Problem 76 - Contemporary Precalculus

Question

The doubling function \(D(x)=\frac{\ln 2}{\ln (1+x)}\) gives the years required to double your money when it is invested at interest rate \(x\) (expressed as a decimal), compounded annually. (a) Find the time it takes to double your money at each of these interest rates: \(4 \%, 6 \%, 8 \%, 12 \%, 18 \%, 24 \%\) \(36 \%\) (b) Round the answers in part (a) to the nearest year and compare them with these numbers: \(72 / 4,72 / 6,72 / 8\) \(72 / 12,72 / 18,72 / 24,72 / 36 .\) Use this evidence to state a rule of thumb for determining approximate doubling time, without using the function \(D .\) This rule of thumb, which has long been used by bankers, is called the rule of 72 .

Step-by-Step Solution

Answer

Answer: The "rule of 72" is a rule of thumb used to approximate the doubling time for a given interest rate. To use the rule, divide 72 by the interest rate (as a percentage) to get the approximate doubling time in years. For the given interest rates, the rule of 72 provides a close approximation of the actual doubling times calculated using the more complex doubling function, D(x) = ln(2) / ln(1+x).

Step 1: Calculate Doubling Time for Each Interest Rate

For each interest rate, we need to input the value into the doubling function and calculate the doubling time in years. Note that the interest rate should be expressed as a decimal. For example, for an interest rate of \(4\%\), we should use \(0.04\). For each interest rate, we will use the following formula to solve: $$D(x) = \frac{\ln 2}{\ln (1+x)},$$ where x = interest rate (as a decimal).

Step 2: Calculate Doubling Time for \(4\%\) Interest Rate

Calculate \(D(0.04)\): $$D(0.04) = \frac{\ln 2}{\ln (1+0.04)} \approx 17.67$$ Doubling time at \(4\%\): approximately \(17.67\) years.

Step 3: Calculate Doubling Time for \(6\%\) Interest Rate

Calculate \(D(0.06)\): $$D(0.06) = \frac{\ln 2}{\ln (1+0.06)} \approx 11.90$$ Doubling time at \(6\%\): approximately \(11.90\) years.

Step 4: Calculate Doubling Time for \(8\%\) Interest Rate

Calculate \(D(0.08)\): $$D(0.08) = \frac{\ln 2}{\ln (1+0.08)} \approx 9.01$$ Doubling time at \(8\%\): approximately \(9.01\) years.

Step 5: Calculate Doubling Time for \(12\%\) Interest Rate

Calculate \(D(0.12)\): $$D(0.12) = \frac{\ln 2}{\ln (1+0.12)} \approx 6.12$$ Doubling time at \(12\%\): approximately \(6.12\) years.

Step 6: Calculate Doubling Time for \(18\%\) Interest Rate

Calculate \(D(0.18)\): $$D(0.18) = \frac{\ln 2}{\ln (1+0.18)} \approx 4.19$$ Doubling time at \(18\%\): approximately \(4.19\) years.

Step 7: Calculate Doubling Time for \(24\%\) Interest Rate

Calculate \(D(0.24)\): $$D(0.24) = \frac{\ln 2}{\ln (1+0.24)} \approx 3.25$$ Doubling time at \(24\%\): approximately \(3.25\) years.

Step 8: Calculate Doubling Time for \(36\%\) Interest Rate

Calculate \(D(0.36)\): $$D(0.36) = \frac{\ln 2}{\ln (1+0.36)} \approx 2.32$$ Doubling time at \(36\%\): approximately \(2.32\) years. ##Part (b):##

Step 9: Round the Doubling Times and Compare with Provided Numbers

We will round the calculated doubling times to the nearest year and compare them with the numbers provided (\(72/4, 72/6, 72/8, 72/12, 72/18, 72/24, 72/36\)). To round to the nearest year: - \(4\%: 18\) years (vs. \(72/4 = 18\)) - \(6\%: 12\) years (vs. \(72/6 = 12\)) - \(8\%: 9\) years (vs. \(72/8 = 9\)) - \(12\%: 6\) years (vs. \(72/12 = 6\)) - \(18\%: 4\) years (vs. \(72/18 = 4\)) - \(24\%: 3\) years (vs. \(72/24 = 3\)) - \(36\%: 2\) years (vs. \(72/36 = 2\)) All the rounded doubling times match the provided numbers.

Step 10: State the Rule of Thumb

Since the rounded doubling times are matching the numbers provided, we can observe a pattern. The evident pattern is that the provided numbers are all multiples of \(72\). Based on this evidence, we can state the rule of thumb for approximate doubling time: Divide \(72\) by the interest rate (as a percentage) to get the approximate doubling time in years. This rule of thumb is called the "rule of 72".

Key Concepts

Doubling Time

Understanding the concept of 'doubling time' is essential when dealing with investments or any context where growth is involved. Doubling time refers to the period it takes for a quantity to double in size or value at a constant growth rate. When dealing with financial investments, the doubling time can give investors a clear view of how quickly their money will grow.

To calculate doubling time in the context of compound interest, we often use the rule of 72, a simple formula that estimates the number of years required to double the invested money at a fixed annual rate of return. You simply divide 72 by the interest rate percentage. This approximation is powerful because it does not require detailed calculations and can be done without a calculator, providing a quick and easy method to understand the growth of your investment over time.

Compound Interest

The concept of 'compound interest' is a fundamental financial principle, crucial for understanding investment growth. It works on the premise that interest is not only calculated on the initial principal amount but also on the accumulated interest over previous periods.

As a result, compound interest grows exponentially over time, as each interest payment adds to the initial principal, creating a snowball effect. The formula to calculate compound interest involves the principal amount, the annual interest rate, the number of times that interest is compounded per year, and the total number of years the money is invested. This exponential increase is why understanding compound interest is vital for long-term financial planning and illustrates the power of time in growing investments.

Exponential Functions

At the heart of both doubling time and compound interest lies 'exponential functions'. These functions are mathematical expressions where a constant base is raised to a variable exponent.

In the context of compound interest, exponential functions describe how investment balances grow over time. Unlike linear growth, where increases by a constant amount, exponential growth accelerates as time progresses, illustrating a rapid increase in growth potential. Financial models leverage exponential functions to predict investment performance, assess compound interest outcomes, and understand how doubling time impacts the growth of money. These functions are not just limited to finance but are also seen in population growth, radioactive decay, and other natural phenomena.

Logarithms

The notion of 'logarithms' is tightly interwoven with exponential functions and is fundamental when solving for variables in exponential equations such as the doubling function. A logarithm answers the question, 'to what exponent must the base be raised, to produce a given number?'.

Within financial contexts, logarithms allow us to isolate the interest rate or time, given a certain rate of growth. In the step-by-step solution presented for calculating doubling time, logarithms were used to reverse-engineer the interest rate, revealing how long it would take for an investment to double. This type of calculation is essential for financial planning, as it gives investors the ability to predict future value of investments. Logarithms thus serve as a bridge between the theoretical and the practical applications of exponential growth in real-world scenarios.