Mr. Naresh has bought 200 shares of City Look Company at ₹100 each in 2015. After selling them he has received ₹30,000 which accounts for 22.47% CAGR. Calculate the number of years for which he was holding the shares.

Question

Accepted Answer

> [!TLDR]
> The CAGR formula $CAGR = \left(\frac{Final}{Initial}ight)^{1/n} - 1$ is used to find the holding period $n$. Given initial investment ₹20,000, final amount ₹30,000, and CAGR 22.47%, solving gives $n = 2$ years.

The key here is understanding what CAGR (Compound Annual Growth Rate) actually measures. It’s the constant annual rate at which an investment would have grown if it compounded smoothly over the entire period — ignoring any ups and downs in between. The formula is:

> [!FORMULA]
> $$CAGR = \left( \frac{Final\ Value}{Initial\ Value} ight)^{\frac{1}{n}} - 1$$

Where $n$ is the number of years. We know the CAGR, the initial and final values, and we need $n$. So we’ll rearrange the formula to solve for $n$.

Let’s work through it step by step.

1. **Find the initial investment.**  
   Mr. Naresh bought 200 shares at ₹100 each.  
   Initial value $= 200 	imes 100 = ₹20,000$.

2. **Identify the final amount.**  
   He received ₹30,000 after selling. So final value $= ₹30,000$.

3. **Write the CAGR equation.**  
   CAGR is given as 22.47%, which as a decimal is $0.2247$.  
   $$0.2247 = \left( \frac{30000}{20000} ight)^{\frac{1}{n}} - 1$$

4. **Simplify the ratio.**  
   $$\frac{30000}{20000} = 1.5$$  
   So the equation becomes:  
   $$0.2247 = (1.5)^{\frac{1}{n}} - 1$$

5. **Isolate the exponential term.**  
   Add 1 to both sides:  
   $$1.2247 = (1.5)^{\frac{1}{n}}$$

6. **Take natural logarithms on both sides.**  
   This is the cleanest way to bring the exponent $1/n$ down.  
   $$\log(1.2247) = \frac{1}{n} \cdot \log(1.5)$$

7. **Compute the logs (or use given values).**  
   You can use approximate values: …

Q.Mr. Naresh has bought 200 shares of City Look Company at ₹100 each in 2015. After selling them he has received ₹30,000 which accounts for 22.47% CAGR. Calculate the number of years for which he was holding the shares.

Concept understanding — Compound Annual Growth Rate

Compound Annual Growth Rate (CAGR)

The Intuition First

The Precise Statement

Applying It to Our Example

Why CAGR Matters in Exams

Step 1 — identify endpoints and periods

Step 2 — apply the CAGR formula

Step 3 — evaluate

Step 1 — CAGR formula

Step 2 — Compute the growth ratio

Step 3 — Take the 5th root

Step 1 — Form the growth ratio

Step 2 — Apply the CAGR formula

Step 3 — Subtract 1

Step 1 — Write the CAGR relation

Step 2 — Take logarithms

Step 3 — Compute

Step 1 — Set up the CAGR relation

Step 2 — Halve the exponent

Step 3 — Match the given value

Step 1 — Use only the endpoints

Step 2 — Apply the CAGR formula

Step 3 — Compute

Step 1 — Identify begin, end and n

Step 2 — Substitute

Step 3 — Evaluate the fourth root

Q.Mr. Naresh has bought 200 shares of City Look Company at ₹100 each in 2015. After selling them he has received ₹30,000 which accounts for 22.47% CAGR. Calculate the number of years for which he was holding the shares.

🎓Unlock everything free for 14 days

Unlock everything free for 14 days