Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Kevin has deposited money in a bank account that compounds interest quarterly. If the nominal interest rate is \(5 \%\), what is the effective interest rate? (b) Ama has deposited money in a bank account that compounds interest quarterly. If the effective interest rate is \(5 \%\) per year, what is the nominal rate of interest?

Step-by-Step Solution

Answer

The effective interest rate for Kevin's account is 5.094%, and the nominal interest rate for Ama's account is 4.888%

Step 1: Calculation of the Effective Interest Rate

First, we calculate the effective interest rate for Kevin's account using the formula \( Effective Interest = (1 + \frac{r}{n})^n - 1 \). Here, 'r' is the nominal interest rate which is 5% or 0.05, and 'n' is the number of times the interest is compounded in a year. Since the interest is compounded quarterly, n = 4.

Step 2: Calculation of the Nominal Interest Rate

Next, we calculate the nominal interest rate for Ama's account using the formula \( Nominal Interest = m * (1+\frac{r}{m})^{1/m} - 1 \). Here, 'r' is the effective interest rate which is 5% or 0.05, and 'm' is the number of times the interest is compounded in a year. Since the interest is compounded quarterly, m = 4.

Key Concepts

Nominal Interest Rate

The nominal interest rate is the stated interest rate associated with a savings account, loan, or investment without taking into account the effects of compounding within a specific period. It represents the simple interest rate annually and doesn't reflect the actual financial charges incurred over time. For instance, a bank might advertise a nominal interest rate of 5% per year on their savings accounts. This is the rate before compounding interest comes into play.

In the case of Kevin's bank account, the given nominal interest rate is 5%. However, this isn't the rate that truly depicts what Kevin earns on his deposit after compounding interest quarterly. The real yield on his investment is measured by the effective interest rate, which will always be higher than the nominal rate whenever compounding occurs more than once per year.

Compound Interest Quarterly

When interest is compounded quarterly, it's calculated and added to the principal amount four times a year. After each quarter, the account balance increases by a fraction of the annual interest rate, and subsequent interest computations are based on the new balance, which includes the previously earned interest.

For Kevin's account with quarterly compounding, the interest is calculated using one-fourth of the annual nominal rate every three months. The formula used is \[ Effective Interest = (1 + \frac{r}{n})^n - 1 \] where 'r' is the annual nominal interest rate (in decimal form) and 'n' is the number of compounding periods per year. With each quarter, the investment grows, and by the end of the year, thanks to the power of compounding, the amount will be more substantial than with simple interest calculated using the nominal rate.

Interest Rate Conversion

Interest rate conversion is the process by which we transform one rate into another to reflect the actual earning or cost of interest over time. Typically, it involves converting a nominal rate into an effective rate and vice versa.

In Ama's scenario, to find the nominal rate from the effective rate, we invert the formula mentioned before. The equation \[ Nominal Interest = m * (1+\frac{r}{m})^{1/m} - 1 \] leads us to the nominal interest that, if compounded quarterly, would result in the given effective annual interest rate. The ability to convert between different types of interest rates is an essential financial skill, allowing investors and borrowers alike to compare different financial products on a like-for-like basis.