BFB1001 Chap.3 Annuities and Perpetuities
Annuities and Perpetuities
Annuities and Perpetuities values repeated cash flows by classifying their pattern before using a factor. An ordinary annuity is finite and begins one period after valuation, an annuity due begins immediately, a deferred annuity starts after an additional delay, and a perpetuity continues indefinitely.
Equal amounts alone do not establish the model: payments must be regularly spaced, and the first and final dates determine where the standard factor places value. The chapter shows how an annuity factor compresses a sum of single-payment present values, why an annuity due is worth more than the corresponding ordinary stream, and how irregular cash flows can be split into valid pieces.
Payment count, rate period and valuation date must remain aligned throughout.
What this chapter covers
- 01
Ordinary annuities
- 02
Annuities due
- 03
Deferred payment streams
- 04
Perpetuity valuation
- 05
Payment and period counts
- 06
Splitting irregular streams
Worked example · free
Value a finite equal-payment stream
- 1Classify the stream as an ordinary annuity beginning at time one.
- 1Use four payments and a 6% annual rate.
- 2Apply the ordinary-annuity present-value factor.
- 2Calculate $10,395.32 and explain why it is below the $12,000 nominal total.
Key terms
- Ordinary annuity
- Equal regular payments for a finite period beginning one period after valuation.
- Annuity due
- Equal regular payments for a finite period beginning at the valuation date.
- Deferred annuity
- A finite equal-payment stream whose first payment begins after an additional delay.
- Perpetuity
- An equal regular payment stream continuing indefinitely.
Annuities and Perpetuities FAQ
Why is an annuity due worth more?
Every payment arrives one period earlier, so each is discounted less and can be used sooner. Multiplying the otherwise identical ordinary-annuity value by one plus the periodic rate captures that one-period shift.
Can unequal payments use an annuity formula?
Not as one stream. Discount each amount separately or divide the cash flows into pieces that each match a valid model, such as a level annuity plus a separate terminal amount. Averaging unequal payments destroys their timing information.
Where does the perpetuity formula value the stream?
One period before its first payment, so a deferred perpetuity must be discounted further back to today. The fact that payments continue forever does not remove the need to locate the first payment accurately.
How do I count periods in an annuity?
Count actual payments rather than the number of date labels on the timeline. A stream paying at times one through five has five payments even though the line also shows time zero.
What rate belongs in an annuity formula?
Use the return per payment interval. Monthly payments need a monthly rate and number of months; annual payments need an annual rate and years. Both inputs must be converted together.
Why is a perpetuity so sensitive to its rate?
Value equals periodic cash divided by required return, so a small denominator produces a large value. Enter a percentage as a decimal and remember that lower required return raises the value of the entire infinite stream.
Can an annuity contain a payment at time zero?
Yes, but that timing makes it an annuity due rather than an ordinary annuity. One approach is to value the matching ordinary stream and multiply by one plus the periodic rate. Another is to value the immediate payment separately and the remaining payments from their dates.
How do I value a delayed annuity?
First use the ordinary-annuity formula at the date one period before the first payment. Then discount that amount through the additional waiting periods to time zero. Marking the formula's valuation point on the timeline prevents one extra or missing discount period.
Why not add all annuity payments and discount once?
Each payment arrives at a different date, so it faces a different number of discount periods. Discounting the total as though every dollar arrived together destroys that timing. The annuity factor works because it adds the correctly dated present values in a compact form.
Assessment move
Write four labels for every stream: payment amount, interval, first payment and final payment. Use them to classify ordinary, due, deferred or perpetuity before touching a formula. Draw time zero even when the question does not show it, circle each actual payment and identify the date one period before the first cash flow, because that is where standard present-value formulas usually locate the result.
Verify the periodic rate matches payment spacing. For irregular streams, separate the level portion from one-off amounts instead of forcing an average payment. Build comparisons by valuing the same cash flow as ordinary and due, and explain why the earlier stream is worth more. Check finite-stream present value against its undiscounted total and test perpetuity answers for percentage-entry errors.
End every alternative-choice question by comparing amounts at the same date and naming the required-return assumption that controls the decision.
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