Problem 46 - Calculus Early Transcendentals

Question

Suppose that \(P\) dollars is invested at an annual interest rate of \(r \times 100 \%\). If the accumulated interest is credited to the account at the end of the year, then the interest is said to be compounded annually; if it is credited at the end of each 6 -month period, then it is said to be compounded semiannually; and if it is credited at the end of each 3-month period, then it is said to be compounded quarterly. The more frequently the interest is compounded, the better it is for the investor since more of the interest is itself earning interest. (a) Show that if interest is compounded \(n\) times a year at equally spaced intervals, then the value \(A\) of the investment after \(t\) years is $$ A=P\left(1+\frac{r}{n}\right)^{n t} $$ (b) One can imagine interest to be compounded each day, each hour, each minute, and so forth. Carried to the limit one can conceive of interest compounded at each instant of time; this is called continuous compounding. Thus, from part (a), the value \(A\) of \(P\) dollars after \(t\) years when invested at an annual rate of \(r \times 100 \%,\) compounded continuously, is $$ A=\lim _{n \rightarrow+\infty} P\left(1+\frac{r}{n}\right)^{n t} $$ Use the fact that \(\lim _{x \rightarrow 0}(1+x)^{1 / x}=e\) to prove that \(A=P e^{r t} .\) (c) Use the result in part (b) to show that money invested at continuous compound interest increases at a rate proportional to the amount present.

Step-by-Step Solution

Answer

(a) \(A = P\left(1 + \frac{r}{n}\right)^{n t}\); (b) \(A = Pe^{rt}\); (c) Deep exponential growth: \( \frac{dA}{dt} = rA\).

Step 1: Formula Derivation for Compounded Interest

The formula for the value of an investment with interest compounded \(n\) times per year is derived from the principle of compound interest. The initial investment \(P\) is multiplied by a factor that accounts for interest gained each period. If compounded \(n\) times a year, then each period accrues interest at rate \(\frac{r}{n}\).As there are \(n\) compounding periods in a year, the formula becomes:\[A = P \left(1 + \frac{r}{n}\right)^{n t}\] where \(t\) is the number of years the money is invested or borrowed.

Step 2: Continuous Compounding Concept

Continuous compounding refers to a scenario where the interest is compounded an infinite number of times per year. Mathematically, this is represented by taking the limit of the compounded interest formula as \(n\) approaches infinity:\[A = \lim_{n \to \infty} P\left(1 + \frac{r}{n}\right)^{n t}\]

Step 3: Apply the Limit for Continuous Compounding

To solve the limit problem, we use the fact that:\[\lim_{x \to 0} (1 + x)^{1/x} = e\]By substituting \(x = \frac{r}{n}\), we get \(n = \frac{r}{x}\). As \(n\to\infty\), \(x\to 0\). Hence,\[\lim_{n \to \infty}\left(1 + \frac{r}{n}\right)^{n} = e^{r}\]Thus, applying this to our equation for continuous compounding, we have:\[A = P e^{r t}\]

Step 4: Show Exponential Growth in Continuous Compounding

The formula \(A = Pe^{rt}\) implies that the growth of the investment is exponential. The differential equation that describes this growth is:\[\frac{dA}{dt} = rA\]This shows that the rate of increase in the amount \(A\) is directly proportional to the amount itself, a characteristic of exponential growth.

Key Concepts

Compound Interest Calculations

Compound interest calculations play a crucial role in understanding how investments grow over time. When we talk about compound interest, we mean the process of earning interest not just on the initial amount that you invest (the principal), but also on the interest that accumulates over time. This is different from simple interest, which is calculated only on the principal amount. The formula for compound interest when it is compounded annually, semiannually, quarterly, or at any regular interval is given by:

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

Here, \(A\) represents the accumulated amount after \(t\) years, \(P\) is the principal amount, \(r\) is the annual interest rate (expressed as a decimal), \(n\) is the number of times interest is compounded per year, and \(t\) is the number of years the investment is held.
The more frequently interest is compounded, the more effective the earning potential. This is because with each compounding, the interest amount is recalculated and added to the principal, thus earning interest on interest. Compounded interest can significantly increase the total amount over time if invested properly.

Exponential Growth

Exponential growth is a key outcome of continuously compounding interest. When we say the growth is exponential, we mean that the amount grows increasingly faster over time. The mathematical representation of this concept when interest is compounded continuously is \(A = Pe^{rt}\), where \(e\) is the base of natural logarithms, approximately equal to 2.718.This formula is indicative of exponential growth because the amount of investment grows in proportion to its current value. In real-world terms, this means that as your investment grows, it does so at an accelerating rate due to the compound interest effect.
Exponential growth in finance is powerful, making it possible to see substantial increases in investment over longer periods, especially when combined with higher interest rates. This growth pattern is why consistent investing, even with relatively small amounts, can lead to significant financial outcomes due to the compounding effect.

Interest Compounding Frequency

Interest compounding frequency refers to how often the accumulated interest is calculated and added to the principal balance of an investment. The frequency of compounding can vary greatly:

• Annually: once a year
• Semiannually: twice a year
• Quarterly: four times a year
• Monthly: twelve times a year
• Continuously: at an infinitely small interval

The more frequently interest is compounded, the more frequently interest earnings are added to the principal, which in turn results in more interest being calculated on interest already earned. This is a desirable outcome for investors because it results in greater wealth accumulation over time.
Understanding the impact of interest compounding frequency is essential in evaluating the benefits of different savings accounts and investment products. A common rule is the more frequent the compounding, the better it is for wealth accumulation, assuming the interest rate remains constant.