25400 · Financial Literacy
Time Value of Money II: Annuities & Perpetuities
Week 2 extends the time value of money to streams of cash flows. You value an ordinary annuity (equal payments at the end of each period) with PV = C·[1 − (1 + i)^−n]/i and FV = C·[(1 + i)^n − 1]/i, solve those for the payment C, and price a perpetuity (PV = C/i), a growing perpetuity (PV = C/(i − g)) and a deferred perpetuity. These tools drive retirement, superannuation and mortgage problems and are core to Quiz 1; the annuity payment formula also reappears as the loan instalment in Week 4.
What this chapter covers
- 01Ordinary annuity PV = C·[1 − (1 + i)^−n] / i (payments at end of each period)
- 02Ordinary annuity FV = C·[(1 + i)^n − 1] / i (accumulation of a savings plan)
- 03Solving for the payment: C = PV·i / [1 − (1 + i)^−n] or C = FV·i / [(1 + i)^n − 1]
- 04Annuity due (payments at start): PV_due = PV_ordinary × (1 + i)
- 05Level perpetuity PV = C / i and growing perpetuity PV = C / (i − g), valid for i > g
- 06Deferred perpetuity: value at PV = (C / i) × (1 + i)^−t, discounting back over the deferral t
- 07Uneven cash flows: discount each C_t individually and sum, PV = Σ C_t / (1 + i)^t
- 08Payment frequency: i = APR/m and n = years × m; Excel PV, PMT, NPER
How much super is needed to fund a monthly pension (PV of an annuity)
- +1Convert to the monthly basis. Periodic rate i = APR/m = 6%/12 = 0.5% = 0.005; number of payments n = 25 × 12 = 300. Payments are at the end of each month, so this is an ordinary annuity.
- +1Choose the PV-of-annuity formula, because we want today's lump sum that funds a stream of equal future withdrawals: PV = C · [1 − (1 + i)^−n] / i.
- +1Value the annuity factor: [1 − 1.005^−300] / 0.005 = (1 − 0.22397) / 0.005 = 0.77603 / 0.005 = 155.207.
- +1Multiply by the payment: PV = 3,000 × 155.207 = $465,621. That is the minimum fund balance today that can sustain $3,000/month for 25 years at 6% p.a.
Key terms
- Ordinary annuity
- A series of equal payments made at the end of each period for a fixed number of periods. PV = C·[1 − (1 + i)^−n]/i and FV = C·[(1 + i)^n − 1]/i. Retirement income streams and level loan repayments are annuities.
- Annuity due
- An annuity whose payments fall at the start of each period rather than the end. Each cash flow is discounted one period less, so PV_due = PV_ordinary × (1 + i) and likewise for FV.
- Perpetuity
- An equal payment stream that continues forever. Its present value is PV = C/i for a level perpetuity, and PV = C/(i − g) for one growing at rate g (requires i > g). Preference shares are valued as perpetuities.
- Deferred perpetuity
- A perpetuity whose payments begin only after a waiting (deferral) period. Value the perpetuity one period before its first payment as C/i, then discount that back over the deferral: PV_0 = (C/i)(1 + i)^−t. A longer deferral lowers today's value.
- Annuity factor
- The bracketed term [1 − (1 + i)^−n]/i that converts a level payment into its present value. It recurs throughout the subject as the engine behind pensions, loan instalments and bond coupon strips.
- Uneven cash flows
- A stream where the amounts differ period to period. There is no shortcut: discount each cash flow at its own (1 + i)^t and sum, PV = Σ C_t/(1 + i)^t. The spreadsheet NPV function automates this.
Time Value of Money II: Annuities & Perpetuities FAQ
Ordinary annuity or annuity due — how do I tell?
Read when the payment occurs. End-of-period payments (most pensions, standard loans) are an ordinary annuity; start-of-period payments (many leases, rent paid in advance) are an annuity due, which is just the ordinary value multiplied by (1 + i). If the problem is silent, assume ordinary.
What is the difference between a perpetuity and a deferred perpetuity?
A level perpetuity pays forever starting next period and is worth C/i today. A deferred perpetuity waits a number of periods before the payments begin, so you first find C/i at the moment payments start, then discount that back over the deferral: (C/i)(1 + i)^−t. The longer the wait, the lower today's value.
Why does the growing perpetuity need i > g?
If the growth rate g equalled or exceeded the discount rate i, the cash flows would grow at least as fast as they are discounted and the sum would not converge — the formula C/(i − g) would give a negative or infinite value. The model is only meaningful when i > g.
How does this connect to loans and bonds?
The annuity PV formula is identical to the loan-principal relationship in Week 4 (the loan is the PV of its instalment annuity) and to the coupon strip of a bond in Week 5. Master the annuity factor here and those chapters become substitution exercises.
Assessment move
Make the annuity factor [1 − (1 + i)^−n]/i automatic — it is the most reused expression in 25400. For every stream, first classify it (single sum, ordinary annuity, annuity due, level perpetuity, growing perpetuity, deferred perpetuity, or fully uneven) because the classification chooses the formula. Drill the payment-solving rearrangements C = PV·i/[1 − (1 + i)^−n] and C = FV·i/[(1 + i)^n − 1], since they return as the loan instalment. Practise the frequency conversion i = APR/m, n = years × m every time. Learn the Excel PV, PMT and NPER functions with the sign convention for the group financial model. Rehearse under time pressure for Quiz 1, and ask Sia to generate fresh annuity and perpetuity problems and check each setup.
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