Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that after 1 year you have \(\$ 1000\) in the bank. If the interest was compounded continuously at \(5 \%\), how much money did you put in the bank one year ago? This is called the present value.

Step-by-Step Solution

Answer

You originally deposited approximately $951.23.

Step 1: Understand the Formula for Continuous Compounding

For continuous compounding, the formula to calculate the future value is given by \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount (initial deposit), \( r \) is the rate of interest, and \( t \) is the time in years.

Step 2: Plug in Given Values

We know from the problem that \( A = 1000 \), \( r = 0.05 \), and \( t = 1 \) year. Substitute these values into the continuous compounding formula: \( 1000 = Pe^{0.05 \times 1} \).

Step 3: Solve for the Principal \( P \)

The equation from Step 2 is \( 1000 = Pe^{0.05} \). To solve for \( P \), divide both sides by \( e^{0.05} \):\[ P = \frac{1000}{e^{0.05}} \]

Step 4: Calculate Using Exponentials

Calculate \( e^{0.05} \) using a calculator, which is approximately \( 1.05127 \). Substitute this back into the equation to find \( P \):\[ P = \frac{1000}{1.05127} \approx 951.23 \]

Step 5: Interpret the Result

The calculated principal \( P \approx 951.23 \) is the amount of money that was deposited in the bank one year ago.

Key Concepts

Present Value

The concept of Present Value is fundamental in finance because it helps us understand the worth of money deposited or invested before it grows through interest. Think of it as the way to "peek" into the past value that grows over time to what we currently have.
By knowing the present value, we can make informed decisions about investments. The formula for present value involves working backward using the interest already accrued. In our exercise, we found the principal amount, or the present value, by dividing the future value by the exponential part of the compounding formula.
This principal is the amount originally placed in the bank before interest worked its magic!

Exponential Functions

Exponential functions are a key player in the world of continuous compounding. They involve raising the mathematical constant "e" to the power of our interest rate times time. This function models situations of continuous growth, like bacteria growth or financial investments.
In our exercise, we use the exponential function to find how much a principal grows into a future amount. The formula we used, \( A = Pe^{rt} \), shows this relationship.
Breaking it down:

• "e" is approximately 2.71828, a constant that appears naturally in various growth processes.
• "r" is the rate expressed as a decimal.
• "t" is time in years over which growth occurs.

Understanding exponentials helps you grasp how investments grow continuously rather than just at discrete intervals.

Interest Rates

Interest Rates are a vital part of every financial transaction involving growth and compounding. They determine how much extra money we earn over time from our deposits or investments. Simply put, the interest rate is the cost of borrowing or the reward for saving.
When dealing with continuous compounding, interest is added an infinite number of times in very small amounts throughout the period. This process results in more growth than any other form of compounding.
In the given exercise, a 5% interest rate was used. In mathematical terms, this means multiplying the initial amount by an exponential factor that grows over time. By understanding how interest rates work, we can predict how various rates affect our investment outcomes.

Financial Mathematics

Financial Mathematics is an extensive field that applies mathematical techniques to solve financial problems, often involving interest, investment, and exponential growth calculations. It's like having a toolbox to make sense of the time value of money, investment growth, and risk assessments.
In our specific exercise, financial mathematics helped us deduce the original deposit amount from a future value after continuous interest compounding. It combines simple algebraic manipulations with exponential calculations to inform investment decisions.
By mastering these mathematical tools, you can assess how money grows over time, determine the benefits of saving versus spending, and be prepared for financial forecasts and decision-making. Whether dealing with loans, savings, or investments, a grasp of financial mathematics is crucial.