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Exam FM · Interest and Annuity Applications · Tutorial 895 of 998

Finding the Deferral Period in a Level Annuity

Set up a timeline and solve the present-value equation to find when a level annuity begins, without confusing its deferral period with its first payment date.

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What You'll Learn

  • Define the deferral period for immediate and due annuities
  • Build a present-value equation from an annuity and dated lump sums
  • Solve for a deferral period using logarithms
  • Check whether a calculated period is valid for the payment schedule
  • Avoid confusing the deferral period with the first payment date

Finding an Annuity’s Start Date

In a staged exchange, the amount of each annuity payment may be known while the start date is not. The key is to value the annuity at a clearly identified date, then discount that value to the date used to value the other cash flows. A timeline helps prevent a particularly common error: treating the deferral period as the date of the first payment.

This tutorial assumes level payments at regular annual intervals and a constant effective annual interest rate \(i>0\). All cash flows are valued at the same interest rate. The examples use annual payments; the same timing logic applies to other regular payment intervals if the interest rate and period units are made consistent.

Definition: For an \(n\)-payment deferred annuity-immediate with deferral period \(m\), the payments occur at times \(m+1,m+2,\ldots,m+n\). Its first payment is at time \(m+1\), not time \(m\). For a deferred annuity-due, the payments occur at times \(m,m+1,\ldots,m+n-1\), so its first payment is at time \(m\). In both cases, \(m\) is the number of full periods before the annuity’s value date, not automatically the date of its first payment.

Set Up the Valuation Equation

Let \(R\) be the level payment and let \(v=1/(1+i)\). An annuity-immediate with payments at times \(m+1\) through \(m+n\) has value \(R a_{\overline{n}|}\) at time \(m\). Discounting this value \(m\) periods to time 0 gives its present value. For an annuity-due with payments at times \(m\) through \(m+n-1\), the value at time \(m\) is \(R\ddot{a}_{\overline{n}|}\).

$$ \begin{aligned} \text{Immediate:}\quad PV_0 &= R a_{\overline{n}|}v^m,\\ \text{Due:}\quad PV_0 &= R\ddot{a}_{\overline{n}|}v^m. \end{aligned} $$

Suppose the other side of the exchange consists of lump sums \(C_j\) at times \(t_j\), all expressed from the perspective of the same party. Their time-0 value is \(\sum_j C_jv^{t_j}\). Set that value equal to the annuity’s present value. If the annuity is the unknown side of an exchange, the equality equates the present values of the two sides; cash-flow signs must be assigned consistently.

Once the annuity factor and the lump-sum value are known, isolate \(v^m\). Taking logarithms then gives the candidate deferral period:

$$ m=\frac{\ln\!\left(PV_0/(R a_{\overline{n}|})\right)}{\ln(v)} \quad\text{for an immediate annuity,} \qquad m=\frac{\ln\!\left(PV_0/(R\ddot{a}_{\overline{n}|})\right)}{\ln(v)} \quad\text{for a due annuity.} $$

Here \(PV_0\) is the present value of the lump-sum side. The denominator \(\ln(v)\) is negative when \(i>0\), so signs matter. With a nonnegative deferral, the ratio \(PV_0/(R a_{\overline{n}|})\) for an immediate annuity must lie between 0 and 1; the corresponding condition for a due annuity uses \(\ddot{a}_{\overline{n}|}\). If the calculated \(m\) is required to be a whole number of payment periods, a noninteger result does not describe an exact start date on that schedule.

Why Discounting by \(v^m\) Works

The factor \(a_{\overline{n}|}\) already values payments one through \(n\) periods after the date on which the annuity-immediate begins. The deferred stream has exactly that same pattern of payment times, shifted forward by \(m\) periods. Each payment is therefore an additional \(m\) periods away from time 0.

$$ \begin{aligned} PV_0 &=Rv^{m+1}+Rv^{m+2}+\cdots+Rv^{m+n}\\ &=v^m\left(Rv+Rv^2+\cdots+Rv^n\right)\\ &=R a_{\overline{n}|}v^m. \end{aligned} $$

The due formula follows by the same argument, with payments at times \(m\) through \(m+n-1\):

$$ \begin{aligned} PV_0 &=Rv^m+Rv^{m+1}+\cdots+Rv^{m+n-1}\\ &=v^m\left(R+Rv+\cdots+Rv^{n-1}\right)\\ &=R\ddot{a}_{\overline{n}|}v^m. \end{aligned} $$

Because \(0<v<1\), the value \(v^m\) decreases as \(m\) increases. For fixed payments and a fixed positive interest rate, increasing the deferral period lowers the annuity’s present value. That monotonicity also means there can be at most one nonnegative real solution for \(m\) in this constant-rate model.

Worked Examples

Worked Example: Immediate Annuity Matched to Two Lump Sums

An exchange offers a level annuity-immediate of $1,000 per year for two years, with unknown deferral period \(m\). The other side consists of $400 immediately and $652 one year from now. The effective annual interest rate is 25%. Find \(m\), and identify the annuity’s payment dates.

Method. Find the time-0 value of the lump sums, equate it to the deferred immediate annuity’s present value, and solve for \(m\).

The discount factor is \(v=1/1.25=0.8\). The annuity-immediate factor is \(a_{\overline{2}|}=v+v^2=0.8+0.64=1.44\). The lump-sum present value is

$$ 400+652(0.8)=400+521.6=921.6. $$

The present-value equation is \(1{,}000(1.44)(0.8)^m=921.6\). Thus \(1{,}440(0.8)^m=921.6\), and dividing by 1,440 gives \((0.8)^m=0.64\). Since \(0.64=(0.8)^2\), \(m=2\).

The payment dates are \(m+1=3\) and \(m+2=4\). Therefore the annuity payments occur at times 3 and 4. Check: their present value is \(1{,}000(0.8^3+0.8^4)=1{,}000(0.512+0.4096)=921.6\), equal to the lump-sum value. The common mistake this setup avoids is calling time 2 the first payment date: time 2 is the annuity’s value date, not its first payment date.

Worked Example: Annuity-Due Timing

A hypothetical arrangement offers three annual payments of $500, paid as an annuity-due after an unknown deferral period \(m\). It is exchanged for $100 immediately and $1,095 one year from now. At an effective annual interest rate of 25%, find \(m\) and the first payment date.

Method. Value the annuity-due at time \(m\), then discount by \(v^m\). The timing convention is different from that for an annuity-immediate: a due annuity’s first payment is at time \(m\).

Again, \(v=0.8\). The due-annuity factor is \(\ddot{a}_{\overline{3}|}=1+v+v^2=1+0.8+0.64=2.44\). The lump-sum present value is

$$ 100+1{,}095(0.8)=100+876=976. $$

Equating values gives \(500(2.44)(0.8)^m=976\), or \(1{,}220(0.8)^m=976\). Hence \((0.8)^m=976/1{,}220=0.8\), so \(m=1\).

The payments occur at times \(1,2,3\), and the first payment is at time 1. Check: their present value is \(500(0.8+0.64+0.512)=500(1.952)=976\), matching the lump sums. The mistake to avoid is adding one to the due annuity’s first payment date: for this annuity-due, the first payment is at \(m\), not \(m+1\).

Worked Example: Combining Lump Sums at Different Dates

A level annuity-immediate pays $1,000 at each of two annual payment dates. Its unknown deferral period is \(m\). The other side consists of $300 immediately and $683.25 at time 2. The effective annual interest rate is 25%. Find \(m\) and verify the exchange.

Method. First consolidate the dated lump sums at time 0. Do not treat a lump sum at time 2 as though it were already a time-0 amount.

Here \(v=0.8\) and \(a_{\overline{2}|}=0.8+0.64=1.44\). The present value of the lump sums is \(300+683.25(0.8)^2=300+683.25(0.64)=300+437.28=737.28\).

The annuity equation is \(1{,}000(1.44)(0.8)^m=737.28\). Dividing by 1,440 gives \((0.8)^m=737.28/1{,}440=0.512\). Since \(0.512=(0.8)^3\), \(m=3\).

The annuity-immediate pays at times 4 and 5. Check: its present value is \(1{,}000(0.8^4+0.8^5)=1{,}000(0.4096+0.32768)=737.28\), matching the discounted lump sums. The main error this method avoids is discounting or accumulating one lump sum in the wrong direction on the timeline.

Interpreting the Result and Avoiding Traps

The logarithm gives a candidate number of periods. Its interpretation depends on the payment convention and on the problem’s timing requirements. If payments occur annually and the start date must be an annual boundary, \(m\) must be an integer. If the calculation instead gives, for example, \(m=2.4\), that is not an exact annual deferral period. It may describe a real-valued solution in a model that allows fractional timing, but it does not place the annuity on an annual payment schedule without further information.

A frequent exam trap is to use the right present-value equation but attach the wrong payment dates to the result. For an immediate annuity, \(m=2\) means the payments start at time 3; for a due annuity, \(m=2\) means they start at time 2. Another trap is to solve using the value of the lump sums at a date other than time 0 without shifting the annuity value to that same date. Both sides of an equation of value must be expressed at the same focal date.

Key takeaway: Identify whether the level stream is immediate or due, write its payment dates, and value both sides at one focal date. Solve for \(v^m\), check that the resulting period fits the allowed schedule, then report the first payment date separately from the deferral period.

Check Your Understanding

For each problem, use the stated effective annual rate and assume the required deferral is in whole annual periods unless the question asks otherwise.

  1. An annuity-immediate pays $1,000 annually for two years. At an effective annual rate of 25%, its value equals a lump sum of $921.60 at time 0. Find the deferral period and payment dates. (Answer: \(m=2\); payments at times 3 and 4.)
  2. A three-payment annuity-due has level payments of $200. The effective annual rate is 25%, and its time-0 value must equal $312.32. Find the deferral period and first payment date. (Answer: \(m=2\); first payment at time 2.)
  3. A two-payment annuity-immediate has level payments of $600. At 25% effective annual interest, its time-0 value must equal $442.368. Find the deferral period and payment dates. (Answer: \(m=3\); payments at times 4 and 5.)
  4. A two-payment annuity-immediate pays $500 per year. At 25% effective annual interest, it is matched to $100 immediately and $451 at time 1. Find the deferral period and payment dates. (Answer: \(m=2\); payments at times 3 and 4.)
  5. A two-payment annuity-immediate pays $500 per year. At 25% effective annual interest, the required time-0 value is $400. Find the real-valued solution for \(m\), then decide whether an exact whole-year deferral exists. (Answer: \(m=\ln(5/9)/\ln(0.8)\approx2.634\); no exact whole-year deferral exists.)

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