Recovering Term Insurance from Endowment Values

Why Separate the Maturity Benefit?

An endowment insurance contract can pay a benefit in either of two ways: on death during the term, or at the end of the term if the insured survives. If the goal is to find the value of the death benefit alone, the maturity payment must be separated from the total endowment value. The key is to value that maturity payment as a pure endowment, which includes both survival to the term’s end and discounting back to the valuation date.

This distinction matters because survival probability by itself is not the present value of a maturity benefit. The benefit is paid in the future, so its value must also reflect interest. The identity in this tutorial makes that adjustment explicit and is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: For a life aged $x$, an $n$-year term insurance pays 1 at the end of the year of death if death occurs within $n$ years, and pays nothing otherwise. An $n$-year endowment insurance pays 1 at the end of the year of death if death occurs within $n$ years, or pays 1 at time $n$ if the insured survives. An $n$-year pure endowment pays 1 at time $n$ only if the insured survives to time $n$.

The Identity and Its Assumptions

Use the following notation for unit benefits: $A^1_{x:n}$ is the actuarial present value of the $n$-year term insurance, $A^{\mathrm{end}}_{x:n}$ is the actuarial present value of the $n$-year endowment insurance, and ${}_nE_x$ is the actuarial present value of the $n$-year pure endowment. The superscript “1” indicates a unit death benefit; the endowment value also assumes a unit maturity benefit.

Assume an effective annual interest rate $i$, with $v=1/(1+i)$. Let ${}_np_x$ be the probability that a life aged $x$ survives $n$ years. Benefits are paid at the end of the year of death, or at time $n$ for a maturity benefit. Under those conventions, the pure endowment value is:

{}_nE_x=v^n\,{}_np_x

The endowment value includes the same death benefits as the term insurance, plus the maturity benefit on the survival-to-$n$ event. Therefore, the term value is obtained by subtracting the pure endowment value:

Theorem: For unit benefits with the payment timing just described, $$ A^{\mathrm{end}}_{x:n}=A^1_{x:n}+{}_nE_x \qquad\text{and hence}\qquad A^1_{x:n}=A^{\mathrm{end}}_{x:n}-{}_nE_x. $$

Why the Identity Holds

The two contracts have identical benefits if death occurs within the $n$-year term. They differ only if the insured survives to time $n$: the endowment contract then pays 1, while the term contract pays 0. The present value of that additional payment is $v^n$ on survival and 0 otherwise.

Taking the expected present value of this additional maturity payment gives:

E\!\left[v^n I(\text{survive to }n)\right] =v^n P(\text{survive to }n) =v^n\,{}_np_x ={}_nE_x.

Thus the endowment value is the term value plus the value of the maturity payment. Rearranging gives the recovery formula. This reasoning depends on matching benefit amounts and timing: the endowment’s death benefit must be the same as the term insurance’s death benefit, and the maturity benefit must be the one represented by ${}_nE_x$.

Identify the benefit basis.
Check that the endowment and term values use the same unit death benefit and the same death-benefit payment timing.

Find the pure endowment value.
Use ${}_nE_x=v^n\,{}_np_x$, or use a pure endowment value supplied on the same interest and mortality basis.

Subtract, then scale if needed.
Calculate $A^1_{x:n}=A^{\mathrm{end}}_{x:n}-{}_nE_x$. For a different level benefit, multiply the unit value by that benefit amount.

Worked Applications

Worked Example: Recovering a Two-Year Term Value

A hypothetical life aged $x$ has an annual death probability of 0.10 in each of the next two years, conditional on being alive at the start of that year. The effective annual interest rate is $i=25\%$. A unit two-year endowment insurance pays at the end of the year of death or at time 2 on survival. Find the corresponding unit two-year term value.

Method. Find the pure endowment component from the survival and interest assumptions, then subtract it from the endowment value. The assumptions give $v=1/1.25=0.8$. Survival to time 2 requires survival through both years, so:

{}_2p_x=(1-0.10)(1-0.10)=0.9(0.9)=0.81

The pure endowment value is:

{}_2E_x=v^2\,{}_2p_x=(0.8)^2(0.81)=0.64(0.81)=0.5184

To make the subtraction transparent, calculate the death-benefit portion. Death in year 1 has probability 0.10 and is paid at time 1; death in year 2 has probability $0.9(0.10)=0.09$ and is paid at time 2. Hence:

A^1_{x:2} =0.8(0.10)+(0.8)^2(0.09) =0.08+0.0576 =0.1376

The endowment value is therefore $0.1376+0.5184=0.6560$. Recovering the term value from it gives $0.6560-0.5184=0.1376$, which agrees with the direct death-year calculation. Common mistake avoided: subtracting the survival probability $0.81$ instead of its discounted value $0.5184$.

Worked Example: Different Mortality Probabilities by Year

For a hypothetical three-year contract, the effective annual interest rate is $i=25\%$. The conditional death probabilities are 0.10 in year 1, 0.20 in year 2, and 0.25 in year 3. A unit endowment benefit is paid at the end of the year of death or at time 3 on survival. Find the unit term value by separating the maturity value.

Here $v=0.8$. The probability of surviving all three years is:

{}_3p_x=(0.90)(0.80)(0.75)=0.54

The pure endowment value is:

{}_3E_x=v^3\,{}_3p_x=(0.8)^3(0.54)=0.512(0.54)=0.27648

Now calculate the death-benefit value by year as a check. The probabilities of death in years 1, 2, and 3 are respectively $0.10$, $0.90(0.20)=0.18$, and $0.90(0.80)(0.25)=0.18$. Their present values are:

\begin{aligned} \text{Year 1: }&0.8(0.10)=0.08000,\\ \text{Year 2: }&0.8^2(0.18)=0.64(0.18)=0.11520,\\ \text{Year 3: }&0.8^3(0.18)=0.512(0.18)=0.09216. \end{aligned}

Thus the term value is $0.08000+0.11520+0.09216=0.28736$. The endowment value is $0.28736+0.27648=0.56384$, so the identity recovers:

A^1_{x:3}=0.56384-0.27648=0.28736.

The direct and subtraction methods agree. Common mistake avoided: using the year-3 death probability alone as the maturity probability. A maturity payment occurs only if the life survives all three years, with probability $0.54$.

Worked Example: Scaling a Table Value to a Benefit Amount

A hypothetical actuarial table, using an effective annual interest rate of $i=4\%$ and its stated mortality basis for a life aged $x$, gives a 10-year unit endowment value of $0.412$ and a 10-year pure endowment value of $0.157$. A contract pays a benefit of $80,000 on death during the term or on survival to year 10. Find the actuarial present value of the death-benefit portion.

Method. The supplied values are on the same age, term, interest, and mortality basis, so subtract the unit pure endowment value from the unit endowment value:

A^1_{x:10}=0.412-0.157=0.255

This is the term value per dollar of benefit. The contract’s death benefit is $80,000, so the present value of that portion is:

$$ 80{,}000(0.255)=20{,}400. $$

The death-benefit portion has an actuarial present value of $20,400. As a check, the full endowment value for this benefit amount is $80{,}000(0.412)=$32{,}960$, and its maturity component is $80{,}000(0.157)=$12{,}560$. Their difference is $$32{,}960-$12{,}560=$20{,}400$. Common mistake avoided: treating the unit factors as dollar amounts without multiplying by the stated benefit.

Checks and Common Traps

A useful numerical check is that the endowment value must be at least as large as the pure endowment value, because it includes the maturity benefit plus a nonnegative death-benefit value. Therefore the recovered term value should not be negative. If it is negative, first verify the subtraction order, then check that both inputs refer to the same duration, age, benefit amount, interest rate, mortality basis, and payment timing.

For example, in the first worked example, the incorrect calculation $0.6560+0.5184=1.1744$ adds the maturity component a second time. Another incorrect calculation is $0.6560-0.81=-0.1540$: it subtracts the probability of survival rather than the present value of the maturity payment. The correct calculation subtracts $v^n\,{}_np_x$, giving $0.6560-0.5184=0.1376$.

The identity also does not automatically apply if the contract has a different maturity benefit, an additional payment on death, or a different benefit timing from the term insurance being compared. In those cases, first express each payment on a consistent benefit basis; subtracting a unit pure endowment is valid only for the unit maturity benefit at time $n$.

Key takeaway: A unit endowment value consists of the unit term-insurance value plus the discounted probability of surviving to the end of the term. Recover the term value by subtracting ${}_nE_x=v^n\,{}_np_x$, not by subtracting survival probability alone.

Check Your Understanding

For each problem, use the stated benefit timing and show the maturity component before subtracting.

A two-year unit endowment has value $0.70$. The effective annual interest rate is $20\%$, and the probability of surviving two years is $0.80$. Find the unit two-year term value. (Answer: $0.1444$)
A hypothetical life has conditional death probabilities 0.20 in year 1 and 0.10 in year 2. At $i=25\%$, a unit two-year endowment value is $0.6720$. Find the unit term value. (Answer: $0.2112$)
On a hypothetical actuarial basis with $i=4\%$, a five-year endowment factor is $0.354$, and the matching pure endowment factor is $0.119$. Find the death-benefit present value for a $75,000 benefit. (Answer: $17,625)
A six-year unit term value is $0.082$, and the matching six-year pure endowment value is $0.218$, both on the same hypothetical actuarial basis. Find the corresponding unit endowment value. (Answer: $0.300$)
A hypothetical four-year endowment factor is $0.610$. At $i=25\%$, the probability of survival to year 4 is $0.75$. Find the pure endowment factor, the unit term factor, and the death-benefit present value for a $40,000 benefit. (Answer: pure endowment $0.3072$; unit term $0.3028$; death-benefit present value $12,112)

What You'll Learn