Pure Endowment Recursion Across One Year

Pure Endowments and One-Year Steps

A pure endowment pays only if the insured is alive at a specified future time. Its value therefore depends on two factors: the probability of surviving to the payment date and the discounting of a payment made in the future. When the term spans several years, a one-year recursion lets us build the value one step at a time, using the survival probability appropriate to each age.

Assume a life aged $x$ is covered by a contract that pays 1 at the end of $n$ years if the life is then alive, and pays nothing on death before that date. Let $i$ be the effective annual interest rate, $v=1/(1+i)$, and ${}_np_x$ the probability that the life survives $n$ years from age $x$. The actuarial present value of this unit benefit is denoted ${}_nE_x$.

Definition: For a pure endowment paying 1 at time $n$ if the life aged $x$ is then alive, with no benefit payable on earlier death, $$ {}_nE_x = v^n\,{}_np_x. $$ For a benefit of amount $B$, the actuarial present value is $B\,{}_nE_x$. The zero-year value is ${}_0E_x=1$.

This definition assumes a constant effective annual interest rate for the valuation, and that survival probabilities are measured over annual intervals. The benefit is paid at the end of the term, not at its beginning. The recursion below also applies when survival probabilities vary by age; it does not assume a constant mortality rate.

Deriving the One-Year Recursion

To value an $n$-year pure endowment from age $x$, first consider the outcome over the coming year. The insured must survive that year, and the value at age $x+1$ must then be discounted back one year. The conditional survival identity is ${}_np_x=p_x\,{}_{n-1}p_{x+1}$: survival for $n$ years from age $x$ requires survival for the first year and then survival for the remaining $n-1$ years from age $x+1$.

\begin{aligned} {}_nE_x &=v^n\,{}_np_x\\ &=v^n p_x\,{}_{n-1}p_{x+1}\\ &=v p_x\left(v^{n-1}\,{}_{n-1}p_{x+1}\right)\\ &=v p_x\,{}_{n-1}E_{x+1},\qquad n\geq 1. \end{aligned}

Thus, the one-year recursion is

{}_nE_x=vp_x\,{}_{n-1}E_{x+1}.

The factor $p_x$ accounts for survival during the first year, $v$ discounts the value at the end of that year to time 0, and ${}_{n-1}E_{x+1}$ is the value of the remaining pure endowment conditional on reaching age $x+1$. The recursion moves the valuation forward by one age and reduces the remaining term by one year.

Set the horizon.
Write down the starting age $x$, term $n$, benefit amount, and annual effective interest rate.

Identify each one-year survival probability.
Use $p_x$ for the year from age $x$, then $p_{x+1}$ for the following year, and so on.

Apply the recursion successively.
At each step, multiply by $v p_{\text{current age}}$ and reduce the remaining term by one.

Scale by the benefit.
Multiply the resulting unit value by the amount payable at maturity. Round only the final monetary result.

Successive Steps and the Product Form

Applying the recursion repeatedly makes its structure explicit. For $n\geq 1$,

{}_nE_x =(vp_x)(vp_{x+1})\cdots(vp_{x+n-1})\,{}_0E_{x+n} =v^n\prod_{j=0}^{n-1}p_{x+j}.

Since ${}_0E_{x+n}=1$, this is the same value as $v^n\,{}_np_x$. The product form is especially useful when a problem gives annual survival probabilities separately. Each factor must correspond to the age at the start of its own year. If the probabilities change with age, replacing them all with the first year’s probability generally changes the answer.

Worked Example: A Three-Year Benefit with the Same Survival Probability Each Year

A hypothetical policy pays \$10,000 at the end of three years if the insured is alive then; it pays nothing on earlier death. Assume an effective annual interest rate of $i=4\%$, and assume the one-year survival probability is $0.95$ at each of the three successive ages. Find the actuarial present value to the nearest cent.

Method. First calculate the unit pure endowment value by applying the one-year recursion three times. Here $v=1/1.04=25/26$, so each one-year factor is

vp=\frac{25}{26}(0.95)=\frac{95}{104}=0.9134615385\ldots

Starting from ${}_0E_{x+3}=1$, the successive values are

\begin{aligned} {}_1E_{x+2}&=\frac{95}{104}=0.9134615385\ldots,\\ {}_2E_{x+1}&=\frac{95}{104}\left(\frac{95}{104}\right) =\frac{9025}{10816}=0.834\ldots,\\ {}_3E_x&=\frac{95}{104}\left(\frac{9025}{10816}\right) =\frac{857375}{1124864}=0.762203\ldots. \end{aligned}

The benefit amount is \$10,000, so the present value is

\$10{,}000\left(\frac{857375}{1124864}\right) =\$7{,}622.032\ldots \longrightarrow \boxed{\$7{,}622.03}.

Check. The same calculation is $10{,}000(0.95)^3/(1.04)^3$, which gives the same value. The common mistake avoided here is to account for survival but omit the three years of discounting.

Worked Example: Age-Specific Survival over Three Years

A hypothetical plan pays \$20,000 at the end of three years if the insured is alive. The effective annual interest rate is $i=25\%$. The one-year survival probabilities at the successive ages are $p_x=0.80$, $p_{x+1}=0.75$, and $p_{x+2}=0.90$. Calculate the actuarial present value.

Method. The discount factor is $v=1/1.25=0.80$. Work backward from the maturity date, using the survival probability for each year. At age $x+2$, the one-year value is

{}_1E_{x+2}=vp_{x+2}(1)=(0.80)(0.90)=0.72.

For the remaining two years from age $x+1$,

{}_2E_{x+1}=vp_{x+1}\,{}_1E_{x+2} =(0.80)(0.75)(0.72)=0.432.

Finally, from age $x$,

{}_3E_x=vp_x\,{}_2E_{x+1} =(0.80)(0.80)(0.432)=0.27648.

The actuarial present value is therefore

\$20{,}000(0.27648)=\boxed{\$5{,}529.60}.

Check. The three one-year factors are $0.64$, $0.60$, and $0.72$; their product is $0.27648$, as obtained recursively. The common mistake avoided is using $p_x=0.80$ for all three years even though the later survival probabilities differ.

Worked Example: Extending a Known Value by Two Years

For a hypothetical life, $i=25\%$, $p_x=0.90$, $p_{x+1}=0.80$, $p_{x+2}=0.80$, and $p_{x+3}=0.9765625$. Find the actuarial present value of a \$100,000 pure endowment payable at the end of four years. Use successive one-year steps.

Method. First value the remaining two years from age $x+2$. Since $v=0.80$,

{}_2E_{x+2} =(0.80)(0.80)(0.80)(0.9765625) =(0.64)(0.78125)=0.50.

Now extend the value one year backward to age $x+1$:

{}_3E_{x+1}=vp_{x+1}\,{}_2E_{x+2} =(0.80)(0.80)(0.50)=0.32.

Extend it one more year backward to age $x$:

{}_4E_x=vp_x\,{}_3E_{x+1} =(0.80)(0.90)(0.32)=0.2304.

Thus the four-year benefit’s actuarial present value is

\$100{,}000(0.2304)=\boxed{\$23{,}040.00}.

Check. The survival product is $(0.90)(0.80)(0.80)(0.9765625)=0.5625$, and $v^4=(0.80)^4=0.4096$; their product is $0.2304$. The common mistake avoided is treating ${}_2E_{x+2}$ as an undiscounted survival probability: it already includes two years of discounting.

Common Errors and Practical Checks

A pure endowment has no payment on death before maturity. Consequently, the probability used is survival to the payment date, not the probability of death during the term. Its value also includes discounting for the full time until payment. These features distinguish the calculation from simply multiplying a benefit by a survival probability.

For example, in the first worked example, omitting discounting would give the incorrect calculation

\$10{,}000(0.95)^3=\$8{,}573.75,

which is the expected benefit measured at the maturity date, not its actuarial present value at time 0. The correct calculation discounts each year as well:

\$10{,}000\left(\frac{0.95}{1.04}\right)^3=\$7{,}622.03 \quad\text{to the nearest cent.}

A second check is to compare the successive recursion with the direct product. The two must agree because the recursion simply factors the full-term survival probability into one-year conditional probabilities and separates the discount factor into annual factors. If they do not agree, check the starting age attached to each $p$, the number of discount periods, and whether the stated amount is a unit benefit or a larger maturity benefit.

Key takeaway: At each step, multiply by $v$ and the one-year survival probability at the current age, then reduce the remaining term by one. After $n$ steps, the unit value is $v^n\,{}_np_x$; multiply by the maturity benefit only after the survival-and-discount value is established.

Check Your Understanding

Use the one-year recursion or its product form. Treat each stated rate as an effective annual rate, and round monetary answers to the nearest cent where needed.

A unit pure endowment runs for two years. If $i=5\%$ and the one-year survival probability is $0.90$ at each of the two ages, find its actuarial present value. (Answer: $0.7347$.)
A hypothetical contract pays \$30,000 after two years if the insured is alive. Let $i=4\%$, $p_x=0.98$, and $p_{x+1}=0.97$. Find the actuarial present value. (Answer: \$26,366.49.)
For a three-year unit pure endowment, let $i=10\%$ and the successive annual survival probabilities be $0.90$, $0.80$, and $0.95$. Find ${}_3E_x$ to four decimal places. (Answer: $0.5139$.)
Suppose $i=5\%$, $p_x=0.95$, and ${}_2E_{x+1}=0.64$. Use one recursion step to find ${}_3E_x$ to four decimal places. (Answer: $0.5790$.)
A hypothetical policy pays \$100,000 at the end of three years if the insured is alive. The effective annual rate is $2\%$, and the successive annual death probabilities are $0.01$, $0.02$, and $0.03$. Find the actuarial present value to the nearest cent. (Answer: \$88,681.39.)

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