Problem 87 - College Algebra

Question

In Exercises \(87-90,\) determine whether cach statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. As the number of compounding periods increases on a fixed investment, the amount of money in the account over a fixed interval of time will increase without bound.

Step-by-Step Solution

Answer

The statement is false. As the number of compounding periods increases, the amount of money in the account over a fixed interval of time increases but not without bound. It tends towards a finite limit, given by the formula \(Pe^{rt}\).

Step 1: Understand the Concept of Compounding

In compound interest calculation, interest is added back to the account and this accumulated sum (principal + interest) earns more interest in the next period. This can be done over different intervals - annually, semi-annually, quarterly, monthly, weekly, daily, and so on. The frequency of these intervals is the number of compounding periods.

Step 2: Analyze the Statement

The statement claims that as the number of compounding periods 'increases without bound' (increases indefinitely), the amount of money in the account will also increase indefinitely. However, this claim is not accurate due to the fact that the compounding formula \(A = P (1 + r/n) ^ {nt}\) (where P is the principal, r is the annual interest rate, n is the number of times that interest is compounded per unit t, and t is the time the money is invested for), implies that there is a limit to the amount A as n approaches infinity. This limit is \(Pe^{rt}\), where e is the base of natural logarithms.

Step 3: Rectify the Statement

With the above analysis in mind, the necessary corrections to the statement can be made. The corrected version of this statement could be: As the number of compounding periods increases on a fixed investment, the amount of money in the account over a fixed interval of time will increase upto a certain limit, and not without bound.

Key Concepts

Understanding Compounding Periods

When we talk about compounding periods in the context of investments or savings, we are referring to the frequency at which interest is applied to the original amount of money, known as the principal. This could range from annual to even daily compounding. The core principle is that the more frequently the interest is compounded, the more often your earnings will generate their own earnings.

For instance, imagine you have \(1,000 in an account with an annual interest rate of 5%. If the interest is compounded annually, at the end of the year, you'll have \)1,050. However, if that interest is compounded semi-annually, you will have slightly more, because interest will be applied to your principal twice within the year, after each 6-month period. The general rule of thumb is that the higher the number of compounding periods, the more interest you will earn as your money reinvests itself more frequently.

Breaking Down the Compound Interest Formula

The compound interest formula is at the heart of understanding how investments grow over time. It's mathematically represented by the equation
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:

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

By plugging the right values into this formula, you can calculate how much your investment will be worth after a certain number of years, taking into account the number of compounding periods. Importantly, as you divide your interest rate by the number of compounding periods, and multiply the exponent by the number of periods, you're adjusting for the increased frequency of compounding which compounds the interest more often, ultimately leading to a higher return.

The Limit of Compound Interest

While it's true that increased compounding periods typically lead to higher returns, there's a misconception that as the number of compounding periods reaches infinity, the interest will increase indefinitely. However, there's actually a mathematical limit to compound interest growth. It's defined by the formula

\[ \lim_{n\to\infty} P\left(1+\frac{r}{n}\right)^{nt} = Pe^{rt} \]
where

• \( e \) is Euler's number - the base of natural logarithms, approximately equal to 2.71828

As the number of compounding periods increases to infinity, the compound interest formula approaches this constant e, multiplied by the initial principal and raised to the product of the interest rate and time. In simpler terms, there is a cap to how much compound interest can increase an investment's value over time—even with continuous compounding, which is theoretically compounding at every possible moment. This ceiling is an important concept in finance, indicating that investments cannot grow beyond a certain point solely through the mechanics of compounding interest, no matter how frequently the interest is compounded.