For four annual $6,000 end-of-year payments and a 5% annual periodic rate, the
present value at t = 0 is first the sum of four individually discounted
amounts:
$6,000/1.05 + $6,000/(1.05)^2 + $6,000/(1.05)^3 + $6,000/(1.05)^4
= $21,275.7030…
The ordinary-annuity factor is a compact algebraic equivalent of that same four-term sum, not a new valuation rule:
$6,000 × [1 - (1.05)^(-4)] / 0.05 = $21,275.70…
The first payment is discounted one period and the fourth four periods. A timeline should make that pattern visible before the factor is used.
For future value at t = 4, the first payment compounds three periods, while
the last payment is already at the valuation date and compounds zero periods.
Giving the last payment another period shifts the valuation date and overstates
the result. The compact future-value factor must reproduce the same four dated
cash flows.
At a zero periodic rate, the present and future values both equal payment times
payment count. The usual factor produces 0/0 at zero, so a calculation engine
must handle the limiting case explicitly rather than report an error or invent
a rate.
The value is conditional on the payments and supplied rate. It is not a product price, recommendation, or accounting recognition amount without additional authority and facts. If every payment occurs one period earlier, use the annuity-due timeline and reconcile its value to this stream.
Put the concept to work
Understand this concept
- Explain why an ordinary annuity's first payment is one period after the valuation date and how that timing shapes its present- and future-value factors.
Apply this concept
- Compute present and future values of a bounded ordinary annuity, verify payment count and timing, and retain full precision until the final display.
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Choose a lesson, try an application, or inspect the sources behind this concept.
Build on these ideas
- Annuity — Apply
To apply this concept: Required. The payment stream must be classified and mapped before valuation.
- Annuity — Understand
To understand this concept: Required. The stream must first satisfy the general annuity conditions.
- Discounting — Apply
To apply this concept: Helpful. Present value is the sum of discounted end-of-period payments.
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Show 2 more related concepts
Use this idea next
- Annuity due — Apply
Required level here: apply. Required. The due identity reuses the matching ordinary-annuity value.
- Annuity due — Understand
Required level here: understand. Required. The one-period shift is defined relative to ordinary-annuity timing.
- Level payment — Apply
Required level here: apply. Required. Payment computation reuses the ordinary-annuity factor.
Show 1 more next steps
- Level payment — Understand
Required level here: understand. Required. The payment equation is the ordinary-annuity present-value relation solved for payment.