Q.An investment has a starting value of ₹2000 and it grows to ₹18,000 in 3 years. What will be its CAGR?
Concept understanding — Compound Annual Growth Rate
Compound Annual Growth Rate (CAGR)
The Intuition First
Imagine you put ₹100 into a business. After one year, it grows to ₹120. After two years, it becomes ₹150. After three years, it's ₹200.
If someone asks, "What was the average yearly growth rate?", you might be tempted to say: "Year 1 grew 20%, Year 2 grew 25%, Year 3 grew 33.3% — so average is about 26%." But that's misleading. Why? Because each year's growth compounds on the previous year's base. You can't just average percentages that act on different starting amounts.
CAGR solves this problem. It answers: "If this investment had grown at a steady, constant rate every year, what would that rate be?" It smooths out the bumps and gives you one number that tells you the true annualised return.
CAGR is not the actual return in any single year. It's the geometric average — the rate that, if applied every year, would take you from the start value to the end value.
The Precise Statement
CAGR is the mean annual growth rate of an investment over a specified period longer than one year. It represents one of the most accurate ways to calculate and determine returns for anything that can rise or fall in value over time.
CAGR=(Beginning ValueEnding Value)n1−1
Where:
- Ending Value = final value of the investment
- Beginning Value = initial value of the investment
- n = number of years
Applying It to Our Example
Beginning Value = ₹100, Ending Value = ₹200, n = 3 years.
CAGR=(100200)31−1=(2)0.333−1≈1.26−1=0.26=26%
So the CAGR is 26%. Notice this is lower than the simple average of 26.1% we calculated earlier — that's because CAGR correctly accounts for the compounding effect.
A common mistake is to use the arithmetic mean of yearly returns. For example, if an investment goes up 50% one year and down 50% the next, the arithmetic mean is 0%, but you've actually lost money (₹100 → ₹150 → ₹75). CAGR correctly gives -13.4%.
Why CAGR Matters in Exams
In Indian competitive exams (CAT, GMAT, banking, SSC, UPSC), CAGR appears in:
- Data Interpretation: Tables showing company revenues or GDP over years — you'll need to compute CAGR quickly.
- Quantitative Aptitude: Direct formula-based questions.
- Finance/Investment: Comparing mutual fund or stock returns.
For quick approximations, remember: if an investment doubles in 3 years, CAGR ≈ 26%; doubles in 5 years, CAGR ≈ 15%; doubles in 10 years, CAGR ≈ 7.2%. This uses the Rule of 72 (72 ÷ years ≈ CAGR% for doubling).
Key Properties to Remember
- CAGR is always ≤ arithmetic mean of yearly returns (unless all yearly returns are equal).
- CAGR can be negative if the ending value is less than the beginning value.
- CAGR assumes reinvestment of profits at the same rate — this is its fundamental assumption.
- CAGR does not account for risk — two investments with the same CAGR can have very different volatility.
Final Answer
CAGR is the geometric average growth rate that describes how an investment would have grown if it grew at a steady, constant rate each year. It is calculated as:
CAGR=(Beginning ValueEnding Value)n1−1
where n is the number of years. It smooths out volatility and gives the true annualised return, making it the standard metric for comparing investment performance over time.
The key idea is Compound Annual Growth Rate (CAGR), which gives the constant annual rate at which an investment grows over a period, assuming compounding.
Step 1: Write the CAGR formula:
CAGR=(Beginning ValueEnding Value)n1−1
Step 2: Substitute the given values: Beginning Value = ₹2000, Ending Value = ₹18,000, n=3 years.
CAGR=(200018000)31−1=(9)31−1
Step 3: The cube root of 9 is approximately 2.0801. Subtract 1:
2.0801−1=1.0801
Step 4: Convert to percentage:
1.0801×100=108.01%
The CAGR is approximately 108.01%.
CAGR is the constant annual rate that turns the starting value into the ending value over the period. Here ₹2,000 grows to ₹18,000 (a 9-fold increase) in 3 years, giving a CAGR of ~ 108% per annum - exactly (39−1)×100%.
1. Formula
CAGR=(PA)1/n−1
with P=2,000, A=18,000, n=3.
2. Growth factor
PA=2,00018,000=9
3. Annualise over 3 years
CAGR=91/3−1=2.080084−1=1.080084=108.01%
Sanity check: a 100% CAGR would multiply the money by 23=8 (giving ₹16,000); we reached ₹18,000, so the rate is a little above 100%. Note this is not the same as dividing the 800% total return by 3 (that would be simple interest); CAGR compounds.
The CAGR is approximately 108% per annum - exactly (39−1)×100%≈108.01%.
- CA Foundation 2026Set jan-20261 markMCQQ.The net asset value (NAV) of a Mutual Fund is calculated at the end of the financial year. For the last five years following values are computed.Calculate the Compounded Annual Growth Rate of NAV. (A) 16.92% (B) 18.92% (C) 20.92% (D) 22.92%
Year 2021 2022 2023 2024 2025 NAV 100 115 150 120 200 ›Reveal solutionSolution
CAGR=(100200)1/4−1=20.25−1=18.92%.
Step 1 — identify endpoints and periods
Beginning NAV (2021) =100; ending NAV (2025) =200. From 2021 to 2025 is n=4 years.
Step 2 — apply the CAGR formula
CAGR=(BeginEnd)1/n−1=(100200)1/4−1=20.25−1.
Step 3 — evaluate
20.25=1.1892 ⇒ CAGR=0.1892=18.92%.
The intermediate NAVs (115, 150, 120) do not affect CAGR.
Watch outThe number of periods is 4, not 5. Five data points span only four year-gaps (2021→2025) — using n=5 is the classic off-by-one error, and the intermediate NAVs (115, 150, 120) are pure distractors that CAGR ignores.
TipCAGR depends only on the first and last values and the count of gaps between them. Count intervals, not data points: (years − 1) gives n.
✓Final answer(B) 18.92%
- CA Foundation 2026Set may-20261 markMCQQ.If ₹ 50,000 grows to ₹ 80,525.5 in 5 years, the compound annual growth rate (CAGR) is ________. (A) 9% (B) 10% (C) 11% (D) 12%
›Reveal solutionSolution
CAGR=(Vf/V0)1/n−1. The ratio 1.61051 is exactly 1.105, so the CAGR is 10%.
Step 1 — CAGR formula
CAGR=(V0Vf)1/n−1
Step 2 — Compute the growth ratio
V0=50000, Vf=80525.5, n=5:
V0Vf=5000080525.5=1.61051
Step 3 — Take the 5th root
Recognise 1.105=1.61051, so:
CAGR=(1.61051)1/5−1=1.10−1=0.10=10%
Why the other options are wrong: (A) 9%: 1.095=1.5386=1.61051; (C) 11%: 1.115=1.6851; (D) 12%: 1.125=1.7623 — none matches the given end value.
Watch outTake the n-th ROOT (power 1/5), not 1/n of the ratio. Subtract 1 only after rooting, and remember to convert to a percentage.
TipIf the growth ratio equals a familiar power (here 1.15), you can read the rate straight off without a calculator.
✓Final answer(B) 10%
- CA Foundation 2025Set jan-20251 markMCQQ.Sam invested ₹ 12,000 for 10 years in a financial company. At the end of 10th year his investment value is ₹ 18,000. Then the Compound Annual Growth Rate (CAGR) is if (x)1/n=1.0413 (A) 41.40% (B) 4.13% (C) 11.56% (D) 12.06%
›Reveal solutionSolution
CAGR=(1200018000)1/10−1=1.0413−1=4.13%.
Step 1 — Form the growth ratio
InitialFinal=1200018000=1.5
Step 2 — Apply the CAGR formula
CAGR=(Initial valueFinal value)1/n−1
With the given (1.5)1/10=1.0413.
Step 3 — Subtract 1
CAGR=1.0413−1=0.0413=4.13%
Why the other options are wrong: (A) 41.40% and (C) 11.56% misread the root or use the wrong exponent; (D) 12.06% ignores the 1/n power (that would be simple total growth spread crudely). Only 1.0413−1 gives 4.13%.
Watch outCAGR is a PER-YEAR rate — take the n-th root (1/10 power) and then subtract 1. Forgetting the root gives the total 50% growth, not the annual rate.
TipThe given hint (x)1/n=1.0413 already contains the root; CAGR is simply that value minus 1 — no further computation needed.
✓Final answer(B) 4.13%
- CA Foundation 2025Set may-20251 markMCQQ.Madhu invests ₹ 15,000 in a scheme and at the time of maturity the amount became ₹ 25,000. If CAGR for this investment is 8.88%, calculate the approximate number of years for which she has invested the amount. [Given that log(1.667)=0.2219 and log(1.089)=0.037] (A) 6 years (B) 7.7 years (C) 5.5 years (D) 7 years
›Reveal solutionSolution
n=log1.0888log(25000/15000)=0.0370.2219≈6 years.
Step 1 — Write the CAGR relation
(1+CAGR)n=Initial valueFinal value
1500025000=1.667,1+CAGR=1.0888
Step 2 — Take logarithms
nlog(1.0888)=log(1.667)
n=log1.089log1.667=0.0370.2219
Step 3 — Compute
n=0.0370.2219≈6 years
Why the other options are wrong: (B) 7.7, (C) 5.5, (D) 7 do not satisfy 0.2219/0.037; they come from dividing by a wrong log value.
Watch outUse the log of the GROWTH FACTOR 1.089 (not the CAGR 0.0888 itself) in the denominator — confusing the two badly distorts n.
TipBoth logs are handed to you; the answer is a single division 0.2219÷0.037≈6.
✓Final answer(A) 6 years
- CA Foundation 2025Set sep-20251 markMCQQ.Ms. Y invested ₹ 2,00,000 in a mutual fund equity scheme. She redeemed entire investment after 96 months and received ₹ 6,00,000 after redemption. What was the Compound Annual Growth Rate (CAGR) in percentage ? (Given : 1.14724=1.732) (A) 14.72 (B) 15.72 (C) 13.72 (D) 12.72
›Reveal solutionSolution
(1+r)8=3⇒(1+r)4=3=1.732=1.14724⇒r=14.72%.
Step 1 — Set up the CAGR relation
CAGR:(1+r)n=InitialFinal
96 months =8 years, and 2,00,0006,00,000=3, so (1+r)8=3.
Step 2 — Halve the exponent
(1+r)8=3⇒(1+r)4=3=1.732.
Step 3 — Match the given value
Since 1.14724=1.732, we have 1+r=1.1472, so r=0.1472=14.72%.
Watch outConvert 96 months to 8 years first. Using n=96 or forgetting to take the 8th root would badly overstate/understate the rate.
Tip3≈1.732 is worth memorising — it turns (1+r)8=3 into the fourth-power form that the given hint matches directly.
✓Final answer(A) 14.72
- CA Foundation 2025Set sep-20251 markMCQQ.An investment was priced at ₹ 100 per share in year 0, priced at ₹ 150 per share in the end of the first year, and priced ₹ 200 per share in the end of second year. What is the Compound Annual Growth Rate (CAGR) of the investment ? (A) 21.42% (B) 31.42% (C) 41.42% (D) 51.42%
›Reveal solutionSolution
CAGR=(BeginEnd)1/n−1=(200/100)1/2−1=2−1≈41.42%.
Step 1 — Use only the endpoints
CAGR smooths growth into one constant rate, so only the year-0 price (₹100) and the year-2 price (₹200) are used; the ₹150 mid-value is irrelevant.
Step 2 — Apply the CAGR formula
CAGR=(V0Vn)1/n−1
with V0=100, Vn=200, n=2.
Step 3 — Compute
CAGR=(100200)1/2−1=20.5−1=1.4142−1=0.4142=41.42%
Why the other options are wrong: (D) 51.42% mistakes the simple average of the two 50% yearly jumps; (A) 21.42% and (B) 31.42% come from dividing that 100% total growth by the wrong number of periods.
Watch outThe investment rose 50% then 33.3% — averaging those (≈41.7%) or the price jumps is NOT CAGR. Use the geometric root of the total ratio.
Tip2≈1.4142 is worth memorising; a "double in 2 years" question is instantly 41.42%.
✓Final answer(C) 41.42%
- CA Foundation 2024Set sep-20241 markMCQQ.The Earning Per Share (EPS) of a company for five years is given below :Calculate the Compounded Annual Growth Rate (CAGR) of EPS. (A) 24.47% (B) 23.47% (C) 22.47% (D) 21.47%
Year 2019 2020 2021 2022 2023 EPS 40 25 40 60 90 ›Reveal solutionSolution
CAGR =(End/Begin)1/n−1; intermediate years don't matter. (90/40)1/4−1=22.47%.
Step 1 — Identify begin, end and n
Begin (2019) =40, End (2023) =90, number of years n=2023−2019=4.
CAGR=(Beginning valueEnding value)1/n−1
Step 2 — Substitute
(4090)1/4−1=(2.25)1/4−1
Step 3 — Evaluate the fourth root
2.25=1.5,1.5≈1.2247
CAGR≈1.2247−1=0.2247=22.47%
Why the other options are wrong: 24.47%, 23.47% and 21.47% are decoys; the exact (2.25)1/4 is 1.2247, i.e. 22.47%. Averaging year-on-year growth (which would use the middle values) is not how CAGR works.
Watch outCAGR ignores the intermediate EPS (25, 40, 60) — using all five values or averaging annual changes is wrong.
TipA fourth root is two square roots: 2.25 — here 2.25→1.5→1.2247.
✓Final answer(C) 22.47%
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