Question

You deposit in an account with an annual interest rate of \(r\) (in decimal form) compounded monthly. At the end of 5 years, the balance is \(A=1000\left(1+\frac{r}{12}\right)^{60}\) Find the rates of change of \(A\) with respect to \(r\) when (a) \(r=0.08\), (b) \(r=0.10\), and (c) \(r=0.12\).

Short answer

After performing the calculations, we obtain the rates of change of \(A\) with respect to \(r\): (a) for \(r=0.08\), \(A' = approximately dollars per year\); (b) for \(r=0.10\), \(A' = approximately dollars per year\); (c) for \(r=0.12\), \(A' = approximately dollars per year\). This means that for each unit increase in the annual interest rate, \(A\) increases by the corresponding \(A'\) value.

Step-by-step solution

Define the function of \(A\) with respect to \(r\)

The total deposit account balance after 5 years is given by the formula \(A=1000\left(1+\frac{r}{12}\right)^{60}\).

Differentiate the function

To find the rates of change of \(A\) with respect to \(r\), we differentiate \(A\) with respect to \(r\). Recall the rule that the derivative of a constant times a function is the constant times the derivative of the function. Similarly, recall the power rule of differentiation, \((u^n)' = n \cdot u^{n-1} \cdot u'\), and the chain rule of differentiation, \((f(g(x)))' = f'(g(x)) \cdot g'(x)\). Use these rules to find the derivative \(A'\). This gives us \(A' = 1000 \cdot 60 \cdot \left(1+\frac{r}{12}\right)^{59} \cdot \frac{1}{12}\).

Evaluate the derivative at the given interest rates

Now, we evaluate this derivative at the given annual interest rates \(r\): (a) \(r=0.08\), (b) \(r=0.10\), and (c) \(r=0.12\). Plug these values into the \(A'\) expression to find their respective rates of change.

Obtain the rates of change of \(A\) with respect to \(r\)

Perform the calculations to determine the rates of change of \(A\) with respect to \(r\) at the given interest rates. These calculations give us the following results: (a) For \(r=0.08\), \(A' = 1000 \cdot 60 \cdot \left(1+\frac{0.08}{12}\right)^{59} \cdot \frac{1}{12}\); (b) For \(r=0.10\), \(A' = 1000 \cdot 60 \cdot \left(1+\frac{0.10}{12}\right)^{59} \cdot \frac{1}{12}\); (c) For \(r=0.12\), \(A' = 1000 \cdot 60 \cdot \left(1+\frac{0.12}{12}\right)^{59} \cdot \frac{1}{12}\)

Key concepts

Rate of Change

The concept of rate of change is central to understanding how variables affect each other within a mathematical relationship. In finance, this is particularly relevant when we examine how the balance of an account evolves as the interest rate changes. The rate of change in this context tells us how sensitive our account balance is to fluctuations in the interest rates.

When dealing with functions, the rate of change is represented by the derivative of the function. In the exercise provided, by finding the derivative of the account balance with respect to the interest rate, we are looking for how a small change in the interest rate, say from 0.10 to 0.11, would affect the balance of the account. Using calculus, we can calculate this rate of change precisely, which is what the differentiation process in Step 2 achieves.

Compound Interest Formula

Compound interest is the concept of earning interest on top of interest that has already been earned. The formula typically used to calculate compound interest 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 (the initial amount of money).\
• \( r \) is the annual interest rate (decimal).\
• \( n \) is the number of times that interest is compounded per year.\
• \( t \) is the time the money is invested for, in years.

Understanding this formula allows you to see how the balance grows over time. In our exercise, the balance after 5 years is determined by the compound interest formula with monthly compounding, which is reflected in the '12' in the denominator of the interest rate fraction.

Calculus in Finance

Calculus plays an important role in finance, particularly through the use of differentiation to understand how different financial variables interact with one another. For example, differentiating the compound interest formula with respect to the interest rate helps us understand how changes in the interest rate can affect the growth of an investment over time.

By taking the derivative of the balance with respect to the interest rate, we can see the sensitivity of the final balance to changes in the interest rate. If the derivative is large, a small change in the interest rate results in a large change in the balance, indicating higher sensitivity. This information can be critical when making investment decisions or when evaluating the risk associated with interest rate fluctuations.

Differentiation Rules

There are several key differentiation rules that are instrumental in solving calculus problems. The power rule \((x^n)' = n \times x^{n-1}\) simplifies taking derivatives of power functions. The product rule and the chain rule are also vital when functions are multiplied or composed, respectively.

In the context of our interest rate problem, we employ the power rule and the chain rule to differentiate the compound interest formula. This allows us to express the derivative in a form that shows how the balance responds to a change in the interest rate, as demonstrated in Step 2. In finance, these rules enable us to model complex relationships and to make precise calculations that are essential for financial analysis and decision-making.