Term Insurance Reserves With a Maturity Benefit

Why the Maturity Condition Matters

A term contract may pay a benefit if the insured dies during the term and a separate benefit if the insured survives to the end. The reserve must reflect both possible payments. In particular, when a maturity benefit is due, the reserve immediately before the contract matures is not zero: it must cover that payment. Applying the death-benefit amount at the boundary, or setting the reserve to zero as if the contract had no maturity payment, changes the reserve throughout the term.

This tutorial considers a single-state model: the insured is either alive and covered or has died, after which the contract ends. Benefits and premiums are continuous-time cash flows under constant forces. The formulas below are specific to those assumptions; discrete premium timing, expenses, or changing forces require an appropriately modified reserve equation.

Definition: Consider a contract issued for a term ending at time $n$. While the insured is alive, a continuous premium is paid at rate $p$ per year. If death occurs before $n$, benefit $B$ is paid immediately at death. If the insured survives to $n$, maturity benefit $M$ is paid. Let $V(t)$ be the reserve at time $t$, conditional on being alive and covered immediately before that time. The terminal condition is $V(n^-)=M$, the reserve just before payment of the maturity benefit. Immediately after that payment, the contract ends and its reserve is zero.

Assume the force of interest is $\delta$ and the force of mortality is constant at $\mu$ throughout the remaining term. Premiums are paid continuously only while the insured is alive and the contract is in force. There are no expenses, surrender payments, or other benefits. For a time $t$ before maturity, write the remaining term as $\tau=n-t$. Under constant mortality, survival for $s$ years from time $t$ has probability $e^{-\mu s}$, and discounting for $s$ years gives $e^{-\delta s}$.

Thiele’s Equation and the Terminal Condition

With the stated cash-flow convention, Thiele’s equation for the reserve while the insured is alive is

\frac{dV(t)}{dt} = \delta V(t)+p-\mu\bigl(B-V(t)\bigr) = (\delta+\mu)V(t)+p-\mu B, \qquad V(n^-)=M.

The terms have direct interpretations. Interest contributes $\delta V(t)$; premium income contributes $p$; and death at intensity $\mu$ produces an expected net outgo of $\mu(B-V(t))$. The death benefit is $B$, whereas the reserve released when the insured dies is $V(t)$. The boundary condition uses $M$, not $B$, because the payment at maturity is triggered by survival, not death.

Theorem: Under constant $\delta$, constant $\mu$, a continuous premium rate $p$, immediate death benefit $B$, and maturity benefit $M$, define $a=\delta+\mu$. For $0\le t\le n$, $$ V(t)=\frac{\mu B-p}{a}\left(1-e^{-a(n-t)}\right)+M e^{-a(n-t)}. $$ This gives $V(n^-)=M$. It applies to the stated continuous-time cash flows and does not, without adjustment, apply to discrete premiums or time-varying forces.

The formula separates the two sources of future liability. The first term is the net contribution from future death benefits and premiums, weighted over the remaining term. The second is the maturity benefit discounted for interest and survival to maturity. The latter discount uses $\delta+\mu$, because the insured must both survive and the payment must be discounted.

Deriving the Closed-Form Reserve

A short integrating-factor argument explains how the terminal amount enters the solution. Let $a=\delta+\mu$ and set $c=(\mu B-p)/a$. Then the differential equation becomes $V'(t)=a(V(t)-c)$. Multiply by $e^{-at}$. The product rule gives

\frac{d}{dt}\left(e^{-at}(V(t)-c)\right) = e^{-at}\left(V'(t)-a(V(t)-c)\right)=0.

Thus $e^{-at}(V(t)-c)$ is constant over the term. Evaluating that constant at $n$, where $V(n^-)=M$, gives

e^{-at}(V(t)-c)=e^{-an}(M-c), \qquad V(t)=c+(M-c)e^{-a(n-t)}.

Expanding the last expression yields the theorem’s formula. In particular, the maturity benefit appears through the terminal value $M$; it is not included in the death benefit term.

The same result can be checked from expected future cash flows. Conditional on being alive at time $t$, death at elapsed time $s$ has density $\mu e^{-\mu s}$. The probability of survival to elapsed time $s$ is $e^{-\mu s}$. Therefore, for $\tau=n-t$, the prospective reserve is

V(t) = \int_0^\tau B\mu e^{-(\delta+\mu)s}\,ds +M e^{-(\delta+\mu)\tau} -\int_0^\tau p e^{-(\delta+\mu)s}\,ds.

Evaluating the integrals gives the same formula. This check also clarifies the signs: future benefits increase the reserve, while future premiums reduce it.

Worked Reserves

Worked Example: Five-Year Endowment Reserve at Issue

A hypothetical five-year contract pays $B=\$100{,}000$ immediately on death during the term and $M=\$120{,}000$ on survival to year 5. The continuous premium rate is $p=\$2{,}000$ per year while alive. Assume constant force of interest $\delta=0.04$ and constant force of mortality $\mu=0.06$. Calculate the reserve at issue, rounding the final amount to the nearest cent.

Method. Apply the constant-force solution with $t=0$, $n=5$, and $a=\delta+\mu$. The combined force is $a=0.04+0.06=0.10$, and the remaining term is $\tau=5$. The exponential factor is $e^{-a\tau}=e^{-0.5}=0.60653066$, so $1-e^{-a\tau}=0.39346934$.

The coefficient on the net death-benefit and premium term is

\frac{\mu B-p}{a} = \frac{0.06(100{,}000)-2{,}000}{0.10} = \frac{6{,}000-2{,}000}{0.10} = 40{,}000.

Hence

V(0) = 40{,}000(0.39346934)+120{,}000(0.60653066) = 15{,}738.77+72{,}783.68 = \$88{,}522.45.

Check. The prospective components are death benefits $60{,}000(0.39346934)=\$23{,}608.16$, maturity benefit $\$72{,}783.68$, and premiums $20{,}000(0.39346934)=\$7{,}869.39$. Their net is $23{,}608.16+72{,}783.68-7{,}869.39=\$88{,}522.45$. The check agrees. The common mistake avoided here is omitting the survival payment from the reserve.

Worked Example: Reserve After Four Years

A hypothetical ten-year endowment policy pays $\$80{,}000$ on death during the term and $\$100{,}000$ at maturity if the insured survives. A continuous premium of $\$1{,}500$ per year is paid while alive. Suppose $\delta=0.05$, $\mu=0.05$, and the insured is alive at time $t=4$. Find the reserve at that time, rounding to the nearest cent.

Method. The remaining term is $\tau=10-4=6$ years. The combined force is $a=0.05+0.05=0.10$. Thus $e^{-a\tau}=e^{-0.6}=0.54881164$ and $1-e^{-a\tau}=0.45118836$. The net coefficient is

\frac{\mu B-p}{a} = \frac{0.05(80{,}000)-1{,}500}{0.10} = \frac{4{,}000-1{,}500}{0.10} = 25{,}000.

Substitution gives

V(4) = 25{,}000(0.45118836)+100{,}000(0.54881164) = 11{,}279.71+54{,}881.16 = \$66{,}160.87.

Check. The death-benefit value is $40{,}000(0.45118836)=\$18{,}047.53$. The maturity-benefit value is $\$54{,}881.16$, and the value of future premiums is $15{,}000(0.45118836)=\$6{,}767.83$. Therefore $18{,}047.53+54{,}881.16-6{,}767.83=\$66{,}160.86$, a one-cent difference from the formula caused by rounding the displayed components. Using unrounded values gives $\$66{,}160.87$. This check illustrates why intermediate values should not be rounded in the calculation.

Worked Example: A Smaller Maturity Benefit Than the Net Coefficient

A hypothetical five-year contract pays $\$50{,}000$ on death during the term and $\$20{,}000$ on survival to maturity. Its continuous premium rate is $\$1{,}000$ per year. Assume $\delta=0.02$, $\mu=0.08$, and calculate the reserve at time $t=2$, rounding to the nearest cent.

Method. There are $\tau=5-2=3$ years remaining and $a=0.02+0.08=0.10$. Therefore $e^{-a\tau}=e^{-0.3}=0.74081822$ and $1-e^{-a\tau}=0.25918178$. The coefficient is

\frac{\mu B-p}{a} = \frac{0.08(50{,}000)-1{,}000}{0.10} = \frac{4{,}000-1{,}000}{0.10} = 30{,}000.

The reserve is

V(2) = 30{,}000(0.25918178)+20{,}000(0.74081822) = 7{,}775.45+14{,}816.36 = \$22{,}591.82.

Check. The expected present value of death benefits is $40{,}000(0.25918178)=\$10{,}367.27$, the maturity benefit contributes $\$14{,}816.36$, and future premiums have value $10{,}000(0.25918178)=\$2{,}591.82$. The net is $\$10{,}367.27+\$14{,}816.36-\$2{,}591.82=\$22{,}591.81$, with the one-cent discrepancy again due only to displayed rounding; the unrounded result is $\$22{,}591.82$. Here the reserve is below the coefficient $30{,}000$ because the maturity amount is below that coefficient and the remaining-term exponential weight is positive.

Boundary Conditions and Common Traps

The terminal condition is the most important modeling choice in this calculation. The boundary is determined by the cash flow at maturity and by whether the reserve is measured immediately before or after that payment. If the benefit $M$ is paid at time $n$, the pre-payment reserve is $M$. If instead a question defines the reserve immediately after the payment and the contract has ended, the post-payment reserve is zero. These are different instants, not contradictory answers.

A common error is to use the terminal condition for a death-only term policy even though the contract pays a maturity benefit. For the first worked example, incorrectly setting $V(5^-)=0$ would give

V_{\mathrm{wrong}}(0) = 40{,}000(1-e^{-0.5}) = \$15{,}738.77.

That is the result for a different contract with no maturity payment. The correct contract has $V(5^-)=\$120{,}000$, so its reserve is $\$88{,}522.45$. Another trap is to insert $M$ into the death term: death during the term triggers $B$, while survival to the end triggers $M$. The two benefits must remain separate unless the contract explicitly makes them equal.

Key takeaway: For a continuous-premium endowment contract under constant forces, solve Thiele’s equation backward from the pre-maturity boundary $V(n^-)=M$. The death benefit $B$ enters through the mortality term, and the maturity benefit enters through the survival-to-maturity term.

Check Your Understanding

For each problem, assume continuous premiums while alive, immediate payment of the death benefit, no expenses, and constant forces of interest and mortality. Round monetary answers to the nearest cent.

A four-year contract pays $\$60{,}000$ on death and $\$90{,}000$ at maturity. The premium rate is $\$1{,}000$ per year, $\delta=0.03$, and $\mu=0.07$. Find the reserve at issue. (Answer: $\$70{,}878.56$)
A three-year contract has death benefit $\$150{,}000$, maturity benefit $\$50{,}000$, and continuous premium rate $\$3{,}000$ per year. Given $\delta=0.04$ and $\mu=0.06$, calculate the reserve at issue. (Answer: $\$52{,}591.82$)
A contract pays a maturity benefit of $\$125{,}000$ at time $n=8$. What is the reserve immediately before that payment, irrespective of the death benefit and premium before maturity? (Answer: $\$125{,}000.00$)
A five-year contract pays $\$75{,}000$ on death and $\$110{,}000$ on survival to maturity, with continuous premium rate $\$2{,}000$ per year. If $\delta=0.06$ and $\mu=0.04$, find the reserve at issue. (Answer: $\$70{,}653.07$)

What You'll Learn