How long should a delivery van, a printing press or a generator be kept before it is replaced? Replace it too early and capital is wasted on a machine with years of life left. Replace it too late and the company pays rising repair and running costs and loses efficiency.
When an asset will be replaced by an identical one again and again, the question is the length of the best replacement cycle. The answer is the cycle with the lowest cost per year. The difficulty is comparing a two-year cycle with a three-year cycle, because the costs cover different periods.
This topic explains the two methods of putting them on a common footing, the equivalent annual cost (EAC) and the lowest common multiple (LCM) method, with worked examples, the decision to replace an existing machine now or later, and the limits of the analysis. The next topic covers capital rationing.
The syllabus asks candidates to evaluate asset replacement decisions using equivalent annual cost and lowest common multiple approaches. Read lease or buy and borrow-to-buy decisions first.
In 30 seconds
An asset replacement decision decides how often an asset in regular use should be replaced by an identical one: choose the least-cost replacement cycle. Relevant cash flows: purchase cost, running and maintenance costs (usually rising with age), tax effects, and the scrap or resale value. Equivalent annual cost (EAC) = present value of cost over one cycle ÷ annuity factor for the cycle’s years; choose the lowest EAC.
The lowest common multiple (LCM) method compares the present value of cost of each policy over a common period (for 2-year and 3-year cycles, 6 years), repeating each cycle; choose the lowest. EAC is quicker; the LCM method copes better with changing costs but becomes long when the cycles do not share a small multiple.
To decide whether to replace an existing asset now, compare the cost of keeping it one more year with the EAC of the new asset.
The decision
An asset replacement decision concerns an asset in regular use that will be replaced by an identical asset when it reaches the end of its chosen life, and so on for ever. It is not a one-off decision to buy an asset; it is a choice of the replacement cycle. The rule: choose the cycle that minimises the present value of cost per unit of time.
The relevant cash flows are the capital cost of the asset, the running and maintenance costs (which usually increase each year as the asset ages), tax relief on the running costs and on the capital allowances, and the scrap or resale value at the end of the cycle.
A present value of cost over two years cannot simply be compared with one over three years, because the three-year figure covers a longer period and will usually be larger.
The equivalent annual cost method
- For each possible cycle length, calculate the present value of cost over one cycle: the purchase at time 0, the running costs each year, the resale value at the end.
- Convert it to an annual figure: EAC = present value of cost ÷ annuity factor for the number of years in the cycle, at the cost of capital.
- Choose the cycle with the lowest EAC.
Example: Brightway buys a machine for ₦12.0m. It can be kept for up to three years. Running costs are ₦1.5m, ₦2.4m and ₦3.6m in years 1, 2 and 3; the resale value is ₦8.0m after one year, ₦5.5m after two and ₦3.0m after three. The cost of capital is 12% (factors 0.893, 0.797, 0.712; annuity factors 0.893, 1.690, 2.402). Taxation is ignored.
| Cycle | Workings (₦m) | Present value (PV) of cost ₦m | Annuity factor | EAC ₦m |
|---|---|---|---|---|
| Replace every year | 12.000 + 1.5 × 0.893 − 8.0 × 0.893 | 6.196 | 0.893 | 6.94 |
| Replace every two years | 12.000 + 1.340 + 1.913 − 4.384 | 10.869 | 1.690 | 6.43 |
| Replace every three years | 12.000 + 1.340 + 1.913 + 2.563 − 2.136 | 15.680 | 2.402 | 6.53 |
The lowest EAC is ₦6.43m, so the machine should be replaced every two years. A yearly replacement wastes capital on frequent purchases (high EAC); a three-year cycle has rising running costs and a low resale value that outweigh the saving in purchases.
The lowest common multiple method
The lowest common multiple (LCM) method compares policies over a period that is a whole number of cycles of each. For cycles of 2 and 3 years the LCM is 6 years. The steps are to find the LCM, set out the cash flows of each policy over that period (repeating the cycle), discount them and compare the total present values of cost.
For Brightway: replacing every two years means three cycles in six years, with purchases at years 0, 2 and 4; the present value is 10.869 × (1 + 0.797 + 0.636) = 10.869 × 2.433 = ₦26.44m. Replacing every three years means two cycles, with purchases at years 0 and 3: 15.680 × (1 + 0.712) = ₦26.84m. The two-year policy is cheaper, as the EAC method showed.
Advantages of LCM: a consistent comparison over a common time, and it can accommodate cash flows that differ between cycles (for example if inflation raises costs in later cycles). Limitations: it is longer to calculate, and where the cycles have no small common multiple (say 4 and 7 years: 28 years) the horizon becomes impractical, so a finite horizon is used instead. EAC is the preferred method when the same cycle repeats unchanged.
Replacing an existing asset
Suppose the company already owns an old machine that could be sold now for ₦5.0m, or run one more year with running costs of ₦4.0m and a resale value of ₦3.5m. The cost of keeping it for one more year, measured at the end of the year, is the running cost plus the interest lost on the cash not received now, less the resale value: 4.0 + 5.0 × 1.12 − 3.5 = ₦6.1m.
This is less than the EAC of the new machine (₦6.43m), so keep the old machine for another year. When the cost of keeping the old machine for a further year exceeds the EAC of the new one, replace it.
The ideas in more depth
Why dividing by the annuity factor works
An annuity factor converts a level annual payment into a present value. Dividing the present value of cost by the factor reverses this: it gives the level annual payment that would have the same present value as the cycle’s costs.
Cycles of different length become comparable because each is expressed as a cost per year, and the same cycle can be repeated for ever with the same cost per year.
What the analysis leaves out
The method assumes that the replacement is identical and that costs and prices stay the same, so that each cycle repeats. If technology changes, so that the replacement will be cheaper to run or more productive, or if inflation raises costs, the assumption fails; the LCM method, or a calculation in money terms, may then be better.
Tax relief on capital allowances and running costs should be included when the question gives the rate. Non-financial factors such as reliability, breakdowns and the effect on customers also matter.
Sources and references
- No accounting standard governs the choice of replacement cycle. The equivalent annual cost method is a standard technique of capital budgeting.
These points are stated as at 5 November 2026.
Worked examples
Example 1: the present value of a cycle. Machine ₦12.0m kept two years; running costs ₦1.5m and ₦2.4m; resale ₦5.5m; factors 0.893 and 0.797. Required: the present value of cost.
- Costs.
12.000 + 1.340 + 1.913 = 15.253. - Net of resale.
15.253 − 5.5 × 0.797 (4.384) = ₦10.87m.
Example 2: the equivalent annual cost. Present value ₦10.87m; annuity factor for 2 years 1.690. Required: the EAC.
- EAC.
10.87 ÷ 1.690 = ₦6.43m.
Example 3: the LCM. Cycles of 2 and 3 years. Required: the common period.
- LCM. The smallest number divisible by both is 6 years: three two-year cycles or two three-year cycles.
How to answer in the exam
- State the relevant cash flows for each cycle and set them out by year.
- Calculate the present value of cost and the EAC for each cycle, then name the lowest.
- If the method is stated, follow it; for LCM show the repeated cash flows and the common period.
- Comment on the assumptions: identical replacements, constant prices, tax and inflation.
Common mistakes in the exam
- Comparing present values of cost for cycles of different length.
- Dividing by the discount factor instead of the annuity factor.
- Forgetting the resale value or discounting it in the wrong year.
- Not repeating the purchase in later cycles in the LCM method.
- Including the cost of the asset already owned that is sunk.
- Ignoring the tax effects when the tax rate is given.
Quick revision
- Choose the least-cost replacement cycle.
- EAC = present value of cost over one cycle ÷ annuity factor; lowest wins.
- LCM: compare policies over the lowest common multiple of the cycles.
- Existing asset: compare the cost of one more year with the EAC of the new one.
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. An asset replacement decision is about:
- Whether to buy a first asset
- The sale price
- How often to replace an asset used regularly
- The loan rate
Show the answer
C. The length of the replacement cycle.
Question 2. The preferred replacement cycle is the one with the:
- Highest present value of cost
- Lowest equivalent annual cost
- Longest life
- Highest resale value
Show the answer
B. It minimises the cost per year.
Question 3. The equivalent annual cost equals:
- Present value of cost ÷ annuity factor for the cycle
- Present value × annuity factor
- Purchase price ÷ years
- Running cost × years
Show the answer
A. It converts the cycle to a level annual cost.
Question 4. Present values of cost for cycles of different lengths cannot be compared directly because:
- They use different rates
- They have no tax
- They are always equal
- They cover different periods
Show the answer
D. A longer cycle has more cost in total.
Question 5. The lowest common multiple of 2 and 3 years is:
- 5
- 6
- 4
- 12
Show the answer
B. The smallest number divisible by both.
Question 6. Running costs usually:
- Rise as the asset ages
- Fall as the asset ages
- Stay constant
- Are ignored
Show the answer
A. Which is why very long cycles cost more.
Question 7. A limitation of the LCM method is that:
- It ignores cash
- It cannot use discounting
- It becomes long when cycles have a large common multiple
- It ignores resale value
Show the answer
C. For example 4 and 7 years give 28 years.
Question 8. The relevant cash flows include:
- Allocated overheads
- Depreciation
- Past costs
- Capital cost, running costs, tax effects and resale value
Show the answer
D. Only future cash flows that differ.
Question 9. To decide whether to replace an existing asset now, compare:
- Its book value and cost
- Its age and the new asset’s age
- The cost of keeping it one more year with the EAC of the new asset
- Its price with tax
Show the answer
C. Keep it while its extra year costs less.
Question 10. The method assumes that replacements are:
- Identical
- Cheaper each time
- Different each time
- Free
Show the answer
A. So each cycle repeats with the same cost.
Section B style: written questions
Question 11 (10 marks). (a) Define an asset replacement decision. (3 marks) (b) State the relevant cash flows. (4 marks) (c) State the formula for the EAC and the decision rule. (3 marks)
Question 12 (15 marks). A machine costs ₦5.0m and can be kept for one or two years. Running costs are ₦0.8m and ₦1.2m; resale values are ₦3.2m after one year and ₦2.0m after two. The cost of capital is 10% (factors 0.909 and 0.826; annuity factors 0.909 and 1.736). (a) Calculate the present value of cost for each cycle. (8 marks) (b) Calculate the EAC of each. (4 marks) (c) State the best cycle. (3 marks)
A further written question
Question 13 (20 marks). A machine costs ₦8.0m. Running costs are ₦1.0m, ₦1.6m and ₦2.6m in years 1 to 3; resale values are ₦5.0m, ₦3.5m and ₦2.0m at the ends of years 1 to 3. The cost of capital is 10% (year factors 0.909, 0.826, 0.751, 0.683; annuity factors 0.909, 1.736, 2.487).
(a) Calculate the present value of cost and the EAC of each cycle. (10 marks) (b) Using the LCM method over 6 years, compare the two-year and three-year policies. (7 marks) (c) State the decision. (3 marks)
Model answers
Show the model answer to Question 11
(a) A decision on how often an asset in regular use should be replaced by an identical asset (3).
(b) Capital cost; running and maintenance costs; tax relief on costs and allowances; scrap or resale value (1 each).
(c) EAC = present value of cost over one cycle ÷ annuity factor for the cycle (2); choose the lowest EAC (1).
Show the model answer to Question 12
(a)
- One year.
−5.000 − 0.8 × 0.909 + 3.2 × 0.909 = −5.000 − 0.727 + 2.909 = ₦2.818m cost (4). - Two years.
−5.000 − 0.727 − 1.2 × 0.826 (0.991) + 2.0 × 0.826 (1.652) = ₦5.066m cost (4).
(b)
- One year.
2.818 ÷ 0.909 = ₦3.10m (2). - Two years.
5.066 ÷ 1.736 = ₦2.92m (2).
(c) Replace every two years, the lowest EAC (3).
Show the model answer to Question 13
(a)
- One year.
−8 − 0.909 + 4.545 = −4.364, EAC 4.364 ÷ 0.909 = ₦4.80m (3). - Two years.
−8 − 0.909 − 1.322 + 2.891 = −7.340, EAC 7.340 ÷ 1.736 = ₦4.23m (4). - Three years.
−8 − 0.909 − 1.322 − 1.953 + 1.502 = −10.681, EAC 10.681 ÷ 2.487 = ₦4.29m (3).
(b)
- Two-year policy.
three cycles, purchases at years 0, 2 and 4: 7.340 × (1 + 0.826 + 0.683) = ₦18.42m (4). - Three-year policy.
two cycles, years 0 and 3: 10.681 × (1 + 0.751) = ₦18.70m (3).
(c) Replace every two years: the lowest EAC and the lowest cost over the common period (3).
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