Pure Endowment as a Survival-Discount Factor

Why Survival and Discounting Both Matter

A pure endowment pays a benefit at a specified future time only if the insured person is alive then. Its value therefore depends on two distinct events: the benefit must be discounted from its payment date, and the insured must survive long enough for it to be paid. The pure endowment factor combines these effects.

This tutorial considers an integer duration of $n$ years, a life aged $x$ at issue, and a benefit paid at the end of year $n$. The interest rate is a deterministic effective annual rate $i$, and the mortality model supplies the probability of survival to that date. Unless a problem states otherwise, there is no payment if the life dies before the benefit date.

Definition: A unit pure endowment issued to a life aged $x$ pays 1 at time $n$ if the life survives to time $n$, and pays 0 otherwise. Its actuarial present value is denoted by ${}_nE_x$. For a benefit of amount $B$, the actuarial present value is $B\,{}_nE_x$.

Deriving the Pure Endowment Factor

Let $I_n$ equal 1 if the insured survives $n$ years and 0 otherwise. The benefit payment is $B I_n$ at time $n$. With a fixed effective annual interest rate $i$, the discount factor for a payment at that time is $v^n$, where $v=1/(1+i)$. Thus the random present value is $Bv^n I_n$. Taking its expected value gives the pure endowment value.

\begin{aligned} \text{Actuarial present value} &=E[Bv^n I_n]\\ &=Bv^nE[I_n]\\ &=Bv^n{}_np_x. \end{aligned}

The last equality holds because $I_n$ is an indicator: its expected value is the probability that it equals 1, namely ${}_np_x$, the probability that a life aged $x$ survives $n$ years. For a unit benefit, this proves

{}_nE_x=v^n{}_np_x=\frac{{}_np_x}{(1+i)^n}.

The formula applies when the benefit is paid at time $n$ only if the insured is alive then, the interest rate is fixed and effective annually, and the survival probability refers to that same duration. No assumption that interest and mortality are independent is needed here: the discount factor is deterministic, so it can be taken outside the expectation. If interest were random, or the payment date depended on the time of death, this calculation would need to be reconsidered.

Key interpretation: ${}_nE_x$ is the expected present value of a unit payment, not the payment amount conditional on survival. Conditional on receiving a unit benefit, its present value is $v^n$. The probability of receiving it is ${}_np_x$.

Getting the Survival Probability

The survival probability can be supplied directly, built from annual conditional survival probabilities, or obtained from a life table. If the conditional probability of surviving each year is given for the relevant attained age, multiply the successive conditional probabilities. For example,

{}_np_x={}_1p_x\,{}_1p_{x+1}\cdots{}_1p_{x+n-1}.

The factors are conditional probabilities at successive ages; they are not generally equal. If a life table gives numbers $\ell_x$ and $\ell_{x+n}$, the corresponding survival probability is

{}_np_x=\frac{\ell_{x+n}}{\ell_x}, \qquad {}_nE_x=v^n\frac{\ell_{x+n}}{\ell_x}.

This ratio uses the number alive at age $x$ as the starting group and the number alive at age $x+n$ as the survivors. It is a probability conditional on being alive at age $x$, not the proportion of the original population that survives from birth.

Worked Valuations

Worked Example: Two-Year Benefit With Annual Survival Probabilities

A hypothetical contract issued to a life aged 40 pays $10,000 at the end of two years if the life is then alive, with no payment otherwise. The effective annual interest rate is 25%. The probability of surviving each of the next two years, conditional on being alive at the start of that year, is 0.90. Find the probability of payment, the present value conditional on payment, and the actuarial present value.

Method. First multiply the conditional survival probabilities to obtain the two-year survival probability; then discount the benefit and weight that present value by the probability of survival.

The two-year survival probability is

{}_2p_{40}=0.90(0.90)=0.81.

Therefore, the probability that no benefit is paid is $1-0.81=0.19$. The two-year discount factor is

v^2=\frac{1}{(1.25)^2}=\frac{1}{1.5625}=0.64.

If the life survives, the present value of the $10,000 payment is

10{,}000(0.64)=\$6{,}400.

The expected present value is the survival probability times this conditional present value:

0.81(\$6{,}400)=\$5{,}184.00.

Equivalently, $10{,}000v^2{}_2p_{40}=10{,}000(0.64)(0.81)=\$5{,}184.00$. As a check, the value is below the $6,400 conditional present value because the payment is not certain. Common mistake avoided: reporting $6,400 as the actuarial present value ignores the 0.81 probability that the insured survives to the payment date.

Worked Example: Using Life-Table Counts

A hypothetical life table has $\ell_{52}=8{,}000$ and $\ell_{54}=6{,}000$. A policy issued at age 52 pays $24,000 at the end of two years if the insured is alive. The effective annual interest rate is 25%. Calculate the actuarial present value and the probability that the benefit is not paid.

Method. Obtain the two-year survival probability as the ratio of the life-table counts, and then apply the two-year discount factor.

The probability of surviving from age 52 to age 54 is

{}_2p_{52}=\frac{\ell_{54}}{\ell_{52}} =\frac{6{,}000}{8{,}000}=0.75.

Thus the probability of no payment is $1-0.75=0.25$. At 25% effective annual interest,

v^2=\frac{1}{(1.25)^2}=0.64.

The present value of the benefit conditional on survival is

\$24{,}000(0.64)=\$15{,}360.

Weighting by the probability of survival gives

\$15{,}360(0.75)=\$11{,}520.00.

The direct check is $\$24{,}000(0.64)(0.75)=\$11{,}520.00$. The computed value is less than the conditional present value, as expected when payment is contingent on survival. Common mistake avoided: using $\ell_{54}/\ell_{52}$ in the wrong order would produce a ratio greater than 1, which cannot be a survival probability.

Worked Example: Recovering the Interest Rate

A hypothetical unit pure endowment lasts two years. The conditional probability of survival in each year is 0.90, and the actuarial present value is 0.5184. Find the effective annual interest rate. Assume the same survival probability applies in both years and interest is fixed.

Method. Calculate the two-year survival probability, substitute it into the pure endowment identity, and solve for the positive effective annual rate.

The probability of surviving both years is

{}_2p_x=0.90(0.90)=0.81.

Let $i$ be the unknown effective annual interest rate. The given actuarial present value must satisfy

0.5184=\frac{0.81}{(1+i)^2}.

Solving,

(1+i)^2=\frac{0.81}{0.5184}=1.5625, \qquad 1+i=\sqrt{1.5625}=1.25, \qquad i=0.25.

The effective annual interest rate is therefore 25.00%. Check by substitution: $0.81/(1.25)^2=0.81/1.5625=0.5184$, the stated value. Common mistake avoided: using the one-year survival probability 0.90 instead of the two-year probability 0.81 would mismatch mortality duration and benefit timing.

Timing and Other Common Traps

The duration in the survival probability and the exponent in the discount factor must both match the benefit date. A benefit payable at the end of year $n$ uses ${}_np_x$ and $v^n$. Discounting by $v^{n-1}$ moves the payment one year too early; using ${}_{n-1}p_x$ fails to require survival through the full coverage period.

It is also important to distinguish a unit pure endowment factor from a contract benefit amount. The factor ${}_nE_x$ values a unit payment. For a fixed benefit $B$, multiply the factor by $B$. If the problem asks for the present value conditional on survival, do not multiply by the survival probability; if it asks for the actuarial present value, do.

Wrong timing: applying $v^n$ to a payment at a different date. Use the actual time from issue to payment.
Wrong survival period: using survival to an earlier age. The insured must be alive on the stated benefit date.
Wrong quantity: confusing the present value if paid with its expected present value across all outcomes.
Wrong life-table ratio: reversing the starting and ending counts. For survival from age $x$ to $x+n$, use $\ell_{x+n}/\ell_x$.

Key takeaway: For a fixed benefit $B$ payable at time $n$ only if a life aged $x$ survives to that time, the actuarial present value is $Bv^n{}_np_x$. Check the benefit date, survival duration, and whether the requested value is conditional or expected.

Check Your Understanding

For each problem, identify the relevant survival probability before applying the discount factor.

A hypothetical one-year pure endowment pays $5,000 if the insured survives one year. The effective annual interest rate is 20%, and the one-year survival probability is 0.80. Find the actuarial present value. (Answer: $3,333.33)
A hypothetical three-year unit pure endowment has conditional survival probability 0.80 in each of the three years and an effective annual interest rate of 25%. Find its actuarial present value. (Answer: 0.262144)
A life table gives $\ell_{54}=9{,}000$ and $\ell_{57}=6{,}300$. A $8,000 benefit is payable at the end of three years if a life aged 54 survives. At an effective annual interest rate of 5%, find the actuarial present value to the nearest cent. (Answer: $4,837.49)
A two-year unit pure endowment has actuarial present value 0.4096 at an effective annual interest rate of 25%. The first-year survival probability is 0.80. Find the conditional probability of surviving the second year, given survival to the start of that year. (Answer: 0.80)
A hypothetical $16,000 pure endowment is payable after three years. The probability of survival to that date is 0.75, and the effective annual interest rate is 10%. Find both the probability of no payment and the actuarial present value to the nearest cent. (Answer: probability of no payment $=0.25$; actuarial present value $=\$9{,}015.78$)

What You'll Learn