Question

Suppose you make a deposit of \(S P\) into a savings account that earns interest at a rate of \(100 \mathrm{r} \%\) per year. a. Show that if interest is compounded once per year, then the balance after \(t\) years is \(B(t)=P(1+r)^{t}\) b. If interest is compounded \(m\) times per year, then the balance after \(t\) years is \(B(t)=P(1+r / m)^{m t} .\) For example, \(m=12\) corresponds to monthly compounding, and the interest rate for each month is \(r / 12 .\) In the limit \(m \rightarrow \infty,\) the compounding is said to be continuous. Show that with continuous compounding, the balance after \(t\) years is \(B(t)=P e^{n}\)

Short answer

Answer: a. For yearly compounding: The balance after t years is calculated using the formula \(B(t) = P(1 + r)^t\). b. For continuous compounding: The balance after t years is calculated using the formula \(B(t) = Pe^{rt}\).

Step-by-step solution

a. Yearly compounding formula

In case of yearly compounding, the interest is added to the principal once every year. Since the given interest rate is 100r% per year, we can express it as a decimal by dividing it by 100 (r=r/100). To find the balance B(t) after t years, the generalized formula for compound interest is used: \(B(t) = P(1 + r)^t\) In this problem: Principal Amount = P Rate of interest per year (in decimal) = r Number of years = t Compounded once per year. So, the balance after t years can be calculated by the given formula.

b. Continuous compounding formula

If interest is compounded m times per year, then the balance after t years is given by the general formula: \(B(t) = P(1+ \frac{r}{m})^{mt}\). When \(m \rightarrow \infty\), it means that the compounding is continuous. To show that with continuous compounding, the balance after t years is \(B(t) = Pe^{rt}\), we start by taking the limit as \(m \rightarrow \infty\) in the general formula: \(B(t) = \lim_{m \rightarrow \infty} P(1+ \frac{r}{m})^{mt}\) We know that as \(m \rightarrow \infty\), \((1+ \frac{1}{m})^{m} \rightarrow e\). Now, let \(x = \frac{r}{m}\). Then as \(m \rightarrow \infty\), \(x \rightarrow 0\). Also, we have \(mx = r\), and \((1+x)^{mx} \rightarrow e^{r}\). So, \(B(t) = \lim_{m \rightarrow \infty} P(1+ \frac{r}{m})^{mt} = P \lim_{x \rightarrow 0} (1+x)^{xt} = P e^{rt}\) This is the desired result for continuous compounding.

Key concepts

Interest Compounding Formulas

Understanding the way your money can grow over time is crucial when you're saving or investing. Compound interest is one of the marvels that can work in your favor in the long run. Specifically, compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods.

The formula for compound interest when compounded yearly is quite straightforward and given by: \[B(t) = P(1 + r)^t\]where:

• \(B(t)\) represents the balance after \(t\) years,
• \(P\) is the principal amount (initial investment),
• \(r\) is the annual interest rate in decimal form,
• \(t\) is the time the money is invested for in years.

This formula shows how the investment grows every year by a certain percentage, allowing one to quickly calculate the future value of their savings.

Continuous Compounding

Let's take compounding a step further into an advanced concept known as continuous compounding. Unlike traditional compounding where the interest is added at regular intervals (annually, semi-annually, etc.), continuous compounding assumes that the interest is added an infinite number of times. It leverages the mathematical constant \(e\), which is approximately 2.71828, to describe this scenario.

Mathematically, the formula for continuous compounding is: \[B(t) = Pe^{rt}\]where \(e\) represents the exponential function. As the number of compounding periods \(m\) heads to infinity, we see that compound interest aligns with expotential growth. The takeaway is, with more compounding periods, the quicker your money grows, and with continuous compounding, growth is maximized as it's constantly accruing interest.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a curve on a graph that becomes steeper and steeper. It's incredibly powerful in the context of compound interest as it illustrates how investments can grow faster as time goes on.

An example of exponential growth can be seen in the formula for continuous compounding. When interest is computed continuously, the principal amount grows exponentially at a rate directly proportional to the amount present at any point in time. Thus, the key characteristic of exponential growth is that the growth rate - in this case, the interest - is always proportional to the current value resulting in the balance growing at an ever-accelerating rate. This concept provides a bedrock understanding for not just financial models, but a myriad of natural phenomena and is pivotal in many aspects of science and economics.