Q16
1 markMCQSection A

If an investment of Rs 10,000 becomes Rs 60,000 in 4 years, then the Compound Annual Growth Rate (CAGR) is :

Financial Mathematics
Compound Annual Growth Rate (CAGR)

Options

(A)64−1100\frac{\sqrt[4]{6} - 1}{100}
(B)64+1100\frac{\sqrt[4]{6} + 1}{100}
(C)(64−1)×100(\sqrt[4]{6} - 1) \times 100
(D)(64+1)×100(\sqrt[4]{6} + 1) \times 100
Official Answer

The correct option is (C) (64−1)×100(\sqrt[4]{6} - 1) \times 100.


Calculation

  • CAGR = [(Final/Initial)1/n−1]×100[(\text{Final}/\text{Initial})^{1/n} - 1] \times 100.
  • Final/Initial = 60000/10000=660000/10000 = 6, n=4n = 4.
  • CAGR = (61/4−1)×100(6^{1/4} - 1) \times 100 = (64−1)×100(\sqrt[4]{6} - 1) \times 100 %.
CAGRcompound annual growth ratefourth root of 6final by initialsubtract 1 times 100percentage growthfinancial mathematicsinvestment growth

Marking Scheme

  • 11 mark: correct option (C) (64−1)×100(\sqrt[4]{6} - 1) \times 100.
  • 2No marks for options that divide by 100 or add 1 instead of subtracting.

Hint

CAGR = [(final/initial)(1/years)−1]×100[(\text{final}/\text{initial})^{(1/\text{years})} - 1] \times 100.

Quick Oral Answer

CAGR is the ratio of final to initial value raised to the power one over the number of years, minus one, multiplied by a hundred, giving the fourth root of six minus one times a hundred.

Analysis & Explanation

CAGR is the constant yearly rate that grows the initial value to the final value over the given number of years.


Concept

  • CAGR = [(End Value/Beginning Value)1/n−1]×100[(\text{End Value} / \text{Beginning Value})^{1/n} - 1] \times 100, expressed as a percentage.

Working

  • Ratio = 60000/10000=660000/10000 = 6.
  • Number of years n=4n = 4.
  • CAGR = (61/4−1)×100(6^{1/4} - 1) \times 100 = (64−1)×100(\sqrt[4]{6} - 1) \times 100 %.

Why the key is right

  • (C) uses the correct fourth-root ratio, subtracts 1, and multiplies by 100 to give a percentage.

Why the distractors are wrong

  • (A) divides by 100 instead of multiplying, giving a value far too small.
  • (B) also divides by 100 and adds 1 (should subtract).
  • (D) adds 1 instead of subtracting, which would give a growth rate above 100% incorrectly.

Common Mistakes

  1. 1Adding 1 instead of subtracting 1 inside the bracket.
  2. 2Dividing by 100 rather than multiplying to convert to a percentage.
  3. 3Using the arithmetic average growth instead of the compound (geometric) growth formula.

Interesting Facts

CAGR smooths out volatility by reporting one steady growth rate, which is why it is the standard metric for comparing mutual-fund and business returns over multiple years.

Because CAGR uses a geometric mean, it always equals or is below the simple average of yearly returns whenever the returns fluctuate.

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Frequently Asked Questions

Why is CAGR multiplied by 100 and not divided?

The bracket (ratio1/n−1)(\text{ratio}^{1/n} - 1) gives the growth as a decimal fraction. Multiplying by 100 converts it to a percentage. Dividing by 100 would make it a hundred times too small.

Why subtract 1 in the CAGR formula?

The term (final/initial)1/n(\text{final}/\text{initial})^{1/n} gives the growth factor per year (e.g. 1.5 means 50% growth). Subtracting 1 removes the original principal portion, leaving just the growth rate.