Question

Rule of 70 Bankers use the Rule of 70 . which says that if an account increases at a fixed rate of \(p \% / \mathrm{yr}\), its doubling time is approximately \(70 / p .\) Use linear approximation to explain why and when this is true.

Short answer

Question: Explain the Rule of 70 using linear approximation. Answer: The Rule of 70 states that if an account increases at a fixed rate of \(p\%/\text{yr}\), its doubling time is approximately \(70/p\). To explain this using linear approximation, we first analyze the exponential growth formula for an account and then apply a linear approximation using the first-order Taylor series expansion. After substituting our linear approximation into the doubling time equation, we find that the doubling time \(t\) is approximately equal to \(\frac{100\ln(2)}{p} \approx \frac{70}{p}\). This shows that the Rule of 70 is valid and provides a decent approximation of doubling time when the interest rate, \(p\), is relatively small.

Step-by-step solution

1. Exponential Growth Formula

The value of an account that grows at a fixed annual interest rate of \(p\%\) is given by the exponential growth formula \(A(t) = A(0)(1+\frac{p}{100})^t\), where \(A(t)\) is the value of the account at time \(t\) years and \(A(0)\) is the initial value. To estimate the time it takes for an account to double, we must solve for \(t\) in the equation \(A(t) = 2A(0)\), while taking into account the interest rate \(p\).

2. Natural Logarithm Transformation

To solve for \(t\) in the equation \(2A(0) = A(0)(1+\frac{p}{100})^t\), first cancel out the initial value \(A(0)\) on both sides, which gives us \((1+\frac{p}{100})^t = 2\). Then, apply the natural logarithm to both sides: \(\ln ((1+\frac{p}{100})^t) = \ln(2)\). This simplifies to \(t\cdot\ln (1+\frac{p}{100}) = \ln(2)\).

3. Linear Approximation using the first-order Taylor series expansion

Recall that the linear approximation of a function \(f(x)\) around \(x=a\) is \(L(x) = f(a) + f'(a)(x - a)\). We want to find the linear approximation of \(\ln (1+\frac{p}{100})\) around \(p = 0\). Let \(f(x) = \ln (1 + \frac{x}{100})\) with \(a=0\). In this case, \(f(a)=\ln (1 + \frac{0}{100}) = \ln(1) = 0\). Now, find the derivative of \(f(x)\): \(f'(x) = \frac{\text{d}(\ln (1+\frac{x}{100}))}{\text{d}x} = \frac{1}{1+\frac{x}{100}} \cdot \frac{1}{100} = \frac{1}{100 + x}\). At \(x=0\), \(f'(0) = \frac{1}{100}\). The linear approximation then becomes \(L(x) = f(0) + f'(0)(x - 0)\), which simplifies to \(L(p) = 0 + \frac{1}{100}(p - 0) = \frac{p}{100}\).

4. Applying the Linear Approximation to the Doubling Time Equation

Now, let's substitute our linear approximation for \(\ln (1+\frac{p}{100})\) in the doubling time equation. Recall that in the second step, we found \(t\cdot\ln (1+\frac{p}{100}) = \ln(2)\). Replace \(\ln (1+\frac{p}{100})\) with our linear approximation, obtaining \(t\cdot\frac{p}{100} = \ln(2)\).

5. Solving for Doubling Time

Solve for \(t\) in the equation \(t\cdot\frac{p}{100} = \ln(2)\). Divide both sides of the equation by \(\frac{p}{100}\), obtaining \(t = \frac{\ln(2)}{\frac{p}{100}}\), which further simplifies to \(t = \frac{100\ln(2)}{p}\). Since \(\ln(2)\approx 0.693\), the doubling time can be approximated as \(t \approx \frac{100(0.693)}{p} \approx \frac{70}{p}\), which shows that the Rule of 70 is valid and provides a decent approximation of doubling time when the interest rate, \(p\), is relatively small.

Key concepts

Exponential Growth Formula

Understanding the Exponential Growth Formula is crucial when dealing with investments or savings that increase at a specific percentage rate over time. It is represented by the equation \(A(t) = A(0)(1+\frac{p}{100})^t\), where \(A(t)\) is the future value of the account, \(A(0)\) is the initial value, \(p\) is the annual growth rate in percent, and \(t\) is the time in years.

The formula reflects how, over time, the initial amount is compounded, growing faster as time passes, which is characteristic of exponential behaviors. When applied to financial contexts, this concept allows us to determine how long it will take for an investment to reach a certain value; for example, calculating the doubling time of an investment.

Natural Logarithm Transformation

The Natural Logarithm Transformation is a mathematical technique used to linearize exponential equations, which makes them easier to handle. By taking the natural logarithm of both sides of our growth equation \((1+\frac{p}{100})^t = 2\), we transform it into \(t \cdot \ln(1+\frac{p}{100}) = \ln(2)\).

This is a critical step towards solving for \(t\), as the logarithmic function is the inverse of the exponential function, allowing us to untangle the variable \(t\) from the exponent. This transformation is used in various disciplines, including economics and biology, to analyze growth rates and other phenomena that exhibit exponential behavior.

Linear Approximation

Linear Approximation is a mathematical method to estimate the value of a function near a given point using its derivative. It is based on the First-order Taylor Series Expansion, where the approximation of a function \(f(x)\) around a specific point \(a\) is given by \(L(x) = f(a) + f'(a)(x - a)\).

In the context of the Rule of 70, linear approximation is used to simplify the natural logarithm of \(1 + \frac{p}{100}\) when \(p\) is small. By using the function's value and its slope at the point of interest, we create a simpler linear relationship that is easier to solve. This technique is essential in fields that require quick, real-world estimations where exact calculations are complex or unnecessary.

First-order Taylor Series Expansion

The First-order Taylor Series Expansion provides a foundation for the linear approximation method. It approximates a function around a point \(a\) using the function's value and its first derivative at that point. Mathematically, it is expressed as the function's value plus the product of its derivative and the difference between the variable and the point \(L(x) = f(a) + f'(a)(x - a)\).

This expansion is vital for creating a simplified version of more complex functions, such as the natural logarithm function we encounter in exponential growth scenarios. It reduces a non-linear equation to a linear one, which significantly simplifies the process of finding solutions to the equation.

Doubling Time Calculation

Doubling Time Calculation is a concept used to measure the time required for a quantity to double in size or value at a constant growth rate. In financial terms, it helps investors understand how long it will take for their investment to grow to twice its original amount, given a fixed interest rate. This calculation becomes straightforward after applying the Rule of 70, which is derived using linear approximation of the logarithmic function.

After transforming the exponential growth equation and applying linear approximation, the doubling time is found to be approximately \(t = \frac{100 \cdot \ln(2)}{p}\), where \(\ln(2)\) is a constant approximately equal to 0.693. Therefore, for small interest rates, \(t\) can be approximated by \(\frac{70}{p}\), making the Rule of 70 a handy tool for quick estimations.