Choosing an Identity from a Contract Payment Pattern

Start with the Contract’s Cash Flows

Insurance-annuity identities are useful only when the benefit and annuity cash flows match the contract’s timing. A benefit paid at the end of the year of death is not the same cash flow as a benefit paid immediately at death. Similarly, an annuity-due makes payments at the beginnings of periods, while a continuous annuity accumulates value throughout time. Choosing a familiar formula before writing down those timings is a common source of errors.

A reliable approach is to identify four features first: when the contract starts, when it ends, what happens on death, and what happens if the insured survives to the end. Then choose the insurance value and annuity value that represent those exact cash flows.

Definition: A term insurance benefit is paid on death during a specified coverage period and has no survival benefit at the end of that period. An endowment insurance pays on death during the period or pays a maturity benefit if the insured survives to the end. A pure endowment pays only if the insured survives to the stated time.

For the discrete identities below, assume an effective annual interest rate \(i\), with \(v=1/(1+i)\) and \(d=1-v\). A death benefit is paid at the end of the policy year of death. An annuity-due pays at times \(0,1,\ldots,n-1\), provided the insured is alive at each payment time. The notation \({}_np_x\) is the probability of surviving \(n\) years from age \(x\).

For continuous identities, assume a constant force of interest \(\delta\) over the period considered. Death benefits are paid at the moment of death, and continuous annuity payments are made at a constant rate while the insured is alive. These timing assumptions matter: the continuous formulas do not value a benefit paid at the end of the year of death.

The Main Identity Map

The following table is a selection guide. Here, \(\ddot a_{x:\overline n|}\) is an \(n\)-year temporary life annuity-due, and \(\bar a_{x:\overline n|}\) is its continuous-payment counterpart. The deferred forms begin payments at time \(m\) and cover the following \(n\) years.

Contract cash flows	Matching identity	What to check
Discrete whole life: death benefit at end of year of death	\(A_x+d\ddot a_x=1\)	There is no fixed maturity date.
Discrete \(n\)-year term: death benefit during the term	\(A^1_{x:\overline n\|}+d\ddot a_{x:\overline n\|}+v^n{}_np_x=1\)	The terminal survival value is a separate pure endowment.
Discrete \(n\)-year endowment: death benefit during the term or benefit at time \(n\)	\(A_{x:\overline n\|}+d\ddot a_{x:\overline n\|}=1\)	The endowment value includes the maturity benefit.
Continuous whole life: benefit at death and payments continuously while alive	\(\bar A_x+\delta\bar a_x=1\)	Both cash flows are continuous in timing.
Continuous \(n\)-year endowment	\(\bar A_{x:\overline n\|}+\delta\bar a_{x:\overline n\|}=1\)	The insurance value includes payment at time \(n\) on survival.
Deferred \(n\)-year coverage, beginning at time \(m\)	Compare deferred insurance with the deferred annuity and surviving endpoint.	Do not include cash flows during the deferment interval.

The superscript \(1\) in \(A^1_{x:\overline n|}\) identifies \(n\)-year term insurance. In contrast, \(A_{x:\overline n|}\) denotes \(n\)-year endowment insurance, which includes the survival benefit. For continuous contracts, a bar indicates continuous benefit timing. Writing the cash flows beside the symbols helps prevent confusing these values.

Why the Discrete Identity Works

The key relationship comes from a one-period accounting step. At the start of a year, the insured is alive. At the end of that year, either a death benefit has been paid during the year (valued at year-end under the discrete convention), or the insured survives and the next year’s cash flows remain. The annuity-due term accounts for the value released by discounting each year’s survival-contingent payment.

For an \(n\)-year term contract, write the expected present value of the term benefit and the temporary annuity-due as sums. The term benefit is paid at time \(k+1\) when death occurs in year \(k+1\); the annuity payment at time \(k\) is made if the insured is alive then.

A^1_{x:\overline n|}=\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}, \qquad \ddot a_{x:\overline n|}=\sum_{k=0}^{n-1}v^k{}_kp_x.

Since \(d=1-v\), the annuity term can be written as a difference of discounted survival probabilities:

d\ddot a_{x:\overline n|} =\sum_{k=0}^{n-1}(v^k-v^{k+1}){}_kp_x.

In the sum of this expression and the term insurance value, the year-\(k+1\) part combines as follows:

v^{k+1}{}_kp_xq_{x+k} +(v^k-v^{k+1}){}_kp_x =v^k{}_kp_x-v^{k+1}{}_{k+1}p_x.

Summing over \(k=0,\ldots,n-1\) makes the intermediate survival terms cancel. The remaining terms are \(1-v^n{}_np_x\). Therefore the identity for term insurance is:

A^1_{x:\overline n|}+d\ddot a_{x:\overline n|} =1-v^n{}_np_x, \qquad A^1_{x:\overline n|}+d\ddot a_{x:\overline n|}+v^n{}_np_x=1.

The last term is the pure endowment value: it pays \(1\) at time \(n\) only if the insured survives. Endowment insurance is term insurance plus that pure endowment, so its identity follows directly:

A_{x:\overline n|}=A^1_{x:\overline n|}+v^n{}_np_x, \qquad A_{x:\overline n|}+d\ddot a_{x:\overline n|}=1.

Worked Examples

Worked Example: Discrete Whole-Life Timing

A hypothetical whole-life contract pays \(1\) at the end of the year of death. Assume an effective annual interest rate of \(5\%\), and assume the probability of surviving each year is \(0.8\), independently of previous years. Find the expected present value of the insurance benefit using the matching annuity identity.

Method and timing. This is discrete whole-life insurance, so the matching identity is \(A_x=1-d\ddot a_x\). The matching annuity is a whole-life annuity-due, with payments at times \(0,1,2,\ldots\) while alive. The annual survival probability is \(p=0.8\), and the annual death probability is \(q=0.2\).

First calculate the interest quantities:

v=\frac{1}{1.05}=\frac{20}{21}, \qquad d=1-v=\frac{1}{21}.

The probability of being alive at time \(k\) is \(0.8^k\). Thus the annuity-due value is a geometric series:

\ddot a_x=\sum_{k=0}^{\infty}(v\cdot0.8)^k =\frac{1}{1-\frac{20}{21}\cdot0.8} =\frac{1}{1-\frac{16}{21}} =\frac{21}{5}=4.2.

Apply the identity:

A_x=1-\frac{1}{21}\cdot\frac{21}{5} =1-\frac{1}{5} =0.8.

Check. Directly, the benefit present value is \(\sum_{k=0}^{\infty}v^{k+1}0.8^k(0.2)\). This equals \(0.2v/(1-0.8v)=0.8\), agreeing with the identity. The common mistake avoided here is using an annuity-immediate: the contract’s annuity counterpart pays at the beginning of each year, not at the end.

Worked Example: Separating Term Insurance from Endowment Insurance

A hypothetical two-year contract pays \(1\) at the end of the year of death if death occurs during the term. If the insured survives two years, a separate maturity benefit of \(1\) is payable at time \(2\). Assume an effective annual interest rate of \(25\%\), with annual survival probability \(0.75\) and death probability \(0.25\). Calculate the term insurance value, pure endowment value, and endowment insurance value.

Method and timing. The death benefit is two-year term insurance. The maturity benefit is a pure endowment. Their sum is the endowment insurance. Here, \(v=1/1.25=0.8\) and \(d=0.2\).

For term insurance, death in year 1 has probability \(0.25\), and death in year 2 has probability \(0.75(0.25)=0.1875\). Discounting each payment to time 0 gives:

A^1_{x:\overline 2|} =(0.8)(0.25)+(0.8^2)(0.1875) =0.20+0.12 =0.32.

The pure endowment pays at time 2 if the insured survives both years:

v^2{}_2p_x=(0.8)^2(0.75)^2 =0.64(0.5625) =0.36.

Therefore the endowment insurance value is \(0.32+0.36=0.68\). As a check using the annuity identity, the two-year temporary annuity-due value is:

\ddot a_{x:\overline 2|} =1+v\,{}_1p_x =1+(0.8)(0.75) =1.6.

Then \(A_{x:\overline 2|}+d\ddot a_{x:\overline 2|}=0.68+(0.2)(1.6)=1\), as required. The common mistake avoided is treating the term value \(0.32\) as the endowment value: it omits the \(0.36\) maturity benefit.

Worked Example: Selecting a Deferred Continuous Identity

A hypothetical contract has continuous death coverage from time \(2\) through time \(5\), with a benefit of \(1\) paid immediately on death during that interval. There is no payment for death before time \(2\) and no maturity benefit at time \(5\). Assume a constant force of mortality \(\mu=\ln 2\) and a constant force of interest \(\delta=\ln 2\). Find the expected present value of the deferred term benefit and check it using the matching deferred continuous annuity.

Method and timing. Since the benefit is paid immediately at death, use continuous insurance. Since coverage starts at time \(2\), use a deferred term value, not whole-life insurance from time 0. The matching continuous annuity is payable while alive only from time \(2\) to time \(5\).

Under the constant force of mortality, the probability of survival to time \(t\) is \(e^{-\mu t}\), and the density of death at time \(t\) is \(\mu e^{-\mu t}\). The discounted expected benefit is therefore:

\bar A^1_{x,\,{}_{2|}\overline 3|} =\int_2^5 e^{-\delta t}\mu e^{-\mu t}\,dt =\mu\int_2^5e^{-2\mu t}\,dt =\frac12\left(e^{-4\ln2}-e^{-10\ln2}\right).

Since \(e^{-4\ln2}=1/16\) and \(e^{-10\ln2}=1/1024\), the value is:

\bar A^1_{x,\,{}_{2|}\overline 3|} =\frac12\left(\frac{64-1}{1024}\right) =\frac{63}{2048} \approx 0.0308.

The deferred continuous annuity value is \(\bar a_{x,\,{}_{2|}\overline 3|}=\int_2^5 e^{-(\delta+\mu)t}\,dt\). Multiplying by \(\delta=\mu\) gives \(\delta\bar a_{x,\,{}_{2|}\overline 3|}=63/2048\), the same value as the insurance benefit. Their sum is \(63/1024\), which equals the discounted probability of being alive at time \(2\) less the discounted probability of being alive at time \(5\):

e^{-(\delta+\mu)2}-e^{-(\delta+\mu)5} =\frac{1}{16}-\frac{1}{1024} =\frac{63}{1024}.

Check. The insurance value and the annuity value multiplied by \(\delta\) sum to the endpoint difference, as required for deferred term coverage. The common mistake avoided is starting the integral at time \(0\), which would include deaths during the deferment period even though the contract provides no coverage then.

A Selection Procedure for Mixed Timing Details

When a description contains several timing conditions, translate each condition into its own cash flow before combining values. For example, an endowment contract has a death benefit and a survival benefit; the death-benefit part is term insurance, and the survival-benefit part is a pure endowment. Present-value linearity permits adding those values, but it does not permit changing their payment times.

Mark the coverage interval.
Record the start time and end time. A deferred contract has no covered death benefit before its start time.

Mark the death-benefit payment time.
End-of-year-of-death coverage points to a discrete insurance value; immediate payment at death points to a continuous value.

Check survival at the endpoint.
If the contract pays at maturity on survival, include the pure endowment. If not, do not add one.

Match the annuity period and timing.
Use an annuity-due for discrete payments at the beginnings of periods, or a continuous annuity for continuous payments. Defer it if its payment period is deferred.

Verify the boundary cash flows.
Check the start and endpoint terms in the identity. They reveal whether the survival benefit has been included correctly.

For a deferred discrete \(n\)-year term contract beginning at time \(m\), the matching identity makes the endpoint adjustment explicit:

{}_{m|}A^1_{x:\overline n|} +d\,{}_{m|}\ddot a_{x:\overline n|} +v^{m+n}{}_{m+n}p_x =v^m{}_mp_x.

The right side is the value at time 0 of being alive at the beginning of the covered interval. The endpoint term is the value of surviving to the end of that interval. If the contract also pays at the end on survival, the term insurance plus that pure endowment is the deferred endowment value. The continuous version uses \(\delta\) and continuous annuities, with the same start and endpoint survival values under the specified continuous timing.

Common Mistakes and a Final Check

Using the right identity for the wrong payment time. Discrete death benefits are paid at the end of the year of death; continuous benefits are paid at death. These values generally differ.
Leaving out the pure endowment. A term insurance identity has an endpoint survival term when written with the temporary annuity. An endowment insurance value already includes that maturity benefit.
Using a whole-life annuity for a temporary contract. An \(n\)-year contract calls for an annuity over the matching \(n\)-year period, not payments extending indefinitely.
Ignoring the deferment period. Deferred coverage and its matching annuity begin at the deferment time, not at time 0.
Mixing \(d\) and \(\delta\). The discrete identities use \(d=1-v\); the continuous identities use the force of interest \(\delta\). They reflect different payment patterns.

Key takeaway: Choose the identity from the contract’s cash flows, not from its product label alone. Match the coverage interval, death-benefit timing, maturity payment, and annuity timing; then check that the boundary survival terms agree with the contract.

Check Your Understanding

For each problem, identify the relevant timing and use it to check the requested value.

A hypothetical three-year discrete term policy pays \(1\) at the end of the year of death. Assume \(i=25\%\), annual survival probability \(0.8\), and annual death probability \(0.2\). Find the term insurance value and the pure endowment value at year 3. (Answer: term value \(0.327936\); pure endowment value \(0.262144\).)
A hypothetical two-year discrete endowment pays \(1\) at the end of the year of death or at time 2 if the insured survives. Assume \(i=10\%\), annual survival probability \(0.9\), and annual death probability \(0.1\). Find its value by separating term insurance and pure endowment. (Answer: \(0.834711\), to 6 decimal places.)
A continuous whole-life contract pays \(1\) immediately on death. Assume constant \(\mu=0.08\) and \(\delta=0.04\). Find the continuous insurance value and verify it using the continuous whole-life annuity value. (Answer: \(\bar A_x=2/3\); \(\bar a_x=25/3\); \(\bar A_x+\delta\bar a_x=1\).)
A discrete deferred two-year term policy begins at time 2 and pays at the end of the year of death. Assume \(i=25\%\), annual survival probability \(0.8\), and annual death probability \(0.2\). Find the deferred term value and check the deferred identity, including survival to time 4. (Answer: deferred term value \(0.10747904\); \(d\,{}_{2|}\ddot a_{x:\overline 2|}=0.1343488\); \(v^4{}_4p_x=0.16777216\); the sum is \(v^2{}_2p_x=0.4096\).)

What You'll Learn