Question

Use a spreadsheet to complete the table, which shows the time \(t\) necessary for \(P\) dollars to triple if the interest is compounded continuously at the rate of \(r\). $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% & 14 \% \\ \hline t & & & & & & & \\ \hline \end{array} $$

Short answer

The time it takes for the initial amount to triple under continuous compounding depends on the interest rate: for 2%, it takes approx. 54.59 years; for 4%, approx. 27.29 years; for 6%, approx. 18.20 years; for 8%, approx. 13.65 years; for 10%, approx. 10.92 years; for 12%, approx. 9.10 years; and for 14%, approx. 7.78 years.

Step-by-step solution

Computation for r = 2%

Substitute 0.02 for r in the formula, as the formula needs the rate in decimal form. The calculation is \(t = \frac{1}{0.02} * log(3) \approx 54.59\) years.

Computation for r = 4%

Substitute 0.04 for r in the formula and solve for t. The formula is \(t = \frac{1}{0.04} * log(3) \approx 27.29\) years.

Computation for r = 6%

Substitute 0.06 for r in the formula and solve for t. As per our computation, \(t = \frac{1}{0.06} * log(3) \approx 18.20\) years.

Computation for r = 8%

Substitute 0.08 for r in the formula and solve for t. The calculation is \(t = \frac{1}{0.08} * log(3) \approx 13.65\) years.

Computation for r = 10%

Substitute 0.1 for r in the formula and solve for t. This gives \(t = \frac{1}{0.1} * log(3) \approx 10.92\) years.

Computation for r = 12%

Substitute 0.12 for r in the formula and solve for t. This gives \(t = \frac{1}{0.12} * log(3) \approx 9.10\) years.

Computation for r = 14%

Substitute 0.14 for r in the formula and solve for t. This gives \(t = \frac{1}{0.14} * log(3) \approx 7.78\) years.

Key concepts

Continuous Compounding

Continuous compounding is an aspect of compounding interest where the frequency of compounding is essentially infinite. Instead of calculating interest at regular intervals such as monthly or annually, continuous compounding calculates interest in every infinitesimal moment in time. To understand this, imagine your interest being calculated at every possible instant—this is the core of continuous compounding.

Mathematically, continuous compounding is represented by the formula \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), with continuous compounding at an interest rate \( r \), and \( P \) is the principal amount. The number \( e \) is the base of the natural logarithm, approximately equal to 2.71828, and it's here where we see the profound connection between continuous compounding and the next concept, exponential growth.

Time Value of Money

The Time Value of Money (TVM) is a financial principle stating that money available now is worth more than the identical sum in the future due to its potential earning capacity. This core concept is the basis for finance and investing; it's the reason people invest or deposit money. TVM is intrinsically linked to the concept of interest—the reward for forgoing the use of money now.

Continuous compounding magnifies this by assuming that the money can earn interest continuously, leading to faster growth compared to standard compounding. TVM is crucial for understanding why and how investments grow over time, and why the rate of return (interest rate) and time are essential components in growing wealth.

Exponential Growth

Exponential growth refers to an increase that is constant in percentage but accelerates in absolute terms over time. It's a process by which the quantity grows by multiplying by a fixed number (greater than one) at fixed time intervals, such as population growth and radioactive decay.

For investments, money grows exponentially with compound interest because as the money earns interest, that interest then earns interest itself, and so on. This leads to the growth curving upwards steeply as time passes. When it comes to continuous compounding, the function \( A = Pe^{rt} \) perfectly describes this type of growth, indicating how investment balances swell exponentially as time and the rate of interest increases.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is used to determine the time taken for an investment to reach a certain level under continuous compounding. In simple terms, a logarithm answers the question: 'To what exponent must we raise a certain base to obtain a given number?'

In the context of our problem, the logarithmic function allows us to solve for \( t \), the time it takes for the initial amount to reach a target amount under continuous compounding. The formula used in the problem is \( t = \frac{1}{r} * log(A/P) \), where \( A/P \) is the ratio of the amount accumulated to the principal, and here, it's 3 since we're looking at how long it takes to triple the investment. This is where we use a logarithmic function to 'unpack' the exponential growth equation and solve for time, illustrating the clear connection between logarithms and exponential growth in financial contexts.