Continuous Term Insurance Reserves

Why the Remaining Term Matters

A continuous term insurance benefit is paid at the moment of death, provided death occurs before the policy’s term expires. At an intermediate duration, the expected present value of that benefit depends on the remaining coverage interval, not the original term. A prospective reserve also subtracts the present value of premiums still due, conditional on the insured being alive at the valuation time.

This tutorial considers a level benefit $B$ payable immediately on death during an $n$-year term, with premiums $P$ payable annually in advance at times $0,1,\ldots,n-1$, while the policy is in force. The reserve at duration $t$ is valued immediately after any premium due at $t$ has been paid. At a non-integer duration, the future premium dates are simply the anniversaries after $t$ and before expiry.

Definition: The prospective reserve at duration $t$, conditional on the policy being in force, is the expected present value at time $t$ of future benefits minus the expected present value at time $t$ of future premiums. For a continuous term benefit $B$, annual premium $P$, and future premium dates $j$, this is $$ {}_tV_x = B\int_0^{n-t} e^{-\delta s}\,{}_sp_{x+t}\,\mu_{x+t+s}\,ds - P\sum_{\substack{j\in\{0,1,\ldots,n-1\}\\j>t}} e^{-\delta(j-t)}\,{}_{j-t}p_{x+t}, \qquad 0\le t<n. $$ Here $\delta$ is the force of interest, ${}_sp_{x+t}$ is the probability of surviving $s$ years from age $x+t$, and $\mu_{x+t+s}$ is the force of mortality at the attained age $x+t+s$. The formula assumes level benefits and premiums, death benefits paid at the moment of death, and premiums paid at the stated annual dates only while the insured is alive and the policy remains in force.

Why the Benefit Is an Integral

Conditional on survival to duration $t$, the probability of surviving a further $s$ years and then dying in the next short interval of length $ds$ is approximately ${}_sp_{x+t}\mu_{x+t+s}\,ds$. Thus the conditional density of future lifetime at $s$ is ${}_sp_{x+t}\mu_{x+t+s}$. If death occurs at that time, the benefit is discounted back $s$ years by $e^{-\delta s}$. Multiplying the benefit by this discounted density and integrating over all covered death times gives its expected present value.

The upper integration limit is $n-t$: deaths after the term expires do not generate a benefit. For premiums, each future payment is included only if it is due before expiry. Its present value is multiplied by the probability the insured survives from $t$ until that payment date.

Set the valuation time and remaining term.
At duration $t$, the maximum covered future lifetime is $n-t$ years.

Value the death benefit.
Integrate its amount times the discounted conditional density of death over $0\le s\le n-t$.

List future premium dates.
Include only dates $j$ with $t<j<n$; a premium already paid at $t$ is not future.

Subtract the premium value.
Discount each future premium from its payment date to $t$, allowing for survival to that date.

Constant Force of Mortality

A useful exam model assumes a constant force of mortality $\mu$ over the relevant ages, so that ${}_sp_{x+t}=e^{-\mu s}$ and $\mu_{x+t+s}=\mu$. With force of interest $\delta$, the benefit component then has a closed form:

B\int_0^{n-t}e^{-(\delta+\mu)s}\mu\,ds = B\frac{\mu}{\delta+\mu} \left(1-e^{-(\delta+\mu)(n-t)}\right).

For a future annual premium due $j-t$ years from the valuation time, its conditional expected present value is $P e^{-(\delta+\mu)(j-t)}$. The mortality and interest forces combine in the exponent because the payment is received only if the insured both survives to that date and the payment is discounted to time $t$.

Worked Example: Find the Net Premium and a Midterm Reserve

A hypothetical three-year continuous term policy pays $B=\$100{,}000$ immediately on death during the term. The force of mortality and force of interest are both $\ln 2$ per year, so the annual effective interest rate is $i=e^{\ln 2}-1=1$. Premiums are payable annually in advance at times 0, 1, and 2. Find the level net premium $P$ by the equivalence principle, then find the reserve at $t=1.5$, immediately after any premium due at that time.

Method. First equate the issue-date expected present values of benefits and premiums. Then apply the prospective reserve formula using only the time remaining after duration 1.5.

At issue, the benefit value is

100{,}000\int_0^3 e^{-2(\ln 2)s}\ln 2\,ds =100{,}000\cdot\frac12(1-2^{-6}) =100{,}000\cdot\frac{63}{128} =\$49{,}218.75.

The expected present value of the three premiums per dollar of premium is

1+e^{-2\ln 2}+e^{-4\ln 2} =1+\frac14+\frac1{16} =\frac{21}{16}.

Therefore the net premium is

P=\frac{49{,}218.75}{21/16}=\$37{,}500.

At duration $1.5$, the remaining term is $3-1.5=1.5$ years. The benefit value is

100{,}000\int_0^{1.5}e^{-2(\ln 2)s}\ln 2\,ds =100{,}000\cdot\frac12(1-2^{-3}) =\$43{,}750.

Of the original premium dates 0, 1, and 2, only date 2 is still in the future and before expiry. It is $0.5$ years away, so its conditional expected present value is

37{,}500e^{-2(\ln 2)(0.5)} =37{,}500\cdot\frac12 =\$18{,}750.

Thus the reserve is $\$43{,}750-\$18{,}750=\$25{,}000$. Check: both terms are valued at duration 1.5; the benefit covers only the remaining 1.5 years, and only the premium at time 2 remains payable.

Worked Reserves at Intermediate Durations

The general formula is not limited to a net premium calculated at issue. If the premium is specified, value the remaining benefit and remaining premiums separately. In particular, do not assume the number of future premiums is the same as the number of years left in the coverage: annual premiums stop at their specified dates, and no premium is due at expiry in the model used here.

Worked Example: Several Premiums Remain After the Valuation Date

A hypothetical four-year continuous term policy has benefit $B=\$64{,}000$, annual premium $P=\$12{,}000$ due at times 0, 1, 2, and 3, and constant forces $\mu=\delta=\ln 2$ per year. Find the reserve at $t=1.5$, immediately after any premium due then.

Method. Integrate the death-benefit density over the remaining $4-1.5=2.5$ years, then value the premiums due at times 2 and 3.

The benefit value is

64{,}000\cdot\frac12(1-2^{-5}) =64{,}000\cdot\frac{31}{64} =\$31{,}000.

The time from the valuation date to premium date 2 is $0.5$ years, and the time to premium date 3 is $1.5$ years. The premiums’ combined expected present value is

12{,}000\left(e^{-2(\ln 2)(0.5)}+e^{-2(\ln 2)(1.5)}\right) =12{,}000\left(\frac12+\frac18\right) =\$7{,}500.

Therefore, $$ {}_ {1.5}V_x=\$31{,}000-\$7{,}500=\$23{,}500. $$ Check: the premium at time 1 has already been paid, and time 4 is expiry rather than a premium date. The two remaining premiums are exactly those at times 2 and 3.

Worked Example: A Reserve After the Final Premium

A hypothetical four-year continuous term policy pays $B=\$80{,}000$ on death during the term. Its annual premium of $\$10{,}000$ is due at times 0, 1, 2, and 3. Assume constant forces $\mu=\delta=\ln 2$ per year. Find the reserve at $t=3$, immediately after the premium at time 3 has been paid.

Method. The remaining coverage is one year. Since all scheduled premiums have now been paid, the future premium value is zero.

The conditional death-benefit value is

80{,}000\int_0^1 e^{-2(\ln 2)s}\ln 2\,ds =80{,}000\cdot\frac12(1-2^{-2}) =80{,}000\cdot\frac38 =\$30{,}000.

There are no premium dates $j$ satisfying $3<j<4$, so the premium component is $\$0$. The reserve is therefore $\$30{,}000$. Check: one year of term coverage remains, but the last premium was paid at the valuation time and must not be subtracted again.

Common Timing Errors

A frequent mistake is to use the original term $n$ as the upper limit of the death-benefit integral at duration $t$. That values a benefit period beginning at issue, not the period still covered at the valuation date. For example, in the first worked example, using three years instead of the remaining 1.5 years would give a benefit value of $\$49{,}218.75$, rather than the correct $\$43{,}750$. This alone overstates the prospective benefit value by $\$5{,}468.75$.

A related mistake is subtracting premiums that have already been paid or counting a premium at expiry. At an anniversary, state whether the reserve is just before or just after the premium due at that time. The examples use the value immediately after any due premium. At a non-anniversary duration, list the future premium dates explicitly before calculating.

Finally, keep the time units consistent. If $\delta$ and $\mu$ are annual forces, then $s$, $t$, and every premium interval must be measured in years. For varying mortality, use the actual survival probabilities and forces in the integral; the constant-force simplification does not apply unless mortality is constant over the interval.

Key takeaway: A continuous term reserve is the conditional expected present value of death benefits over the remaining coverage interval, less the conditional expected present value of premiums still due before expiry. The integration limit and premium dates must both be set from the valuation duration.

Check Your Understanding

For each problem, use the stated timing convention: reserves are measured immediately after any premium due at the valuation time.

A two-year continuous term policy has benefit $\$40{,}000$, constant forces $\mu=\delta=\ln 2$, and no future premiums after the valuation time. What is the reserve with one year of coverage remaining? (Answer: $\$15{,}000$)
A three-year policy has benefit $\$32{,}000$, annual premium $\$6{,}000$ at times 0, 1, and 2, and $\mu=\delta=\ln 2$. Find the reserve at $t=0.5$. (Answer: $\$11{,}750$)
A continuous term policy has two years of coverage remaining, benefit $\$50{,}000$, no future premiums, constant mortality force $\mu=0.20$, and force of interest $\delta=0.05$. Find the reserve, to the nearest cent. (Answer: $\$15{,}738.78$)
A four-year policy has benefit $\$100{,}000$, annual premium $\$10{,}000$ at times 0, 1, 2, and 3, and $\mu=\delta=\ln 2$. Find the reserve at $t=2.25$. (Answer: $\$42{,}045.05$)
A two-year policy has benefit $\$80{,}000$, annual premiums due at times 0 and 1, and $\mu=\delta=\ln 2$. First find the level net premium by the equivalence principle; then find the reserve at $t=1.25$. (Answer: net premium $\$30{,}000$; reserve $\$25{,}857.86$)

What You'll Learn