Question

Consider a certificate of deposit that pays \(10 \%\) (annual percentage rate) on an initial deposit of \(\$ 1000\). The balance \(A\) after 10 years is \(A=1000(1+0.1 x)^{10 / x}\) where \(x\) is the length of the compounding period (in years). (a) Use a graphing utility to graph \(A\), where \(0 \leq x \leq 1\). (b) Use the zoom and trace features to estimate the balance for quarterly compounding and daily compounding. (c) Use the zoom and trace features to estimate \(\lim _{x \rightarrow 0^{+}} A\) What do you think this limit represents? Explain your reasoning.

Short answer

The balance in the account after ten years alters when the length of the compounding period changes. For quarterly compounding (x=1/4) and daily compounding (x=1/365), the balances can be estimated from the graph. As x approaches 0 (infinitesimal compounding period), the limit of the function A represents the maximum growth of the account with continuous compounding. However, exact balances require exact computations, and the graphed results can only give an estimation.

Step-by-step solution

Analyze the problem and concerned variables

The problem is about a certificate of deposit with an annual interest rate of 10% for an amount of $1000. The balance \(A\) after 10 years is represented by the function \(A=1000(1+0.1x)^{10 / x}\), where \(x\) represents the length of the compounding period. The involved variables are balance \(A\), compounding period \(x\) and initial amount.

Graph of the function

Using a graphing utility to graph \(A=1000(1+0.1x)^{10 / x}\) with the interval \(0 <= x <= 1\). This graph represents how the balance changes with different compounding periods.

Estimation of Balance for different compounding periods

Use the zoom and trace features of the graphing utility to estimate the balance for quarterly compounding and daily compounding. For quarterly compounding \(x=1/4\) and for daily compounding \(x=1/365\). The values read from the graph are the estimated account balances when compounding is done quarterly and daily.

Estimation of limit

Next, the limit of the function as \(x\) approaches zero also needs to be estimated. This limit, \(\lim_{x \rightarrow 0^{+}} A\), can be found by focusing on the part of the graph where \(x\) gets very close to zero. If there is a value that the graph seems to most closely 'stick to' as it gets closer and closer to zero on the x-axis, that is the estimate for the limit.

Interpretation of the limit

The estimated limit represents the balance in the account after 10 years, given a compounding period that is infinitesimally small (continuous compounding as opposed to discrete compounding). This mathematical concept reflects the most accurate prediction of compound growth.

Key concepts

Compound Interest

Compound interest is a key concept in finance, particularly when discussing financial products like certificates of deposit (CDs). It involves earning interest not only on the initial principal but also on the accumulated interest from previous periods. This means that as time goes on, the amount of interest earned in each period can grow, leading to exponential growth of the account balance.

For example, if you have a CD with an annual interest rate of 10%, the interest you earn is added to your account balance, and in the next period, you earn interest on the new, higher balance. This continual process of compounding increases the total balance at a faster rate than simple interest, which is only calculated on the initial principal.

Key factors affecting compound interest include the rate of interest, the frequency of compounding (such as annually, quarterly, or daily), and the length of time the money is invested or borrowed. The more frequently interest is compounded, the greater the total amount will be by the end of the investment period. Understanding these components can help you make informed financial decisions.

Graphing Utility

When dealing with mathematical functions involving variables and constants, a graphing utility becomes an indispensable tool. It allows us to visually interpret and analyze functions like the one for the balance of a certificate of deposit over time. Graphing utilities can be software or calculators that plot graphs based on equations and provide various features to explore the details of these graphs.

In our exercise, using a graphing utility helps us to plot the function \(A = 1000(1+0.1x)^{10/x}\). This function shows how the balance changes depending on different compounding periods represented by variable \(x\). By graphing this equation, we can clearly see how the balance increases as the compounding period becomes shorter, reinforcing our understanding of compound interest.

Moreover, these tools often have features like zoom and trace, which let us get a closer look at specific parts of the graph and trace along the curve to find precise values for different \(x\) values. This capability makes graphing utilities powerful aids in comprehending how complex functions behave across different ranges.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the value that a function approaches as the input approaches a certain point. In financial contexts, understanding limits helps us accurately describe behavior over extremely small intervals.

In the exercise, we are interested in the limit \(\lim_{x \rightarrow 0^{+}} A\), which indicates what happens to the balance \(A\) as the compounding period \(x\) becomes infinitesimally small. This represents continuous compounding.

To estimate this limit using a graph, we look at the graph of the function \(A\) as \(x\) gets closer to zero. If the graph tends to settle around a specific value, that is our approximate limit. The role of the limit here is to signify the highest possible balance achieved under continuous compounding conditions, where interest is effectively being compounded at every possible moment.

Continuous Compounding

Continuous compounding takes the concept of compound interest to its most extreme case. Instead of interest being added at regular intervals (annual, quarterly, or daily), continuous compounding assumes that the interest is being added at every possible moment. This is a theoretical situation that provides the maximum possible growth for an investment.

In our example, the limit of the function \(A = 1000(1+0.1x)^{10/x}\) as \(x\) approaches zero reflects continuous compounding. As \(x\) decreases, the compounding intervals become smaller, eventually mimicking continuous compounding. The balance reaches its maximum because interest is compounding constantly.

This theoretical model is significant because it provides a benchmark for the "best-case scenario" in terms of growth due to compounding. By understanding this concept, you can more accurately predict and appreciate the impact of frequent compounding on your investments. It also highlights the power of compounding, which Albert Einstein reportedly referred to as the "eighth wonder of the world."