Problem 86 - Precalculus with Limits

Question

Compound Interest Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to calculate the balance of an investment when \(P=\$ 3000\) , \(r=6 \%,\) and \(t=10\) years, and compounding is done (a) by the day, (b) by the hour, (c) by the minute, and (d) by the second. Does increasing the number of compoundings per year result in unlimited growth of the balance? Explain.

Step-by-Step Solution

Answer

Increase in the number of times the compound interest is compounded per year does not result in unlimited balance growth. After running the calculations, you'd see that the balance does increase as you compound more frequently, but it never approaches infinity.

Step 1: Calculate compound interest compounded daily

Here, \(n\) is the number of times interest compounds in a year. Since this is being compounded daily, \(n = 365\). Using the formula \(A=P(1+\frac{r}{n})^{nt}\), the yearly balance will be \$3000(1+\frac{0.06}{365})^{365*10}\.

Step 2: Calculate compound interest compounded hourly

Since this is being compounded hourly, \(n = 24*365\). Then the yearly balance will be \$3000(1+\frac{0.06}{24*365})^{24*365*10}\.

Step 3: Calculate compound interest compounded every minute

In the case of being compounded every minute, \(n = 24*60*365\). The yearly balance will turn out to be \$3000(1+\frac{0.06}{24*60*365})^{24*60*365*10}\.

Step 4: Calculate compound interest compounded every second

When compounded every second, \(n = 24*60*60*365\). Using the same formula, the yearly balance is determined by \$3000(1+\frac{0.06}{24*60*60*365})^{24*60*60*365*10}\.

Step 5: Does increasing the number of compoundings per year result in unlimited balance growth?

The answer is no. As is evident from the above calculations, the balance does increase with a higher frequency of compounding, but the rate of increase diminishes, and the balance eventually tapers off, gaining very little with each additional increase in compounding frequency. This is due to the limiting factor in the formula which is the \(1 + r/n\), even though \(n\) increases, the total amount cannot reach infinity.

Key Concepts

Compounding Frequency

Compounding frequency refers to the number of times interest is added to the principal balance of an investment within a given year. Imagine placing your savings in a bank account, and the bank calculates interest several times a year.

Depending on when and how often this interest is added, you can have different compounding frequencies: annually, semi-annually, quarterly, monthly, daily, and so on. Each increase in compounding frequency means the accumulated interest is being reinvested more often. For example:

• Daily compounding assumes interest is calculated and added every day of the year.
• Compounding hourly, minute by minute, or even by the second increases this frequency even more.

Generally, the more frequently interest is compounded, the greater the total balance will be over time.

Exponential Growth

Exponential growth describes how investments increase over time when they are subject to regularly compounding interest. This is different from simple linear growth, where the balance grows by fixed amounts.

In exponential growth, each period's interest calculation builds on the accumulated balance from previous periods, leading to a larger and larger increase. This effect is known as "interest on interest."

With very frequent compounding, such as minute-by-minute or second-by-second, the accumulation can be quite significant. However, it's critical to understand that despite the multiplicative nature of exponential growth, it doesn’t lead to infinite amounts. The total growth eventually slows down due to limitations imposed by other factors.

Interest Formula

The compound interest formula used here is a mathematical representation of how interest adds up over time. The formula is

\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]

where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (initial investment).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the number of years the money is invested for.

This formula underscores the effects of compounding frequency, as changing \(n\) affects how quickly the total amount \(A\) grows. The fraction \(\frac{r}{n}\) adjusts for how often the interest is applied, and the exponent \(nt\) multiplies out to accommodate the total number of compounding periods.

Limiting Factor

When considering compound interest, we reach limitations, known as a limiting factor, even with frequent compounding.

This is because as we make \(n\), the frequency of compounding, larger and larger, increments added to the balance become smaller and smaller. Though still increasing, the growth isn't infinite.

In the compound interest formula, this is reflected in the \(\left(1 + \frac{r}{n}\right)\) part. As \(n\) approaches infinity, the expression begins to resemble something known as the natural exponential function. In financial terms, this means there is a practical cap on how much can be gained, despite higher n-values or more frequent compounding.