Ordinary Annuities: Future Value of Regular Deposits

What is an annuity?

An annuity is a sequence of equal payments at equal time intervals. In an ordinary annuity, payments occur at the end of each period — the typical model for most loans, pensions, and systematic savings plans.

The future-value formula

$FV = PMT \cdot \dfrac{(1 + r)^n - 1}{r}$

where

• $PMT$ = payment per period,
• $r$ = interest rate per period,
• $n$ = total number of payments,
• $FV$ = future value (balance right after the last payment).

The fraction $\dfrac{(1+r)^n - 1}{r}$ is the future-value annuity factor — it grows faster than $n$ because each deposit earns compound interest until the end.

Step-by-step solution

Setup: $PMT = 200/\text{month}$, annual rate 6% ⟹ monthly rate $r = 0.06/12 = 0.005$. 20 years ⟹ $n = 240$ payments.

Step 1 — Compute the growth factor. $(1 + 0.005)^{240} = (1.005)^{240} \approx 3.31020$

Step 2 — Subtract 1 and divide by $r$: $\dfrac{3.31020 - 1}{0.005} = \dfrac{2.31020}{0.005} = 462.04$

Step 3 — Multiply by the payment: $FV = 200 \cdot 462.04 \approx \boxed{92{,}408}$

Step 4 — What you actually deposited vs. what you earned:

• Total deposits: $240 \cdot 200 = 48{,}000$
• Interest earned: $92{,}408 - 48{,}000 \approx 44{,}408$

Nearly half of the final balance is interest — the power of 20 years of compounding.

Why the factor works

Each deposit at the end of month $k$ earns interest for $n - k$ remaining months. The total is a geometric series: $FV = PMT \sum_{k=1}^{n} (1+r)^{n-k} = PMT \cdot \dfrac{(1+r)^n - 1}{r}$

which closes using the geometric-sum formula.

Useful cross-calculations

Given any three of $PMT, r, n, FV$, solve for the fourth:

• $PMT = \dfrac{FV \cdot r}{(1+r)^n - 1}$ — how much to contribute to hit a target.
• $n = \dfrac{\ln(1 + FV \cdot r / PMT)}{\ln(1 + r)}$ — how many periods you need.

Solving for $r$ generally requires numerical methods.

Common mistakes

• Using the annual rate as $r$ when payments are monthly. The per-period rate and period count must match the payment frequency.
• Confusing ordinary annuities with annuities due. Annuity due (payments at the start of each period) multiplies the FV by an extra $(1 + r)$.
• Forgetting that the last payment earns no interest in an ordinary annuity — the FV is measured the moment that final deposit lands.

Try it in the visualization

A stacked bar chart adds each month's deposit (constant height) plus the interest it earns going forward (growing triangle). Over 240 months you see the principal stack grow linearly while the interest stack grows curved.