Problem 362 - Calculus Volume 1

Question

For the following exercises, use \(y=y_{0} e^{k t}\). If a bank offers annual interest of \(7.5 \%\) or continuous interest of \(7.25 \%\), which has a better annual yield?

Step-by-Step Solution

Answer

Both options yield approximately 1.075 times the initial amount annually.

Step 1: Understand the Problem

We are given two forms of interest: annual and continuous. We need to determine which yields a higher return over one year.

Step 2: Calculate Annual Interest Yield

For annual interest, the amount after 1 year with a rate of 7.5% is calculated using simple interest. The formula is:\[ y = y_0 (1 + r) \]where \( r = 0.075 \). Hence, the yield after 1 year is \( 1.075y_0 \).

Step 3: Calculate Continuous Interest Yield

For continuous interest, use the formula \[ y = y_0 e^{kt} \]where \( k = 0.0725 \) and \( t = 1 \). Thus, the yield after 1 year is \( y = y_0 e^{0.0725} \).

Step 4: Calculate and Compare the Values

Compute the exponential value with \( k = 0.0725 \):\[ e^{0.0725} \approx 1.075 \].Now, compare the yields: the continuous compounding result is approximately \( 1.075y_0 \), which matches the annual yield of \( 1.075y_0 \).

Step 5: Conclude the Comparison

Since both annual and continuous interest yield approximately the same value, neither option provides a higher annual yield.

Key Concepts

Continuous Compounding Interest

Continuous compounding refers to the process where interest is calculated and added to the account balance infinitely many times per period. This concept assumes that interest is being compounded an infinite number of times within any given period, leading to a theoretical scenario where your principal grows constantly. Continuous compounding is described mathematically by the formula:\[ y = y_0 e^{kt} \]

• \(y\) represents the balance after time \(t\).
• \(y_0\) is the initial principal amount.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(k\) is the growth rate, specifically the continuous interest rate.
• \(t\) is the time the money is invested or borrowed for.

The beauty of continuous compounding lies in its ability to maximize investment gains by compounding interest as frequently as theoretically possible. As shown in the example, the formula used \( k = 0.0725 \) and \( t = 1 \). This results in exponential growth, precisely calculated as \( y = y_0 e^{0.0725} \), with an approximate answer of \(1.075y_0\).

Understanding Annual Interest

Annual interest is a simpler and more traditional method of calculating interest, compared to continuous compounding. At its core, annual interest calculates how much interest is added to the principal after each year. The formula for annual interest calculation is:\[ y = y_0 (1 + r) \]

• \(y\) is your investment balance after one year.
• \(y_0\) is the starting principal amount.
• \(r\) is the annual interest rate, expressed as a decimal.

In the example, the annual interest rate is \( 7.5\% \). This means that after one year, the formula computes it to become \( y = 1.075y_0 \). This represents a straightforward annual growth, easily understood by considering what's gained by simply adding interest once at the end of each year.

Exponential Growth in Interest Calculations

Exponential growth is a process where the quantity grows at a rate proportional to its current value. This creates a sharp upward curve on a graph and is a key concept in both continuous and annual compounding of interest.For both continuous and annual interest, the growth is exponential, albeit calculated differently:

• **Continuous compounding:** This uses Euler's number \(e\) to create smooth, unbroken exponential growth. The value \(e^{0.0725}\) approximately equals \(1.075\), demonstrating exponential increase over a period of one year.
• **Annual interest:** This adds interest at regular intervals, showing exponential increase by applying compounding effects after every fixed period. With a 7.5% rate annually, it grows to \(1.075y_0\).

Exponential growth in interest calculations is what makes investments potentially grow so rapidly over time, underscoring the compounding effects in finance. Whether interest is calculated continuously or on an annual basis, the fundamental idea remains that of growth accelerating over time.