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Increasing annuities

Payments 1, 2, 3, …, n. The closed form (äₙ − n vⁿ)/i turns an arithmetic series into two annuity factors you already know.

The formulas

Present value

$$(Ia{)}_{\overline{n}|}=\frac{{\ddot{a}}_{\overline{n}|}-n{v}^n}{i}$$

Accumulated value

$$(Is{)}_{\overline{n}|}=\frac{{\ddot{s}}_{\overline{n}|}-n}{i}$$

Increasing perpetuity

$$(Ia{)}_{\overline{\infty }|}=\frac{1}{i}+\frac{1}{i^2}$$

General arithmetic

$$PV=P{a}_{\overline{n}|}+Q\frac{a_{\overline{n}|}-n{v}^n}{i}$$

payments P, P+Q, P+2Q, …

Where it comes from

1. Write the payment $k$ at time $k$ as a sum of $k$ unit payments, one starting at each of times $1,2,\dots ,k$.
2. Regrouping turns the double sum into ${\sum }_{j=1}^n{v}^{j-1}{a}_{\overline{n-j+1}|}$, which telescopes to $\frac{{\ddot{a}}_{\overline{n}|}-n{v}^n}{i}$.
3. The general arithmetic form is a level annuity of $P$ plus $Q$ times a unit increasing annuity that starts at 0.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• (Ia)ₙ uses ä on top, not a. The due factor is the signature of the increasing annuity.
• (Ia)∞ = 1/i + 1/i² — one perpetuity for the level part, one more for the growth.

Traps

• Using aₙ instead of äₙ in the numerator.
• Applying (Ia) to payments that grow by a PERCENTAGE — that is a geometric annuity, not an arithmetic one.

Related

• Decreasing annuities
• Geometrically increasing annuities
• Perpetuities

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