level and varying annuities

Not every stream of payments is flat. A pension might rise each year to keep pace with prices; a loan might be repaid in shrinking instalments; a salary-based contribution might grow as wages grow. A level annuity pays the same amount every period, while a varying annuity changes the payment from period to period — increasing, decreasing, or growing by a fixed percentage. The valuation idea stays the same; only the payment pattern changes.

The most common patterns have tidy formulas. An increasing annuity that pays 1, then 2, then 3, and so on, written (I a)-angle-n, has present value equal to (a-double-dot-angle-n minus n times v to the n) divided by i. A decreasing annuity paying n, n minus 1, down to 1, written (D a)-angle-n, has present value (n minus a-angle-n) divided by i. Payments that grow by a fixed percentage g each period form a geometric annuity, and you value it by discounting at an adjusted rate that nets out the growth: roughly, discount at (1 + i) over (1 + g). The principle behind all of them is identical — present-value each payment and sum.

Varying annuities matter because real obligations rarely stay flat: pensions are often indexed to inflation, insurance benefits may step up over time, and rental incomes escalate. Modelling these correctly is essential for pricing and reserving; treating an inflation-linked pension as if it were level would dangerously understate the liability. When no closed formula fits the pattern, actuaries fall back on the always-reliable method of valuing each cash flow individually and adding the present values.