(a) A person invested ₹ 5,000 in a fund for 5 years. The value of the investment was ₹ 4,800 at the end of the second year, ₹ 6,000 at the end of the third year, ₹ 6,700 at the end of the fourth year and on maturity, the final investment sold at ₹ 8,000. Find the CAGR. [Use $(1\cdot6)^{\frac{1}{5}} = 1\cdot098$] OR (b) The annual depreciation of an asset is ₹ 50,000 and its scrap value after useful life of 10 years is ₹ 60,000. Find the original cost of the asset, using linear depreciation method.

> [!TLDR] > (a) CAGR $=\left(\tfrac{8000}{5000}\right)^{1/5}-1=(1.6)^{1/5}-1=0.098=9.8\%$. > (b) From $D=\tfrac{C-S}{n}$, $50000=\tfrac{C-60000}{10}\Rightarrow C=$ ₹ 5,60,000. > [!FORMULA] > CAGR $=\left(\dfrac{V_{f}}{V_{0}}\right)^{1/n}-1$, where $V_0$ = beginning value, $V_f$ = ending value, $n$ = number of years. > Linear depreciation: $D=\dfrac{C-S}{n}$, where $C$ = original cost, $S$ = scrap value, $n$ = useful life. **(a) CAGR of the investment** 1. Only the initial and maturity values matter: $V_0=5000$, $V_f=8000$, $n=5$ years. (The intermediate year-end values ₹4,800, ₹6,000, ₹6,700 are not used in CAGR.) 2. $\text{CAGR}=\left(\dfrac{8000}{5000}\right)^{1/5}-1=(1.6)^{1/5}-1$. 3. Using the given $(1.6)^{1/5}=1.098$: $\text{CAGR}=1.098-1=0.098$. …

(a) A person invested ₹ 5,000 in a fund for 5 years. The value of the investment was ₹ 4,800 at the end of the

Question

(1\cdot6)^{\frac{1}{5}} = 1\cdot098

(OR)

(b) The annual depreciation of an asset is ₹ 50,000 and its scrap value after useful life of 10 years is ₹ 60,000. Find the original cost of the asset, using linear depreciation method.

CBSECBSE Class XII Board 2025Subjective· 3mImportance★★★★★

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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.

Note

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.

\text{CAGR} = \left( \frac{\text{Ending Value}}{\text{Beginning Value}} \right)^{\frac{1}{n}} - 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.

\text{CAGR} = \left( \frac{200}{100} \right)^{\frac{1}{3}} - 1 = (2)^{0.333} - 1 \approx 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.

Watch out

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. …

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CAGR $=\left(\tfrac{8000}{5000}\right)^{1/5}-1=(1.6)^{1/5}-1=0.098=9.8\%$ .
From $D=\tfrac{C-S}{n}$ , $50000=\tfrac{C-60000}{10}\Rightarrow C=$ ₹ 5,60,000.

=\left(\dfrac{V_{f}}{V_{0}}\right)^{1/n}-1

(a) CAGR of the investment

Only the initial and maturity values matter: $V_0=5000$ , $V_f=8000$ , $n=5$ years. (The intermediate year-end values ₹4,800, ₹6,000, ₹6,700 are not used in CAGR.)
$\text{CAGR}=\left(\dfrac{8000}{5000}\right)^{1/5}-1=(1.6)^{1/5}-1$ .
Using the given $(1.6)^{1/5}=1.098$ : $\text{CAGR}=1.098-1=0.098$ . …

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