Financial Management · Investment appraisal techniques

Annuity and Perpetuity Factors in DCF for ACCA FM

Updated 11 October 2026 · Fact-checked

An annuity is a level cash flow received for a fixed number of years. A perpetuity is a level cash flow that lasts forever. Multiply the cash flow by an annuity factor or by 1 ÷ r to get present value in one step, instead of discounting each year separately.

Understand Discounted Cash Flow Annuities and Perpetuities

Discounting means converting future cash into today's value. For one cash flow you multiply by 1 ÷ (1 + r)ⁿ. When the same amount arrives every year, doing this year by year is slow and error-prone.

An annuity is a constant cash flow at the end of each year for n years. The annuity factor is simply the sum of the discount factors for those years. You multiply the cash flow by that one number. The tables ACCA gives you list annuity factors, so you often do not need to calculate them.

A perpetuity is a constant cash flow that continues for ever. Its present value is the cash flow divided by the discount rate. It works because the discount factors for later years get smaller and smaller, and their total settles at 1 ÷ r.

A deferred annuity starts later than year 1. You find the annuity factor for the full period, subtract the annuity factor for the years before the start, and use the difference. A growing perpetuity has cash flows that rise at a constant rate g. Its value is the first cash flow divided by (r − g), provided r is greater than g.

All these shortcuts assume cash flows arise at the end of each year. If the first payment is made now (time 0), you must treat it separately.

Key rules to remember

Present value of a single cash flow: PV = CF × 1 ÷ (1 + r)ⁿ; Discount factor for year n at rate r. Use tables where possible.
Annuity factor: AF = [1 − (1 + r)⁻ⁿ] ÷ r; For n years of equal end-year cash flows. Tables give the same value.
PV of an annuity: PV = annual cash flow × annuity factor (n years, r%); First cash flow is at the end of year 1.
PV of a perpetuity: PV = cash flow ÷ r; First cash flow is one year from the valuation date.
Deferred annuity: PV = cash flow × (AF to final year − AF to the year before the start); Example: cash flows in years 4 to 10 use AF(10) − AF(3).
Deferred perpetuity: PV = (cash flow ÷ r) × discount factor for the year before the first cash flow; If the first flow is in year 5, discount the perpetuity value back 4 years.
Growing perpetuity: PV = first cash flow ÷ (r − g); Needs r > g. The first cash flow is the one arising at the end of year 1.
Annuity due (first payment at time 0): PV = cash flow × (1 + AF for n − 1 years); The time 0 payment has a factor of 1.0.

How to solve Discounted Cash Flow Annuities and Perpetuities questions

Use this method for any question with regular cash flows. Draw a timeline first. It shows when the flows start and end.

1Write down the cash flow, the discount rate and the years in which the flows occur.
2Decide the pattern: fixed term (annuity), never-ending (perpetuity), starting later (deferred) or growing.
3Check the timing. The standard formulas assume the first flow is one year after the point you are valuing from.
4Choose the factor. For annuities, read the table or use the formula. For perpetuities, use 1 ÷ r, or 1 ÷ (r − g) if growing.
5For a deferred annuity, subtract the annuity factor for the years before the start from the annuity factor to the final year.
6For a deferred perpetuity, calculate the perpetuity value, then discount it back to today using the discount factor of the year before the first flow.
7Multiply the cash flow by the factor. Add any other flows, such as initial investment at time 0, to get NPV.
8Check the answer is sensible: the present value must be less than the total of the undiscounted cash flows.

Quickest way: Table subtraction and one-line perpetuity

When to use it: Use in Section A and OT case questions where you have a few minutes and tables are provided.

Mark the first and last year of the flows on a quick timeline.
For an annuity, take the table factor for the last year. If the flows start after year 1, subtract the factor for the year before the first flow.
For a perpetuity, divide by r. For a growing one, divide the year 1 flow by (r − g).
For a deferred perpetuity, divide by r, then multiply by the single discount factor for the year before the first flow.
Scan the options. If two answers differ by one year of timing, recheck your start year.

Common mistakes in Discounted Cash Flow Annuities and Perpetuities

Using the wrong annuity factor for a deferred annuity, such as using AF(7) for flows in years 4 to 10.
Students count the number of payments and forget the delay.
Fix: Use AF(10) − AF(3). The subtracted year is always the year before the first flow.
Discounting a deferred perpetuity by the wrong number of years.
Perpetuity value is found one year before the first flow, so students discount by the year of the first flow instead.
Fix: If the first flow is in year n, discount the perpetuity value back n − 1 years.
Using the year 0 flow in a growing perpetuity formula.
The formula needs the first future flow, not the latest one.
Fix: If the latest flow is C₀, the first flow is C₀ × (1 + g). Then divide by (r − g).
Applying the annuity factor to a payment made now.
Tables assume year-end flows, but some payments start at time 0.
Fix: Take the time 0 payment at face value and add an annuity for the remaining years.
Using the cost of capital as a percentage in the formula, such as dividing by 8 instead of 0.08.
Rushing and mixing percentage and decimal forms.
Fix: Convert r to a decimal before dividing. Check the answer is large for a perpetuity.
Using an annuity factor when cash flows change each year.
Students apply shortcuts to every pattern.
Fix: Only use factors for flows that are equal. Split the timeline into level blocks.

Worked examples

Example 1

A project costs $120,000 now. It produces net cash inflows of $30,000 a year in years 1 to 6, then $20,000 a year in years 7 to 10. The cost of capital is 10%. Annuity factors at 10%: years 1–6 = 4.355, years 1–10 = 6.145. Calculate the NPV.

Show the solution

Years 1 to 6: 30,000 × 4.355 = 130,650.
Years 7 to 10: factor = 6.145 − 4.355 = 1.790.
20,000 × 1.790 = 35,800.
Total PV of inflows = 130,650 + 35,800 = 166,450.
NPV = 166,450 − 120,000 = 46,450.

Answer: NPV = $46,450 positive, so accept the project on this measure.

Example 2

A company expects to receive $50,000 a year for ever, with the first receipt in year 4. The cost of capital is 8%. The discount factor at 8% for year 3 is 0.794. Calculate the present value. Then find the value if instead the first receipt of $50,000 is in year 1 and receipts grow at 3% a year.

Show the solution

Perpetuity value one year before the first receipt: 50,000 ÷ 0.08 = 625,000.
This value is at the end of year 3. Discount back 3 years: 625,000 × 0.794 = 496,250.
Growing case: first cash flow is 50,000 in year 1, r = 8%, g = 3%.
PV = 50,000 ÷ (0.08 − 0.03) = 50,000 ÷ 0.05 = 1,000,000.

Answer: Deferred perpetuity PV = $496,250. Growing perpetuity PV = $1,000,000.

Exam tips

Read when the first cash flow occurs before choosing any factor. Many wrong options in OT questions are built on timing errors.
Learn the subtraction rule for deferred annuities: last year minus the year before the start.
In perpetuity questions, check whether the given flow is the next one or the one just received. Growth formulas need the next one.
In Section C, show the factor you used and the years it covers. This earns method marks even if you misread a table.
Use the tables provided. If the rate is not in them, use the formula and show your working.

Practice questions from Investment appraisal techniques

Discounted Cash Flow Annuities and Perpetuities in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Discounted Cash Flow Annuities and Perpetuities: frequently asked questions

What is the difference between an annuity factor and a discount factor?

A discount factor values one cash flow in one year. An annuity factor is the total of the discount factors over several years. Use it when the same cash flow occurs every year.

How do you calculate a deferred annuity present value?

Take the annuity factor for the final year and subtract the annuity factor for the year before the flows start. Multiply the result by the annual cash flow. For flows in years 4 to 10, use AF(10) − AF(3).

When can I use the growing perpetuity formula?

Use it when cash flows grow at a constant rate for ever and the discount rate is higher than the growth rate. The value is the next year's cash flow divided by (r − g).

Does the perpetuity formula work if the first payment is now?

Not directly. The formula C ÷ r values flows starting one year ahead. If a payment is made now, add it separately at face value, and then add C ÷ r for the later payments.