Problem 68 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

RULE OF 70 Investors are often interested in knowing how long it takes for a particular investment to double. A simple means for making this determination is the "rule of 70 ," which says: The doubling time of an investment with an annual interest rate \(r \%\) compounded continuously is given by \(d=\frac{70}{r}\). a. For interest rate \(r\), use the formula \(B=P e^{r t}\) to find the doubling time for \(r=4,6,9,10\), and 12. In each case, compare the value with the value obtained from the rule of 70 . b. Some people prefer a "rule of \(72^{*}\) and others use a "rule of 69 ." Test these alternative rules as in part (a), and write a paragraph on which rule you would prefer to use.

Step-by-Step Solution

Answer

The Rule of 70 closely matches the exact calculations and can be used reliably. The Rule of 69 may be slightly more precise for lower rates, while the Rule of 72 underestimates doubling times.

Step 1 - Use the formula for continuous compounding

The formula for continuous compounding of an investment is given by \(B = P e^{rt}\), where \(B\) is the final amount, \(P\) is the principal amount, \(r\) is the interest rate, and \(t\) is time. To find the doubling time, set \(B = 2P\) and solve for \(t\).

Step 2 - Set up the equation for doubling

Using \(B = 2P\), the equation becomes \(2P = P e^{rt}\). Divide both sides by \(P\) to get \(2 = e^{rt}\).

Step 3 - Solve for time \(t\)

Take the natural logarithm of both sides to get \(\ln(2) = rt\). Thus, the doubling time \(t\) is \(t = \frac{\ln(2)}{r}\).

Step 4 - Compare with the Rule of 70

The Rule of 70 states that the doubling time \(d\) is given by \(d = \frac{70}{r}\). Calculate \(\frac{\ln(2)}{r}\) for each given rate and compare it with \(\frac{70}{r}\).

Step 5 - Calculate doubling time for \(r = 4\)

Using \(t = \frac{\ln(2)}{0.04}\), \(t \approx 17.33\) years. Rule of 70 gives \(\frac{70}{4} = 17.5\) years.

Step 6 - Calculate doubling time for \(r = 6\)

Using \(t = \frac{\ln(2)}{0.06}\), \(t \approx 11.55\) years. Rule of 70 gives \(\frac{70}{6} \approx 11.67\) years.

Step 7 - Calculate doubling time for \(r = 9\)

Using \(t = \frac{\ln(2)}{0.09}\), \(t \approx 7.7\) years. Rule of 70 gives \(\frac{70}{9} \approx 7.78\) years.

Step 8 - Calculate doubling time for \(r = 10\)

Using \(t = \frac{\ln(2)}{0.10}\), \(t \approx 6.93\) years. Rule of 70 gives \(\frac{70}{10} = 7\) years.

Step 9 - Calculate doubling time for \(r = 12\)

Using \(t = \frac{\ln(2)}{0.12}\), \(t \approx 5.78\) years. Rule of 70 gives \(\frac{70}{12} \approx 5.83\) years.

Step 10 - Test alternative rules

Alternative Rule of 72: \(\frac{72}{r}\). For \(r = 4\): \(18\), \(r = 6\): \(12\), \(r = 9\): \(8\), \(r = 10\): \(7.2\), \(r = 12\): \(6\). Alternative Rule of 69: \(\frac{69}{r}\). For \(r = 4\): \(17.25\), \(r = 6\): \(11.5\), \(r = 9\): \(7.67\), \(r = 10\): \(6.9\), \(r = 12\): \(5.75\).

Step 11 - Compare and conclude

The Rule of 70 is very close to the calculated values for all interest rates. The Rule of 69 is slightly more accurate for lower rates, while the Rule of 72 underestimates doubling time. Preferential use might involve considering which rule gives the closest approximation for the specific interest rates of interest.

Key Concepts

Doubling Time

Doubling time is a crucial concept in finance and growth analytics. It refers to the amount of time it takes for an investment or quantity to double in size. The formula used to calculate the doubling time when interest is compounded continuously is derived from the natural exponential growth equation: \(B = Pe^{rt}\). Here, \(B\) represents the final amount, \(P\) is the principal amount, \(r\) is the annual interest rate, and \(t\) is time.
To find the doubling time, we set \(B = 2P\) and solve for \(t\), leading to the equation \(\frac{\text{ln}(2)}{r}\). This calculation gives us an accurate measure of how long it takes for an investment to double. Another simpler method is the Rule of 70, which approximates doubling time by dividing 70 by the annual interest rate. For example, if the interest rate is 5%, the doubling time is roughly \(\frac{70}{5} = 14\) years.

Continuous Compounding

Continuous compounding refers to the mathematical limit where interest is calculated and added to the principal balance an infinite number of times per year. The formula for an investment compounded continuously is \(B = Pe^{rt}\), where

• \(B\) is the final amount.
• \(P\) is the principal amount.
• \(r\) is the annual interest rate.
• \(t\) is the time in years.

By using continuous compounding, an investment grows at the highest possible rate because interest is being calculated constantly. This can be extremely useful for calculating expected returns over a long period.

Interest Rate Calculation

Calculating the interest rate is essential for investment planning. The interest rate helps determine how much money an investment will yield over a specific period. There are different interest rate types:

• **Simple Interest**: Not common in high growth investments; calculated as \(I = Prt\).
• **Compounded Interest**: More common and practical; can be calculated quarterly, monthly, or even continuously.

Continuous compounding is the most aggressive form, represented as \(B = Pe^{rt}\). This calculation assumes money is compounded an infinite number of times per period, resulting in maximum growth.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are essential in solving for time in doubling time problems. For continuous compounding, the doubling equation \(2P = Pe^{rt}\) simplifies to \(2 = e^{rt}\). To isolate \(t\), we use the natural logarithm function: \(t = \frac{\text{ln}(2)}{r}\).
This method leverages the properties of logarithms to simplify complex exponential equations, helping us easily find the required time for investments or quantities to grow by a specific factor.