Endowment Reserves from Insurance and Pure Endowment Values

Why Decompose an Endowment Reserve?

An endowment insurance policy pays a benefit if the insured dies during a stated term, and also pays a benefit at the end of the term if the insured survives. These are two distinct payment contingencies. Separating them makes it easier to value the benefit at a reserve date, especially when the death and survival payments have different timing.

This tutorial considers a level benefit payable at the end of the year of death, or at the end of the term on survival. Premiums are level and due at the start of each policy year while the insured is alive. Values are conditional on the insured being alive at the valuation date. The formulas below use an effective annual interest rate $i$, with $v=1/(1+i)$.

Definition: An $n$-year endowment insurance with benefit $B$ can be decomposed into (1) $n$-year term insurance, paying $B$ at the end of the year of death if death occurs within the term, and (2) a pure endowment, paying $B$ at time $n$ if the insured survives to time $n$. The endowment benefit is the sum of these two benefit components.

At duration $k$, let $m=n-k$ be the number of years remaining, and suppose the insured is alive at that duration. The conditional probability of surviving $j$ more years is ${}_jp_{x+k}$; the probability of dying in the following year, conditional on being alive at its start, is ${}_1q_{x+k+j}$. The term component pays for death in one of the next $m$ years; the pure endowment component pays only if the insured survives all $m$ years.

Conditional Benefit and Premium Values

For a discrete endowment with the payment timing specified above, the conditional value of the remaining benefits at duration $k$ is

B\left[\sum_{j=0}^{m-1}v^{j+1}{}_jp_{x+k}{}_1q_{x+k+j} +v^m{}_mp_{x+k}\right], \qquad m=n-k.

The sum is the term insurance component. The final term is the pure endowment component. The discount exponent is $j+1$ for a death benefit in the $j+1$st future year, and $m$ for the maturity benefit.

If premiums of amount $P$ are due at the start of each remaining policy year while the insured is alive, their conditional value at duration $k$ is

P\sum_{j=0}^{m-1}v^j{}_jp_{x+k}.

The first premium in this sum is due immediately at duration $k$, so its discount factor is $v^0=1$. Each later premium is paid only if the insured survives to its due date.

Theorem: Immediately before the premium due at duration $k$, the prospective reserve is the conditional value of future benefits minus the conditional value of future premiums. For the endowment described here, $$ {}_kV_x = B\left[\sum_{j=0}^{m-1}v^{j+1}{}_jp_{x+k}{}_1q_{x+k+j} +v^m{}_mp_{x+k}\right] -P\sum_{j=0}^{m-1}v^j{}_jp_{x+k}, \qquad m=n-k. $$

The formula assumes annual discrete payments with the stated timing, a level benefit, and level premiums due only while the insured is alive. If the valuation is immediately after a premium has been paid, that premium is no longer a future cash flow and must be omitted. Continuous payment timing or different benefit timing requires a corresponding change in the cash-flow values.

Why the Decomposition Works

The death-benefit events and the survival-to-maturity event divide all possible outcomes over the remaining term into mutually exclusive cases. A death in future year $j+1$ has conditional probability ${}_jp_{x+k}{}_1q_{x+k+j}$, and the probability of surviving all $m$ years is ${}_mp_{x+k}$. Each probability is multiplied by the value, at duration $k$, of the payment made in that case. This gives the benefit expression above.

The same reasoning applies to premiums: the premium due $j$ years from duration $k$ is paid exactly when the insured survives those $j$ years. Its expected discounted value is therefore $P v^j{}_jp_{x+k}$. Summing those values gives the premium annuity value. Subtracting it from the benefit value follows directly from the prospective definition of a policy value.

Set the valuation date and timing.
Identify the duration, whether it is before or after a premium, and how many years remain.

Value the death benefit.
For each remaining year, multiply its conditional death probability by the benefit discounted to that year.

Value the pure endowment.
Multiply the maturity benefit by the probability of surviving the full remaining term and its discount factor.

Subtract future premiums.
Include a premium due immediately only if the valuation is before that premium is paid.

Worked Endowment Reserve Calculations

Worked Example: Three-Year Endowment with Zero Interest

A hypothetical three-year endowment pays $B=\$5{,}000$ at the end of the year of death or at maturity. The effective annual interest rate is $i=0\%$. The conditional probabilities of death during years 1, 2, and 3 are respectively $0.10$, $0.20$, and $0.25$. Premiums are level, due at the start of each year while alive, and calculated by the net premium equivalence principle. Find the reserve at duration 1 immediately before the premium due then.

Method. Calculate the net premium at issue, then apply the prospective reserve formula using the two years and two premiums remaining. This also checks that survival probabilities are conditioned on reaching each year.

At issue, the probability of death in year 1 is $0.10$. The probability of death in year 2 is $0.90(0.20)=0.18$. The probability of death in year 3 is $0.90(0.80)(0.25)=0.18$. The probability of surviving three years is $0.90(0.80)(0.75)=0.54$. Since $i=0\%$, each benefit payment has discount factor 1, so the benefit value is

\$5{,}000(0.10+0.18+0.18+0.54)=\$5{,}000.

The expected number of premiums at issue is $1+0.90+0.90(0.80)=2.62$. Thus the unrounded net premium is

P=\frac{\$5{,}000}{2.62}=\$1{,}908.396946\ldots.

At duration 1, two years remain. Conditional on being alive then, the death probability in year 2 is $0.20$, the probability of death in year 3 is $0.80(0.25)=0.20$, and the probability of surviving to maturity is $0.80(0.75)=0.60$. The term component and pure endowment component together have value

\$5{,}000(0.20+0.20)+\$5{,}000(0.60)=\$2{,}000+\$3{,}000=\$5{,}000.

There is a premium due immediately and another one year later if the insured survives the next year. Their value is $P(1+0.80)=1.80P$. Using the unrounded net premium,

{}_1V_x=\$5{,}000-1.80(\$1{,}908.396946\ldots) =\$1{,}564.885496\ldots.

Rounded to the nearest cent, the reserve is \$1,564.89. If the stated premium is first rounded to \$1,908.40, the reserve is \$5,000-\$3,435.12=\$1,564.88. The one-cent difference is solely due to premium rounding.

Worked Example: Interest and a Two-Year Endowment

A hypothetical two-year endowment pays $B=\$4{,}000$. The effective annual interest rate is $i=5\%$, and the conditional probability of death in either year is $0.25$. Premiums are annual, due at the start of each year while alive. Find the net premium and the reserve at duration 1 immediately before the second premium.

Method. At issue, separate the two death-payment possibilities from survival to maturity. Use the same benefit timing for both death and maturity: payment at the end of the year in which the event occurs.

Here $v=1/1.05=20/21$. The probability of death in year 1 is $0.25$, death in year 2 is $0.75(0.25)=0.1875$, and survival to maturity is $0.75^2=0.5625$. The term component per dollar of benefit is

0.25v+0.1875v^2 =\frac{5}{21}+\frac{25}{147} =\frac{20}{49}.

The pure endowment component per dollar is

0.5625v^2=\frac{25}{49}.

Therefore, the benefit value at issue is $\$4{,}000(20/49+25/49)=\$4{,}000(45/49)$. The premium annuity value is $1+0.75v=12/7$. Equating the benefit and premium values gives

P=\frac{\$4{,}000(45/49)}{12/7} =\$4{,}000\left(\frac{15}{28}\right) =\$2{,}142.857142\ldots.

At duration 1, one year remains. The death benefit is paid at the end of that year with probability $0.25$, and the maturity benefit is paid at the same time with probability $0.75$. Their combined value is $\$4{,}000v=\$3{,}809.523809\ldots$. The second premium is due immediately, so its value is $P$. Thus

{}_1V_x=\$3{,}809.523809\ldots-\$2{,}142.857142\ldots =\$1{,}666.666666\ldots.

The net premium is \$2,142.86, and the reserve is \$1,666.67, each rounded to the nearest cent. The reserve computation uses the unrounded premium.

Worked Example: A Two-Year Remaining Term with Unequal Mortality

A hypothetical three-year endowment pays $B=\$12{,}000$. At duration 1, the insured is alive and the conditional death probabilities in the next two years are $0.20$ and $0.25$. The effective annual interest rate is $5\%$. The remaining annual premiums are each \$2,400, due at the start of a year while alive. Find the reserve immediately before the premium at duration 1.

Method. Value the two possible death payments and the maturity payment separately, then subtract the two premiums still scheduled. The given premium is used as stated; no issue-pricing calculation is needed.

The probability of death in the first remaining year is $0.20$. The probability of death in the second remaining year is $0.80(0.25)=0.20$, and the probability of surviving both years is $0.80(0.75)=0.60$. With $v=20/21$, the term component per dollar of benefit is

0.20v+0.20v^2 =\frac{84}{441}+\frac{80}{441} =\frac{164}{441}.

The pure endowment component per dollar is $0.60v^2=240/441$. Consequently, the total benefit value is

\$12{,}000\left(\frac{164}{441}+\frac{240}{441}\right) =\$12{,}000\left(\frac{404}{441}\right) =\$10{,}993.197278\ldots.

The premium due immediately has value \$2,400. The later premium is due one year from now only if the insured survives the first remaining year, so its value is $\$2{,}400(0.80)v=\$1{,}828.571428\ldots$. The total premium value and reserve are therefore

\$2{,}400+\$1{,}828.571428\ldots =\$4{,}228.571428\ldots, $$ $$ {}_1V_x=\$10{,}993.197278\ldots-\$4{,}228.571428\ldots =\$6{,}764.625850\ldots.

The reserve is \$6,764.63, rounded to the nearest cent. As a check, the premium value includes exactly two installments, with the second weighted by survival and discounted for one year.

Common Timing and Calculation Traps

A frequent error is to treat the pure endowment as though survival to maturity were certain. For example, with a two-year term, $B=\$1{,}000$, $i=10\%$, and conditional death probability $0.10$ in each year, the pure endowment value at issue is not just $\$1{,}000v^2$. The insured must survive both years:

\$1{,}000(0.90)^2\left(\frac{1}{1.10}\right)^2 =\$669.421487\ldots =\$669.42.

The incorrect calculation $\$1{,}000v^2=\$826.45$ omits the survival probability. Another common error is to omit a premium due immediately when the reserve is defined immediately before that premium. The correction is to label the valuation instant first: before payment, include the installment; after payment, exclude it.

These distinctions matter because an endowment reserve combines both mortality-contingent benefit values and premiums contingent on survival. This decomposition is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the remaining death and survival benefit values and apply the stated premium timing.

A two-year endowment pays \$2,000 at death during the term or at maturity. Interest is $0\%$, and the conditional death probabilities are $0.10$ in year 1 and $0.20$ in year 2. The premium due at duration 1 immediately before valuation is \$700. Find the reserve then. (Answer: \$1,300)
A hypothetical two-year policy has a \$1,000 pure endowment benefit, effective annual interest $i=10\%$, and conditional death probability $0.10$ in each year. Find the pure endowment component at issue, to the nearest cent. (Answer: \$669.42)
A three-year endowment has benefit \$5,000, interest $0\%$, and conditional death probabilities $0.10$, $0.20$, and $0.25$. At duration 1 immediately before the premium, two annual premiums remain; each is \$1,500 and is due while alive. Find the reserve. (Answer: \$2,300)
A two-year endowment pays \$4,000; $i=5\%$, and the conditional death probability is $0.25$ in each year. Level annual premiums are due at the start of each year while alive. Find the net premium, rounded to the nearest cent. (Answer: \$2,142.86)
A three-year endowment pays \$10,000, has $i=0\%$, and has conditional death probabilities $0.10$, $0.20$, and $0.25$. Premiums are due at the start of each year while alive. Find the issue net premium, rounded to the nearest cent, and split the issue benefit value into the term and pure endowment components. (Answer: net premium \$3,816.79; term value \$4,600; pure endowment value \$5,400)

What You'll Learn