Problem 36 - Precalculus Mathematics for Calculus

Question

Compound Interest Which of the given interest rates and compounding periods would provide the better investment? (a) \(5 \frac{1}{4} \%\) per year, compounded semiannually (b) \(5 \%\) per year, compounded continuously

Step-by-Step Solution

Answer

Option (a) provides a better investment with a higher effective annual rate of 5.306%.

Step 1: Convert Percentage to Decimal

First, convert the annual interest rates from percentage to decimal form. For (a), the interest rate is \(5\frac{1}{4}\%\), which is \(\frac{21}{4}\%\) or 0.0525 in decimal form. For (b), the interest rate is \(5\%\), which is 0.05 in decimal form.

Step 2: Calculate Effective Annual Rate for Option (a)

Option (a) is compounded semiannually, meaning it compounds twice a year. Use the formula for effective annual rate (EAR): \[EAR = \left(1 + \frac{r}{n}\right)^n - 1\]where \(r = 0.0525\) and \(n = 2\):\[EAR = \left(1 + \frac{0.0525}{2}\right)^2 - 1 = 0.05306\]Thus, the effective annual rate for (a) is 5.306%.

Step 3: Calculate Effective Annual Rate for Option (b)

For continuous compounding, use the formula:\[A = e^{rt}\]For a 1-year period with \(r = 0.05\):\[A = e^{0.05} \approx 1.05127\]The effective annual rate is therefore 1.05127 - 1 = 0.05127 or 5.127%.

Step 4: Compare Effective Annual Rates

Compare the effective annual rates calculated. For option (a), the rate is 5.306%, and for option (b), it is 5.127%. Since 5.306% is greater than 5.127%, option (a) with an effective annual rate of 5.306% provides a better investment.

Key Concepts

Effective Annual Rate

The Effective Annual Rate (EAR) is a crucial concept that helps investors understand the real profitability of their investments when interest is compounded. It converts the nominal interest rate, which may compound more than once a year, to an annualized figure that makes it easier to compare different investment options.

To calculate the EAR, you apply the formula: \[EAR = \left(1 + \frac{r}{n}\right)^n - 1\]where:

• \(r\) is the nominal interest rate expressed in decimal form.
• \(n\) is the number of compounding periods per year.

For semiannual compounding (twice a year), EAR accounts for the effect of compounding at each half-year, providing you a figure that is the equivalent to being compounded annually. Thus, offering a number directly comparable with other annual rates. Understanding this helps you make well-informed decisions by highlighting which investment option gives a better return over the same period.

Semiannual Compounding

Semiannual Compounding involves interest being added to the principal twice a year. This means the interest is calculated and added to the initial investment every six months. Therefore, money starts earning additional interest earlier than with annual compounding.

Let's break it down:

• Each time the interest is compounded, it is calculated on the new total, including any interest compounded previously.
• This leads to a slightly higher amount of interest earned as compared to annual compounding at the same nominal rate.

For example, with a rate of 5.25% compounded semiannually, you would split the rate into two compounding periods, each consisting of 2.625%. As a result, your Effective Annual Rate turns out to be slightly more than the nominal rate, in this case, 5.306%, because of the extra compounding effect over the year. This shows the advantage of more frequent compounding, enhancing your returns with the same nominal rate.

Continuous Compounding

Continuous Compounding represents an ideal mathematical concept where interest is compounded infinitely within a specified time period. This means interest is being added to the principal constantly, every moment.

The formula used for determining the future value with continuous compounding is:\[A = e^{rt}\]where:

• \(A\) is the amount of money accumulated after \(n\) years, including interest.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(r\) is the annual interest rate in decimal form.
• \(t\) is the time the money is invested for in years.

For a rate of 5% compounded continuously, you can see a slightly lower return compared to the semiannual or other frequent compounding methods, because it doesn't include the extra boost provided by periodically higher compounded interest as seen in some discrete compounding methods. Yet, its application is quite popular in sophisticated financial scenarios for its continuous effect.

Investment Comparison

Investment Comparison becomes crucial when deciding between different interest rates and compounding methods. The EAR helps provide a fair basis for comparison, as it reveals the actual annual interest yield when compounding is taken into full account.

Using the EAR, you can make side-by-side comparisons:

• For semiannual compounding, a nominal rate might end up with a higher effective annual rate once extra compounding periods are considered.
• For continuous compounding, you often get slightly lower returns unless the difference in nominal rates is substantial enough to swing the results.

In real-world applications, the comparison is crucial in choosing the best option for maximizing returns. When deciding between two investment options like in the present case, option (a) with semiannual compounding ultimately turns more profitable with an EAR of 5.306% compared to 5.127% from option (b). Understanding this comparison is key to optimizing investment strategies.