Suppose a person invested ₹15,000 in a mutual fund and the value of investment at the time of redemption was ₹25000. If CAGR for this investment is 8.88%, calculate the number of years for which he has invested the amount?

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> [!TLDR]
> The number of years is found by rearranging the CAGR formula $CAGR = \left(\frac{FV}{PV}ight)^{\frac{1}{n}} - 1$ and solving for $n$ using logarithms. The investment period is **5 years**.

The Compound Annual Growth Rate (CAGR) is the single rate that describes the year-over-year growth of an investment over a specified period, assuming profits are reinvested at the end of each year. It smooths out volatility and gives a "geometric average" annual return.

The core idea is that the initial principal ($PV$) grows at a constant annual rate $r$ for $n$ years to reach the final value ($FV$). This is exactly the compound interest formula:

$$FV = PV 	imes (1 + r)^n$$

Here, $r$ is the CAGR expressed as a decimal. We know $PV$, $FV$, and $r$, and we need $n$. Since $n$ is an exponent, we'll use logarithms to bring it down.

Let's work through it step by step.

1.  **Identify the known values.**  
    Present Value (investment amount), $PV = ₹15,000$  
    Future Value (redemption amount), $FV = ₹25,000$  
    CAGR, $r = 8.88\% = 0.0888$  
    Number of years, $n = ?$

2.  **Write the CAGR formula.**  
    The relationship is:
    $$FV = PV 	imes (1 + r)^n$$

3.  **Substitute the known values.**
    $$25,000 = 15,000 	imes (1 + 0.0888)^n$$
    $$25,000 = 15,000 	imes (1.0888)^n$$

4.  **Isolate the exponential term.**  
    Divide both sides by 15,000:
    $$\frac{25,000}{15,000} = (1.0888)^n$$
    $$\frac{5}{3} = (1.0888)^n$$
    $$1.6667 \approx (1.0888)^n$$

5.  **Apply logarithms to solve for $n$.**  
    Taking the natural logarithm (or log base 10 — any base works as long as you're consistent) on both sides:
    $$\ln\left(\frac{5}{3}ight) = \ln\left((1.0888)^night)$$
    Using the power rule of logarithms ($\ln(a^b) = b \cdot \ln(a)$):
    $$\ln\left(\frac{5}{3}ight) = n \cdot \ln(1.0888)$$

6.  **Solve for $n$.**
    $$n = \frac{\ln\left(\frac{5}{3}ight)}{\ln(1.0888)}$$

Now compute the values.  
    $\ln(5/3) = \ln(1.6667) \approx 0.

Q.Suppose a person invested ₹15,000 in a mutual fund and the value of investment at the time of redemption was ₹25000. If CAGR for this investment is 8.88%, calculate the number of years for which he has invested the amount?

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.Suppose a person invested ₹15,000 in a mutual fund and the value of investment at the time of redemption was ₹25000. If CAGR for this investment is 8.88%, calculate the number of years for which he has invested the amount?

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