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Common Indexing Errors in Life Annuity Values

Use payment dates and survival conditions to build life annuity values accurately and catch common indexing errors.

Beginner 11 min read

What You'll Learn

  • Match annuity-due and annuity-immediate payments to their correct dates
  • Attach the appropriate survival probability to every life-contingent payment
  • Index deferred annuities from the valuation date, not from the end of deferment
  • Separate guaranteed payments from later survival-contingent payments
  • Check the final payment date in a temporary annuity

Why Annuity Indexing Matters

A life annuity value is a sum of discounted payments, but each payment must also be weighted by the probability that it is payable. A correct formula therefore depends on two details: when each payment occurs and what must be true for it to be made. An off-by-one error in either detail can change both the discount factor and the survival probability.

This tutorial focuses on discrete annual payments. The examples use hypothetical contracts and deliberately simple mortality assumptions so that the indexing, rather than arithmetic, is the main issue. An annual effective interest rate \(i\) gives discount factor \(v=1/(1+i)\). For a life aged \(x\), \({}_kp_x\) is the probability of surviving \(k\) years from age \(x\), with \({}_0p_x=1\).

Definition: An annual life annuity-due makes a payment at time \(0\), and then at times \(1,2,\ldots\) while the annuitant is alive. An annual life annuity-immediate makes payments at times \(1,2,\ldots\), provided the annuitant is alive at each payment date. A payment at time \(k\) is discounted by \(v^k\); if it is contingent on survival to time \(k\), its expected value is also multiplied by \({}_kp_x\).

Set the Payment Dates Before Writing the Value

For a temporary annuity-due with \(n\) possible payments, the payment dates are \(0,1,\ldots,n-1\). For a temporary annuity-immediate with \(n\) possible payments, they are \(1,2,\ldots,n\). This difference affects the final payment date as well as the first.

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

Here each annuity pays one unit at each scheduled date, subject to the stated survival condition. The due annuity’s time-zero payment is made immediately, so it has discount factor \(v^0=1\) and survival probability \({}_0p_x=1\). The immediate annuity has no time-zero payment. Its \(n\)th and final possible payment is at time \(n\), not at time \(n-1\).

For a whole-life annuity-due, the same indexing continues for every integer payment date while the annuitant is alive:

$$ \ddot{a}_x=\sum_{k=0}^{\infty}v^k\,{}_kp_x. $$

Worked Example: Due Versus Immediate Payments

A hypothetical four-payment annuity has annual effective interest \(i=25\%\). Assume the probability of surviving each year is \(p=0.8\), independently from year to year, so \({}_kp_x=0.8^k\). Find the value of (a) an annuity-due paying at times \(0,1,2,3\), and (b) an annuity-immediate paying at times \(1,2,3,4\).

Method. List the payment dates, then use the discount and survival factors for each date. This avoids treating the two four-payment schedules as if they had the same final date.

The discount factor is \(v=1/(1.25)=0.8\). At time \(k\), the product of discount and survival factors is \(v^k{}_kp_x=(0.8)^k(0.8)^k=(0.64)^k\).

(a) For the due annuity, the four dates are \(0,1,2,3\):

$$ \ddot{a}_{x:\overline{4}|} =1+0.64+0.64^2+0.64^3 =1+0.64+0.4096+0.262144 =2.311744. $$

(b) For the immediate annuity, the four dates are \(1,2,3,4\):

$$ a_{x:\overline{4}|} =0.64+0.64^2+0.64^3+0.64^4 =0.64+0.4096+0.262144+0.16777216 =1.47951616. $$

Rounded to four decimals, the values are \(2.3117\) and \(1.4795\). As a check, the immediate schedule is the due schedule shifted one year later, so each corresponding term is multiplied by \(v p=0.64\); indeed \(0.64(2.311744)=1.47951616\). The common mistake is to put the immediate annuity’s last payment at time \(3\), which would incorrectly omit the payment at time \(4\).

Why Each Term Uses Its Own Survival Probability

The expected present value of a payment at time \(k\) follows directly from its possible outcomes. If a unit payment is made only when the annuitant survives to time \(k\), its present value is \(v^k\) on survival and \(0\) otherwise. Its expected value is therefore \(v^k{}_kp_x\). Summing these expected values over the scheduled payment dates gives the annuity value.

$$ E[\text{present value of payment at }k] =v^k\cdot P(\text{payment is made}) =v^k\,{}_kp_x. $$

The survival term is tied to the payment date, not to the number of payments that have already been made. In particular, a due annuity payment at time \(0\) does not require one year of survival; a deferred payment at time \(m\) does require survival to time \(m\) if it is life-contingent.

For a deferred life annuity-due whose first payment is at time \(m\), the payment dates are \(m,m+1,\ldots\). Its value at time \(0\) is:

$$ {}_{m|}\ddot{a}_x =\sum_{j=0}^{\infty}v^{m+j}\,{}_{m+j}p_x. $$

The index \(j\) counts payments from the start of the deferred period, while \(m+j\) is the actual time from issue. Thus the discount and survival indices both include the deferment. Equivalently, conditional survival to time \(m\) gives the value \(v^m{}_mp_x\ddot{a}_{x+m}\), when the annuitant’s future survival probabilities after time \(m\) are represented from age \(x+m\).

Worked Example: A Deferred Annuity-Due

A hypothetical annuity has a two-year deferment and then makes three annual life-contingent payments at times \(2,3,4\). The annual effective interest rate is \(25\%\), and the probability of surviving each year is \(0.8\), independently from year to year. Find its expected present value at issue.

Method. Use the actual payment dates from issue. The three payments are due at the start of the payment period, but none occurs at time \(0\); the deferment moves the first payment to time \(2\).

Here \(v=0.8\) and \({}_kp_x=0.8^k\). For a payment at time \(k\), the combined factor is \(v^k{}_kp_x=0.64^k\). Therefore:

$$ \text{Value} =0.64^2+0.64^3+0.64^4 =0.4096+0.262144+0.16777216 =0.83951616. $$

Rounded to four decimals, the value is \(0.8395\). As a check, the three terms correspond exactly to survival and discounting to times \(2,3,4\). A common mistake is to value the payments at times \(0,1,2\) and then multiply by a deferment factor; that misses the required survival and discounting during the two-year deferment. The first payment here is at time \(2\), not at time \(0\).

Guaranteed Payments and the First Contingent Payment

A guarantee changes which payments depend on survival. Suppose an annuity-due guarantees the first \(n\) payments, at times \(0,1,\ldots,n-1\), regardless of whether the annuitant survives. After those payments, it continues annually only while the annuitant is alive. The first payment that is contingent on survival is at time \(n\).

$$ \text{Value} =\sum_{k=0}^{n-1}v^k +\sum_{k=n}^{\infty}v^k\,{}_kp_x. $$

The first sum contains certain payments and therefore has no survival probabilities. In the second sum, each payment at time \(k\) is weighted by the probability of survival to \(k\). The formula can also be written using the life annuity-due from age \(x+n\):

$$ \text{Value} =\ddot{a}_{\overline{n}|} +v^n\,{}_np_x\,\ddot{a}_{x+n}, \qquad \ddot{a}_{\overline{n}|}=\sum_{k=0}^{n-1}v^k. $$

The factor \(v^n{}_np_x\) brings the value at time \(n\) back to issue and accounts for survival to the first contingent payment date. The annuity-due \(\ddot{a}_{x+n}\) includes a payment at that date, so it does not add another year of deferment.

Worked Example: Two Guaranteed Payments Followed by Life Payments

A hypothetical annuity-due makes two guaranteed annual payments at times \(0\) and \(1\), followed by annual payments at times \(2,3,\ldots\) only if the annuitant is alive. Assume annual effective interest \(i=25\%\) and a constant annual survival probability of \(0.8\). Find the expected present value.

Method. Separate the certain payments from the survival-contingent payments. This prevents applying survival probabilities to payments that are guaranteed.

The discount factor is \(v=0.8\). The guaranteed payments have value \(1+0.8=1.8\). For a life-contingent payment at time \(k\), the discount-survival factor is \(0.64^k\). Thus the remaining payments have value:

$$ \sum_{k=2}^{\infty}0.64^k =\frac{0.64^2}{1-0.64} =\frac{0.4096}{0.36} =1.137777\ldots. $$

The total value is \(1.8+1.137777\ldots=2.937777\ldots\), or \(2.9378\) rounded to four decimals. As a direct check, the full expression is \(1+0.8+0.4096+0.262144+\cdots\), which has the same first two certain terms and the same contingent tail. The common mistake is to make the payment at time \(2\) guaranteed as well; a two-payment guarantee here covers only times \(0\) and \(1\).

A Reliable Indexing Procedure

When a problem contains more than one timing feature, write the payment schedule before selecting an annuity symbol. The following procedure makes the start date, survival condition, and final payment explicit.

1
Write the actual payment dates.
For example, an \(n\)-payment due annuity starts at time \(0\); a deferment of \(m\) years moves its first due payment to time \(m\).
2
Mark which payments are certain.
A guaranteed payment is not multiplied by a survival probability. For other payments, identify the date to which survival is required.
3
Attach factors to each date.
A payment at time \(k\) receives discount factor \(v^k\); if payable only on survival, it also receives \({}_kp_x\).
4
Check the number and endpoint of terms.
Count the scheduled payments, then verify that the first and final dates match the contract wording.

This procedure is especially useful for temporary and deferred annuities, where an incorrect endpoint can silently add or remove a payment. It also helps distinguish a guarantee period from a deferment: a guarantee identifies payments made regardless of survival, while a deferment identifies how long the contract waits before payments begin.

Common Indexing Traps

  • Shifting an annuity-due by one year. A due annuity starts at time \(0\). Do not assign its first payment a factor \(v{}_1p_x\).
  • Ending an annuity-immediate too early. With \(n\) payments, an immediate annuity’s dates are \(1\) through \(n\), inclusive.
  • Resetting time at deferment. A payment \(j\) years after deferment is at time \(m+j\) from issue, so its discount and survival indices use \(m+j\).
  • Applying survival to guaranteed payments. A payment expressly guaranteed is certain under the contract, even if the annuitant has died.
  • Counting the guarantee endpoint twice. If the first \(n\) due payments occur at \(0\) through \(n-1\), the first payment after the guarantee is at \(n\).
Key takeaway: Treat each payment as its own dated cash flow. Its time determines the discount factor, and the contract’s condition for payment determines whether a survival probability belongs in that term.

Check Your Understanding

For each problem, identify the payment dates before evaluating or describing the value.

  1. A temporary annuity-due makes three annual payments while the annuitant is alive. Assume \(i=10\%\) and annual survival probability \(p=0.9\), constant by year. Find its value at issue, rounded to four decimals. (Answer: \(2.4876\))
  2. A three-payment annuity-immediate pays at times \(1,2,3\). Assume \(i=20\%\) and annual survival probability \(p=0.75\), constant by year. Find its value at issue, rounded to four decimals. (Answer: \(1.2598\))
  3. A hypothetical two-payment deferred annuity-due has its payments at times \(1\) and \(2\). Assume \(i=25\%\) and annual survival probability \(p=0.8\), constant by year; both payments require survival to their payment dates. Find the value, rounded to four decimals. (Answer: \(1.0496\))
  4. A life annuity-due guarantees its first three payments, at times \(0,1,2\), and then pays annually while the annuitant is alive, beginning at time \(3\). Assume \(i=25\%\) and constant annual survival probability \(p=0.8\). Find the value, rounded to four decimals. (Answer: \(3.1682\))
  5. A four-payment temporary annuity is described as “due” and pays only if the annuitant is alive at the payment date. Assume \(i=25\%\) and constant annual survival probability \(p=0.8\). State its payment dates and value, rounded to four decimals. (Answer: Dates \(0,1,2,3\); value \(2.3117\))

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