Pure Endowment Reserves with Survival Benefits

Why Condition on Survival at the Valuation Date?

A pure endowment pays a benefit only if the insured is alive at a specified future date. To value the contract at an intermediate duration, the relevant question is not the probability of surviving from issue. It is the probability of surviving from the valuation date to the maturity date, given survival to the valuation date. That conditional view is the foundation of its prospective reserve.

This distinction matters in practice. A reserve is calculated for a policy that is in force at the valuation date. Its future cash flows are therefore valued conditional on the insured being alive then. The same principle applies whether the contract has a single premium or premiums still due.

Definition: A pure endowment issued at age $x$ for $n$ years pays a benefit $B$ at time $n$ if the insured is then alive, and pays no death benefit. Its benefit at maturity is therefore contingent on survival through the full term.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Write ${}_tp_y$ for the probability that a life aged $y$ survives for $t$ years. Suppose the policy is valued at duration $k$, where $0\leq k<n$, and the insured is alive at that date. The remaining term is $n-k$ years, and the insured’s age is $x+k$.

Key result: For a pure endowment with no future premiums, the prospective reserve at duration $k$, conditional on the insured being alive, is $$ {}_kV_x = Bv^{n-k}{}_{n-k}p_{x+k}. $$ This is the maturity benefit discounted for the remaining term and multiplied by the probability of surviving that remaining term.

Deriving the Benefit Value

At duration $k$, there are two relevant possibilities before maturity: the insured survives to time $n$, or does not. In the first case, the benefit $B$ is paid at time $n$, which is $n-k$ years in the future. In the second case, no benefit is paid. Thus the expected present value at duration $k$ is the discounted benefit times the conditional probability of receiving it.

\operatorname{EPV}_k(\text{benefit}) = v^{n-k}\left(B\,P(\text{survive from }k\text{ to }n\mid\text{alive at }k)\right) = Bv^{n-k}{}_{n-k}p_{x+k}.

The conditioning is also visible in the survival-probability identity ${}_np_x={}_kp_x\,{}_{n-k}p_{x+k}$, when ${}_kp_x>0$. Dividing by ${}_kp_x$ gives the probability of surviving the remaining term conditional on being alive at duration $k$. Multiplying the reserve by ${}_kp_x$ again would value the policy as though it might not be in force at the valuation date.

Worked Example: A Single-Premium Pure Endowment

A hypothetical policy pays $12,000 at the end of year 4 if the insured is alive. It has no death benefit and no future premiums. At duration 2, the insured is alive. The annual effective interest rate is 25%, and the probability of surviving each of the next two years is 0.80. Find the reserve at duration 2.

Method. Value the one possible future payment conditional on being alive at duration 2. The remaining term is $4-2=2$ years, and the two-year survival probability is $0.80\times0.80=0.64$.

The discount factor is $v=1/1.25=0.80$, so $v^2=0.64$. Therefore,

{}_2V_x = 12{,}000(0.64)(0.64)=12{,}000(0.4096)=\$4{,}915.20.

Check. The expected payment at maturity, conditional on survival to duration 2, is $12{,}000(0.64)=\$7{,}680$. Discounting that amount for two years gives $7{,}680(0.64)=\$4{,}915.20$, as required. The common mistake avoided here is using survival from issue rather than survival over the remaining two years.

Reserves When Premiums Are Still Due

A level-premium pure endowment has both a future benefit and, while the contract remains in force, future premium payments. The prospective reserve is the expected present value of future benefits minus the expected present value of future premiums. To make the timing explicit, suppose premiums $P$ are payable annually in advance at times $0,1,\ldots,n-1$, provided the insured is alive at each payment date.

In the formula below, the reserve is measured immediately after the premium at duration $k$ has been paid. Consequently, that premium is no longer a future payment. If $k<n$, the remaining premiums are due at durations $k+1,\ldots,n-1$. A premium due at duration $k$ is instead included when valuing the contract immediately before that premium is paid.

{}_kV_x = Bv^{n-k}{}_{n-k}p_{x+k} - P\sum_{j=1}^{n-k-1}v^j{}_jp_{x+k}, \qquad 0\leq k<n.

Each term in the sum values one future premium: $j$ years from the valuation date, discounted by $v^j$, and payable only if the insured survives to that payment date. If $n-k=1$, the sum is empty: after the premium at duration $k$, there are no later scheduled premiums before maturity.

The net level premium at issue can be found by equating the expected present values of benefits and premiums at time 0, before the initial premium is paid. Under the stated premium timing and assuming no expenses, this gives

P\sum_{j=0}^{n-1}v^j{}_jp_x = Bv^n{}_np_x, \qquad P= \frac{Bv^n{}_np_x}{\displaystyle\sum_{j=0}^{n-1}v^j{}_jp_x}.

This issue-date equation is useful as a check, but the intermediate reserve still uses the probabilities conditional on survival to duration $k$. The premium denominator represents the expected number of discounted premium payments, not a fixed number of payments regardless of survival.

Worked Example: A Net-Premium Pure Endowment Reserve

A hypothetical three-year pure endowment pays $10,000 at time 3 if the insured is alive. Annual premiums are payable in advance at times 0, 1, and 2 while the insured is alive. The annual effective interest rate is 25%, and the probability of surviving each year is 0.80. Find the net annual premium and the reserve at duration 1, immediately after the premium at that date has been paid.

Step 1: Find the one-year discounted survival factor. For each year, $v=1/1.25=0.80$, so $v\,{}_1p=0.80(0.80)=0.64$. The corresponding factors for two and three years are $0.64^2=0.4096$ and $0.64^3=0.262144$.

Step 2: Calculate the net premium at issue. The benefit’s expected present value is $10{,}000(0.262144)=\$2{,}621.44$. The premium value factor is $1+0.64+0.4096=2.0496$. Hence

P=\frac{2{,}621.44}{2.0496} = \$1{,}279.00078\ldots \approx \$1{,}279.00.

Step 3: Value future cash flows at duration 1. There are two years remaining. The benefit value is $10{,}000(0.64^2)=\$4{,}096.00$. Since the premium at time 1 has just been paid, the only future premium is at time 2. Its value at duration 1 is $P(0.64)$, because it is due in one year and is paid only if the insured survives that year.

Using the unrounded premium, that future-premium value is $1{,}279.00078\ldots(0.64)=\$818.56050\ldots$. Thus the reserve is

{}_1V_x =4{,}096.00-818.56050\ldots =\$3{,}277.44.

Check. The net premium was set by equivalence at issue, using all three possible premium dates. At duration 1, the already-paid premiums are not future cash flows; only the time-2 premium is subtracted. The result is rounded to the nearest cent.

Premium Timing and a Useful Reserve Check

A reserve can differ depending on whether it is measured just before or just after a premium date. Immediately before the premium at duration $k$, that premium is still due if the insured is alive. Under the same annual-in-advance schedule, the prospective value immediately before that payment is

{}_kV_x^{\text{before}} = Bv^{n-k}{}_{n-k}p_{x+k} - P\sum_{j=0}^{n-k-1}v^j{}_jp_{x+k}.

The $j=0$ term equals 1, representing the premium payable immediately, conditional on being alive at that date. Immediately after that premium has been received, this term is no longer included. Under a level premium $P$, therefore, the before-premium value is the after-premium value minus $P$. This timing difference is not a change in mortality or interest assumptions.

Worked Example: Changing the Valuation Side of a Premium Date

A hypothetical four-year pure endowment pays $10,000 at time 4 if the insured is alive. At duration 2, annual premiums of $1,000 are due in advance while the insured is alive, at times 0 through 3. The annual effective interest rate is 25%, and each future year’s survival probability is 0.80. Find the reserve at duration 2 immediately after the premium there has been paid. Then find the value immediately before that premium.

After the premium. There are two years to maturity, so the benefit value is $10{,}000(0.64^2)=\$4{,}096.00$. Only the premium at time 3 remains. Its value is $1{,}000(0.80)(0.80)=\$640.00$: the first factor is discounting for one year, and the second is survival to that payment date. The reserve immediately after the time-2 premium is therefore

{}_2V_x=4{,}096.00-640.00=\$3{,}456.00.

Before the premium. The time-2 premium of $1,000 is now still payable. The future premium values total $1{,}000+640=\$1{,}640.00$, so

{}_2V_x^{\text{before}} =4{,}096.00-1{,}640.00 =\$2{,}456.00.

Check. The before-premium value is $3{,}456-1{,}000=\$2{,}456$, exactly one premium below the after-premium value. Subtracting the time-2 premium in the after-premium calculation would count it twice.

Common Errors to Avoid

Using survival from issue. At duration $k$, the policy is already in force. Use ${}_{n-k}p_{x+k}$, not ${}_np_x$, in the conditional reserve calculation.
Discounting for the original term. A valuation at duration $k$ is $n-k$ years before maturity, so the benefit is discounted by $v^{n-k}$, not $v^n$.
Subtracting every scheduled premium. Premiums paid before the valuation date are past cash flows. Include only premiums due after the chosen valuation instant.
Ignoring whether a premium is due at the valuation instant. State “before” or “after” the premium date and include the due premium only in the before-premium value.
Adding a death benefit that the contract does not have. A pure endowment pays only on survival to maturity. If the insured dies earlier, its stated benefit is zero.

Key takeaway: At an intermediate duration, first value the maturity benefit conditional on the insured being alive at that duration. Then subtract only the premiums that remain due, using their correct payment dates and survival conditions.

Check Your Understanding

For each problem, state which cash flows remain after the valuation instant before calculating.

A single-premium pure endowment pays $5,000 in two years if the insured survives. At the valuation date, the insured is alive; the effective annual interest rate is 25%, and survival probability is 0.80 in each remaining year. Find the reserve. (Answer: $2,048.00)
A pure endowment pays $15,000 in two years. At the valuation date, the insured is alive; the annual effective interest rate is 5%, and the survival probabilities for the first and second remaining years are 0.90 and 0.80. There are no future premiums. Find the reserve to the nearest cent. (Answer: $9,795.92)
A four-year pure endowment pays $10,000 at maturity if the insured is alive. At duration 2, immediately after the premium at that date, one premium of $1,000 remains, due in one year. The effective annual interest rate is 25%, and the survival probability to that premium date is 0.80. The probability of surviving both years to maturity is 0.64. Find the reserve. (Answer: $3,456.00)
Use the same contract and assumptions as in Question 3, but value it immediately before the premium due at duration 2. Find the reserve. (Answer: $2,456.00)
At a valuation date, an insured is alive and has three years until a $20,000 pure endowment benefit is due. The effective annual interest rate is 25%. The successive one-year survival probabilities from the valuation date are 0.90, 0.80, and 0.75. Premiums of $500 are due at the ends of years 1, 2, and 3 if the insured is alive. Find the reserve immediately after the valuation-date premium, if any, has been paid. (Answer: $4,800.96)

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