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Exercise 7.5 · Q1

Q.An investment has a starting value of ₹5000 and it grows to ₹25,000 in 4 years. What will be its CAGR?

Sikkim CbseNCERTSubjective· 3mImportance★★★★★
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CAGR is the constant annual growth rate that would turn ₹5000 into ₹25000 over 4 years. Using the formula CAGR=(250005000)1/4−1\text{CAGR} = \left(\frac{25000}{5000}\right)^{1/4} - 1, we get approximately 49.53%.

Why CAGR? The Concept

When an investment grows unevenly over time, we often want a single number that captures its average annual performance — as if it grew at a steady rate each year. That's exactly what Compound Annual Growth Rate (CAGR) does. It's not the arithmetic average of yearly returns; it's the geometric mean rate that compounds the initial value to the final value over the given period.

Think of it this way: if you had a bank account that gave you the same interest rate rr every year, and you started with ₹5000, after 4 years you'd have:

Final=5000×(1+r)4\text{Final} = 5000 \times (1+r)^4

CAGR simply solves for rr in that equation, using the actual starting and ending values. It assumes smooth, steady growth — even if the real investment bounced around.

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

where nn is the number of years.

Step-by-Step Solution

1. Identify the known quantities.

We have:

  • Beginning value, BV=₹5000BV = ₹5000
  • Ending value, EV=₹25000EV = ₹25000
  • Time period, n=4n = 4 years

2. Write the CAGR formula and substitute.

The CAGR rr satisfies:

EV=BV×(1+r)nEV = BV \times (1+r)^n

So:

25000=5000×(1+r)425000 = 5000 \times (1+r)^4

3. Isolate the growth factor.

Divide both sides by 5000:

250005000=(1+r)4\frac{25000}{5000} = (1+r)^4

5=(1+r)45 = (1+r)^4

This tells us that the investment multiplied by a factor of 5 over 4 years. The CAGR is the annual multiplier that, when compounded 4 times, gives 5.

4. Take the 4th root to solve for 1+r1+r.

Raise both sides to the power 14\frac{1}{4}:

1+r=51/41+r = 5^{1/4}

Now, 51/45^{1/4} means the fourth root of 5. You can compute this using a calculator or by successive approximation.

Tip

To compute 51/45^{1/4} without a calculator, note that 51/2≈2.2365^{1/2} \approx 2.236, and then take the square root of that: 2.236≈1.495\sqrt{2.236} \approx 1.495. So 51/4≈1.4955^{1/4} \approx 1.495.

5. Subtract 1 to get the CAGR.

r=51/4−1r = 5^{1/4} - 1

Using a more precise calculation:

51/4=e14log⁡5≈e14×1.60944=e0.40236≈1.495355^{1/4} = e^{\frac{1}{4} \log 5} \approx e^{\frac{1}{4} \times 1.60944} = e^{0.40236} \approx 1.49535

Thus:

r≈1.49535−1=0.49535r \approx 1.49535 - 1 = 0.49535

6. Convert to percentage.

r≈0.49535×100%=49.535%r \approx 0.49535 \times 100\% = 49.535\%

Watch out

A common mistake is to simply divide the total return by the number of years: 25000−50005000×4=100%\frac{25000-5000}{5000 \times 4} = 100\% per year. That ignores compounding entirely. CAGR is always less than the simple average return when growth is positive, because compounding accelerates growth — the same rate applied to a growing base yields a larger final amount.

✓Final answer

The CAGR is approximately 49.53% per annum.

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