FINM7006 Chap.4 Annuities and Perpetuities
Annuities and Perpetuities
Annuities are finite equal payments, while perpetuities continue without a terminal date. The formulas become reliable when their timing boundary is explicit: an ordinary stream is valued one period before its first payment. This chapter uses that boundary to handle ordinary, due and deferred cases without memorising disconnected equations.
Loan payments, scholarship funds, leases and level preference dividends all become variations on the same timing decision.
What this chapter covers
- 01
Ordinary annuity timing
- 02
Annuity-due left shifts
- 03
Deferred annuity boundary values
- 04
Ordinary and due perpetuities
- 05
Level-payment loan amortisation
- 06
Timing and direction controls
Worked example · free
Solve a level annual loan payment
- +1Classify: the five end-of-year payments form an ordinary annuity.
- +1Set up: $30,000 = C[1 - (1.07)-5]/0.07.
- +2Solve: C = $30,000(0.07)/[1 - (1.07)-5] = $7,316.72.
- +1Check: payment exceeds first-year interest because it must also reduce principal.
Key terms
- Ordinary annuity
- An ordinary annuity pays equal amounts at the end of each period.
- Annuity due
- An annuity due pays equal amounts at the beginning of each period.
- Deferred annuity
- A deferred annuity postpones the first payment beyond the ordinary first-payment date.
- Perpetuity due
- A perpetuity due begins with a payment at the valuation date and continues indefinitely.
Annuities and Perpetuities FAQ
What is the simplest way to distinguish ordinary and due payments?
Locate the first payment relative to the valuation date. If it arrives one period later, the stream is ordinary. If it arrives immediately, it is due. Do not classify from the product name alone; rent, loan and scholarship wording can describe different timing depending on the exact dates supplied.
Why is a deferred perpetuity valued in two stages?
The perpetuity formula creates a boundary value one period before the first endless payment. When that boundary lies in the future, the boundary value must then be discounted to today. Discounting from the first payment date instead would add one period too many because the formula already bridges back one period.
How can I check a level loan payment?
Calculate first-period interest from the opening balance. The payment should cover that interest and reduce principal for a standard amortising loan. Subtract principal reduction to obtain the next balance, then compare that balance with the present value of remaining payments. Both routes should agree apart from rounding.
Exam move
Create three payment strips with the same amount and count but different first-payment dates. Value the ordinary strip, then shift it left for due and right for deferred timing. Repeat the boundary method for perpetuities. For loans, split the first payment into interest and principal and verify the remaining balance using the annuity formula.
Always state whether the requested value is before or after a payment, because that single word changes inclusion. Make the first-payment date the centre of every solution. Mark payment one, place the ordinary formula one period earlier, and label that point as the boundary value. Only after computing there should you move the completed value to the requested date.
This single habit handles deferred annuities and perpetuities without a collection of unrelated formulas. Use identical amounts and counts to compare ordinary and due streams. Shift all dates left by one period and verify that the due value equals the ordinary value multiplied by one plus the periodic rate.
Count payments separately from elapsed time: four end-of-period payments occur at times one through four, while four beginning-of-period payments occupy times zero through three. For perpetuities, state the assumptions explicitly: level payments, regular spacing, no terminal date and a positive required return.
Practise a due perpetuity by separating the immediate payment from the ordinary perpetuity that begins one period later. For a deferred perpetuity, compute the value one period before its first payment and then discount that boundary value to today. Loan questions need a second representation. Solve the annuity for the level payment, then create the first two rows of an amortisation schedule.
Interest equals the opening balance times the periodic rate; the remainder of the payment reduces principal. Compare the new balance with the present value of the remaining payments. If the two routes disagree beyond rounding, revisit timing or rate conversion. Add exercises with one exceptional cash flow so you practise decomposition: value the level strip with the shortcut and the extra amount separately.
In final preparation, classify each question aloud as finite or infinite, level or uneven, ordinary or due, immediate or deferred. Do not enter numbers until all four labels are settled. Close every response with a direction check covering earlier payment, higher rate, more payments or longer deferral.
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