Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) A certain amount of money is put in an account with a fixed nominal annual interest rate, and interest is compounded continuously. If 70 years later the money in the account has doubled, what is the nominal annual interest rate? (b) Answer the same question if the interest is compounded only once a year.

Step-by-Step Solution

Answer

For continuous compounding, the nominal annual interest rate, r, is approximately 0.99% or 0.0099 in decimal form. For annual compounding, the nominal annual interest rate, r, is approximately 1.02% or 0.0102 in decimal form.

Step 1: Understand the Concept

The formula for continuous compounding is \( A = Pe^{rt} \) where:\n- A is the amount of money accumulated after n years, including interest. - P is the principal amount (the initial amount of money). - r is the annual interest rate (in decimal).- t is the time the money is invested for, in this case 70 years.

Step 2: Formula for Continuous Compounding

We know that A is double the principal, P, so A=2P. We can substitute A in the formula:\n\( 2P = Pe^{r(70)} \).\nThen we solve for r: \( r = \frac{ln(2)}{70} \).

Step 3: Formula for Annual Compounding

The formula for annual compounding is \( A = P(1 + r)^t \). Again, we know that A is double the principal, P, so A=2P. We substitute A in the formula:\n\( 2P = P(1 + r)^{70} \).\nAfter simplifying, we get \( (1 + r) = 2^{1/70} \). Then solving for r:

Step 4: Solve for r

For continuous compounding, \( r = \frac{ln(2)}{70} \). We can approximate this to 0.0099 or 0.99% as the nominal annual interest rate.\nFor annual compounding, \( r = 2^{1/70} - 1 \). We can approximate this to 0.0102 or 1.02% as the nominal annual interest rate.

Key Concepts

Continuous Compounding Formula

Understanding the continuous compounding formula is crucial for anyone studying finance and interest calculations. It represents a scenario where the interest accumulates at every possible moment. This is the mathematical representation of exponential growth in a financial context. The formula is written as
\[ A = Pe^{rt} \]
where A is the final amount in the account, P is the original principal balance, r is the nominal annual interest rate (expressed as a decimal), and t is the time in years. Continuous compounding can be considered as letting the number of compounding periods per year grow without bound. This is where the power of the exponential function comes into play. Using natural logarithms, as shown in the step-by-step solution, we can isolate and solve for r when we know the other variables.

Annual Compounding Formula

In contrast to the previous method, the annual compounding formula applies when interest is compounded once per year. It's less intensive computationally than continuous compounding and is given by:
\[ A = P(1 + r)^t \]
Here the variables hold the same meaning, but the compounding effect is clearly visible as the principal is multiplied by (1 + r) raised to the power of t. When the final amount (A) is known to be double the principal (P), we set A to 2P and solve for the rate (r). This scenario, as outlined in the textbook exercise, is a solid example of how the compounding frequency can affect the rate needed to achieve a financial goal. This concept extends beyond just annual compounding to other frequencies like quarterly or monthly, changing the formula slightly.

Solving for Interest Rate

A pivotal skill in finance is solving for the interest rate, which can determine the growth of an investment or the cost of a loan. Whether dealing with continuous or annual compounding, the process involves isolating r in the compounding formula. In the context of continuous compounding, the natural logarithm can be employed to derive r given the known variables. For annual compounding, it involves more arithmetic but follows the same basic principle – rearrange the compounding formula to make r the subject and solve. Remember to express the nominal annual interest rate as a percentage by converting the decimal answer accordingly. Techniques for solving these equations include using logarithmic functions and following algebraic principles.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. In finance, this concept is mirrored in the compound interest formulas where the investment grows exponentially over time. Both continuous and annual compounding formulas exhibit exponential growth, but the continuous compounding formula with its base e (Euler's number) displays the most rapid growth, as it compounds incessantly. These formulas show how money can increase dramatically over time due to the power of exponential growth, emphasizing the importance of understanding and calculating interest rates accurately in financial planning and investment strategies.