Varying Annuity Calculations

A retiree wants income that keeps pace with inflation. Pricing a payment that grows 3% a year for 20 years uses a different engine than a level annuity, and confusing the two misvalues the plan by six figures.

There are two ways a payment grows. Arithmetic adds a fixed amount $Q$ each period (100, 150, 200, and so on). Geometric multiplies by a fixed factor (100, 103, 106.09 at 3%). Inflation-linked income is geometric; a fixed dollar step-up is arithmetic. The formulas differ, so the first decision on any varying-annuity problem is which one you have.

The building blocks are the unit increasing and decreasing annuities, payments $1, 2, \ldots, n$ and $n, n-1, \ldots, 1$:

$$(Ia)_{\overline{n\vert}} = \frac{\ddot{a}_{\overline{n\vert}} - nv^n}{i}, \quad (Da)_{\overline{n\vert}} = \frac{n - a_{\overline{n\vert}}}{i}$$

They satisfy $(Ia)_{\overline{n\vert}} + (Da)_{\overline{n\vert}} = (n+1)a_{\overline{n\vert}}$, a quick check on your two factors.

For a general arithmetic stream $P, P+Q, \ldots, P+(n-1)Q$, split it into a level annuity of $P-Q$ plus $Q$ copies of the unit increasing annuity:

Common mistakes

• Using $P$ instead of $P-Q$ in the decomposition. Payments 1,000, 1,200,... with $P=1000$, $Q=200$: the level piece is $800\,a$, not $1000\,a$. Using 1,000 overcounts by $Q\,a = 200(6.4632) = 1{,}293$. Trap: $11,950 instead of $10,657.
• Using $a$ instead of $\ddot{a}$ in $(Ia)_n$. The numerator is $\ddot{a}_{\overline{n\vert}} - nv^n$, not $a_{\overline{n\vert}} - nv^n$. At $n=8$, $i=5\%$: correct $(Ia)=27.43$, wrong $=20.97$. Trap: 20.97.
• Confusing arithmetic and geometric growth. "Increasing by $200" is arithmetic; "increasing by 3%" is geometric. Applying the wrong formula gives a completely different present value.

Bottom line

• Arithmetic $P, P+Q, P+2Q, \ldots$: $PV = (P-Q)\,a_{\overline{n\vert}} + Q\,(Ia)_{\overline{n\vert}}$. The level coefficient is $P-Q$, not $P$.
• $(Ia)_{\overline{n\vert}} = (\ddot{a}_{\overline{n\vert}} - nv^n)/i$, $(Da)_{\overline{n\vert}} = (n - a_{\overline{n\vert}})/i$, and they sum to $(n+1)a_{\overline{n\vert}}$.
• Accumulated value: multiply the PV by $(1+i)^n$, or use $(Is)_{\overline{n\vert}} = (\ddot{s}_{\overline{n\vert}} - n)/i$.
• Increasing perpetuity: $(Ia)_{\overline{\infty\vert}} = 1/(id) = (1+i)/i^2$, due $1/d^2$. General stream $P, P+Q, \ldots$: $PV = P/i + Q/i^2$.

Exam shortcut

First decision: arithmetic (adds $Q$) or geometric (multiplies by $1+g$)? Arithmetic uses the level-plus-ramp decomposition; geometric uses the direct ratio formula. Write the type before you write any factor. "Level plus ramp" for arithmetic: $(P-Q)$ level plus $Q$ copies of $(Ia)$. "Ratio to the n, over the gap" for geometric: numerator $1$ minus the ratio raised to $n$, denominator $i-g$.

The full lesson (about 3,505 words, 23 min read) adds 5 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b

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