Continuous Endowment Reserves

Why Continuous Endowment Reserves Have Two Benefit Components

A continuous endowment policy combines insurance protection with a payment on survival. If the insured dies during the term, the death benefit is paid immediately at death. If the insured survives to the end of the term, a maturity benefit is paid then. At an intermediate duration, the prospective reserve must allow for both possible benefits, weighted by their probabilities and discounted to the valuation time.

This tutorial assumes a single death benefit amount and a single maturity benefit amount, with level premiums payable annually in advance while the insured is alive. The death benefit is continuous in its timing; the premiums are not. Keeping these cash-flow timings separate is essential to a correct reserve.

Definition: A continuous endowment policy issued to age $x$ for $n$ years pays $B$ immediately on death during the term and pays $M$ at time $n$ if the insured survives to that time. Here $B$ is the death benefit and $M$ is the maturity benefit. A prospective reserve at duration $t$, conditional on survival to $t$, is the expected present value of benefits still payable minus the expected present value of premiums still payable.

Use a constant effective annual interest rate $i$, with $v=1/(1+i)$, unless a different interest model is stated. Let ${}_sp_{x+t}$ be the probability that a life aged $x+t$ survives a further $s$ years, and let $\mu_{x+t+s}$ be the force of mortality at that future age. These survival and mortality quantities are conditional on the insured being alive at duration $t$.

Valuing Benefits After a Duration

Suppose the reserve is calculated at an integer duration $k<n$, immediately before the premium due at that duration. Write $m=n-k$ for the remaining term. From the valuation time, a death at future time $s$, where $0\leq s<m$, has conditional density ${}_sp_{x+k}\mu_{x+k+s}$. Discounting the resulting payment by $e^{-\delta s}$, where $\delta=\ln(1+i)$, gives the present value of the death benefit. The maturity benefit is paid only if the insured survives the full remaining term.

\operatorname{EPV}(\text{benefits}\mid \text{alive at }k) = B\int_0^m e^{-\delta s}{}_sp_{x+k}\mu_{x+k+s}\,ds + M e^{-\delta m}{}_mp_{x+k}.

The first term is not the value of a benefit paid at the end of the term: it accumulates the discounted death benefit over every possible death time. The second term is the discounted maturity benefit, weighted by the probability of reaching maturity alive.

Prospective reserve before a premium: At integer duration $k<n$, if annual premiums of amount $P$ are due at durations $k,k+1,\ldots,n-1$ while the insured is alive, then

{}_kV_x = B\int_0^{n-k} e^{-\delta s}{}_sp_{x+k}\mu_{x+k+s}\,ds + M e^{-\delta(n-k)}{}_{n-k}p_{x+k} - P\sum_{r=0}^{n-k-1}v^r{}_rp_{x+k}.

The premium sum includes a payment at the valuation time because this version of the reserve is defined immediately before the premium due at $k$. Immediately after that premium is paid, the current premium is no longer outstanding: omit the $r=0$ term. Between policy anniversaries, include only premiums whose scheduled dates are still in the future.

Why the Formula Holds

A prospective reserve is a conditional expected present value of future cash flows. The death-benefit part follows from the conditional density of death; the maturity-benefit part follows from survival to the end of the remaining term. Future premiums reduce the insurer’s net liability, so their expected present value is subtracted.

Condition on being alive at duration $k$.
Only future survival and death probabilities from age $x+k$ are relevant to the reserve.

Value death payments over time.
At future time $s$, the conditional probability density of death is ${}_sp_{x+k}\mu_{x+k+s}$. The payment $B$ is discounted by $e^{-\delta s}$, and integration over $0\leq s<n-k$ gives the death-benefit present value.

Value the maturity payment.
The payment $M$ occurs only if the insured survives $n-k$ more years, so its expected present value is $M e^{-\delta(n-k)}{}_{n-k}p_{x+k}$.

Subtract premiums not yet paid.
For each scheduled premium, multiply $P$ by its discount factor and the probability the insured is alive to pay it. Include a premium due at the valuation instant only when valuing immediately before it is paid.

This argument also explains the key contrast with a pure endowment: a continuous endowment has a death-benefit term as well as a survival-benefit term. Omitting either one changes the contract being valued.

Constant Forces of Mortality and Interest

A useful exam model assumes a constant force of mortality $\mu$ over the remaining term and a constant force of interest $\delta$. Then ${}_sp_{x+k}=e^{-\mu s}$, so each death-benefit payment is weighted by $e^{-(\mu+\delta)s}$. For $m$ years remaining, the benefit value becomes:

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

The integral is evaluated using $\int_0^m e^{-cs}\,ds=(1-e^{-cm})/c$, with $c=\mu+\delta$. The maturity factor includes both survival and discounting. This constant-force formula does not apply if mortality or interest varies over the remaining term; in that case, use the general integral and survival probability.

Worked Example: One-Year Reserve Before the Premium

A hypothetical one-year endowment policy has one year remaining. It pays $\$60{,}000$ immediately on death during the year and $\$100{,}000$ at maturity if the insured survives. The constant forces are $\mu=\delta=\ln 2$. A premium of $\$40{,}000$ is due immediately. Find the reserve just before that premium.

Method. Calculate the expected present value of death and maturity benefits separately, then subtract the premium due at the valuation time. Under the stated forces, $\mu+\delta=2\ln 2$, so $e^{-(\mu+\delta)}=1/4$.

The death-benefit factor for one year is $\frac{\mu}{\mu+\delta}(1-e^{-(\mu+\delta)})=\frac12(1-\frac14)=\frac38$. Thus the death-benefit value is $\$60{,}000(3/8)=\$22{,}500$.

The maturity-benefit factor is $e^{-(\mu+\delta)}=1/4$. Its value is $\$100{,}000(1/4)=\$25{,}000$.

Total benefit value is $\$22{,}500+\$25{,}000=\$47{,}500$. Subtract the premium due immediately: $\$47{,}500-\$40{,}000=\boxed{\$7{,}500}$.

Check and trap avoided. Both benefit components are positive, and the premium is subtracted once because the reserve is just before it is paid. The reserve is not zero: the benefit value exceeds the premium by $\$7{,}500$.

Net Premiums and Duration Reserves

At issue, the net annual premium is chosen so that the expected present value of premiums equals the expected present value of benefits, under the same mortality and interest assumptions. With annual premiums due at times $0,1,\ldots,n-1$, provided the insured is alive at each due date, the net premium is:

P= \frac{ B\displaystyle\int_0^n e^{-\delta s}{}_sp_x\mu_{x+s}\,ds + M e^{-\delta n}{}_np_x }{ \displaystyle\sum_{r=0}^{n-1}v^r{}_rp_x }.

The denominator is an expected present value of premiums, not simply the number of premium dates. Survival to each date and discounting both matter. At a later duration, use only the future portion of the premium schedule. A reserve immediately before an anniversary premium includes that premium; a reserve immediately after it does not.

Worked Example: Two-Year Net Premium and Anniversary Reserve

A hypothetical two-year policy pays $\$40{,}000$ immediately on death during the term and $\$80{,}000$ at the end of year 2 if the insured survives. The forces are constant, with $\mu=\delta=\ln 2$. Annual premiums are due at times 0 and 1 while the insured is alive. Find the net annual premium and the reserve at duration 1 just before the premium due then.

Method. First value the full two-year benefit stream at issue. Then divide by the expected present value of the two premiums. Finally, at duration 1 value only the remaining year’s benefits and subtract the premium due at that duration.

For one year, the death factor is $3/8$. For a death during year 2, the factor from issue is the probability-and-discount factor to reach year 1, $e^{-(\mu+\delta)}=1/4$, multiplied by the one-year death factor $3/8$. Thus the two-year death factor is $3/8+(1/4)(3/8)=3/8+3/32=15/32$. The death-benefit value is $\$40{,}000(15/32)=\$18{,}750$.

The maturity factor is $e^{-2(\mu+\delta)}=1/16$, so the maturity-benefit value is $\$80{,}000(1/16)=\$5{,}000$. Total benefit value at issue is $\$18{,}750+\$5{,}000=\$23{,}750$.

The premium present-value factor is $1+e^{-(\mu+\delta)}=1+1/4=5/4$. Therefore $P=\$23{,}750/(5/4)=\$19{,}000$.

At duration 1, one year remains. The death-benefit value is $\$40{,}000(3/8)=\$15{,}000$; the maturity-benefit value is $\$80{,}000(1/4)=\$20{,}000$. The total is $\$35{,}000$. Since the reserve is just before the premium due at duration 1, subtract $\$19{,}000$: the reserve is $\boxed{\$16{,}000}$.

Check and trap avoided. The issue premium is verified by $\$19{,}000(5/4)=\$23{,}750$, matching the issue benefit value. At duration 1, do not subtract the original two premiums or omit the premium currently due.

Reserves Between Premium Dates

A valuation time need not be an integer duration. The benefit formula still uses the exact remaining time $m=n-t$. The premium calculation must follow the actual schedule: a premium is included only if its scheduled date is after the valuation time, or at that time when the reserve is defined just before payment. In particular, do not treat a fractional remaining term as a whole number of annual premiums.

Worked Example: Fractional Duration with One Premium Remaining

Consider a hypothetical two-year policy with death benefit $\$32{,}000$, maturity benefit $\$64{,}000$, and constant forces $\mu=\delta=\ln 2$. Annual premiums are payable at times 0 and 1 while alive. Find the reserve at duration $0.5$, after the time-0 premium has been paid and before any premium at time 1.

Method. First find the net annual premium at issue, then value the remaining benefits and the single premium still scheduled at time 1 from duration $0.5$.

At issue, the two-year death factor is $15/32$, so the death-benefit value is $\$32{,}000(15/32)=\$15{,}000$. The maturity factor is $1/16$, so the maturity-benefit value is $\$64{,}000(1/16)=\$4{,}000$. Total benefits are $\$19{,}000$. The premium factor is $1+1/4=5/4$, giving $P=\$19{,}000/(5/4)=\$15{,}200$.

From duration $0.5$, the remaining term is $1.5$ years. The death factor is $\frac12(1-e^{-3\ln 2})=\frac12(1-\frac18)=7/16$, giving a death-benefit value of $\$32{,}000(7/16)=\$14{,}000$. The maturity factor is $e^{-3\ln 2}=1/8$, giving a maturity-benefit value of $\$64{,}000(1/8)=\$8{,}000$. Total remaining benefit value is $\$22{,}000$.

The premium at time 1 is $0.5$ years after the valuation time. Its expected present-value factor is $e^{-(\mu+\delta)(0.5)}=e^{-\ln 2}=1/2$. Therefore its value is $\$15{,}200(1/2)=\$7{,}600$. The reserve is $\$22{,}000-\$7{,}600=\boxed{\$14{,}400}$.

Check and trap avoided. There is exactly one future premium date, at time 1. The premium is discounted and weighted for survival over half a year; it is not treated as due immediately at duration $0.5$.

Common Mistakes and Boundary Checks

Leaving out the maturity payment. An endowment includes a survival benefit, even if the death benefit is paid continuously. Include $M e^{-\delta m}{}_mp_{x+k}$.
Using only the maturity payment. The death benefit is payable at every possible time before maturity, so its value is an integral over the remaining term.
Misreading premium timing. A reserve just before a premium includes that premium among future inflows; a reserve just after it does not.
Using the wrong remaining term. At duration $k$, the remaining term is $n-k$, not $n$. At a fractional duration $t$, use $n-t$.
Mixing annual and continuous timing. Continuous death-benefit timing does not imply continuously paid premiums. Use the schedule specified by the contract.

A useful boundary check is maturity. Immediately before maturity, if the insured is alive and the maturity payment is still due, the prospective benefit value approaches $M$, with no future premiums if the premium term has ended. Immediately after the maturity benefit is paid, that benefit is no longer outstanding. The reserve convention and cash-flow timing must therefore be stated when interpreting a boundary value.

Key takeaway: A continuous endowment reserve is the present value of future continuous death benefits plus the discounted survival benefit at maturity, less the present value of premiums still due. Condition on survival to the valuation duration, use the exact remaining term, and match premium inclusion to the before-or-after-payment convention.

Check Your Understanding

For each question, state the benefit values and apply the premium timing described.

A one-year policy has one year remaining, death benefit $\$24{,}000$, maturity benefit $\$40{,}000$, and $\mu=\delta=\ln 2$. A $\$12{,}000$ premium is due immediately. Find the reserve just before payment. (Answer: $\$7{,}000$)
At a duration with one year remaining, a policy pays $\$20{,}000$ on death and $\$32{,}000$ at maturity. The forces are $\mu=\delta=\ln 2$, and a $\$5{,}000$ premium is due immediately. Find the reserve just before the premium. (Answer: $\$10{,}500$)
A two-year policy has death benefit $\$16{,}000$, maturity benefit $\$32{,}000$, and $\mu=\delta=\ln 2$. Annual premiums are due at times 0 and 1 while alive. Find the net annual premium. (Answer: $\$7{,}600$)
A one-year policy has one year remaining, death benefit $\$100{,}000$, maturity benefit $\$50{,}000$, constant $\mu=0.20$, constant $\delta=0.05$, and a $\$20{,}000$ premium due immediately. Find the reserve just before the premium, to the nearest cent. (Answer: $\$36{,}635.98$)

What You'll Learn