Common Errors in Continuous Insurance Valuation

Why Continuous Insurance Values Are Easy to Misread

A continuous insurance benefit is paid at the moment of death, so its value depends on the death time, the contract’s coverage interval, and the discounting from that time back to the valuation date. A maturity benefit is different: it is paid at a specified time only if the insured is alive then. Combining these cash flows without keeping their timing and conditions separate is a common source of errors.

This tutorial focuses on diagnosing those errors. The calculations use a future lifetime $T_x$, where $T_x$ is the time from issue until death for a life aged $x$, and assume a continuous lifetime distribution with force of mortality $\mu_{x+t}$. Let ${}_tp_x=P(T_x>t)$. With force of interest $\delta$, a dollar payable at time $t$ has value $e^{-\delta t}$ at issue.

Definition: If a death benefit of $b(t)$ is payable at death only when $T_x$ lies in a specified coverage set $C$, its expected present value at issue is $$ E\left[e^{-\delta T_x}b(T_x)\mathbf{1}_{\{T_x\in C\}}\right] = \int_C e^{-\delta t}b(t)\,{}_tp_x\mu_{x+t}\,dt. $$ The integrand uses the density of death at time $t$, which is ${}_tp_x\mu_{x+t}$. If a separate benefit $M$ is payable at time $n$ only if the insured survives to $n$, its expected present value is $M e^{-\delta n}{}_np_x$.

The formulas separate the events and cash flows: the death benefit is integrated over death times when it is covered, while the maturity benefit is discounted to its fixed payment time and weighted by the probability of being alive then. In the examples below, mortality is modeled by a constant force, so ${}_tp_x=e^{-\mu t}$. The results do not require constant force in general; with a varying force, use the appropriate survival probability and force at each time.

Start With the Coverage Interval

A continuous benefit payable for death during $[m,n)$ does not cover death before $m$ or at or after $n$. Under a continuous lifetime distribution, the probability of death at either exact endpoint is zero, so including or excluding a single endpoint does not change the expected value. The endpoints still matter for setting up the integral: the limits must match the stated coverage.

Result: Under a constant force of mortality $\mu$, a constant death benefit $B$ payable at death during $[m,n)$, and a constant force of interest $\delta$, the value at issue is $$ B\int_m^n e^{-\delta t}e^{-\mu t}\mu\,dt = B\frac{\mu}{\mu+\delta} \left(e^{-(\mu+\delta)m}-e^{-(\mu+\delta)n}\right), \qquad \mu+\delta>0. $$ The factor $e^{-(\mu+\delta)t}$ combines survival to time $t$ and discounting to issue.

The integral follows because the probability of death in a short interval from $t$ to $t+dt$ is approximately ${}_tp_x\mu\,dt=e^{-\mu t}\mu\,dt$. Multiplying this probability by the discounted benefit $Be^{-\delta t}$, then summing over covered times by integration, gives the stated expression. The lower limit is the start of cover; it is not automatically zero just because the valuation date is issue.

Worked Example: Staged Death Benefits and a Maturity Payment

A hypothetical policy is issued to a life subject to a constant force of mortality $\mu=0.04$. Interest is at a constant force $\delta=0.06$. The policy pays $\$100{,}000$ at death during the first two years, $\$150{,}000$ at death during years 2 through 5, and $\$20{,}000$ at time 5 if the insured is alive. Find the issue value of all benefits.

Method. Value each death benefit on its own interval, then value the survival-contingent maturity payment separately. Here $\mu+\delta=0.10$.

For the first death benefit, use $[0,2)$:

100{,}000\frac{0.04}{0.10} \left(1-e^{-0.10(2)}\right) = 40{,}000(1-e^{-0.2}) = 7{,}250.7698769.

For the second death benefit, use $[2,5)$, not $[0,5)$:

150{,}000\frac{0.04}{0.10} \left(e^{-0.2}-e^{-0.5}\right) = 60{,}000(0.8187307531-0.6065306597) = 12{,}732.0056019.

The maturity payment occurs at time 5 only on survival to time 5:

20{,}000e^{-0.06(5)}{}_5p_x = 20{,}000e^{-0.3}e^{-0.04(5)} = 20{,}000e^{-0.5} = 12{,}130.6131943.

Adding the unrounded components gives $7{,}250.7698769+12{,}732.0056019+12{,}130.6131943=32{,}113.3886731$. Thus the issue value is $\$32{,}113.39$, rounded to the nearest cent. The check is that the second death benefit starts at year 2 and the maturity payment includes both discounting and survival to year 5.

Common mistake avoided. Applying the second benefit over $[0,5)$ would incorrectly pay $\$150{,}000$ for deaths during the first two years as well. Also, rounding each component to cents before summing can change the final cent; add unrounded values and round only the total.

Issue Values and Survival-Conditioned Values

A deferred benefit can be valued at issue or at the deferment date conditional on survival to that date. These are different quantities. The issue value includes the chance of surviving the deferment period and the discounting over that period. A value conditional on being alive at deferment does not include the probability of surviving to deferment, because that survival is already part of the condition.

Worked Example: Deferred Cover at Issue Versus Conditional on Survival

A hypothetical policy pays $\$80{,}000$ at death during years 3 through 7. The force of mortality is $\mu=0.05$, and the force of interest is $\delta=0.03$. Find (a) the value at issue and (b) the value at time 3 conditional on survival to time 3.

Method. At issue, integrate over $[3,7)$. At time 3, restart the clock at zero and value four years of cover. The combined force is $\mu+\delta=0.08$.

(a) The issue value is

80{,}000\frac{0.05}{0.08} \left(e^{-0.08(3)}-e^{-0.08(7)}\right) = 50{,}000(e^{-0.24}-e^{-0.56}) = 10{,}770.9398609.

Therefore the value at issue is $\$10{,}770.94$.

(b) Conditional on being alive at time 3, let $s$ denote time since time 3. The covered times are $0\le s<4$, so the conditional value is

80{,}000\frac{0.05}{0.08}(1-e^{-0.08(4)}) = 50{,}000(1-e^{-0.32}) = 13{,}692.5481463.

The issue value is the conditional value multiplied by the probability of surviving three years and the discount factor for three years:

13{,}692.5481463\,e^{-0.05(3)}e^{-0.03(3)} = 13{,}692.5481463\,e^{-0.24} = 10{,}770.9398609.

This reproduces the direct issue calculation. A common wrong calculation is to use $50{,}000(1-e^{-0.56})=\$21{,}439.55$ at issue: it treats the cover as starting immediately and therefore includes deaths before year 3. Another error is to multiply the conditional value by survival to year 3 but omit $e^{-0.03(3)}$, which fails to discount the time-3 value back to issue.

Keep Maturity Benefits Out of the Death Integral

A maturity benefit has a fixed payment time and a survival condition; it is not a death benefit paid at that time. In an endowment-style contract with a death benefit during $[0,n)$ and a maturity benefit at time $n$, value the death benefit over $[0,n)$ and add $M e^{-\delta n}{}_np_x$ for the maturity benefit. The survival probability belongs in the maturity term even though no death integral is used for it.

Under a continuous lifetime model, death exactly at time $n$ has probability zero. The usual contract interpretation is therefore that the death benefit applies before $n$, while the maturity benefit is paid at $n$ if the insured survives. If a model instead allows a point mass of deaths at the boundary, that boundary convention must be stated and the point mass handled explicitly.

Worked Example: Matching the Interest Convention to Continuous Timing

A hypothetical two-year contract pays $\$30{,}000$ at death before time 2 and $\$5{,}000$ at time 2 if the insured survives. The effective annual interest rate is $i=e^{0.04}-1$, so the force of interest is $\delta=\ln(1+i)=0.04$. The force of mortality is $\mu=0.06$. Find the issue value.

Method. For the continuous death benefit, use the instantaneous death time and discount by $e^{-\delta t}$. For maturity, use survival and discount at the fixed time 2. Here $\mu+\delta=0.10$.

The death-benefit value is

30{,}000\frac{0.06}{0.10}(1-e^{-0.10(2)}) = 18{,}000(1-e^{-0.2}) = 3{,}262.8464446.

The maturity-benefit value is

5{,}000e^{-0.04(2)}e^{-0.06(2)} = 5{,}000e^{-0.2} = 4{,}093.6537654.

The total, using unrounded values, is $3{,}262.8464446+4{,}093.6537654=7{,}356.5002100$. The issue value is $\$7{,}356.50$, rounded to the nearest cent.

Since $1+i=e^{0.04}$, using $(1+i)^{-t}$ for a real-valued time $t$ gives exactly $e^{-0.04t}$ in this example. The trap is not the use of an effective rate itself; it is using an inconsistent time convention, such as discounting a payment at a fractional time $t$ as though it were payable only at the end of $\lfloor t\rfloor+1$. Continuous benefits are discounted to their actual payment times.

A Reliable Error-Checking Sequence

For a multi-part continuous contract, a short checklist catches most setup errors before arithmetic begins. Keep the death benefit, maturity benefit, and any survival conditioning in distinct lines of work.

Mark each payment time or interval.
For each death benefit, write its coverage interval explicitly. For a maturity benefit, write its fixed payment time.

State the valuation date and condition.
Decide whether the requested value is at issue or conditional on survival to a later date. Do not mix the two.

Match the discount factor to the time unit.
With a force of interest $\delta$, discount for $t$ years by $e^{-\delta t}$. Convert an effective annual rate using $\delta=\ln(1+i)$ when continuous-time discounting is needed.

Include the correct survival or death probability.
A death-time integral uses ${}_tp_x\mu_{x+t}$; a maturity payment uses ${}_np_x$. A value conditional on survival to a deferment date does not include survival to that date again.

Combine unrounded components and check the interpretation.
Confirm that no benefit applies outside its stated interval, add full-precision values, and round only the final monetary result.

The central exam habit is to translate contract wording into payment timing before selecting a formula. “At death during years 3 through 7” describes a death-time interval; “at year 7 if alive” describes a survival-contingent payment. They may appear in one contract, but they are not the same random cash flow. This distinction is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payment interval or survival condition before calculating. Assume constant forces where specified, and round monetary answers to the nearest cent.

A hypothetical policy pays $\$10{,}000$ at death during the first three years. The forces are $\mu=0.04$ and $\delta=0.06$. Find the value at issue. (Answer: $\$1{,}036.73$)
A deferred benefit of $\$20{,}000$ is payable at death during years 2 through 5. The forces are $\mu=0.03$ and $\delta=0.07$. Find its issue value. (Answer: $\$1{,}273.20$)
A maturity benefit of $\$12{,}000$ is payable at time 4 if the insured survives. With $\mu=0.02$ and $\delta=0.03$, find its issue value. (Answer: $\$9{,}824.77$)
A one-year continuous death benefit is $\$100{,}000$. The effective annual interest rate is $i=e^{0.04}-1$ and the constant force of mortality is $\mu=0.06$. Find the issue value. (Answer: $\$5{,}709.75$)
A hypothetical contract pays $\$10{,}000$ at death in year 1, $\$20{,}000$ at death in year 2, and $\$5{,}000$ at time 2 if alive. The forces are $\mu=0.05$ and $\delta=0.05$. Find the total issue value. (Answer: $\$5{,}430.53$)

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