Classify and value level payment streams

Classify before choosing a factor

A bounded annuity has equal amounts, equal intervals, a finite payment count, and known timing. Check all four. The word “annual” proves only interval length; it does not prove equal amounts or first-payment date.

Both streams contain four payments. The due stream is shifted one period earlier; it does not add a payment at t = 4.

Reconstruct the ordinary-annuity factor

The present value of an ordinary annuity is the sum of discounted payments:

PMT/(1+i)^1 + PMT/(1+i)^2 + … + PMT/(1+i)^n

The compact factor equals that geometric sum. During initial learning, discount the first and last payments separately to keep the timing visible. A factor without a timeline is easy to misapply.

Future value accumulates each payment to t = n. The final ordinary payment is already at t = n and earns zero additional periods. Compounding it once more creates an off-by-one error.

Shift to an annuity due

Every due payment receives one additional period relative to its matching ordinary payment. Therefore, at the same valuation basis:

due value = ordinary value × (1+i)

At a positive rate, the due value is higher. At zero rate, timing does not change the mathematical total because no period carries growth or discount. The ordinary factors divide by i, so a calculation engine must handle the zero-rate limit as PMT × n.

Scope controls

Real leases, subscriptions, insurance, pensions, and service contracts may have escalations, residuals, options, partial periods, taxes, or framework- specific accounting. Do not call every recurring cash flow an annuity, and do not treat this foundation as lease or pension accounting.

Exit check

For three $1,000 payments at 5%, the full-precision ordinary present value is $2,723.24802937048 and the due present value is $2,859.4104308390038; the extended calculator precision is shown only to make the × 1.05 identity auditable, while the displayed currency values are $2,723.25 and $2,859.41.

1. Reconcile the unrounded ordinary value times 1.05 and draw both timelines.
2. Explain the timing relationship without saying the due stream contains more cash.