Direct Integral Valuation of Endowment Insurance

Introduction

A continuous endowment contract has two possible payment events: death during the coverage term, or survival to the end of that term. Valuing the contract directly means calculating the expected present value of each event from its timing and probability, then adding the results. The death payment is valued by integrating over all possible death times; the maturity payment is valued using the probability of surviving to the term.

This approach makes the role of each assumption explicit. The force of interest determines how strongly later payments are discounted, while the survival model determines how probability is distributed across possible death times and the maturity date. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Consider a life aged $x$ and a contract term of $n$ years. Let $T_x$ be the future lifetime. Suppose a benefit of $B$ dollars is paid immediately at death if $0 \leq T_x < n$, and a benefit of $C$ dollars is paid at time $n$ if $T_x \geq n$. With force of interest $\delta$, the present value $Z$ is $B e^{-\delta T_x}$ on death before $n$, and $C e^{-\delta n}$ on survival to $n$.

The benefit amounts $B$ and $C$ need not be equal. If there is no maturity payment, set $C=0$; if there is no death payment, set $B=0$. Unless otherwise stated, assume the lifetime has a continuous distribution, so the probability of death at precisely the term boundary is zero. The distinction between $T_x<n$ and $T_x\leq n$ therefore does not affect the expected present value.

The Direct Valuation Formula

Write ${}_tp_x=P(T_x>t)$ for the probability of surviving $t$ years, and $\mu_{x+t}$ for the force of mortality at duration $t$. Under a continuous survival model, the density of $T_x$ at $t$ is ${}_tp_x\mu_{x+t}$. Thus the probability of dying in a short interval around $t$ is approximately the probability of reaching $t$, multiplied by the force of mortality and the interval length.

Theorem: For a constant death benefit $B$, paid immediately on death before the end of the term, and a maturity benefit $C$, paid at time $n$ on survival, the expected present value is

\operatorname{EPV}(Z) = B\int_0^n e^{-\delta t}{}_tp_x\mu_{x+t}\,dt + C e^{-\delta n}{}_np_x.

The first term accounts for every possible death time in the interval from 0 to $n$. The second term is a single payment at $n$, conditional on survival to that time. This direct formula assumes that the death benefit is paid immediately and that the maturity benefit is paid at the end of the stated term. A benefit paid at a different time requires a different discount factor.

Why the Death-Benefit Integral Works

To justify the integral, divide the term into short intervals. For a small interval of length $\Delta t$ beginning at $t$, the probability of death in that interval is approximately ${}_tp_x\mu_{x+t}\Delta t$. If death occurs at approximately $t$, the present value of the death benefit is approximately $B e^{-\delta t}$. The expected present value contributed by this interval is therefore approximately $B e^{-\delta t}{}_tp_x\mu_{x+t}\Delta t$. Summing over intervals and taking the limit as their widths approach zero gives the integral.

The maturity term follows directly from the payment event. The payment is worth $C e^{-\delta n}$ at time 0 if the life survives $n$ years, and is zero otherwise. Its expected present value is therefore the payment amount times its discount factor times the survival probability:

C e^{-\delta n}P(T_x\geq n) = C e^{-\delta n}{}_np_x.

For a constant force of mortality $\mu$, survival is ${}_tp_x=e^{-\mu t}$, and the death density is $\mu e^{-\mu t}$. If the interest force is also constant, the formula simplifies to

\operatorname{EPV}(Z) = B\frac{\mu}{\delta+\mu} \left(1-e^{-(\delta+\mu)n}\right) + C e^{-(\delta+\mu)n}, \qquad \delta+\mu>0.

The simplification comes from combining the discount factor and survival probability in the death integral: $e^{-\delta t}e^{-\mu t}=e^{-(\delta+\mu)t}$. Integrating $\mu e^{-(\delta+\mu)t}$ from 0 to $n$ gives the death-benefit factor. The maturity payment has the same combined exponent because it is discounted for $n$ years and requires survival for $n$ years.

Worked Valuations

Worked Example: Level Benefits Under Constant Forces

A hypothetical policy pays $100,000 immediately if death occurs during a 10-year term, and $20,000 at the end of year 10 if the insured survives. Assume a constant force of mortality $\mu=0.04$ per year and a constant force of interest $\delta=0.06$ per year. Find the expected present value, rounded to the nearest cent.

Method. Apply the direct formula. The combined exponent is $\delta+\mu=0.06+0.04=0.10$, so over 10 years it is $0.10(10)=1$.

Death benefit. Its expected present value is

100{,}000\frac{0.04}{0.10}(1-e^{-1}) = 100{,}000(0.4)(1-0.3678794412) = 40{,}000(0.6321205588) = 25{,}284.82235.

Maturity benefit. The probability of survival to year 10 is $e^{-0.04(10)}=e^{-0.4}$, and the discount factor is $e^{-0.06(10)}=e^{-0.6}$. Their product is $e^{-1}=0.3678794412$. Therefore,

20{,}000e^{-1} = 20{,}000(0.3678794412) = 7{,}357.58882.

Adding the two event values gives $25{,}284.82235+7{,}357.58882=32{,}642.41117$. The expected present value is $32,642.41, rounded to two decimal places. As a check, both terms are positive, and the death and maturity events together cover all outcomes under the stated model.

Common mistake avoided: The maturity payment is not paid to everyone. Multiplying it only by the discount factor omits the chance of surviving to year 10.

Worked Example: Piecewise-Constant Mortality

A hypothetical 10-year contract pays $80,000 immediately at death during the term and $30,000 at year 10 if the insured survives. The force of interest is $\delta=0.04$ per year. The force of mortality is $0.02$ per year during years 0 through 5, and $0.06$ per year during years 5 through 10. Find the expected present value, rounded to the nearest cent.

Method. Split the death integral where the force of mortality changes. In the first five years, the combined rate $\delta+\mu$ is $0.04+0.02=0.06$. In the second five years it is $0.04+0.06=0.10$. Survival and discounting from time 0 to year 5 contribute the factor $e^{-0.06(5)}=e^{-0.3}$ to the second interval.

For deaths in the first five years, the death-benefit factor is

\frac{0.02}{0.06}(1-e^{-0.06(5)}) = \frac{1}{3}(1-e^{-0.3}) = \frac{1}{3}(1-0.7408182207) = 0.0863939264.

The corresponding present value is $80{,}000(0.0863939264)=6{,}911.51411$.

For deaths during years 5 through 10, first account for reaching year 5, then integrate over the next five years:

e^{-0.3}\frac{0.06}{0.10}(1-e^{-0.10(5)}) = 0.7408182207(0.6)(1-0.6065306597) = 0.1748935541.

The second-interval death benefit has present value $80{,}000(0.1748935541)=13{,}991.48433$. The survival probability to year 10 is $e^{-0.02(5)-0.06(5)}=e^{-0.4}$; discounting the maturity payment contributes $e^{-0.04(10)}=e^{-0.4}$. Its present value is therefore

30{,}000e^{-0.8} = 30{,}000(0.4493289640) = 13{,}479.86892.

The total is $6{,}911.51411+13{,}991.48433+13{,}479.86892=34{,}382.86736$. The expected present value is $34,382.87, rounded to two decimal places.

Common mistake avoided: Applying the second mortality force from time 0 would give the wrong survival probability at year 5. The factor $e^{-0.3}$ correctly reflects both mortality and interest accumulated before that interval.

Worked Example: Solving for the Death Benefit

A hypothetical continuous endowment contract has a term of $n=10\ln 2$ years, a constant force of mortality $\mu=0.05$, and a constant force of interest $\delta=0.05$. It pays $10,000 at maturity if the insured survives. What immediate death benefit $B$ makes the expected present value equal to $20,000?

Method. Use the constant-force formula and solve for $B$. Here $\delta+\mu=0.10$, so $e^{-(\delta+\mu)n}=e^{-0.10(10\ln 2)}=e^{-\ln 2}=0.5$. The death-benefit factor is

\frac{\mu}{\delta+\mu} \left(1-e^{-(\delta+\mu)n}\right) = \frac{0.05}{0.10}(1-0.5) = 0.5(0.5) = 0.25.

The maturity-benefit present value is $10{,}000(0.5)=5{,}000$. Thus the equation for the target value is $0.25B+5{,}000=20{,}000$. Subtracting the maturity value gives $0.25B=15{,}000$, and dividing by $0.25$ gives $B=60{,}000$.

The required death benefit is $60,000. Checking it in the original formula gives $0.25(60{,}000)+5{,}000=15{,}000+5{,}000=20{,}000$, as required. The term is approximately $6.9315$ years, but its exact value is used in the calculation.

Common mistake avoided: The factor $0.25$ applies to the death benefit, not to the total contract value. The maturity component must be calculated separately before solving for $B$.

Checks and Common Traps

A direct valuation is easiest to audit when the payment event, timing, and probability are kept together. For each benefit, identify what must happen for it to be paid, when it is paid, and which model quantity represents that event. Then check that the result has the units of money and that its components are nonnegative.

Forgetting survival in the maturity term. The maturity payment is multiplied by ${}_np_x$, as well as by $e^{-\delta n}$.
Discounting a death benefit at the term end. The death benefit is paid at the actual death time $t$, so its discount factor inside the integral is $e^{-\delta t}$, not $e^{-\delta n}$.
Using a mortality force as a probability. A force is a rate per unit time, not a probability of death over an interval. The death density also includes the probability of surviving to that time.
Applying a piecewise force over the wrong interval. When the force changes, split the integral and carry forward the survival accumulated before each interval.
Mixing time units. If $n$ is measured in years, both $\delta$ and each force of mortality must be stated per year.

Under the usual assumptions, the formula values a fixed death benefit and a fixed maturity benefit. A contract with another payment schedule requires the payment amount and timing to be represented appropriately inside the calculation. Also, the integral uses a continuous lifetime model; if the model assigns positive probability to death exactly at the term boundary, that probability must be handled according to the contract’s boundary-payment rule.

Key takeaway: For a continuous endowment contract, integrate the discounted death payment against the future-lifetime density over the term, then add the discounted maturity payment multiplied by the probability of survival to the term. Keep the death and maturity events distinct until the final addition.

Check Your Understanding

For each contract, use the stated continuous model and round monetary answers to the nearest cent.

A hypothetical four-year contract pays $100,000 immediately at death before the term and nothing at maturity. Assume constant forces $\mu=0.02$ and $\delta=0.03$, both per year. Find its expected present value. (Answer: $7,250.77)
A hypothetical five-year contract pays $40,000 at death during the term and $20,000 at maturity if the insured survives. Assume constant forces $\mu=0.06$ and $\delta=0.04$. Find its expected present value. (Answer: $21,573.88)
A hypothetical six-year contract pays $40,000 at death during the term and $10,000 at maturity if the insured survives. Let $\delta=0.02$; let $\mu=0.04$ during years 0 through 3 and $\mu=0.08$ during years 3 through 6. Find its expected present value. (Answer: $17,508.21)
A hypothetical contract pays $50,000 at death during its term and $20,000 at maturity. Assume constant forces $\mu=0.06$ and $\delta=0.04$. What term $n$ makes the expected present value $25,000? (Answer: $n=10\ln 2$, approximately 6.9315 years)

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