ECON1202 Chap.3 Geometric Progressions and Annuities
Geometric Progressions and Annuities
An annuity formula is a compact geometric sum, so recognition comes before substitution. Equal payments must occur at equal intervals for a finite number of periods. An ordinary annuity pays at each period end; an annuity due pays at each period start. Shifting the entire ordinary stream one interval earlier multiplies its value by one plus the periodic rate.
Present value locates the stream before the ordinary payments begin, while future value locates it at the last payment date. Loan payments reverse the present-value formula: the principal today equals the discounted end-of-period instalments. A perpetuity has no final payment and uses a separate infinite-stream result. Mixed problems often contain an opening lump sum plus regular deposits or withdrawals.
Value each component with its own model, then combine only at their shared date. The timeline should display the first payment because it distinguishes due from ordinary immediately. Contribution totals, payment counts and positive-rate direction provide useful scale checks after the exact calculation.
What this chapter covers
- 01
Finite geometric progressions
- 02
Ordinary annuity present value
- 03
Ordinary annuity future value
- 04
Annuity due timing shift
- 05
Loan repayment formula
- 06
Perpetuity present value
- 07
Mixed lump-sum and annuity streams
- 08
Payment-timing checks
Worked example · free
Value an annuity due at its final payment date
- +1Use r = 0.04/4 = 0.01 per quarter and n = 12 deposits.
- +1First calculate the corresponding ordinary-annuity future value: 250[(1.01)^12 - 1]/0.01.
- +1Shift every payment one quarter earlier by multiplying the ordinary value by 1.01.
- +1The annuity-due value is approximately $3,202.33.
Key terms
- Geometric progression
- A sequence whose consecutive terms share a constant ratio.
- Ordinary annuity
- A finite stream of equal end-of-period payments.
- Annuity due
- A finite stream of equal start-of-period payments.
- Loan instalment
- The equal payment that gives a loan principal its annuity present value.
- Perpetuity
- An equal end-of-period payment stream with no final payment.
- Payment interval
- The time between successive cash flows in a regular stream.
Geometric Progressions and Annuities FAQ
How do I distinguish an ordinary annuity from an annuity due?
Look at the first payment relative to the valuation date. An ordinary annuity begins one period later, while an annuity due begins immediately. Draw the first two dates instead of relying on words such as monthly or regular.
Why is an annuity due multiplied by one plus the rate?
Every due payment occurs one interval earlier than the corresponding ordinary payment. At the same later focal date, each therefore receives one additional growth period, so the entire ordinary value is multiplied by the one-period factor.
When can a payment stream use an annuity formula?
Payments must be equal and equally spaced, with a fixed finite count for an annuity. Unequal amounts need separate dated terms or decomposition into simpler streams. A perpetual stream uses the separate perpetuity result.
How should a mixed retirement account be valued?
Treat the opening balance as one cash flow and the regular contributions as an annuity. Move both to the same retirement date, add them there, and handle any extra isolated contribution separately.
What makes an annuity solution complete rather than formula-only?
Show where the first and last equal payments fall relative to the valuation date. State whether the stream is ordinary or due and whether a present or accumulated value is required. Translate the interest quote to the payment period, then write the selected factor with its payment count. If the stream is due, explain the one-period shift rather than presenting an unexplained extra factor.
In a loan problem, equate principal to the present value of instalments and distinguish the level contractual payment from its changing interest and principal portions. For a mixed stream, value lump sums and repeated payments separately at one date. Check that the result lies on the correct side of the undiscounted payment total for the direction used.
Exam move
Start every stream problem with a timeline showing time zero, the first payment and the requested focal date. Label ORDINARY when the first payment is one interval later and DUE when it is immediate. Practise converting nominal rates to the payment interval and count payments separately from interest gaps. Derive the due adjustment once from a shifted timeline so it is understood rather than memorised.
For loans, write the principal as the present value of instalments and solve algebraically for the payment. For retirement and saving questions, decompose the opening balance, regular contributions and any extra cash flow, then combine only at the shared date. Use two scale checks: a positive-return saving balance should exceed total contributions, and total loan instalments should exceed principal under a positive rate.
Label whether the answer is balance, payment or interest.
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