Solving for a Level Premium From a Reserve Boundary

Why a Reserve Boundary Can Determine the Premium

A level premium is often found by balancing expected present values. The same unknown can instead be determined from the reserve dynamics: specify the reserve at two points in time, then choose the premium that makes the Thiele equation connect those boundary values. This is useful when a problem gives a required initial reserve, a prescribed terminal reserve, or both.

The method below concerns a hypothetical single-state term insurance contract. It uses constant forces of interest and mortality, a constant death benefit, and a continuous level premium payable only while the insured is alive. Expenses, lapses, and other transitions are excluded. Expense-loaded and multi-state equations require additional terms and are not covered here. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and assumptions: Let $t$ measure time from the start of the contract and let $T$ be the term. While the insured is alive, the continuous premium rate is $P$. The force of interest is constant at $\delta$, the force of mortality is constant at $\mu$, and a benefit $B$ is paid immediately upon death before time $T$. There are no expenses or other cash flows. The prospective reserve $V(t)$ is the value, at time $t$, of future benefits less future premiums, conditional on being alive at $t$. Under these assumptions, the single-state Thiele equation is $V'(t)=\delta V(t)+P-\mu(B-V(t))$.

The reserve equation can be written in a form that makes the unknown premium explicit:

V'(t)=(\delta+\mu)V(t)+P-\mu B.

Write $k=\delta+\mu$. The mortality term contains both $-\mu B$, the expected rate of benefit outgo, and $+\mu V(t)$, the release of the reserve when death occurs. Omitting the latter term would change the coefficient from $k$ to $\delta$ and would not represent the stated Thiele equation.

From Two Reserve Boundaries to a Premium

Suppose $V(0)$ and $V(T)$ are specified. The boundary values may be zero or nonzero; they must be interpreted as the reserves required by the problem. A nonzero terminal reserve is not automatically appropriate just because a contract has reached its end. If no liability continues at $T$, the natural terminal reserve is usually zero. A nonzero value must represent a stated terminal liability, continuation value, or imposed reserve condition.

Theorem: Under the constant-force assumptions above, let $k=\delta+\mu>0$. For a level continuous premium $P$, initial reserve $V(0)$, and terminal reserve $V(T)$, $$ V(0)=e^{-kT}V(T)+(\mu B-P)\frac{1-e^{-kT}}{k}. $$ Consequently, the unique premium satisfying both boundaries is $$ P=\mu B+\frac{e^{-kT}V(T)-V(0)}{(1-e^{-kT})/k}. $$

The denominator $(1-e^{-kT})/k$ is positive when $T>0$ and $k>0$. Thus, for fixed boundary reserves, there is exactly one algebraic value of $P$. That value need not be nonnegative; if it is negative, the boundary conditions require a net payment to the policyholder rather than an ordinary premium.

Why the Boundary Formula Holds

The justification is an integrating-factor solution of the constant-coefficient Thiele equation. This is a short way to see why the terminal reserve must be discounted back to time zero and why the benefit outgo and premium enter with opposite signs.

Put the equation in linear form.
With $k=\delta+\mu$, write $V'(t)-kV(t)=P-\mu B$.

Multiply by the integrating factor.
Multiplying by $e^{-kt}$ gives $\frac{d}{dt}(e^{-kt}V(t))=(P-\mu B)e^{-kt}$.

Integrate from zero to $T$.
The left side integrates to $e^{-kT}V(T)-V(0)$, while the right side integrates to $(P-\mu B)(1-e^{-kT})/k$.

Rearrange for the premium.
Solving the resulting equality gives $P=\mu B+(e^{-kT}V(T)-V(0))/((1-e^{-kT})/k)$.

The formula also has a useful forward-time interpretation. Solving the same differential equation from $0$ to $T$ gives

V(T)=e^{kT}V(0)+(P-\mu B)\frac{e^{kT}-1}{k}.

These two forms are equivalent. The first discounts the terminal reserve to time zero; the second accumulates the initial reserve to time $T$. Use one direction consistently rather than mixing an accumulated reserve with a discounted cash-flow term.

Worked Applications

In each example, $P$ is a continuous rate in dollars per year. Unless stated otherwise, reserve boundary values are in dollars and the forces are per year. The method is to calculate $k$, calculate the boundary factor, substitute into the formula, and then verify the original boundary relation.

Worked Example: Solving with Two Nonzero Reserve Boundaries

A hypothetical 10-year term policy has constant force of interest $\delta=0.03$, constant force of mortality $\mu=0.02$, and death benefit $B=\$100{,}000$. The required reserves are $V(0)=\$5{,}000$ and $V(10)=\$2{,}000$. Find the level continuous premium.

Method. Use the terminal-to-initial boundary formula, retaining the specified nonzero terminal reserve. The combined force is $k=0.03+0.02=0.05$, and the annual death-benefit outgo term is $\mu B=0.02(100{,}000)=\$2{,}000$.

The discount factor over the 10-year term is $e^{-kT}=e^{-0.05(10)}=e^{-0.5}=0.60653066$. The boundary denominator is \[ \frac{1-e^{-kT}}{k} =\frac{1-0.60653066}{0.05} =\frac{0.39346934}{0.05} =7.8693868. \] The discounted terminal reserve is $e^{-kT}V(T)=0.60653066(\$2{,}000)=\$1{,}213.0613$.

Substitute these values: \[ P=\$2{,}000+ \frac{\$1{,}213.0613-\$5{,}000}{7.8693868} =\$2{,}000-\$481.224\ldots =\$1{,}518.78. \] The premium is $\$1{,}518.78$ per year, rounded to the nearest cent.

Check. The boundary equation gives $e^{-0.5}V(10)+(\mu B-P)(7.8693868)$. Using $V(10)=\$2{,}000$ and $P=\$1{,}518.776\ldots$, this is $\$1{,}213.0613+(\$481.224\ldots)(7.8693868)=\$1{,}213.0613+\$3{,}786.9387=\$5{,}000$, as required. The common mistake avoided is using the terminal reserve without discounting it.

Worked Example: A Zero Terminal Reserve

A hypothetical 8-year term contract has $\delta=0.025$, $\mu=0.015$, and death benefit $B=\$60{,}000$. Its initial reserve is $\$4{,}500$, and the contract has no continuing liability at the end of the term, so $V(8)=0$. Find the level continuous premium.

Method. Use the terminal boundary $V(T)=0$. First, $k=0.025+0.015=0.04$ and $\mu B=0.015(\$60{,}000)=\$900$. The discount factor and denominator are \[ e^{-kT}=e^{-0.04(8)}=e^{-0.32}=0.72614904, \qquad \frac{1-e^{-kT}}{k} =\frac{1-0.72614904}{0.04} =6.8462741. \]

Since the terminal reserve is zero, its discounted value is also zero. Therefore, \[ P=\$900+\frac{0-\$4{,}500}{6.8462741} =\$900-\$657.2918\ldots =\$242.71 \] per year, rounded to the nearest cent.

Check. The starting-reserve relation gives $0+(\$900-\$242.7082\ldots)(6.8462741) =\$657.2918\ldots(6.8462741)=\$4{,}500$. Thus the initial and terminal reserves are consistent. The common mistake avoided is using $\delta$ alone in $k$; the mortality release of reserve makes $k=\delta+\mu=0.04$.

Worked Example: Using a Chosen Discount Factor

A hypothetical contract has $\delta=0.025$, $\mu=0.015$, and $B=\$100{,}000$. Its term $T$ is chosen so that $e^{-(\delta+\mu)T}=0.5$. The reserve boundaries are $V(0)=\$4{,}000$ and $V(T)=\$3{,}000$. Find the level continuous premium and the term.

Method. Here $k=0.025+0.015=0.04$. From $e^{-0.04T}=0.5$, \[ T=\frac{-\ln(0.5)}{0.04} =\frac{\ln 2}{0.04} =17.3287\ldots\text{ years}. \] The boundary denominator is especially simple: \[ \frac{1-e^{-kT}}{k}=\frac{1-0.5}{0.04}=12.5. \] Also, $\mu B=0.015(\$100{,}000)=\$1{,}500$, and the discounted terminal reserve is $0.5(\$3{,}000)=\$1{,}500$.

The premium is \[ P=\$1{,}500+\frac{\$1{,}500-\$4{,}000}{12.5} =\$1{,}500-\$200 =\$1{,}300 \] per year.

Check. The initial reserve recovered from the terminal boundary is $0.5(\$3{,}000)+(\$1{,}500-\$1{,}300)(12.5) =\$1{,}500+\$2{,}500=\$4{,}000$. The two boundaries agree exactly. The common mistake avoided is treating the specified exponential value as an annual discount factor rather than a discount factor over the entire term.

Common Boundary Errors and Feasibility Checks

A frequent exam trap is to substitute $V(T)$ directly where the formula requires $e^{-kT}V(T)$. The error is not merely notational: it changes the premium by $V(T)(1-e^{-kT})/((1-e^{-kT})/k)$, which generally is not zero. For the first worked example, using the incorrect expression \[ P_{\mathrm{wrong}}=\mu B+\frac{V(T)-V(0)}{(1-e^{-kT})/k} \] would give \[ P_{\mathrm{wrong}}=\$2{,}000+\frac{\$2{,}000-\$5{,}000}{7.8693868} =\$1{,}618.78. \] That value fails the reserve boundary: the correct equation requires the discounted terminal amount $\$1{,}213.0613$, not $\$2{,}000$, and gives $\$1{,}518.78$.

Other checks help catch setup errors before the final arithmetic:

Check the time unit. The forces and $T$ must use compatible units. If forces are annual, enter $T$ in years.
Check the mortality term. The benefit outgo is $\mu B$, and the reserve coefficient is $\delta+\mu$ under this model.
Check the boundary direction. In the formula from $V(T)$ back to $V(0)$, discount $V(T)$ by $e^{-kT}$. In a forward formula, accumulate $V(0)$ by $e^{kT}$.
Check premium feasibility. A negative algebraic result satisfies the boundary equation but is not a nonnegative premium. It signals that the specified boundaries require a different cash-flow interpretation or a subsidy.
Check the product assumptions. The result does not include expenses, time-varying forces, lapse, or transitions to other states. Those changes alter the differential equation and may require a different solution method.

Key takeaway: For a constant-force single-state model, two reserve boundaries determine the level continuous premium through $P=\mu B+(e^{-(\delta+\mu)T}V(T)-V(0))/((1-e^{-(\delta+\mu)T})/(\delta+\mu))$. Discount the terminal reserve to the starting time, keep all forces and durations in consistent units, and substitute the result back into the boundary equation to verify it.

Check Your Understanding

For each hypothetical policy, use the constant-force single-state model in this tutorial. Give the continuous premium to the nearest cent unless the result is exact.

A 10-year contract has $\delta=0.02$, $\mu=0.01$, $B=\$40{,}000$, $V(0)=0$, and $V(10)=0$. Find $P$. (Answer: $\$400.00$ per year)
A 5-year contract has $\delta=0.03$, $\mu=0.02$, $B=\$50{,}000$, $V(0)=\$1{,}000$, and $V(5)=0$. Find $P$. (Answer: $\$773.96$ per year)
A 10-year contract has $\delta=0.04$, $\mu=0.01$, $B=\$120{,}000$, $V(0)=\$2{,}500$, and $V(10)=\$1{,}000$. Find $P$. (Answer: $\$959.39$ per year)
A contract has $\delta=0.04$, $\mu=0.02$, and $B=\$40{,}000$. Its term satisfies $e^{-(\delta+\mu)T}=0.5$, and its reserves are $V(0)=\$1{,}000$ and $V(T)=\$3{,}000$. Find $P$. (Answer: $\$860.00$ per year)
A contract has $\delta=0.06$, $\mu=0.04$, and $B=\$25{,}000$. Its term satisfies $e^{-(\delta+\mu)T}=0.5$, with $V(0)=\$1{,}500$ and $V(T)=\$2{,}500$. Find both $T$ and $P$. (Answer: $T=6.9315\ldots$ years and $P=\$950.00$ per year)

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