Annuities

An annuity is a sequence of equal payments at regular intervals whose value depends on compounding and payment timing.

Cash-flow timing determines how financial quantities accumulate. Dated value markers sit above a timeline and grow across successive comparison dates. The caption beneath the axis requires every cash flow to be moved to one common date before totals are compared.

Annuity formulas add many deposits or payments after moving each one to the same point in time.

What an annuity is

An annuity consists of equal payments made at regular intervals. Examples include monthly savings deposits, loan payments, pensions, and some leases.

The payment interval and compounding period must be aligned or converted carefully.

Worked example: savings plan

Total deposits are $200(60)=\$12\,000$; the difference is interest earned.

Finding a payment

Rearrange the appropriate formula. To reach a target future amount,

R=FV\frac{i}{(1+i)^n-1}.

For a loan,

R=PV\frac{i}{1-(1+i)^{-n}}.

Keep full precision until the final currency rounding.

Timeline method

Draw a timeline showing payment dates, present date, future date, and compounding periods. A timeline prevents beginning/end timing errors and clarifies whether an extra payment occurs at time $0$.

Count intervals, not only calendar labels.

Common mistakes

Using the annual rate as $i$ for monthly payments. Convert to the period rate.

Using years as $n$. Count payments.

Confusing beginning and end payments. Identify ordinary versus due.

Treating total deposits as account value. Interest changes the total.

Rounding the periodic rate early. It affects many compounded periods.

Quick self-check

Are payments equal and equally spaced?
Do they occur at the beginning or end of periods?
Do $i$ and $n$ use the same period as payments?
Is the question asking for present value, future value, payment, or time?
Does a timeline confirm the first and last payment dates?
Are fees, rate changes, and currency rounding distinguished from the ideal model?