Solving for an Unknown Annuity Term or Deferment

Turning an Annuity Value into an Unknown

Life-annuity questions often give a present value and ask for a feature of the payment pattern rather than asking for the value itself. That feature might be the number of annual payments or how long payments are deferred. The key is to express the value as a function of the unknown, then test or solve that equation while keeping the payment dates and survival conditions explicit.

In the examples below, payments are annual and equal to 1 unless stated otherwise. Interest is effective annually. A life annuity-due pays at the start of each payment year, provided the annuitant is then alive. Consequently, the payment at time 0 is certain for a person alive at issue; the payment at time $k$ is contingent on survival to time $k$.

Definition: For an $n$-payment temporary life annuity-due issued at age $x$, with annual effective interest rate $i$, the present value per unit annual payment is $$ \ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k\,{}_kp_x,\qquad v=\frac{1}{1+i}. $$ Here $n$ is a positive integer, and ${}_kp_x$ is the probability that a life aged $x$ survives for $k$ years. The sum includes payment times $0,1,\ldots,n-1$, not $1,2,\ldots,n$.

If the annual payment is $B$, the present value is $B\ddot{a}_{x:\overline{n}|}$. If payments are deferred, each possible payment must also be discounted to the valuation date and weighted by the probability the annuitant survives until that payment. The annuity equation therefore depends on both the cash-flow dates and the survival probabilities.

Solving for the Number of Payments

When the unknown is $n$, evaluate the annuity value for successive integer terms until the given present value is reached or bracketed. Each additional payment adds one nonnegative term to the sum. If the relevant survival probability is positive, that term is positive, so the annuity value increases with the number of payments.

Result: For a fixed interest rate and mortality model, $$ \ddot{a}_{x:\overline{n+1}|} =\ddot{a}_{x:\overline{n}|}+v^n\,{}_np_x. $$ Thus, if ${}_np_x>0$, the value for $n+1$ payments is greater than the value for $n$ payments. For a discrete annual contract, an exact solution for the number of payments must be an integer.

This recursion also gives a practical search method: start with the first payment, add the next payment’s discounted survival contribution, and stop when the target is matched or lies between successive values. Interpolation can help locate an approximate position between those values, but it does not turn a fractional payment count into an exact number of annual payments.

Worked Example: Identifying an Unknown Term Under Constant Survival

A hypothetical annual life annuity-due pays 1 at each payment date while the annuitant is alive. The effective annual interest rate is $5\%$, and the conditional probability of surviving each year is $0.9$, independently of earlier survival. The quoted present value is $1105/343$. Determine the number of annual payments.

Method and assumptions. The annuitant is alive at time 0, and each future year has survival probability $0.9$. Thus ${}_kp_x=(0.9)^k$. With $i=0.05$, $v=1/1.05=20/21$, so $v\,{}_1p_x$ per year of deferral combines to the ratio $v(0.9)=(20/21)(9/10)=6/7$. The annuity value for $n$ payments is therefore $$ \ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}\left(\frac67\right)^k. $$

For two payments, the value is $1+6/7=13/7$. For three payments, it is $$ 1+\frac67+\left(\frac67\right)^2 =\frac{49+42+36}{49} =\frac{127}{49} \approx 2.591837. $$ For four payments, retain the exact terms before rounding: $$ 1+\frac67+\left(\frac67\right)^2+\left(\frac67\right)^3 =\frac{343+294+252+216}{343} =\frac{1105}{343} \approx 3.221574. $$

The target is exactly $1105/343$, so the solution is four annual payments. As a check, the fourth payment adds $(6/7)^3=216/343$ to the three-payment value $889/343$, giving $1105/343$, the stated target.

Common mistake avoided: Do not add rounded decimal approximations of the individual terms and then round again. That can change the last reported digit. Here the exact fraction gives $3.221574$ to six decimal places.

Solving for a Deferral Period

Suppose an annuity-due has $n$ payments, but its first payment is at time $m$, rather than time 0. Under an annual model, its payment times are $m,m+1,\ldots,m+n-1$. If the annuitant must be alive for each payment, the value is a sum of terms $v^{m+j}{}_{m+j}p_x$. The deferral period changes both the discounting and the survival requirement.

Definition: For a deferred $n$-payment annual life annuity-due whose first payment is at time $m$, the present value per unit payment is $$ \sum_{j=0}^{n-1}v^{m+j}\,{}_{m+j}p_x. $$ The first payment is made at time $m$, conditional on survival to that time. This convention is important: a payment at time $m$ requires survival for $m$ years, not $m+1$ years.

When survival is constant from year to year, ${}_kp_x=p^k$, the deferred value factors into a deferral term and the value of the same annuity beginning at time 0. With $r=vp$, that deferral term is $r^m$. The factorization can make the unknown $m$ straightforward to solve.

Derivation: Under constant annual survival probability $p$, the value is $$ \sum_{j=0}^{n-1}v^{m+j}p^{m+j} =(vp)^m\sum_{j=0}^{n-1}(vp)^j. $$ The first equality follows because a payment at time $m+j$ is discounted by $v^{m+j}$ and made only if the life survives $m+j$ years, with probability $p^{m+j}$. Factoring out the common factor $(vp)^m$ leaves the $n$-payment annuity value at time 0. Thus deferring the annuity multiplies its value by $(vp)^m$ in this constant-survival model.

Worked Example: Finding the Deferral Period by Trial and Logs

A hypothetical annuity-due pays 1 annually for three payments, with its first payment at time $m$. The effective annual interest rate is $5\%$, and the conditional annual survival probability is $0.9$ at every future age. Its present value is $4572/2401$. Find $m$, assuming $m$ is a nonnegative integer.

Method and assumptions. As in the stated model, $v=20/21$, and $vp=6/7$. The value of three payments beginning at time 0 is $$ 1+\frac67+\left(\frac67\right)^2 =\frac{127}{49}. $$ For a deferral of $m$ years, the value is therefore $$ \left(\frac67\right)^m\frac{127}{49}. $$

Test integer deferrals. For $m=1$, the value is $$ \frac67\cdot\frac{127}{49} =\frac{762}{343} \approx 2.221574. $$ For $m=2$, it is $$ \left(\frac67\right)^2\frac{127}{49} =\frac{36}{49}\cdot\frac{127}{49} =\frac{4572}{2401} \approx 1.904207. $$ This matches the target exactly, so the deferral is two years.

The equation can also be solved using logarithms: $$ \left(\frac67\right)^m\frac{127}{49}=\frac{4572}{2401} \quad\Longrightarrow\quad \left(\frac67\right)^m=\left(\frac67\right)^2 \quad\Longrightarrow\quad m=2. $$ The trial values and the exact equation agree, which checks the answer.

Common mistake avoided: A deferral of two years means the first payment is at time 2, not time 3. Using ${}_3p_x$ and $v^3$ for that first payment would incorrectly shift every payment one year later.

When Survival Probabilities Vary by Year

A constant annual survival probability is convenient, but it is not necessary for the search. If survival probabilities vary by year, compute each payment contribution from the specified ${}_kp_x$, then accumulate the terms. The same approach works for solving for an unknown deferral: calculate the value at candidate integer deferrals using the survival probabilities at the actual payment times.

Worked Example: Bracketing a Term with a Survival Table

A hypothetical annual annuity-due pays 1 while the annuitant is alive. The effective annual interest rate is $4\%$. The supplied survival probabilities from age $x$ are ${}_0p_x=1$, ${}_1p_x=0.96$, ${}_2p_x=0.90$, and ${}_3p_x=0.82$. The target present value is $3.40$. Determine the number of payments, assuming the target is to be matched by a whole number of annual payments.

Method and assumptions. Here $v=1/1.04=25/26$. The first payment at time 0 contributes 1. The second payment contributes $$ v\,{}_1p_x=\frac{25}{26}(0.96)=\frac{12}{13}\approx0.923077. $$ The third contributes $$ v^2\,{}_2p_x =\left(\frac{25}{26}\right)^2(0.90) \approx0.832101. $$ Thus the three-payment value is $$ 1+\frac{12}{13}+\left(\frac{25}{26}\right)^2(0.90) \approx2.755178. $$

The fourth payment contributes $$ v^3\,{}_3p_x =\left(\frac{25}{26}\right)^3(0.82) \approx0.728977. $$ The four-payment value is therefore approximately $2.755178+0.728977=3.484155$. The target satisfies $2.755178<3.40<3.484155$, so it lies between the three- and four-payment values. Because the number of annual payments must be an integer, four payments is the first term whose value reaches or exceeds the target; the target is not an exact value for any whole payment count in this bracket.

The conclusion is checked by the positive fourth-payment contribution: adding a payment increases the value from about $2.755178$ to about $3.484155$, crossing $3.40$. Values are shown to six decimal places for comparison; the underlying calculations use the unrounded expressions.

Common mistake avoided: Do not interpolate to claim an exact fractional number of annual payments. Interpolation between these values gives a rough location, but a standard annual annuity has a whole-number payment count. If a contract permits a specially defined partial final payment, that cash flow must be modeled explicitly.

Trial, Interpolation, and Checks

Numerical trial is usually the safest first step because it makes the payment timing visible. For an unknown term, calculate successive annuity values until the target is reached or bracketed. For an unknown deferral, calculate the value at successive candidate deferral periods. If the values form a geometric sequence, a logarithm may solve for the unknown directly; otherwise, numerical trial is generally more transparent.

Linear interpolation between two trial values can provide an estimate when a continuous approximation is useful. If $V(m)$ is known at $m=a$ and $m=b$, an estimated location for target $T$ is $$ m\approx a+\frac{T-V(a)}{V(b)-V(a)}(b-a). $$ This is an approximation, not a general exact formula: annuity values need not vary linearly with the unknown. Moreover, a contract requiring an integer number of annual deferrals or payments still requires an integer answer. Verify any proposed integer by substituting it into the original annuity equation.

Before finalizing a solution, check that the valuation date is clear, the first payment date is correctly indexed, each payment is weighted by survival to its own date, and the discount factor uses the stated interest period. These checks catch errors that can otherwise produce plausible-looking values.

Key takeaway: Write the present value as a sum over actual payment dates. For an unknown term, accumulate the payment contributions; for an unknown deferral, shift the payment dates and survival probabilities consistently. Use trial or algebra to identify a candidate, then verify it in the original equation.

Check Your Understanding

For each problem, use the stated timing and assumptions. Answers are given at the end of each question for checking.

A hypothetical annual annuity-due pays 1 while the annuitant is alive. The effective annual interest rate is $4\%$, and the conditional annual survival probability is $0.8$ each year. Its value is $399/169$. Find the whole-number payment term. (Answer: 3 payments.)
A two-payment annual annuity-due is deferred so that its first payment is at time $m$. The effective annual interest rate is $5\%$, annual survival probability is $0.9$, and its present value is $78/49$. Find the nonnegative integer $m$. (Answer: 1 year.)
An annual annuity-due has an effective annual interest rate of $4\%$, with survival probabilities ${}_0p_x=1$, ${}_1p_x=0.96$, ${}_2p_x=0.90$, and ${}_3p_x=0.82$. Its target value is $3.40$. What is the smallest whole-number term whose value is at least the target? (Answer: 4 payments.)
A hypothetical two-payment annual annuity-due is deferred $m$ years. The effective annual interest rate is $5\%$, and the conditional annual survival probability is $0.8$ each year. Its present value is $9472/9261$. Find $m$, assuming a nonnegative integer deferral. (Answer: 2 years.)

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