Application of Logarithmic Functions in Investment Returns

This example intuitively demonstrates how logarithmic functions simplify complex compound growth problems, especially when calculating the annualized rate of return in finance.

1. Traditional Calculation Method: Exponential Equation

Suppose you have an initial investment of P that grows to A after t years. You want to calculate the annualized rate of return, r. This can be expressed with an exponential equation:

$$A = P \times (1+r)^t$$

To solve for r, we need to algebraically rearrange the formula:

$$(1+r)^t = \frac{A}{P}$$ $$1+r = \left(\frac{A}{P}\right)^{\frac{1}{t}}$$ $$r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1$$

This formula involves a fractional exponent, which is very difficult to calculate manually without a calculator.

2. Using Logarithms to Simplify Calculations

Now, let's start with the same formula but simplify it using logarithms. We take the natural logarithm (ln) of both sides of the equation:

$$\ln(A) = \ln(P \times (1+r)^t)$$

Based on two core properties of logarithms: the logarithm of a product is the sum of the logarithms $\ln(xy) = \ln(x) + \ln(y)$ and the logarithm of a power is the exponent times the logarithm $\ln(x^y) = y \cdot \ln(x)$, we can simplify the formula:

$$\ln(A) = \ln(P) + \ln((1+r)^t)$$ $$\ln(A) = \ln(P) + t \cdot \ln(1+r)$$

Now, we solve for $\ln(1+r)$ with simple algebra:

$$\ln(A) - \ln(P) = t \cdot \ln(1+r)$$ $$\ln\left(\frac{A}{P}\right) = t \cdot \ln(1+r)$$ $$\ln(1+r) = \frac{\ln(A/P)}{t}$$

Here, a key mathematical approximation comes into play: when the rate of return r is very small, $\ln(1+r)$ is approximately equal to r. This is an important application of the Taylor series expansion and is widely used in finance for quick estimation.

Using this approximation, our annualized rate of return r can be approximated as:

$$r \approx \frac{\ln(A/P)}{t}$$

3. Interactive Comparison and Visualization

Below, you can adjust the parameters by dragging the sliders to observe the rates of return calculated by both methods and their corresponding growth curves. This will visually demonstrate the convenience of the logarithmic approximation method and the error it introduces.

1000
1500
5 years

Exact Rate of Return:

Logarithmic Approx. Rate:

Practice Question

You invest a principal of $1,000, and after 10 years, your investment grows to $1,350. Using the logarithmic approximation method, what is the approximate annualized rate of return (r)?

(Hint: $\ln(1.35) \approx 0.3$)

Solution

The logarithmic approximation formula for the annualized rate of return is:

$$r \approx \frac{\ln(A/P)}{t}$$

In this problem, we have the following values:

  • Principal $(P) = \$1,000$
  • Final Amount $(A) = \$1,350$
  • Time $(t) = 10$ years

First, calculate the ratio of the final amount to the principal:

$$\frac{A}{P} = \frac{1350}{1000} = 1.35$$

Next, substitute this value and the time into the formula. We are given the hint that $\ln(1.35) \approx 0.3$:

$$r \approx \frac{\ln(1.35)}{10} \approx \frac{0.3}{10}$$

Finally, perform the division to get the approximate rate:

$$r \approx 0.03$$

To express this as a percentage, multiply by 100:

$$r \approx 0.03 \times 100\% = 3.0\%$$

Therefore, the correct answer is B) 3.0%.