Would you rather have ₦1 million today or ₦1 million in five years? Everyone says today, and the reason is not only that the future is uncertain or that prices rise. Money today can be invested and earn a return, so ₦1 million today becomes more than ₦1 million in five years.
This is the time value of money. It is the foundation of every discounted cash flow method: before cash flows from different years can be added up, they must be converted to the same date.
This topic explains compounding and discounting, how to calculate discount factors and use the tables, annuities and the annuity factor, deferred annuities and perpetuities, the growing annuity formula given in the exam, and the effect of compounding more than once a year, with worked calculations. The next topic covers net present value and the case for discounted cash flow.
The paper gives annuity tables with a formula sheet, so the skill is choosing and applying the right factor. Read relevant cash flows in investment appraisal first.
In 30 seconds
Money has a time value: a sum today can be invested to earn a return, so it is worth more than the same sum later. Compounding finds the future value: amount × (1 + r)n. Discounting finds the present value: amount × discount factor, where the discount factor = 1 ÷ (1 + r)n; it is the reverse of compounding.
An annuity is a constant cash flow for a number of periods; its present value = the amount × the annuity factor = [1 − (1 + r)−n] ÷ r, which is the sum of the individual discount factors. A perpetuity is worth the cash flow ÷ r. Cash flows are assumed to arise at the end of each year, and a higher rate gives a lower present value.
Compounding and discounting
The time value of money is the idea that a sum received today is worth more than the same sum later, because it can be invested to earn a return. Two techniques apply it:
Compounding
Compounding finds the future value of an amount invested today. Formula: future value = amount × (1 + r)n, where r is the rate and n is the number of years.
Example. ₦500,000 is invested for 4 years at 9%.
- Work out the growth factor. The money grows by 9% a year, so multiply by 1.09 four times.
1.094 = 1.411582 - Multiply.
500,000 × 1.411582 = ₦705,791
Discounting
Discounting finds the present value (PV) of a future cash flow. Formula: present value = amount × discount factor, where discount factor = 1 ÷ (1 + r)n. It is the reverse of compounding: multiplying by a discount factor is the same as dividing by the compound factor.
Example. ₦800,000 is received in 3 years. The rate is 12%.
- Work out the compound factor.
1.123 = 1.404928 - Divide. Dividing by the compound factor takes the money back to today.
800,000 ÷ 1.404928 = ₦569,424
The present value of a future amount is what an investor would need to invest today to receive that amount later. A higher discount rate makes the present value smaller.
Example. The same ₦800,000 in 3 years, discounted at 15%.
- Work out the compound factor.
1.153 = 1.520875 - Divide. This is lower than the ₦569,424 found at 12%.
800,000 ÷ 1.520875 = ₦526,013
Discount tables
Discount factors can be calculated, or read from a table that lists them by rate and year.
Example. At 10% the discount factors are 0.909 (year 1), 0.826 (year 2), 0.751 (year 3) and 0.683 (year 4). A project has cash flows of ₦200,000, ₦300,000, ₦300,000 and ₦250,000 in years 1 to 4.
| Year | Cash flow ₦ | Discount factor at 10% | Present value ₦ |
|---|---|---|---|
| 1 | 200,000 | 0.909 | 181,800 |
| 2 | 300,000 | 0.826 | 247,800 |
| 3 | 300,000 | 0.751 | 225,300 |
| 4 | 250,000 | 0.683 | 170,750 |
| Total | 825,650 |
- Pick the factor for each year. Read it from the table at 10%.
- Multiply each cash flow by its factor. This gives the present value of each year’s cash flow.
- Add the present values.
181,800 + 247,800 + 225,300 + 170,750 = ₦825,650
Annuities
An annuity is a constant cash flow for a given number of periods, for example ₦30,000 each year for years 1 to 5. The present value could be found by discounting each amount, but for many years it is quicker to use the annuity factor, the sum of the individual discount factors for those years:
Annuity factor = [1 − (1 + r)−n] ÷ r
Example. Find the annuity factor for 4 years at 10%.
- Add the discount factors.
0.909 + 0.826 + 0.751 + 0.683 = 3.170 - Check with the formula. The formula is a shortcut for the same sum.
[1 − 1.1−4] ÷ 0.10 = 0.316987 ÷ 0.10 = 3.170 ✓
Example. ₦150,000 a year for 6 years at 8%. Find the present value.
- Find the annuity factor. Read it from the table, or use the formula.
4.62288 (the table shows 4.623) - Multiply. Cash flow times annuity factor.
150,000 × 4.62288 = ₦693,432
Deferred annuities
An annuity that starts later. Subtract the factors.
Example. ₦100,000 a year for years 3 to 7, discounted at 10%.
- Take the factor for 7 years. This values payments in years 1 to 7.
4.86842 - Take off the factor for 2 years. This removes years 1 and 2, when nothing is paid.
1.73554 - Subtract.
4.86842 − 1.73554 = 3.13288 - Multiply. Rounded table factors (4.868 − 1.736 = 3.132) would give ₦313,200.
100,000 × 3.13288 = ₦313,288
Perpetuities and growth
A perpetuity is a constant cash flow that continues for ever. Its present value is the cash flow divided by the rate, because the annuity factor tends to 1 ÷ r as n becomes large.
Example. ₦90,000 a year for ever at 15%.
- Divide the cash flow by the rate.
90,000 ÷ 0.15 = ₦600,000
For a cash flow that grows at a constant rate g for n years, the exam provides the growing annuity formula:
PV = A ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))n], where A is the first payment, received at the end of year 1.
Example. A first payment of ₦100,000 grows by 5% a year for 5 years. The discount rate is 12%.
- Work out A ÷ (r − g). Here r − g = 12% − 5% = 7%.
100,000 ÷ 0.07 = 1,428,571 - Work out the growth ratio. Divide the growth factor by the discount factor, then raise it to the power n.
(1.05 ÷ 1.12)5 = 0.93755 = 0.72420 - Subtract from 1.
1 − 0.72420 = 0.27580 - Multiply. Rounded figures give 394,000. The unrounded figures give the exact answer.
1,428,571 × 0.27580 = ₦394,005
Compounding within the year
If interest is added more often than once a year, the effective annual rate is higher than the nominal rate.
Example. Interest is 1.5% a month.
- Compound the monthly rate for 12 months.
1.01512 = 1.1956 - Take off 1.
1.1956 − 1 = 19.56% a year, not 18%
In appraisal, cash flows are usually annual and the cost of capital is an annual effective rate, so check that the rate and the period match.
The ideas in more depth
Why the annuity factor is a sum of discount factors
Each payment of an annuity is the same amount, so the present value is the amount multiplied by the total of the discount factors for the years in which it is received. The formula is simply a shortcut for adding them up, and the tables list the totals.
Why the end-of-year convention matters
Real cash flows arise throughout the year, but appraisal treats each year’s flow as arising at its end, and the initial outlay at time 0, as if it were paid today. This keeps the calculation manageable, and it is applied the same way to every project so that comparisons are fair. A project whose flows arrive early in the year is slightly undervalued by it.
Sources and references
- No accounting standard governs the technique. The formulae and tables are given on the formula sheet of the paper, so check the discount rates and periods available in the table before you calculate.
These points are stated as at 5 November 2026.
Worked examples
Example 1: a present value. ₦800,000 received in 3 years; the rate is 12%. Required: the present value.
- Compound factor. The rate is 12% for 3 years.
1.123 = 1.404928 - Discount factor. It is 1 divided by the compound factor.
1 ÷ 1.404928 = 0.71178 - Present value.
800,000 × 0.71178 = ₦569,424
Example 2: an annuity. ₦150,000 a year for 6 years at 8%. Required: the present value.
- Annuity factor. Read the factor for 6 years at 8% from the table.
4.62288 (the table shows 4.623) - Present value. Cash flow times annuity factor.
150,000 × 4.62288 = ₦693,432
Example 3: a perpetuity. ₦90,000 a year for ever at 15%. Required: the present value.
- Divide the cash flow by the rate. The present value of a perpetuity is the cash flow divided by the rate.
90,000 ÷ 0.15 = ₦600,000
How to answer in the exam
- Draw the timeline and mark each cash flow at its year.
- Use the annuity factor for equal flows and show the subtraction for a deferred annuity.
- Match the rate and the period; convert a monthly rate to an annual effective rate.
- Show the factors you used so that a small arithmetic slip still earns method marks.
Common mistakes in the exam
- Discounting the initial outlay at time 0.
- Using the wrong year for the start of a deferred annuity.
- Treating a monthly rate multiplied by 12 as the annual rate.
- Applying the annuity factor to flows that are not equal.
- Using the compounding factor instead of the discount factor.
- Choosing a rate that is not in the table without calculating the factor.
Quick revision
- Future value = amount × (1 + r)^n; present value = amount ÷ (1 + r)^n.
- Annuity factor = [1 − (1 + r)^(−n)] ÷ r = sum of discount factors.
- Perpetuity = cash flow ÷ r.
- Cash flows at year-end; higher rate, lower present value.
Practice questions
These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.
Recall check: multiple choice
Question 1. The time value of money means that:
- Money loses value in a bank
- A later sum is worth more
- A sum today is worth more than the same sum later
- Interest is illegal
Show the answer
C. Because it can be invested to earn a return.
Question 2. The discount factor is:
- (1 + r)n
- 1 ÷ (1 + r)n
- r ÷ n
- 1 − r
Show the answer
B. It is the inverse of the compound factor.
Question 3. Discounting is:
- The reverse of compounding
- The same as compounding
- A tax
- A price cut
Show the answer
A. It finds a present value from a future amount.
Question 4. The annuity factor is:
- (1 + r)n
- r ÷ n
- 1 ÷ r
- [1 − (1 + r)−n] ÷ r
Show the answer
D. The sum of the discount factors.
Question 5. The present value of a perpetuity is:
- Cash flow × rate
- Cash flow ÷ rate
- Cash flow × years
- Cash flow − rate
Show the answer
B. The factor tends to 1 ÷ r.
Question 6. An annuity factor equals:
- The sum of the discount factors for the years
- The product of the discount factors
- The last discount factor
- The first discount factor
Show the answer
A. So it applies to equal flows.
Question 7. An annuity for years 3 to 7 is valued using:
- The 3-year factor
- The 7-year factor only
- The 7-year factor less the 2-year factor
- The 5-year factor times 2
Show the answer
C. The 2-year factor removes years 1 and 2.
Question 8. A higher discount rate gives:
- A higher present value
- The same present value
- A negative future value
- A lower present value
Show the answer
D. Future cash flows are worth less.
Question 9. In appraisal, cash flows are assumed to arise:
- At the start of each year
- Throughout the year
- At the end of each year, with the outlay at time 0
- At random
Show the answer
C. This keeps the calculation consistent.
Question 10. The present value of a future amount is:
- The amount × the discount factor
- The amount ÷ the discount factor
- The amount + interest
- The amount − tax
Show the answer
A. The factor is less than 1.
Section B style: written questions
Question 11 (10 marks). (a) State the formulae for the discount factor and the annuity factor. (4 marks) (b) State how to value an annuity that starts in year 3. (3 marks) (c) State the present value of a perpetuity. (3 marks)
Question 12 (15 marks). (a) Calculate the value of ₦1,200,000 invested for 5 years at 8%. (4 marks) (b) Calculate the present value of ₦3,000,000 received in 4 years at 14%. (4 marks) (c) Calculate the present value of ₦250,000 a year for 5 years at 12% (factor 3.605). (4 marks) (d) State why the answer to (b) is much less than ₦3,000,000. (3 marks)
A further written question
Question 13 (20 marks). A company is offered three options at a discount rate of 12%: ₦6m now; ₦1.4m a year for 6 years (factor 4.111); or ₦9m in 6 years (discount factor 0.507).
(a) Calculate the present value of each option. (9 marks) (b) State which to take and why. (4 marks) (c) State the equivalent perpetuity that would match ₦6m now. (3 marks) (d) State TWO limits of the method. (4 marks)
Model answers
Show the model answer to Question 11
(a) Discount factor = 1 ÷ (1 + r)n (2); annuity factor = [1 − (1 + r)−n] ÷ r (2).
(b) Use the annuity factor for the final year less the factor for the two years before the start (3).
(c) The cash flow divided by the rate (3).
Show the model answer to Question 12
(a)
- Growth factor.
1.085 = 1.469328 - Future value.
1,200,000 × 1.469328 = ₦1,763,194 (4)
(b)
- Compound factor.
1.144 = 1.68896 - Present value.
3,000,000 ÷ 1.68896 = ₦1,776,241 (4)
(c)
- Annuity factor. From the table.
3.605 - Present value.
250,000 × 3.605 = ₦901,250 (about ₦901,194 with the unrounded factor) (4)
(d) A sum received in four years could be replaced by investing a smaller amount today at 14%, so it is worth less now (3).
Show the model answer to Question 13
(a)
- Cash now. No discounting is needed.
₦6.0m (2) - The annuity. Annual amount times the annuity factor.
1.4m × 4.111 = ₦5.76m (4) - The lump sum. Lump sum times the discount factor.
9m × 0.507 = ₦4.56m (3)
(b) Take ₦6m now, because it has the highest present value (4).
(c)
- Annual return on ₦6m. A perpetuity pays the rate on the capital every year.
6m × 12% = ₦0.72m a year for ever (3)
(d) (2 marks each, any two) The rate is an estimate; cash flows are assumed to arise at year-end; the future amounts are uncertain; tax and inflation are not considered.
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