Endowment Premiums from Insurance-Annuity Identities

Endowment Premiums and the Annuity Identity

An endowment contract combines a death benefit during a specified term with a payment at the end of that term if the insured survives. Its net level premium must fund both possible benefit outcomes. The insurance-annuity identity makes the calculation efficient, while a direct present-value calculation provides an independent check.

Assume a discrete contract issued to a life aged $x$, with term $n$ years and a unit benefit. If death occurs during a policy year, the benefit is paid at the end of that year. If the insured survives to time $n$, the benefit is paid at time $n$. Level premiums are paid annually in advance at times $0,1,\ldots,n-1$, provided the insured is alive at each payment time. There are no expenses or other payments in this net-premium model.

Definition: Let $A_{x:\overline{n}|}$ denote the expected present value of the unit $n$-year endowment insurance benefit. Let $\ddot{a}_{x:\overline{n}|}$ denote the expected present value of a temporary life annuity-due that pays 1 at times $0,1,\ldots,n-1$ while the insured is alive. With $v=1/(1+i)$ and $d=1-v$, the endowment insurance-annuity identity is $$ A_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|}=1. $$

The identity applies when the insurance benefit is paid at the end of the year of death or at time $n$ on survival, and the annuity payments are due at the beginning of each year while the insured is alive. The annual effective interest rate is $i$, and both values use the same survival model and term. Changing the benefit timing or premium timing can change the relationship.

Why the Identity Holds

The identity can be justified by comparing present values for each possible outcome, rather than beginning with expected values. Let $K$ be the curtate future lifetime: death in year $K+1$ means $K$ full years are survived. For outcomes in which death occurs during the term, $K=0,\ldots,n-1$. The annuity-due payments then occur at times $0$ through $K$, and the endowment insurance pays at time $K+1$. If the insured survives the term, the annuity payments occur at times $0$ through $n-1$, and the benefit is paid at time $n$.

In either outcome, let $m$ be the number of premium-payment dates: $m=K+1$ if death occurs during the term, and $m=n$ if the insured survives to time $n$. The present value of the annuity payments on that outcome is $1+v+\cdots+v^{m-1}$. Multiplying by $d=1-v$ gives $1-v^m$. The insurance benefit’s present value is $v^m$: the payment is one year after the last premium-payment date. Thus the two amounts sum to 1 on every outcome.

d(1+v+\cdots+v^{m-1})+v^m =(1-v^m)+v^m =1.

Taking expected values preserves this equality. The expected present value of the annuity payments becomes $\ddot{a}_{x:\overline{n}|}$, and that of the insurance benefit becomes $A_{x:\overline{n}|}$. This proves the identity under the stated payment conventions.

From the Identity to a Level Net Premium

Under the equivalence principle, the net level premium $P$ is set so that the expected present value of premiums equals the expected present value of benefits. Since the premium is paid only while the insured is alive and only at the specified dates, the expected present value of premiums is $P\ddot{a}_{x:\overline{n}|}$. Therefore:

P\ddot{a}_{x:\overline{n}|}=A_{x:\overline{n}|}, \qquad P=\frac{A_{x:\overline{n}|}}{\ddot{a}_{x:\overline{n}|}} =\frac{1}{\ddot{a}_{x:\overline{n}|}}-d.

The second form follows by substituting $A_{x:\overline{n}|}=1-d\ddot{a}_{x:\overline{n}|}$ from the identity and dividing by the annuity value. It is a convenient calculation when the temporary annuity-due value is known. The first form is also useful because it states the equivalence principle directly.

This is a net premium: it covers the expected present value of the stated benefit only. It does not include expenses, profit margins, or other loadings. A premium computed from this equation is per unit of benefit; the examples below keep the benefit at one unit.

Worked Example: Three-Year Endowment with Constant Mortality

A hypothetical three-year unit endowment is issued to a life aged $x$. The annual effective interest rate is $i=25\%$, and the conditional probability of death in each policy year, given survival to its start, is $q=0.20$. Find the annual net premium and check it by direct present-value equivalence.

Method. First calculate the survival probabilities and the temporary annuity-due value. Use the identity for the premium, then sum the discounted benefit probabilities directly. This check also avoids omitting the maturity benefit.

The discount factor is $v=1/1.25=0.8$. The one-year survival probability is $p=1-q=0.8$, so the probabilities of surviving to years 1, 2, and 3 are $0.8$, $0.8^2=0.64$, and $0.8^3=0.512$. The temporary annuity-due value is

\ddot{a}_{x:\overline{3}|} =1+(0.8)(0.8)+(0.64)(0.8^2) =1+0.64+0.4096 =2.0496.

Here $d=1-v=0.20$. The endowment insurance value and level premium are

A_{x:\overline{3}|} =1-(0.20)(2.0496) =0.59008, \qquad P=\frac{0.59008}{2.0496} =0.287900\ldots.

For the direct check, the probabilities of death in years 1, 2, and 3 are $0.20$, $(0.80)(0.20)=0.16$, and $(0.64)(0.20)=0.128$. The probability of survival to maturity is $0.512$. Death in year 3 and survival to maturity both result in a payment at time 3, so their probabilities are combined at that payment date:

\begin{aligned} \text{EPV of benefits} &=(0.20)(0.8)+(0.16)(0.8^2)+(0.128+0.512)(0.8^3)\\ &=0.16+0.1024+0.32768\\ &=0.59008. \end{aligned}

The EPV of premiums at the unrounded premium is $0.287900\ldots(2.0496)=0.59008$, agreeing with the direct benefit calculation. The net premium is $0.2879$ per year to four decimal places.

Worked Example: Using a Given Set of Benefit Probabilities

For a hypothetical two-year unit endowment, the effective annual interest rate is $i=4\%$. The probability of death during year 1 is $0.05$, the probability of death during year 2 is $0.05$, and the probability of surviving to time 2 is $0.90$. These are unconditional probabilities from issue. Calculate the annual net premium and verify it directly.

Method. The annuity requires the probability of being alive at the start of each premium year. Use the supplied outcome probabilities to obtain that survival probability, then compare the identity-based premium with the direct benefit EPV.

The discount factor and discount rate are $v=1/1.04=25/26$ and $d=1/26$. The probability of being alive at time 1 is $1-0.05=0.95$. Thus

\ddot{a}_{x:\overline{2}|} =1+(0.95)\left(\frac{25}{26}\right) =\frac{199}{104} =1.9134615\ldots.

The identity gives

A_{x:\overline{2}|} =1-\frac{1}{26}\left(\frac{199}{104}\right) =\frac{2505}{2704} =0.9264053\ldots, \qquad P=\frac{A_{x:\overline{2}|}}{\ddot{a}_{x:\overline{2}|}} =\frac{2505}{5174} =0.4841515\ldots.

For a direct check, the year-1 death benefit is discounted one year. The year-2 death benefit and the maturity benefit are both paid at time 2, and their probabilities sum to $0.05+0.90=0.95$:

\begin{aligned} \text{EPV of benefits} &=(0.05)\left(\frac{25}{26}\right) +(0.05+0.90)\left(\frac{25}{26}\right)^2\\ &=0.0480769\ldots+0.8783284\ldots\\ &=0.9264053\ldots. \end{aligned}

The premium EPV is $(2505/5174)(199/104)=2505/2704$, the same benefit value. The annual net premium is $0.4842$ to four decimal places. Treating the two death probabilities as conditional probabilities would produce a different, incorrect survival pattern.

Worked Example: Checking the Premium from a Mortality Pattern

A hypothetical three-year unit endowment has $i=10\%$. The conditional probabilities of death during years 1, 2, and 3, given survival to the start of the relevant year, are $0.10$, $0.20$, and $0.30$. Find the net level premium, and verify the result using direct expected present values.

Method. Build the unconditional death and survival probabilities from the conditional mortality assumptions. Then calculate the annuity value and benefit value using their payment dates.

The discount factor is $v=10/11$, and $d=1/11$. The survival probabilities to times 1, 2, and 3 are $0.90$, $(0.90)(0.80)=0.72$, and $(0.72)(0.70)=0.504$. Therefore,

\ddot{a}_{x:\overline{3}|} =1+(0.90)\left(\frac{10}{11}\right) +(0.72)\left(\frac{10}{11}\right)^2 =\frac{292}{121} =2.4132231\ldots.

The identity gives the benefit value and premium:

A_{x:\overline{3}|} =1-\frac{1}{11}\left(\frac{292}{121}\right) =\frac{1039}{1331} =0.7806161\ldots, \qquad P=\frac{1039}{3212} =0.3234745\ldots.

For the direct calculation, death probabilities in years 1, 2, and 3 are $0.10$, $(0.90)(0.20)=0.18$, and $(0.72)(0.30)=0.216$. Survival to maturity has probability $0.504$. The two time-3 outcomes are combined:

\begin{aligned} \text{EPV of benefits} &=(0.10)\left(\frac{10}{11}\right) +(0.18)\left(\frac{10}{11}\right)^2 +(0.216+0.504)\left(\frac{10}{11}\right)^3\\ &=\frac{121}{1331}+\frac{198}{1331}+\frac{720}{1331}\\ &=\frac{1039}{1331} =0.7806161\ldots. \end{aligned}

The premium EPV is $(1039/3212)(292/121)=1039/1331$, confirming equivalence. The annual net premium is $0.3235$ to four decimal places. A useful check is that the outcome probabilities sum to $0.10+0.18+0.216+0.504=1$.

Common Timing and Probability Traps

The identity is compact, but using it correctly requires matching the contract’s cash-flow timing. One frequent error is to use an annuity-immediate value, which starts at time 1, even though the premiums are due at time 0. Another is to calculate only the term death-benefit value and omit the payment to a survivor at time $n$.

For example, in the first worked example, using the correct benefit value $0.59008$ but incorrectly dividing by the annuity-immediate value $0.64+0.4096+0.262144=1.311744$ gives a premium of about $0.4501$. This calculation prices a different premium pattern: its first payment is at time 1, not time 0. The specified premiums are in advance, so the correct denominator is $\ddot{a}_{x:\overline{3}|}=2.0496$, giving $P=0.2879$.

A second trap is confusing conditional mortality with unconditional outcome probabilities. A conditional probability of death in year 2 applies only to those who survived year 1; the unconditional probability of death in year 2 must also include the probability of reaching that year alive. Conversely, if probabilities of death and survival from issue are explicitly supplied as unconditional probabilities, they should not be multiplied by earlier survival probabilities again.

These calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. A reliable solution states the benefit timing, premium dates, interest rate, and mortality probabilities before applying the identity, then checks that the expected present values of premiums and benefits agree.

Key takeaway: For a unit $n$-year endowment with annual premiums in advance while alive, $P=A_{x:\overline{n}|}/\ddot{a}_{x:\overline{n}|}=1/\ddot{a}_{x:\overline{n}|}-d$. Check the result by discounting each possible benefit payment at its actual payment time.

Check Your Understanding

For each hypothetical contract, find the annual net premium for a unit endowment. Premiums are payable annually in advance while alive. Use the stated discrete benefit timing.

A two-year contract has $i=5\%$, with conditional mortality $q=0.10$ in each year. Use the identity to find the premium. (Answer: $0.4908$ per year, to four decimal places.)
A two-year contract has $i=25\%$. The unconditional probabilities are $0.15$ for death in year 1, $0.17$ for death in year 2, and $0.68$ for survival to maturity. Find the premium and check that the outcome probabilities sum to 1. (Answer: $0.3952$ per year, to four decimal places; the probabilities sum to $1.00$.)
A three-year contract has $i=10\%$ and survival probabilities to times 1, 2, and 3 of $0.96$, $0.90$, and $0.81$. Find the premium from the annuity value and verify the benefit value using direct payment probabilities. (Answer: $0.2913$ per year, to four decimal places.)
A two-year contract has $i=8\%$, with conditional mortality $q_x=0.25$ in year 1 and $q_{x+1}=0.20$ in year 2. Find the premium using $d=1-v$, and state the survival probability to time 2. (Answer: $0.5161$ per year, to four decimal places; ${}_2p_x=0.60$.)
A two-year contract has $i=5\%$, with unconditional probabilities $0.20$ for death in year 1, $0.30$ for death in year 2, and $0.50$ for survival to maturity. Calculate the benefit EPV by payment time, then find the premium using the annuity-due value. (Answer: benefit EPV $=0.9161$; premium $=0.5199$ per year, to four decimal places.)

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