Pure Endowment Values at a Fixed Term

Introduction

A benefit payable only on a specified future date, and only if the insured is alive then, depends on two separate factors: whether the insured survives to that date and the value today of money paid at that date. A pure endowment isolates those factors. Unlike insurance that pays on death, it pays nothing if the insured dies before maturity.

The calculation is direct once the contract condition and payment time are identified. First determine the probability of being alive at the fixed term. Then discount the maturity benefit to the valuation date. The survival probability is not a discount factor, and neither factor can be omitted.

Definition: A pure endowment issued to a life aged $x$ is a contract that pays a specified benefit at time $n$ if and only if the insured is alive at time $n$. It pays no benefit on death before time $n$. For a unit benefit of 1 payable at time $n$, the expected present value is denoted by ${}_nE_x$: $$ {}_nE_x = v^n\,{}_np_x, $$ where $v=1/(1+i)$ is the annual discount factor at effective annual interest rate $i$, and ${}_np_x$ is the probability that a life aged $x$ survives for $n$ years.

For a maturity benefit $B$, the expected present value at issue is $B\,{}_nE_x$. This formula assumes a fixed benefit, a deterministic interest rate, and a survival probability appropriate to the stated term. It values the benefit alone: expenses, premiums, and other cash flows are not included. The term $n$ is measured in years when $i$ is an effective annual rate; for a different payment interval, use the discount factor corresponding to that interval.

Why Survival and Discounting Are Both Needed

Let $I_n$ equal 1 if the insured is alive at time $n$, and 0 otherwise. The present value of the benefit is $Bv^n I_n$. It is zero on paths where the insured does not survive and equals $Bv^n$ on paths where the insured does survive. Taking its expectation gives the pure endowment value:

E[Bv^n I_n] = Bv^n E[I_n] = Bv^n P(\text{alive at time }n) = Bv^n\,{}_np_x.

The key step is $E[I_n]={}_np_x$: the expected value of an indicator is the probability of the event it indicates. This also explains why the benefit is discounted for the full time to maturity, not just for the time until some expected death date. The contract makes no payment at death.

If annual survival probabilities are provided for each year, multiply them to obtain survival to the end of the term. If a life table gives numbers alive at the relevant ages, use their ratio. Under a continuous model with force of mortality $\mu_{x+s}$, survival is obtained by integrating that force over the term. With force of interest $\delta$, the discount factor over $n$ years is $e^{-\delta n}$.

Theorem: Under a constant force of mortality $\mu$ and a constant force of interest $\delta$, both measured per year over the full term $n$, $$ {}_np_x=e^{-\mu n}, \qquad {}_nE_x=e^{-(\mu+\delta)n}, \qquad E[\text{present value of benefit }B]=Be^{-(\mu+\delta)n}. $$ The constant-force formulas apply only when those forces are constant over the term. If the forces vary with duration, integrate them rather than multiplying a single constant by $n$.

Worked Valuations

Worked Example: Annual Survival Probabilities and Effective Interest

A hypothetical contract pays $20,000 at the end of two years if the insured is alive then, and pays nothing otherwise. The probabilities of surviving each year, conditional on being alive at the start of that year, are both 0.95. The effective annual interest rate is $i=0.05$. Find the expected present value at issue.

Method. Multiply the conditional annual survival probabilities to obtain the probability of surviving the entire two-year term: $$ {}_2p_x=0.95(0.95)=0.9025. $$ The two-year discount factor is $$ v^2=\frac{1}{(1.05)^2}=\frac{1}{1.1025}. $$ Therefore the unit pure endowment value is $$ {}_2E_x=\frac{0.9025}{1.1025} =\frac{361/400}{441/400} =\frac{361}{441} \approx 0.8185941. $$ For the stated benefit, the expected present value is $$ 20{,}000(0.8185941\ldots) =\frac{20{,}000(361)}{441} =\$16{,}371.88. $$ The result is rounded to the nearest cent. A check is that the present value is less than the $20,000 benefit, as both survival risk and positive interest reduce its value. The common mistake avoided here is discounting the benefit but forgetting that it is conditional on survival.

Worked Example: Constant Forces Over a Fixed Term

A hypothetical five-year pure endowment pays $100,000 at the end of year 5 if the insured is alive. Assume a constant force of mortality $\mu=0.03$ per year and a constant force of interest $\delta=0.04$ per year. Calculate the expected present value.

Method. The survival probability to year 5 is $$ {}_5p_x=e^{-0.03(5)}=e^{-0.15}\approx 0.86070798. $$ The discount factor for five years is $$ e^{-0.04(5)}=e^{-0.20}\approx 0.81873075. $$ Multiplying these factors and the benefit gives $$ 100{,}000(e^{-0.15})(e^{-0.20}) =100{,}000e^{-0.35} \approx \$70{,}468.81. $$ The value is rounded to the nearest cent. Equivalently, the combined exponent is $-(\mu+\delta)5=-(0.03+0.04)5=-0.35$. As a check, $e^{-0.35}\approx0.70468809$, so the value is about 70.47% of the maturity benefit. This method avoids treating a force of interest as an effective annual rate: the appropriate discount factor under the stated force model is $e^{-\delta n}$, not $(1+\delta)^{-n}$.

Worked Example: Reading Survival From a Life Table

A hypothetical life table has $\ell_{60}=10{,}500$ and $\ell_{62}=8{,}400$. A two-year pure endowment issued to a life aged 60 pays $50,000 at age 62 if the insured is alive. The effective annual interest rate is $i=0.05$. Find its expected present value at issue.

Method. In a life table, $\ell_y$ is the number of lives surviving to age $y$ in the table’s cohort. The probability of surviving from age 60 to age 62 is the ratio of survivors: $$ {}_2p_{60}=\frac{\ell_{62}}{\ell_{60}} =\frac{8{,}400}{10{,}500}=0.8. $$ The two-year discount factor is $$ v^2=\frac{1}{(1.05)^2} =\frac{1}{1.1025} \approx 0.9070295. $$ Thus $$ E[\text{present value}] =50{,}000(0.8)\frac{1}{1.1025} =\frac{40{,}000}{1.1025} \approx \$36{,}281.18. $$ The answer is rounded to the nearest cent. The intermediate survival probability is exactly 0.8, consistent with the life-table ratio. The common mistake avoided is reversing that ratio: $\ell_{62}/\ell_{60}$, not $\ell_{60}/\ell_{62}$, is the proportion surviving to the later age.

Common Errors and Checks

The pure endowment value is sensitive to both the event described by the contract and the timing of its payment. A useful calculation sequence is to name the survival event, determine its probability over the entire term, apply the appropriate discount factor, and multiply by the benefit. Keeping those steps separate makes errors easier to locate.

Using the probability of death instead of survival. If the contract pays only when the insured survives, use ${}_np_x$, not ${}_nq_x=1-{}_np_x$. The death probability corresponds to the event that produces no pure endowment payment.
Discounting without allowing for survival. For a $20,000 benefit due in two years at 5%, the calculation $20{,}000/(1.05)^2=\$18{,}140.59$ is only the discounted amount conditional on payment. It is not the expected present value when survival probability is 0.9025. Including survival gives $20{,}000(0.9025)/(1.05)^2=\$16{,}371.88$.
Discounting to the wrong date. A benefit payable at the end of year $n$ receives the factor $v^n$, even if the survival probability is supplied directly for the whole term.
Mixing interest conventions. With effective annual rate $i$, use $(1+i)^{-n}$. With constant force of interest $\delta$, use $e^{-\delta n}$. Do not substitute one convention’s formula for the other without converting the rate.
Misreading life-table values. The later-age number of survivors is the numerator and the starting-age number is the denominator. The ratio must be between 0 and 1 for a valid survival probability.

A final reasonableness check is to compare the expected present value with the maturity benefit. With nonnegative interest and a survival probability no greater than 1, the value cannot exceed the benefit. If either assumption is changed, apply the stated model rather than relying on that check alone.

Key takeaway: A pure endowment’s expected present value equals the maturity benefit multiplied by the probability of surviving to maturity and by the discount factor for the full time to maturity. State the survival and interest assumptions explicitly before calculating.

Check Your Understanding

For each hypothetical contract, identify the survival probability for the full term, apply the correct discounting convention, and report the requested value.

A two-year pure endowment pays $12,100. Conditional annual survival probabilities are 0.90 in each year and $i=0.10$. Find the expected present value at issue. (Answer: $8,100.00)
A four-year pure endowment pays $20,000. The force of mortality is constant at 0.02 per year and the force of interest is constant at 0.03 per year. Find the expected present value. (Answer: $16,374.62)
A life table gives $\ell_{50}=5{,}000$ and $\ell_{52}=4{,}000$. A benefit of $12,100 is payable at age 52 if the insured, currently age 50, is alive. Given $i=0.10$, calculate the issue value. (Answer: $8,000.00)
A three-year pure endowment pays $50,000, has an expected present value of $40,000, and is valued at $i=0.04$. What survival probability to maturity is implied by these data? (Answer: ${}_3p_x=0.8999$, to four decimal places)
Two hypothetical contracts each pay $10,000 in two years only if the insured survives. Contract A has annual conditional survival probabilities 0.90 and 0.90 and $i=0.10$. Contract B has annual conditional survival probabilities 0.95 and 0.95 and $i=0.15$. Calculate each expected present value and identify the larger one. (Answer: A = $6,694.21; B = $6,824.20; Contract B is larger)

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