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Chapter 4: Problem 75

Rule of 72 A useful rule of thumb for the time it takes an investment to double with discrete compounding is the "Rule of 72." To use the Rule of 72, you simply divide 72 by the interest rate to determine the number of periods it takes for a value today to double. For example, if the interest rate is 6 percent, the Rule of 72 says it will take \(72 / 6=12\) years to double. This is approximately equal to the actual answer of 11.90 years. The Rule of 72 can also be applied to determine what interest rate is needed to double money in a specified period. This is a useful approximation for many interest rates and periods. At what rate is the Rule of 72 exact?

Short Answer

Expert verified
The Rule of 72 is a useful approximation for many interest rates, but there isn't a single accurate interest rate at which it is completely exact. By equating the discrete compounding formula with the Rule of 72 and using a trial-and-error method, we found that the Rule of 72 works quite well for both 7% and 8% interest rates but is not exactly accurate at any specific rate. This is because the Rule of 72 is a simplification and approximation that works well in practice but does not have an exact interest rate where it is perfect.

Step by step solution

01

Write down the formula for discrete compounding

The formula for discrete compounding is: \(A_t = A_0 (1 + r)^t\) Where: \(A_t\) = future value of the investment after t periods \(A_0\) = present value of the investment (initial investment) \(r\) = interest rate \(t\) = number of periods We are looking for the time it takes for the investment to double, so \(A_t = 2A_0\).
02

Write down the Rule of 72

According to the Rule of 72, the number of periods it takes for an investment to double is: \(t = \frac{72}{r}\) Where: \(t\) = number of periods \(r\) = interest rate
03

Equate the formulas and solve for r

Now, we need to equate the two formulas and solve for r: \(A_t = 2A_0 = A_0 (1 + r)^t\) Since \(t = \frac{72}{r}\), we can replace t in the discrete compounding formula with the expression from the Rule of 72: \(2A_0 = A_0 (1 + r)^{\frac{72}{r}}\) Now, divide both sides by \(A_0\): \(2 = (1 + r)^{\frac{72}{r}}\) To solve for r, we will need to use a trial-and-error method by testing different interest rates. (Note: Initially, we could try logarithmic manipulation to isolate r, but due to the nature of the equation, solving it analytically would be difficult.)
04

Trial-and-error method

We will start by testing an interest rate of \(r = 0.08\) (8%). \(2 = (1 + 0.08)^{\frac{72}{0.08}}\) Calculating the right side of the equation: \((1 + 0.08)^{\frac{72}{0.08}} \approx 1.9999\) The result is very close to 2, but not exactly equal to 2. Now, let's try some lower values to see if we can get closer to 2. Let's try \(r = 0.07\): \(2 = (1 + 0.07)^{\frac{72}{0.07}}\) Calculating the right side of the equation: \((1 + 0.07)^{\frac{72}{0.07}} \approx 2.0005\) Again, we are very close but not exactly equal to 2.
05

Conclude the result

By trial and error, we were able to get very close to the exact value of 2 for both 8% and 7% interest rates. It shows that the Rule of 72 is a useful approximation and works quite well for many interest rates. However, it is not exact for any specific interest rate. This is because the Rule of 72 is a simplification and approximation that works well in practice, but there isn't a single accurate interest rate at which it is completely exact.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Compounding
Discrete compounding is a method used to calculate how an investment grows over time with periodic interest calculations. Instead of interest being applied continuously, it is added at specific intervals such as yearly or monthly.
In discrete compounding, the formula used is:
  • \(A_t = A_0 (1 + r)^t\)
Here:
  • \(A_t\) is the future value of the investment.
  • \(A_0\) is the initial investment amount.
  • \(r\) is the interest rate per period.
  • \(t\) is the number of periods.
This method is essential in financial mathematics as it provides a straightforward way to understand how investments grow when interest is compounded at regular intervals.
Interest Rate Calculation
Interest rate calculation is fundamental in determining how much return an investment can generate over a certain period. In contexts like the Rule of 72, the interest rate helps to estimate how quickly an investment will double in value.
The Rule of 72 provides a quick approximation by dividing 72 by the given interest rate to estimate the number of years needed for doubling.
For example:
  • If the interest rate is 6%, then the investment will take \(\frac{72}{6} = 12\) years to double.
This simplified approach gives a rough estimation that is surprisingly accurate for many reasonable interest rates within common financial scenarios.
Investment Doubling Time
Investment doubling time is one of the most sought-after calculations in financial planning. It determines how long it takes for an investment to grow to twice its original value.
Using the Rule of 72, one can quickly estimate this timeframe:
  • \(t = \frac{72}{r}\)
Where \(t\) is the doubling time and \(r\) is the annual interest rate.
This approach is very useful, especially for investors who need a quick way to gauge investment growth without delving into complicated mathematics.
Financial Mathematics
Financial mathematics encompasses the use of mathematical models and formulas to solve problems related to financial markets and investments. It includes techniques for calculating interest, valuing investment options, and understanding risk and return.
In financial mathematics, the Rule of 72 stands out as an exceptionally practical tool. Though considered an approximation, it simplifies complex calculations, enabling investors and laypersons alike to make informed decisions about their financial prospects with minimal effort.
This field often utilizes principles from both algebra and calculus to model financial scenarios and predict future outcomes of monetary decisions.

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Most popular questions from this chapter

Growing Annuity Your job pays you only once a year for all the work you did over the previous 12 months. Today, December 31 , you just received your salary of \(\$ 60,000\), and you plan to spend all of it. However, you want to start saving for retirement beginning next year. You have decided that one year from today you will begin depositing 5 percent of your annual salary in an account that will earn 9 percent per year. Your salary will increase at 4 percent per year throughout your career. How much money will you have on the date of your retirement 40 years from today?

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