Question

Compound Interest In Exercises 49 and 50 , find the time necessary for \(\$ 1000\) to double when it is invested at a rate of \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$ r=7 \% $$

Short answer

The time necessary for $1000 to double when it is invested at a rate of 7% is approximately (a) 10.24 years if interest is compounded annually, (b) 9.99 years if compounded monthly, (c) 9.96 years if compounded daily, and (d) 9.90 years if compounded continuously.

Step-by-step solution

Determine the rate in decimal form

First, convert the annual interest rate from percentage to decimal form by dividing by 100. Thus, \(r = 7/100 = 0.07\).

Calculate the time for annually compounded interest

Substitute \(P = 1000\), \(A = 2000\), \(r = 0.07\), and \(n = 1\) into the compound interest formula, then solve for \(t\). This gives \[t = log(2) / (1 * log(1 + 0.07))\]. Calculate the value to get approximately 10.24 years.

Calculate the time for monthly compounded interest

Substitute \(P = 1000\), \(A = 2000\), \(r = 0.07\), and \(n = 12\) into the compound interest formula, then solve for \(t\). This gives \[t = log(2) / (12 * log(1 + 0.07/12))\]. Calculate the value to get approximately 9.99 years.

Calculate the time for daily compounded interest

Substitute \(P = 1000\), \(A = 2000\), \(r = 0.07\), and \(n = 365\) into the compound interest formula, then solve for \(t\). This gives \[t = log(2) / (365 * log(1 + 0.07/365))\]. Calculate the value to get approximately 9.96 years.

Calculate the time for continuously compounded interest

Substitute \(P = 1000\), \(A = 2000\), and \(r = 0.07\) into the continuous compound interest formula \(A = Pe^{rt}\), then solve for \(t\). This gives \[t = log(2) / 0.07\]. Calculate the value to get approximately 9.90 years.

Key concepts

Time Value of Money

Understanding the time value of money (TVM) is pivotal in grasping how compound interest works. At its core, TVM is the concept that a sum of money has different values at different points in time due to its potential earning capacity. This principle suggests that money available now is worth more than the same amount in the future because of its ability to earn interest.

Consider you have \(1,000 today. You can invest this sum and earn interest over time, meaning that \)1,000 could grow to \(2,000 or more in future years. If you receive \)1,000 in ten years instead, that money has not been working for you - it hasn't been earning interest - and is, therefore, not as valuable as having the money today. In financial calculations, we use formulas to determine exactly how much a present sum of money will be in the future, taking into account the interest it will accumulate over time.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a curve on a graph that becomes steeper and steeper. In terms of compound interest, it's the phenomenon where the amount of interest earned itself earns interest, resulting in the amount growing at an accelerating rate.

For example, if you invest your money at an interest rate that compounds, not only does your initial amount earn interest, but the earned interest earns interest in following periods as well. This creates a snowball effect where your investment grows exponentially rather than linearly. The longer the time, the more dramatic the compounding effect becomes, as the interest continues stacking upon itself and increasing the total amount at an ever-faster rate.

Compound Interest Formula

The compound interest formula is essential for calculating how much interest an investment will earn over a period when the interest is compounded. The formula is expressed as
\[ A = P \times (1 + \frac{r}{n})^{nt} \]
where:

• \( A \) is the future value of the investment/loan, including interest
• \( P \) is the principal amount (the initial amount of money)
• \( r \) is the annual interest rate (in decimal form)
• \( n \) is the number of times that interest is compounded per year
• \( t \) is the time the money is invested or borrowed for, in years

By adjusting the compounding frequency (\( n \times t \)), you can see how interest compounds annually, monthly, daily, or continuously. The formula serves as a powerful tool to visualize how both the compounding frequency and time affect the growth of an investment.

Continuously Compounded Interest

Continuously compounded interest is a term used when interest is considered to be compounded infinitely. In other words, it assumes that the interest is constantly being calculated and added to the principal at every possible moment.

The formula to compute the future value of an investment with continuously compounded interest is represented as
\[ A = Pe^{rt} \]
where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount.
• \( e \) is Euler's number (approximately equal to 2.71828).
• \( r \) is the annual interest rate as a decimal.
• \( t \) is the time the money is invested in years.

This model assumes that the peaks and falls happen so frequently that they become a smooth exponential curve. The continuous compounding formula is powerful because it shows the upper limit of compound interest - where your investment's growth is maximized.