Skip to content

Quantitative Aptitude · Mathematics of Finance

Depreciation and Compound Annual Growth Rate (CAGR)

Updated 1 October 2026 · Fact-checked

Depreciation is the fall in an asset's value over time. Straight-line deducts the same amount every year: (Cost − Scrap) ÷ Life. Reducing balance deducts a fixed percentage of the opening book value, so book value = Cost × (1 − r)^n. CAGR is the steady yearly growth rate: (End ÷ Start)^(1/n) − 1.

Understand Depreciation and Compound Annual Growth Rate

Depreciation is the loss in value of an asset, such as a machine, because of use and time. In this chapter you only need the arithmetic. The book value is what the asset is worth in the books after depreciation. The scrap value (residual value) is what it is expected to fetch at the end of its life.

In the straight-line method, the same amount is written off every year. The book value falls in a straight line, like a simple interest pattern in reverse. It is an arithmetic progression with a negative common difference.

In the reducing balance method (also called written down value), a fixed percentage is charged on the book value at the start of each year. The charge is large in early years and smaller later. The book value follows a geometric progression with ratio (1 − r). This is compound interest with a negative rate.

CAGR tells you the single steady yearly rate at which a value would have to grow to go from a starting value to an ending value over n years. The real growth may have been uneven. CAGR smooths it out. Population growth, sales growth and investment growth all use the same compound formula, P × (1 + g)^n.

So the whole topic reduces to two ideas. Equal amounts each year means straight line. Equal percentages each year means compounding.

Key formulas to remember

Straight-line depreciation per year
D = (Cost − Scrap value) ÷ n
n is the useful life in years. If there is no scrap value, D = Cost ÷ n.
Book value under straight line
Book value after t years = Cost − t × D
Valid for t up to n. At t = n the book value equals the scrap value.
Book value under reducing balance
Book value after n years = Cost × (1 − r)^n
r is the depreciation rate as a decimal, charged on the opening book value each year.
Rate from scrap value (reducing balance)
r = 1 − (Scrap ÷ Cost)^(1/n)
Use when the scrap value after n years is given and the rate is asked.
Compound growth
Pₙ = P₀ × (1 + g)^n
Use for population, sales or investment growing at a constant rate g per period. For a decline, use (1 − r)^n.
CAGR
CAGR = (Ending value ÷ Starting value)^(1/n) − 1
n is the number of years between the two values, not the number of data points.

How to solve Depreciation and Compound Annual Growth Rate questions

Use this order for any depreciation or growth question. It keeps you from mixing the two methods.

  1. 1Identify the method. Look for 'straight line', 'equal instalments' or 'fixed amount' versus 'reducing balance', 'written down value', 'per annum on book value' or 'growth rate'.
  2. 2List the given values: cost or starting value, scrap or ending value, rate, and number of years n.
  3. 3Check what is asked: depreciation per year, book value after some years, the rate, or the CAGR.
  4. 4For straight line, find D = (Cost − Scrap) ÷ n first. Then book value = Cost − t × D.
  5. 5For reducing balance or growth, convert the rate to a decimal and apply Cost × (1 ± rate)^n.
  6. 6For CAGR or an unknown rate, compute the ratio End ÷ Start, then take the n-th root and subtract 1.
  7. 7Convert the answer to a percentage if a rate is asked, and check that it is reasonable: book value must stay positive and below cost; growth must give a ratio above 1.

Quickest way: Test the options instead of solving the root

When to use it: Use in MCQs where CAGR or the depreciation rate is asked, or where n is 2 or 3 and the numbers are clean.

  1. Find the ratio End ÷ Start, for example 14.4 ÷ 10 = 1.44.
  2. Take each option rate, add 1, and raise it to the power n in your head or on scratch paper. For n = 2, 1.2 × 1.2 = 1.44.
  3. Stop at the first option that matches the ratio. Do not work through the rest.
  4. For reducing balance with n = 2 or 3, multiply the factor repeatedly instead of using logs. For example 0.9 × 0.9 × 0.9 = 0.729.
  5. For straight line, work with the yearly charge only. Subtract t × D from cost. No powers are needed.
  6. Skip a question if it needs a non-clean root with no matching option. Each wrong answer costs 0.25 marks.

Common mistakes in Depreciation and Compound Annual Growth Rate

  • Ignoring scrap value in straight-line depreciation.

    Students divide cost by life out of habit.

    Fix: Always check whether scrap value is given. Use (Cost − Scrap) ÷ n.

  • Using the original cost every year under reducing balance.

    The method gets confused with straight line.

    Fix: Charge the rate on the opening book value each year. Use Cost × (1 − r)^n.

  • Taking the final answer as the depreciation instead of the book value, or the reverse.

    The formula output is the book value, but the question may ask for total depreciation.

    Fix: If total depreciation is asked, subtract the book value from the cost.

  • Using the wrong n in CAGR, such as counting the number of years listed instead of the gaps.

    Data for 2020 to 2024 lists five years but covers four years of growth.

    Fix: n = last year − first year.

  • Forgetting to subtract 1 in the CAGR formula.

    The root gives 1.2, which looks like a finished answer.

    Fix: CAGR = root − 1, so 1.2 − 1 = 0.20 = 20%.

  • Adding up yearly growth rates or dividing total growth by n.

    Students treat growth as simple, not compound.

    Fix: A rise of 44% over 2 years is not 22% per year. Compounding gives 20%.

Worked examples

Example 1

A machine costing ₹5,00,000 has a scrap value of ₹50,000 and a life of 10 years. Using the straight-line method, its book value after 4 years is: (a) ₹3,00,000 (b) ₹3,20,000 (c) ₹2,75,000 (d) ₹3,50,000

Show the solution
  1. Depreciation per year = (5,00,000 − 50,000) ÷ 10 = 4,50,000 ÷ 10 = ₹45,000.
  2. Depreciation for 4 years = 4 × 45,000 = ₹1,80,000.
  3. Book value = 5,00,000 − 1,80,000 = ₹3,20,000.
  4. Option (a) comes from subtracting 4 × 50,000, which wrongly uses scrap value as the yearly charge.

Answer: (b) ₹3,20,000

Example 2

A vehicle bought for ₹2,00,000 depreciates at 10% per annum on the reducing balance method. Its book value after 3 years is: (a) ₹1,40,000 (b) ₹1,45,800 (c) ₹1,60,000 (d) ₹1,48,000

Show the solution
  1. Use Book value = Cost × (1 − r)^n = 2,00,000 × (0.9)^3.
  2. 0.9 × 0.9 = 0.81, and 0.81 × 0.9 = 0.729.
  3. Book value = 2,00,000 × 0.729 = ₹1,45,800.
  4. Check by years: 1,80,000, then 1,62,000, then 1,45,800. This agrees.
  5. Option (a) is what you get with straight-line 10% of cost for 3 years, which is a different method.

Answer: (b) ₹1,45,800

Example 3

The sales of a firm rose from ₹10 lakh to ₹14.4 lakh in 2 years. The compound annual growth rate is: (a) 18% (b) 20% (c) 22% (d) 44%

Show the solution
  1. Ratio = 14.4 ÷ 10 = 1.44.
  2. CAGR = (1.44)^(1/2) − 1.
  3. √1.44 = 1.2, so CAGR = 1.2 − 1 = 0.20 = 20%.
  4. Check: 1.2 × 1.2 = 1.44, so 10 lakh becomes 14.4 lakh.
  5. Option (d) is the total growth over 2 years, not the yearly rate.

Answer: (b) 20%

Exam tips

  • Read the method name first. One word, 'straight line' or 'reducing balance', decides the whole formula.
  • Check the question carefully for what is asked: depreciation, book value, rate or total depreciation. The options often include the wrong ones.
  • Expect clean numbers. If your root is not neat, recheck n and the ratio before choosing an option.
  • Keep powers of common factors ready: 1.1², 1.2², 0.9³, 0.8². Memorising them saves time.
  • Rate questions usually give scrap value after n years. Use r = 1 − (Scrap ÷ Cost)^(1/n) or test the options.

Practice questions from Mathematics of Finance

Depreciation and Compound Annual Growth Rate: frequently asked questions

What is the CAGR formula for CA Foundation?

CAGR = (Ending value ÷ Starting value)^(1/n) − 1, where n is the number of years between the two values. Multiply by 100 to express it as a percentage.

What is the difference between straight-line and reducing balance depreciation?

Straight line charges the same amount every year, calculated on cost less scrap value. Reducing balance charges a fixed percentage on the opening book value, so the yearly charge falls over time. Straight-line book value falls in an arithmetic pattern and reducing balance in a geometric pattern.

Does the reducing balance method ever reach zero?

No. Multiplying by (1 − r) each year makes the book value smaller but never exactly zero, as long as r is less than 100%. Straight line can reach scrap value, or zero if there is no scrap value.

Is CAGR the same as the average of yearly growth rates?

No. The simple average of yearly rates ignores compounding and usually gives a different figure. CAGR is the single constant rate that links the starting and ending values over the period.