t=-(\mu B-c)e^{-k(n-t)}$. Substituting the formula into the Thiele equation gives the same derivative, since $kV_t+c-\mu B=-(\mu B-c)e^{-k(n-t)}$.

Worked Example: Reserve Three Years Before the Term

A hypothetical five-year contract has a continuous premium rate of $p=\$1{,}500$ per year, continuous expenses of $e=\$300$ per year, and an immediate death benefit of $B=\$30{,}000$. Use a constant force of interest $\delta=0.04$ per year and a constant force of mortality $\mu=0.06$ per year. Find the reserve at duration $t=2$. There is no maturity benefit, and the terminal reserve is zero.

Method. Use the closed-form solution with the remaining duration $n-t$. First, the net cash-flow rate is $c=p-e=\$1{,}500-\$300=\$1{,}200$ per year. The mortality-weighted death benefit is $\mu B=0.06(\$30{,}000)=\$1{,}800$ per year, and $k=\delta+\mu=0.04+0.06=0.10$ per year.

The remaining term is $5-2=3$ years. Therefore, $$ V_2=\frac{1{,}800-1{,}200}{0.10}\left(1-e^{-0.10(3)}\right) =6{,}000(1-e^{-0.3}). $$

Using $e^{-0.3}=0.7408182207$, the reserve is $6{,}000(0.2591817793)=\$1{,}555.0906758$. Rounded to the nearest cent, $V_2=\$1{,}555.09$.

Check. The terminal condition is zero because the formula gives $6{,}000(1-1)=0$ at $t=5$. At $t=2$, the equation’s right side is $0.10(1{,}555.0907)+1{,}200-1{,}800=-444.4909$, which agrees with the derivative $-600e^{-0.3}=-444.4909$ dollars per year. This negative derivative is consistent with the reserve declining as the term approaches its endpoint. The common mistake avoided here is using elapsed time $2$ instead of remaining time $3$.

Worked Example: Separating Premium Income from Expenses

A hypothetical six-year policy pays an immediate death benefit of $B=\$40{,}000$. Premiums are received continuously at $p=\$1{,}800$ per year, and expenses are paid continuously at $e=\$250$ per year. The constant forces are $\delta=0.03$ and $\mu=0.07$ per year. Find the reserve at duration $t=1$, assuming no maturity benefit and $V_6=0$.

Method. Combine the continuous premium and expense into their net rate, $c=\$1{,}800-\$250=\$1{,}550$ per year. The mortality-weighted benefit is $\mu B=0.07(\$40{,}000)=\$2{,}800$ per year, and $k=0.03+0.07=0.10$ per year. The remaining term is $6-1=5$ years.

Substitution gives $$ V_1=\frac{2{,}800-1{,}550}{0.10}\left(1-e^{-0.10(5)}\right) =12{,}500(1-e^{-0.5}). $$

Since $e^{-0.5}=0.6065306597$, $V_1=12{,}500(0.3934693403)=\$4{,}918.3667538$. Rounded to the nearest cent, $V_1=\$4{,}918.37$.

Check. The reserve is positive because the mortality-weighted benefit, $\$2{,}800$ per year, exceeds the net premium rate, $\$1{,}550$ per year. The terminal value remains zero. Treating the full premium as net income would incorrectly use $c=\$1{,}800$, omitting the $\$250$ annual expense and understating the reserve.

Worked Example: Checking Reserves at Two Durations

A hypothetical four-year policy has $p=\$2{,}200$ per year, $e=\$400$ per year, and an immediate death benefit of $B=\$35{,}000$. Assume constant forces $\delta=0.02$ and $\mu=0.08$ per year, no maturity benefit, and terminal reserve $V_4=0$. Calculate the reserves at durations 0 and 2.

Method. The net continuous rate is $c=\$2{,}200-\$400=\$1{,}800$ per year. The mortality-weighted benefit is $\mu B=0.08(\$35{,}000)=\$2{,}800$ per year, so $\mu B-c=\$1{,}000$ per year. Also, $k=0.02+0.08=0.10$ per year.

At issue, the remaining term is 4 years: $$ V_0=\frac{1{,}000}{0.10}(1-e^{-0.10(4)}) =10{,}000(1-e^{-0.4}) =\$3{,}296.799535\ldots $$

At duration 2, the remaining term is $4-2=2$ years: $$ V_2=10{,}000(1-e^{-0.2}) =\$1{,}812.692469\ldots $$

Rounded to the nearest cent, $V_0=\$3{,}296.80$ and $V_2=\$1{,}812.69$. The reserve at duration 2 is lower because only two years of coverage remain, compared with four years at issue. As a differential-equation check at duration 2, the right side is $0.10(1{,}812.6925)+1{,}800-2{,}800=-818.7308$, matching $-1{,}000e^{-0.2}$. The check also shows why reserves should not be interpolated linearly between durations: the solution depends on an exponential term.

Common Mistakes and Scope

A plausible but incorrect shortcut is to use only the force of interest in the exponent and denominator. In the first worked example, this would give $$ \frac{600}{0.04}(1-e^{-0.04(3)}) =15{,}000(1-e^{-0.12}) \approx \$1{,}696.19, $$ rather than the correct $\$1{,}555.09$. The shortcut omits the mortality contribution to the reserve coefficient. Because death releases the reserve as well as triggering the benefit, the equation contains $\delta+\mu$, not $\delta$ alone.

A second common error is to put $t$, the elapsed duration, into the exponential where the remaining term $n-t$ belongs. The boundary condition provides a simple diagnostic: whatever expression is used must give zero at $t=n$. A formula that does not meet that condition cannot be the solution for this contract.

This closed form requires constant forces and constant continuous cash-flow rates over the period being valued, along with the stated immediate death benefit and zero terminal reserve. It does not directly apply when, for example, mortality varies with duration, premiums are paid annually, benefits change by year, or a maturity benefit is payable. Those changes alter the differential equation, cash-flow timing, or boundary condition and must be handled using an appropriate model.

Key takeaway: For a single-state contract with constant forces, continuous net cash-flow rate $c$, immediate death benefit $B$, and zero terminal reserve, calculate $k=\delta+\mu$ and use $V_t=(\mu B-c)(1-e^{-k(n-t)})/k$. Check both the sign convention and the remaining term.

Check Your Understanding

For each problem, assume continuous premiums and expenses while the policy is in force, an immediate death benefit, constant forces, no maturity benefit, and a zero terminal reserve unless stated otherwise. Round money to the nearest cent.

A two-year contract has $p=\$1{,}000$ per year, $e=\$100$ per year, $B=\$20{,}000$, $\delta=0.04$, and $\mu=0.06$ per year. Find $V_0$. (Answer: $\$543.81$)
A four-year contract has $p=\$900$ per year, $e=\$150$ per year, $B=\$30{,}000$, $\delta=0.02$, and $\mu=0.03$ per year. Find $V_0$. (Answer: $\$543.81$)
A seven-year contract has $p=\$1{,}400$ per year, $e=\$200$ per year, $B=\$28{,}000$, $\delta=0.05$, and $\mu=0.05$ per year. Find the reserve at duration 3. (Answer: $\$659.36$)
A six-year contract has $p=\$1{,}600$ per year, $e=\$100$ per year, $B=\$40{,}000$, $\delta=0.04$, and $\mu=0.06$ per year. Find the reserve at duration 2 and identify the remaining term used in the exponential. (Answer: $\$2{,}967.12$; 4 years)
A ten-year contract has $p=\$1{,}200$ per year, $e=\$200$ per year, $B=\$30{,}000$, $\delta=0.03$, and $\mu=0.04$ per year. Find the reserve at duration 4. (Answer: $\$979.87$)

What You'll Learn