Worked Problems on Arithmetic-Gradient Annuities

How to Value an Arithmetic Payment Stream

An arithmetic annuity changes by a fixed dollar amount from one payment to the next. The key is to distinguish the first payment from the increment. If the first payment is \(C\) and each later payment increases by \(G\), the payments are \(C,\ C+G,\ C+2G,\ldots\). For a decreasing stream, the same pattern applies with a negative increment, or equivalently with a positive decrement subtracted each period.

A timeline fixes the payment dates; a focal-date equation then discounts or accumulates each payment to one common date. This is particularly important when the annuity is due or deferred: the same payment amounts at different dates do not have the same value. Unless stated otherwise, the examples use annual payments and a constant effective annual interest rate \(i\), with interest effective at each year-end.

The standard factors \(a_{\overline{n}|}\) and \(\ddot{a}_{\overline{n}|}\) are the present values of \(n\) unit payments at the ends and beginnings of successive periods, respectively. The increasing-gradient factor \((Ia)_{\overline{n}|}\) is the present value of payments \(1,2,\ldots,n\) at times \(1,2,\ldots,n\). Thus

These factors describe payment timing as well as payment amounts. The increasing-gradient factor does not by itself represent the increments \(0,G,2G,\ldots\) added to a base payment: its weights are \(1,2,\ldots,n\). That indexing difference must be handled explicitly.

Immediate Payments: Increasing and Decreasing Gradients

For an annuity-immediate whose payments are \(C,\ C+G,\ldots,C+(n-1)G\), split each payment into a level part \(C\) and an increment part. At time \(k\), the level part is \(C\) and the increment part is \((k-1)G\). Therefore the present value is

For a decreasing stream with decrement \(D>0\), payments \(C,\ C-D,\ldots,C-(n-1)D\) have present value

The formulas require a constant effective rate per payment period and payment dates at the ends of successive periods. A decreasing schedule may be valued algebraically even if it eventually has negative payments, but a stream described as nonnegative payments must satisfy \(C-(n-1)D\geq 0\).

Annuity-Due Payments

For an annuity-due, the first payment occurs at time 0, so the payment pattern \(C,\ C+G,\ldots,C+(n-1)G\) has present value \(C\) at time 0, \(C+G\) at time 1, and so on. Its gradient part has weights \(0,1,\ldots,n-1\), not \(1,2,\ldots,n\) at those dates. The due-stream present value is

An equivalent expression for the gradient factor is \((1+i)\bigl((Ia)_{\overline{n}|}-a_{\overline{n}|}\bigr)\). To see why, replace \(j\) by \(k-1\) in the sum: \(\sum_{j=0}^{n-1}jv^j=\sum_{k=1}^n(k-1)v^{k-1}=(1/v)\sum_{k=1}^n(k-1)v^k\), and \(1/v=1+i\). This relationship is useful when factors for annuity-immediate gradients are available, but a timeline is still the safest way to check the indexing.

Deferred Streams and Focal Dates

A deferred annuity is valued most simply by first finding its value immediately before its first payment period, then discounting that value to the requested date. For example, if payments occur at times \(m+1,m+2,\ldots,m+n\), the payment pattern is an annuity-immediate shifted \(m\) years later. At effective annual rate \(i\), the present value at time 0 is

The bracketed amount is the value at time \(m\); multiplying by \(v^m\) moves that value back to time 0. This expression applies to annual payments at the ends of periods and a constant effective annual rate. If the first payment is at a different date, identify its actual time on the timeline rather than assuming the same deferral.

Common Exam Traps and Checks

A frequent error is to use \((Ia)_{\overline{n}|}-a_{\overline{n}|}\) as the increment factor but also treat \((Ia)_{\overline{n}|}\) as if its first gradient payment were zero. In fact, \((Ia)_{\overline{n}|}\) has weights \(1,2,\ldots,n\) at times \(1,2,\ldots,n\). The factor for increments \(0,1,\ldots,n-1\) at those same immediate dates is the difference \((Ia)_{\overline{n}|}-a_{\overline{n}|}\).

For example, in the first worked example the incorrect calculation \(500a_{\overline{4}|}+100(Ia)_{\overline{4}|}\) gives \(1,584.932723+754.798170=2,339.730893\). That calculation adds a \$100 gradient payment at time 1, so it represents payments \$600, \$700, \$800, and \$900—not the stated schedule beginning at \$500. The correction is to subtract the level-annuity factor from the increasing-gradient factor for the increment component, giving \$2,022.74.

• Check whether the first payment is at time 0 or time 1. That determines whether the stream is due or immediate.
• Check the increment index. An immediate stream starting with \(C\) has additions \(0,G,\ldots,(n-1)G\).
• Check the deferral from the actual dates. A stream first paid at time \(m+1\) has a value at time \(m\) that must be discounted \(m\) years to time 0.
• Check decreasing amounts. If every payment must be nonnegative, verify that the last payment is not below zero.

A sound focal-date equation is more reliable than memorizing a factor in isolation. These valuation methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus.