Increasing and Decreasing Annuities for Exam FM

Increasing Annuity-Immediate

An increasing annuity-immediate (Ia)-angle-n pays 1 at time 1, 2 at time 2, ..., n at time n. Its present value is (Ia)-angle-n = (a-double-dot-angle-n - n*v^n) / i. This formula is the most commonly tested version on Exam FM, so commit it to memory. The accumulated value at time n is (Is)-angle-n = (s-double-dot-angle-n - n) / i.

For payments that increase by a constant amount Q (not necessarily 1), the present value is P * a-angle-n + Q * (Ia)-angle-n - Q * a-angle-n, where P is the first payment. Alternatively, use the general formula for an arithmetic annuity: PV = (P + Q/i) * a-angle-n - Q*n*v^n / i. This second form is often more efficient on exam problems because it combines terms neatly.

Example: An annuity pays $100 at time 1, $150 at time 2, $200 at time 3, and so on for 10 payments, with each payment increasing by $50. Here P = 100 and Q = 50. At i = 6%, PV = (100 + 50/0.06) * a-angle-10 - 50*10*v^10 / 0.06 = (100 + 833.33) * 7.3601 - 500 * 0.5584 / 0.06 = 933.33 * 7.3601 - 4653.33 = 6869.43 - 4653.33 = $2216.10. Always double-check your arithmetic by verifying that the answer falls between the present value of 10 payments of $100 and 10 payments of $550 (the average payment amount).

Decreasing Annuity-Immediate

A decreasing annuity-immediate (Da)-angle-n pays n at time 1, n-1 at time 2, ..., 1 at time n. Its present value is (Da)-angle-n = (n - a-angle-n) / i. The accumulated value is (Ds)-angle-n = (n*(1+i)^n - s-angle-n) / i.

A useful identity connects increasing and decreasing annuities: (Ia)-angle-n + (Da)-angle-n = (n+1) * a-angle-n. This identity can save significant time on Exam FM. If you need (Da)-angle-n and already have (Ia)-angle-n and a-angle-n, simply compute (n+1)*a-angle-n - (Ia)-angle-n. The identity works because the sum of payments at each time t is (t) + (n+1-t) = n+1, which is constant.

For a general decreasing annuity where payments start at P and decrease by Q each period, the present value is P * a-angle-n - Q * ((Ia)-angle-n - a-angle-n). Make sure the payments remain positive throughout; if Q is too large, later payments could become negative, which changes the nature of the problem.

Increasing and Decreasing Annuities-Due

The annuity-due versions follow the same pattern with timing shifted. The increasing annuity-due (Ia-double-dot)-angle-n has present value (1+i) * (Ia)-angle-n. Similarly, (Da-double-dot)-angle-n = (1+i) * (Da)-angle-n. These relationships mirror the standard annuity-immediate to annuity-due conversion.

For continuously payable increasing annuities, (I-bar a-bar)-angle-n = (a-bar-angle-n - n*v^n) / delta, where delta is the force of interest. This formula parallels the discrete case with delta replacing i and continuous annuity values replacing discrete ones.

Geometric Annuities

When payments grow geometrically (each payment is (1+g) times the previous), the present value of n payments starting at 1 is: PV = (1 - ((1+g)/(1+i))^n) / (i - g) when i is not equal to g, and PV = n/(1+i) when i = g. This formula handles salary-linked payments and inflation-adjusted annuities, both common on Exam FM.

Always verify whether the growth rate g applies to each payment relative to the previous one and whether the first payment occurs at time 0 (due) or time 1 (immediate). For a geometric annuity-due, multiply the immediate version by (1+i). For a growing perpetuity (n approaches infinity) with i > g, the present value converges to 1/(i - g) for the immediate case and (1+i)/(i - g) for the due case.

Example: An employee receives a salary of $60,000 per year, growing at 3% annually, for 30 years. At i = 7%, the present value of total salary payments is 60,000 * (1 - (1.03/1.07)^30) / (0.07 - 0.03) = 60,000 * (1 - 0.3083) / 0.04 = 60,000 * 17.2925 = $1,037,550.

Combining Arithmetic and Geometric Patterns

Some Exam FM problems present payment patterns that do not fit neatly into a single category. A common technique is to decompose the payments into simpler components. For instance, payments of 10, 12, 14, 16, ... can be written as a level annuity of 8 plus an increasing annuity of 2, 4, 6, 8, ..., which is 2 times an increasing annuity. This decomposition approach is powerful and appears frequently on the exam.

Another technique involves splitting a payment stream into blocks. If payments are $100 per year for years 1 through 5 and then $200 per year for years 6 through 10, treat this as a level annuity of $100 for 10 years plus a deferred level annuity of $100 for 5 years starting at year 6. This avoids the need for specialized formulas.

Study Tips and Common Pitfalls

The biggest source of errors with increasing and decreasing annuities is confusing the formulas. Write them on a formula card and practice until the patterns are second nature. Remember that (Ia)-angle-n uses a-double-dot-angle-n (annuity-due) in its numerator, not a-angle-n (annuity-immediate). This is a frequent source of mistakes.

For geometric annuities, the most common error is using the wrong sign convention for g or forgetting to check whether i equals g. When i = g, the standard formula has a zero denominator, so you must use the special case PV = n*v. Practice at least 8 problems involving each type before the exam, paying special attention to problems that combine arithmetic progressions with deferral periods.