FNCE90065 Chap.4 Annuities, Perpetuities and Loans
Annuities, Perpetuities and Loans
Cash-flow stream formulas are shortcuts for specific timelines. This chapter distinguishes ordinary annuities, annuities due, deferred annuities, perpetuities, growing streams and fixed-rate amortising loans. It shows how the first payment date and stream horizon determine the correct factor, then uses loan amortisation to explain why interest falls and principal repayment rises over time.
The main defence against errors is drawing every payment before selecting a formula.
What this chapter covers
- 01
An ordinary annuity has finite equal payments beginning one period after valuation
- 02
An annuity due shifts every payment one period earlier
- 03
A perpetuity continues indefinitely but has finite present value under positive discounting
- 04
Growing perpetuity value requires required return to exceed long-run growth
- 05
A deferred stream is valued near its first payment and then discounted again
- 06
An amortising loan splits each fixed payment between interest and principal
Worked example · free
Value a finite annual grant
- 1Identify six level payments with the first at the end of year one.
- 1Use the ordinary-annuity present-value factor at 5 percent for six periods.
- 1Multiply the factor of approximately 5.0757 by $4,800.
- 1Present value is approximately $24,363, below the undiscounted total.
Key terms
- Ordinary annuity
- An ordinary annuity is a finite level stream whose first payment arrives one period after valuation.
- Annuity due
- An annuity due is a finite level stream with payments at the beginning of each period.
- Deferred annuity
- A deferred annuity is a finite level stream whose first payment begins after an initial waiting period.
- Growing perpetuity
- A growing perpetuity is an indefinite stream whose payments grow at a constant rate.
- Amortising loan
- An amortising loan is repaid through periodic payments that cover interest and progressively reduce principal.
Annuities, Perpetuities and Loans FAQ
How do I identify an annuity due?
Look for the first payment at time zero or the beginning of each period. An annuity due shifts every cash flow one period earlier than an ordinary annuity, so its present value equals the comparable ordinary-annuity value multiplied by one plus the periodic rate.
Why does a perpetuity have finite value?
Positive discounting makes distant payments contribute progressively less present value. The infinite sequence converges to payment divided by discount rate for a level perpetuity. The formula assumes the first payment is one period ahead and, for a growing perpetuity, required return exceeds growth.
Why does the interest share of a loan payment fall?
Interest each period is calculated on the opening loan balance. As principal is repaid, that balance declines, so interest falls and more of the same fixed payment reduces principal. Extra principal accelerates this process and can shorten the term and reduce total interest.
Exam move
Create a four-box classifier: finite or infinite, level or growing, first payment now or later, immediate or deferred. Put every stream question into those boxes before selecting a formula. For loans, build the first two rows of an amortisation schedule by hand: opening balance, interest, payment, principal repaid and closing balance.
Check that the final balance trends toward zero and that a higher rate raises the required payment. Use two methods for annuity-due questions, shifting the ordinary-annuity value and separately adding the immediate payment, to verify timing. Draw five timelines with identical payment amounts but different structures: ordinary annuity, annuity due, deferred annuity, level perpetuity and growing perpetuity.
Write the valuation date and first payment date beside each drawing. Then explain what changes in the formula without doing arithmetic. This isolates timing from number entry. For deferred streams, value the stream one period before its first payment and draw a second arrow that discounts that lump back to the required date.
For growing perpetuities, write the next payment explicitly and confirm required return exceeds growth before division. In loan practice, calculate payment from present value, then produce the first, middle and final schedule rows. Explain why the interest share falls and why an extra principal payment reduces later interest.
Compare a shorter and longer term at the same rate, distinguishing lower periodic burden from higher total interest. Keep a unit line above every formula with payment interval, periodic rate and total period count. Finish by taking a deliberately irregular cash-flow stream and stating why no annuity shortcut applies; discounting each cash flow separately is better than forcing a familiar factor onto the wrong pattern.
Test one refinancing offer by valuing the remaining old-loan payments at the new periodic rate and including any fee at time zero. State whether the comparison concerns payment relief, present-value saving or both, because a lower payment alone need not imply a cheaper loan.
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