Deferred Arithmetic-Increasing Annuities

Why the Payment Timeline Matters

A deferred annuity does not begin paying immediately, and an arithmetic-increasing annuity does not make the same payment each period. Combining these features requires careful attention to two separate questions: when does the first payment occur, and how does each later payment differ from it? A one-period error in either answer changes the value of the entire stream.

This tutorial considers deterministic payments made at the ends of successive periods under a constant effective rate per period. It shows how to calculate present value and accumulated value when the first payment may differ from the fixed periodic increase. There is no mortality assumption or probability distribution in these examples; each payment is certain and made at its stated date.

Definition: A deferred arithmetic-increasing annuity has $n$ payments at the ends of $n$ consecutive periods, with each payment exceeding the preceding payment by a fixed amount $G$. If the first payment is $P$, the payments are $P,\ P+G,\ P+2G,\ldots,\ P+(n-1)G$. If the first payment is made at time $m+1$, the annuity is deferred for $m$ complete periods, and its payments occur at times $m+1,\ldots,m+n$.

Here $i$ is the effective interest rate per payment period and $v=1/(1+i)$. The formulas below assume $i$ is constant over the deferral period and the payment period, and that each payment is made at the end of its stated period. If payment dates, period lengths, or interest rates differ, the individual cash flows must instead be valued using the rates and dates that apply to them.

Separating the Level Payment and the Gradient

The notation $a_{\overline{n}|}$ denotes the present value at time 0 of payments of 1 at times 1 through $n$. The notation $(Ia)_{\overline{n}|}$ denotes the present value of payments of 1, 2, through $n$ at those same times. Equivalently,

a_{\overline{n}|}=\sum_{k=1}^{n}v^k, \qquad (Ia)_{\overline{n}|}=\sum_{k=1}^{n}k v^k.

For a deferred stream, the value at time $m$ is found by valuing the payments as if the first occurred one period later, at time $m+1$. Discount that value by $v^m$ to obtain the value at time 0.

Theorem: For payments $P+(k-1)G$ at times $m+k$, for $k=1,\ldots,n$, the present value at time 0 is

PV_0 = v^m\left[P a_{\overline{n}|} +G\left((Ia)_{\overline{n}|}-a_{\overline{n}|}\right)\right] = v^m\left[(P-G)a_{\overline{n}|}+G(Ia)_{\overline{n}|}\right].

The expression $(Ia)_{\overline{n}|}-a_{\overline{n}|}$ is the present-value factor for payments $0,1,\ldots,n-1$: the first payment has no increment, the second has one increment, and so on. The second form rewrites each payment as $(P-G)+kG$. Both forms handle an initial payment different from the increase $G$; there is no requirement that $P=G$.

Why the Present-Value Formula Holds

At the $k$th payment date measured from the start of the annuity, the payment is $P+(k-1)G$. Its time-0 value, before accounting for the deferral, is that amount multiplied by $v^k$. Summing over all $n$ payments gives

\sum_{k=1}^{n}\bigl(P+(k-1)G\bigr)v^k = P\sum_{k=1}^{n}v^k + G\sum_{k=1}^{n}(k-1)v^k.

The first sum is $P a_{\overline{n}|}$. In the second sum, the coefficient $k-1$ equals $k-1$, so its factor is $(Ia)_{\overline{n}|}-a_{\overline{n}|}$. This proves the value of the stream at its start. A deferral of $m$ periods shifts every payment date forward by $m$, multiplying every present-value term by $v^m$. Therefore the whole value is multiplied by $v^m$, as stated.

Worked Examples

Worked Example: Present Value with a Two-Period Deferral

A fund will make four annual payments, with the first payment of $500 at the end of year 3. Each later payment is $100 greater than the previous one. Find the present value at time 0 using an effective annual interest rate of 10%. Payments are certain; no mortality or probability distribution is assumed.

Method. There are $m=2$ complete periods of deferral and $n=4$ payments. The payments are $500, $600, $700, and $800 at times 3, 4, 5, and 6. The gradient is $G=100$, not $500: the first payment and the increase are separate quantities.

First calculate the factors for the four payment periods. With $v=1/1.10=0.909091$,

a_{\overline{4}|} =0.909091+0.826446+0.751315+0.683013 =3.169865, $$ $$ (Ia)_{\overline{4}|} =0.909091+2(0.826446)+3(0.751315)+4(0.683013) =7.547981.

At time 2, the value is the value of a $400 level annuity plus a $100 gradient with payments 1 through 4:

$$ 400(3.169865)+100(7.547981) =1{,}267.946+754.798 =2{,}022.744. $$

Discount two periods to time 0:

PV_0=2{,}022.744(0.909091)^2 =2{,}022.744(0.826446) \approx \$1{,}671.69.

Check. Discounting the stated payments directly gives $500v^3+600v^4+700v^5+800v^6$, which is approximately $1,671.69. The direct cash-flow check agrees with the annuity-factor method. The main mistake avoided here is treating the first payment as the gradient.

Worked Example: Accumulated Value at the Final Payment Date

A grant makes three annual payments: $300 at the end of year 3, $350 at the end of year 4, and $400 at the end of year 5. Find the accumulated value at time 5 using an effective annual interest rate of 5%. The payments are certain; no mortality or probability distribution is assumed.

Method. The stream is deferred for two complete years, but the requested focal date is time 5, not time 0. Accumulate each payment only from its own payment date to time 5. The year-3 payment earns interest for two years, the year-4 payment for one year, and the year-5 payment earns none.

$$ AV_5 =300(1.05)^2+350(1.05)+400 $$

The intermediate growth factors are $(1.05)^2=1.1025$ and $1.05$. Thus,

AV_5 =300(1.1025)+350(1.05)+400 =330.75+367.50+400 =\$1{,}098.25.

Check. The payment dates are years 3, 4, and 5, so the exponents must be 2, 1, and 0. The accumulated value is $1,098.25. Using exponents 3, 2, and 1 would incorrectly accumulate the payments one year past the requested focal date.

Worked Example: Solving for an Unknown First Payment

A deferred annuity pays $P$ at the end of year 2 and $P+100$ at the end of year 3. Its present value is $1,203.20 at an effective annual interest rate of 25%. Find $P$. Payments are certain; no mortality or probability distribution is assumed.

Method. The annuity is deferred for one complete year and has two payments, so $m=1$, $n=2$, and $G=100$. With $v=1/1.25=0.8$,

a_{\overline{2}|}=v+v^2=0.8+0.64=1.44, \qquad (Ia)_{\overline{2}|}=v+2v^2=0.8+1.28=2.08.

The gradient-only component has factor $2.08-1.44=0.64$. Apply the deferred present-value equation and solve:

1{,}203.20 =0.8\left[1.44P+100(0.64)\right] =0.8(1.44P+64), $$ $$ 1{,}504=1.44P+64, \qquad 1.44P=1{,}440, \qquad P=\$1{,}000.

Check. The two payments are $1,000 and $1,100. Their present values are $1{,}000(0.8)^2=\$640$ and $1{,}100(0.8)^3=\$563.20$, totaling $1,203.20. This confirms the solution and the placement of the one-period deferral.

Accumulation and Focal Dates

The present-value formula is often the simplest route when the requested value is at time 0. When the requested value is at the final payment date, each payment must instead be accumulated from its own date. For payments $P+(k-1)G$ at times $m+k$, the accumulated value at time $m+n$ is

AV_{m+n} = \sum_{k=1}^{n}\bigl(P+(k-1)G\bigr)(1+i)^{n-k}.

The deferral period $m$ does not appear in this expression: by time $m+n$, the first payment has accumulated for $n-1$ periods, the second for $n-2$, and the final payment for zero periods. If a present value at time 0 is already known, it can also be accumulated to time $m+n$ by multiplying it by $(1+i)^{m+n}$, provided the same constant effective rate applies throughout.

Mark the payment dates.
If the first payment is at time $m+1$, the payments end at time $m+n$. Count complete periods before the first payment to determine $m$.

Identify the payment pattern.
Record the first payment $P$, fixed increase $G$, number of payments $n$, and effective rate per payment period $i$.

Choose the focal date.
For time 0, value the stream at its start and multiply by $v^m$. For the final payment date, accumulate each payment for its remaining number of periods.

Check the cash flows.
Write out the first and last payment and verify the timing exponents. A direct sum is a useful check on an annuity-factor calculation.

Common Exam Trap: Misindexing the Increase

The first payment is $P$, not $P+G$. In the increasing component, the first payment has zero increments; the second has one increment. Therefore the gradient factor is $(Ia)_{\overline{n}|}-a_{\overline{n}|}$, not $(Ia)_{\overline{n}|}$ by itself when it is added to $P a_{\overline{n}|}$.

For the first example, the incorrect expression $v^2[500a_{\overline{4}|}+100(Ia)_{\overline{4}|}]$ counts the payment sequence as $600, $700, $800, and $900 at the local payment dates. It gives approximately $0.826446[500(3.169865)+100(7.547981)]=\$1{,}933.67$, which is too high. The correct sequence is $500, $600, $700, and $800, so use

v^2\left[500a_{\overline{4}|} +100\left((Ia)_{\overline{4}|}-a_{\overline{4}|}\right)\right] \approx \$1{,}671.69.

A second timing check is equally important: a deferral of $m$ complete periods means the first payment is at $m+1$, so the present-value multiplier is $v^m$, not $v^{m+1}$. The first payment's own discounting is already included in $a_{\overline{n}|}$.

Key takeaway: Write the payment sequence and dates before selecting a factor. For a first payment $P$, fixed increase $G$, $n$ payments, and $m$ complete periods of deferral, use $v^m[(P-G)a_{\overline{n}|}+G(Ia)_{\overline{n}|}]$ for present value at time 0. For another focal date, align every payment with that date explicitly.

Check Your Understanding

For each problem, assume certain payments and a constant effective annual interest rate; no mortality or probability distribution is involved.

An annuity is deferred for two complete years. It pays $400 at the end of year 3, with each later annual payment $50 greater than the preceding payment. There are three payments and the effective annual rate is 6%. Find the present value at time 0. (Answer: approximately $1,065.92)
A fund pays $250, $275, $300, and $325 at the ends of years 4, 5, 6, and 7, respectively. At an effective annual rate of 4%, find the accumulated value at time 7. (Answer: $1,215.66)
A two-payment annuity is deferred for one year. It pays $P$ at the end of year 2 and $P+100$ at the end of year 3. At an effective annual rate of 25%, its present value is $1,203.20. Find $P$. (Answer: $1,000)
An annuity is deferred for four complete years and pays $700, $760, and $820 at the ends of years 5, 6, and 7. At an effective annual rate of 8%, find its value at time 4 and its present value at time 0. (Answer: approximately $1,950.67 at time 4 and $1,433.79 at time 0)

What You'll Learn