Decompose Endowment Insurance into Risk and Survival Benefits

Why Decompose an Endowment?

A discrete endowment insurance pays a benefit if the insured dies during a specified term, and it also pays a benefit at the end of the term if the insured survives. These are two different events with different timing. Separating them makes it easier to organize probabilities, discount benefits, and check a calculation.

This tutorial uses the standard annual model: benefits are paid at the end of the year of death, or at the end of the term for survival; interest is an effective annual rate; and premiums, when considered, are paid at the beginning of each year while the insured is alive. A different contract may use different payment timing, so timing must be stated before applying the formulas.

Definition: An $n$-year endowment insurance pays a specified amount if death occurs during the $n$-year term and pays a specified amount at time $n$ if the insured survives to that time. With the same benefit amount for either event, the benefit is paid exactly once: at the end of the year of death, or at the end of year $n$ on survival.

Let $q_{x+j}$ be the conditional probability of death during policy year $j+1$, given survival to the start of that year, and let $p_{x+j}=1-q_{x+j}$. Then ${}_kp_x$ is the probability of surviving $k$ years from age $x$, and ${}_kq_x=1-{}_kp_x$ is the probability of dying within $k$ years. Let $v=1/(1+i)$, where $i$ is the effective annual interest rate.

For an endowment benefit of 1, the term insurance component pays at the end of the year of death if death occurs within $n$ years. The pure endowment component pays at time $n$ only if the insured survives $n$ years. We use $A^1_{x:\overline{n}|}$ for the first component and ${}_nE_x$ for the second. The endowment value is denoted $A_{x:\overline{n}|}$.

Theorem: Under the annual, end-of-year benefit model, $$ A_{x:\overline{n}|}=A^1_{x:\overline{n}|}+{}_nE_x =\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}+v^n{}_np_x. $$ Here $A^1_{x:\overline{n}|}$ is the present value of a unit term insurance benefit, and ${}_nE_x=v^n{}_np_x$ is the present value of a unit pure endowment benefit.

Why the Decomposition Holds

The death-in-year-$(k+1)$ event and survival-to-year-$n$ event are mutually exclusive. Together, they cover every possible outcome over the term: either the insured dies during one of the $n$ policy years, or survives all $n$ years. Their respective probabilities are ${}_kp_xq_{x+k}$ and ${}_np_x$.

On death in year $k+1$, the unit benefit is paid at time $k+1$, so its present value is $v^{k+1}$. On survival to time $n$, the unit benefit is paid at time $n$, so its present value is $v^n$. Taking the expected present value over these disjoint outcomes gives the formula. This also shows why the survival benefit is added once, rather than being included in the term insurance component.

List the possible benefit events.
For each policy year, identify death in that year; separately identify survival to the end of the term.

Find each event probability.
For death in year $k+1$, multiply survival through $k$ years by the conditional death probability in the next year.

Discount each payment at its payment time.
Use $v^{k+1}$ for a benefit paid at the end of year $k+1$, and $v^n$ for the maturity benefit.

Add the components.
Multiply the resulting unit value by the benefit amount. If premiums are present, value their cash flows separately.

Calculating an Endowment Value from Mortality Data

The following example uses conditional mortality probabilities that vary by year. This is a useful format when mortality data are supplied as a short sequence rather than as a complete life table.

Worked Example: Split a Three-Year Benefit into Its Two Components

A hypothetical contract pays $\$100{,}000$ at the end of the year of death if death occurs within three years, or at the end of year 3 if the insured survives. The effective annual interest rate is $5\%$. The conditional death probabilities in years 1, 2, and 3 are $0.10$, $0.20$, and $0.25$, respectively. Find the term insurance value, pure endowment value, and total endowment value.

Method and assumptions. Use discrete end-of-year benefits and the supplied conditional probabilities. The corresponding survival probabilities are $0.90$, $0.80$, and $0.75$. Thus the probabilities of death in years 1, 2, and 3 are

0.10,\qquad 0.90(0.20)=0.18,\qquad 0.90(0.80)(0.25)=0.18.

The probability of surviving all three years is $0.90(0.80)(0.75)=0.54$. The discount factor is $v=1/1.05=20/21$, so $v^2=400/441$ and $v^3=8000/9261$.

Term insurance component. Discount the death benefit according to the year in which it is paid:

A^1_{x:\overline{3}|} =0.10v+0.18v^2+0.18v^3 =0.413994\ldots

Therefore the term insurance value is $\$100{,}000(0.413994\ldots)=\$41{,}399.42$.

Pure endowment component. This component is payable only if the insured survives to year 3:

{}_3E_x=0.54v^3=0.466472\ldots

Its value is $\$100{,}000(0.466472\ldots)=\$46{,}647.23$. Adding the components gives $\$41{,}399.42+\$46{,}647.23=\$88{,}046.65$, rounded to the nearest cent. As a check, the four discounted outcome contributions total the same value, and the four event probabilities sum to $0.10+0.18+0.18+0.54=1$.

Common mistake avoided: Counting the year-3 death probability as though it were the maturity survival probability. Death in year 3 has probability $0.18$; survival to maturity has probability $0.54$. They are distinct outcomes, and each is discounted at $v^3$.

Using the Decomposition to Find a Net Premium

Under the equivalence principle, a level net premium is chosen so that the expected present value of premiums equals the expected present value of benefits at issue. For premiums payable at the beginning of each year while the insured is alive, a premium is due at time 0, and a later premium is due only if the insured survives to its payment date. This timing means the premium present value factor is an annuity-due factor on the same survival basis.

For a level annual premium $P$ payable for at most $n$ years while alive, its expected present value is $P\sum_{k=0}^{n-1}v^k{}_kp_x$. If the endowment benefit is $B$, the benefit value is $B A_{x:\overline{n}|}$, so equating the two gives the net premium. This calculation assumes no expenses, taxes, or profit loading.

Worked Example: Find the Level Net Premium for a Three-Year Endowment

A hypothetical three-year endowment pays $\$50{,}000$ on death during the term or on survival to year 3. The effective annual interest rate is $10\%$, and the conditional death probability is $0.20$ in each year. Premiums are payable at the beginning of each year while the insured is alive, for at most three payments. Find the level annual net premium.

Method and assumptions. Here $v=10/11$. The probability of death in the three policy years is $0.20$, $0.80(0.20)=0.16$, and $0.80^2(0.20)=0.128$. The probability of survival to year 3 is $0.80^3=0.512$.

The term insurance factor is

0.20v+0.16v^2+0.128v^3 =\frac{242+176+128}{1331} =\frac{546}{1331}.

The pure endowment factor is

0.512v^3=\frac{512}{1331}.

Thus the endowment factor is $1058/1331$. The premium factor is the expected present value of one unit of premium at times 0, 1, and 2, conditional on being alive at each payment time:

1+0.80v+0.80^2v^2 =1+\frac{8}{11}+\frac{64}{121} =\frac{273}{121}.

Equating expected present values gives

P\left(\frac{273}{121}\right) =\$50{,}000\left(\frac{1058}{1331}\right), \qquad P=\$50{,}000\left(\frac{1058}{3003}\right) =\$17{,}615.72.

The premium is rounded to the nearest cent. As a check, the benefit value is approximately $\$39{,}744.55$, and the premium value is approximately $\$17{,}615.72(273/121)=\$39{,}744.55$. The two present values agree. The method avoids treating all three premiums as certain: the second and third premiums depend on survival to their payment dates.

Prospective Values at a Later Duration

At a later duration, value the remaining benefits and premiums conditional on the insured being alive at that duration. Use the conditional mortality probabilities for the remaining years; do not use the original issue-age probabilities again. If the valuation is immediately before a premium due at that time, include that premium in the future premium value. If it is immediately after the premium, exclude it.

For an endowment with a common benefit amount, the remaining benefit value at duration $t$ can again be split into term insurance over the remaining period and a pure endowment to the maturity date. The same disjoint-outcome argument applies, now using probabilities conditional on survival to $t$.

Worked Example: Calculate a Prospective Value with Two Years Remaining

A hypothetical three-year endowment pays $\$30{,}000$ on death during the term or on survival to year 3. At duration 1, the insured is alive. The conditional death probabilities for policy years 2 and 3 are $0.25$ and $0.40$. The effective annual interest rate is $25\%$. Annual premiums of $\$8{,}000$ are payable at the beginning of each remaining year while the insured is alive. Find the prospective value immediately before the premium due at duration 1.

Method and assumptions. Two years remain, so the conditional survival probabilities for those years are $0.75$ and $0.60$, and $v=1/1.25=0.80$. The death probability in the first remaining year is $0.25$. The probability of death in the second remaining year is $0.75(0.40)=0.30$, and the probability of surviving both years is $0.75(0.60)=0.45$.

The remaining benefit factor is the sum of term insurance and pure endowment:

$$ 0.25v+0.30v^2+0.45v^2 =0.25(0.80)+0.75(0.80^2) =0.20+0.48 =0.68. $$

The expected present value of remaining benefits is $\$30{,}000(0.68)=\$20{,}400$. Immediately before the current premium, two premiums may be paid: $\$8{,}000$ now, and another $\$8{,}000$ in one year if the insured survives the first remaining year. Their factor is $1+0.75v=1.60$, so their expected present value is $\$8{,}000(1.60)=\$12{,}800$.

The prospective value is the benefit value less the premium value:

{}_1V_x=\$20{,}400-\$12{,}800=\$7{,}600.00.

The value is rounded to the nearest cent. The check is that the benefit factor includes death in either remaining year and survival to maturity exactly once, while the premium factor includes the premium due now and the survival-contingent premium one year later. The method avoids the time-unit error of treating this as a one-year remaining contract: the original term is three years and valuation is at duration 1, so two years remain.

Common Traps and Practical Checks

The decomposition is simple, but many errors arise from mixing event probabilities, benefit times, or valuation dates. Before finalizing a calculation, check the following:

Do not double-count the maturity benefit. The term insurance component covers death within the term; the pure endowment covers survival to the term’s end. These events do not overlap.
Use conditional survival in sequence. Death in year $k+1$ has probability ${}_kp_xq_{x+k}$, not merely $q_{x+k}$ unless survival to that year is already part of the stated condition.
Match the discount exponent to the payment date. A death benefit paid at the end of year $k+1$ is discounted by $v^{k+1}$, even though death occurs during that year.
State whether the valuation is before or after a premium. A premium due at the valuation time is included immediately before payment and excluded immediately after payment.
Use the remaining term at the valuation duration. For a contract originally lasting $n$ years, duration $t$ leaves $n-t$ years, not $n$ years.

A useful final check is to add all outcome probabilities. The probabilities of death in each policy year plus the probability of survival to maturity must sum to 1. Also check that each benefit component is discounted to its own payment time and that premium payments are conditional on survival only when the contract requires it.

Key takeaway: A discrete endowment value is the sum of a term insurance value and a pure endowment value. At any duration, apply the same decomposition using the remaining conditional mortality probabilities, discount benefits to their payment dates, and value premiums on their actual payment schedule.

Check Your Understanding

For each problem, identify the separate benefit events before calculating. Monetary answers are rounded to the nearest cent where applicable.

A two-year endowment pays $\$10{,}000$ on death during the term or survival to maturity. Interest is $0\%$, and the conditional survival probabilities are $0.80$ for year 1 and $0.50$ for year 2. Find the term insurance and pure endowment values. (Answer: term insurance $\$6{,}000.00$; pure endowment $\$4{,}000.00$)
A two-year endowment pays $\$12{,}000$. The effective annual interest rate is $10\%$, with conditional survival probabilities $0.90$ and $0.80$. Find the endowment value. (Answer: $\$10{,}016.53$)
A two-year endowment pays $\$20{,}000$. The effective annual interest rate is $5\%$, and conditional survival probabilities are $0.80$ and $0.75$. Premiums are payable at the beginning of each year while alive. Find the level annual net premium. (Answer: $\$10{,}398.97$)
A three-year endowment pays $\$40{,}000$. At duration 1, the insured is alive, and the conditional death probabilities in the two remaining years are $0.20$ and $0.50$. The effective annual interest rate is $10\%$. Premiums of $\$9{,}000$ are due at duration 1 and one year later if alive. Find the prospective value immediately before the duration-1 premium. (Answer: $\$18{,}173.55$)

What You'll Learn