STAT2032 Chap.2 Annuity Valuation
Annuity Valuation
An annuity formula is a compressed cash-flow schedule, so timing comes before notation. This chapter separates payments in arrears from payments in advance, values a stream at its natural date, and then moves that value if the stream is deferred. It extends the same idea to perpetuities, increasing and decreasing payments, continuous payments and indexed streams.
The governing check is visual: identify the first and last payment dates, count payments rather than gaps, and decide whether each cash flow has moved one period earlier or later.
An immediate annuity first pays at the end of a period and its standard present value is dated immediately ahead of that first cash flow.
A due annuity shifts every payment one period earlier, so its value at the same focal date gains one accumulation factor. Deferral is a separate movement: first calculate the ordinary annuity value at its natural date, then discount that single value across the waiting interval.
Keeping these two movements separate prevents the common error of counting the first payment label as the deferral exponent.
Varying streams demand one more layer of care because both amount and date change. Expand the first few terms of an increasing, decreasing or indexed annuity before using a compact identity.
The expansion confirms which payment is largest, whether growth occurs before the first payment, and which discount power belongs to each amount. A perpetuity removes the final date but not the convergence condition. Continuous payments replace a sum by an integral of payment rate times discount factor.
Across every form, the formula should be treated as shorthand for a visible schedule that can be reconstructed term by term.
What this chapter covers
- 01
Annuity immediate timing
- 02
Annuity due timing
- 03
Deferred level payments
- 04
Perpetuity boundaries
- 05
Increasing and decreasing streams
- 06
Continuous and indexed annuities
Worked example · free
Compare payments in arrears and in advance
- 1Payments in arrears occur at times one through six, so use an annuity immediate.
- 2The immediate value is 1200[1 - (1.05)^(-6)]/0.05 = $6,090.83.
- 1Payments in advance occur one period earlier at times zero through five.
- 1Multiply by 1.05 to obtain $6,395.37; the earlier stream must be worth more.
Key terms
- Annuity immediate
- A payment stream with the first payment at the end of the first valuation period.
- Annuity due
- A payment stream with the first payment at the beginning of the first valuation period.
- Deferred annuity
- A payment stream whose first payment occurs after a waiting interval beyond the ordinary immediate timing.
- Perpetuity
- A payment stream that continues without a final payment date under a convergent discount model.
Annuity Valuation FAQ
Where is the natural valuation date for an annuity immediate?
For a level annuity immediate, the standard present value is dated immediately ahead of its first cash flow. If the first payment is at time one, that date is time zero. If the stream begins later, first place the annuity value at that preceding date and then discount that single value to the required focal date.
Why is an annuity due worth one accumulation factor more?
Relative to an annuity immediate with the same payments, every due payment occurs exactly one period earlier. At a common focal date, each payment is therefore discounted for one fewer period. Multiplying the immediate value by one plus the periodic interest rate applies that one-period shift to the entire stream.
How should I value an increasing annuity?
Write the actual payment pattern first, including whether the first payment is one unit or another amount. You can value each flow directly or use a standard increasing-annuity identity if its timing matches. The direct expansion is an excellent check because it preserves both the payment size and the date of every term.
What does the deferral exponent count?
It counts the periods between the required focal date and the natural valuation date of the ordinary annuity, not necessarily the label on the first payment. If the first payment is at time four, an immediate-annuity value sits at time three. Moving that value to time zero therefore uses three discount periods.
Exam move
Make timing diagrams the centre of revision. For each question, circle the first payment date, underline the number of payments and place the standard annuity value immediately ahead of the first cash flow. Practise converting the same six-payment stream among immediate, due and deferred forms so that each algebraic factor corresponds to a visible shift.
Derive the perpetuity value from the infinite geometric series rather than treating it as a disconnected formula. For varying payments, expand the first three and last terms before using a compressed identity; this exposes whether the sequence increases from one or from another base. Keep payment growth and interest discounting separate in indexed annuities.
After solving, compare values: earlier payments are worth more, a longer positive stream is worth more, and a higher discount rate lowers present value.
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