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Chapter 5: Problem 23

Find the effective rate corresponding to the given nominal rate. \(8 \% /\) year compounded monthly

Short Answer

Expert verified
The effective interest rate corresponding to the given nominal rate of 8% compounded monthly is approximately \(8.30\%\) per year.

Step by step solution

01

Identify the Nominal Rate and Compounding Periods

The nominal rate is 8% per year, and it is compounded monthly. Nominal Rate = 8%, or 0.08 (converting percentage to decimal) Number of Compounding Periods = 12 (since there are 12 months in a year)
02

Calculate the Effective Interest Rate

Now, we will use the formula for converting a nominal interest rate to an effective interest rate: Effective Rate = \((1 + \dfrac{Nominal Rate}{Number\, of\, Compounding\, Periods})^{Number\, of\, Compounding\, Periods} - 1\) Substitute the values of the Nominal Rate and Number of Compounding Periods into the formula: Effective Rate = \((1 + \dfrac{0.08}{12})^{12} - 1\)
03

Evaluate the Expression

First, evaluate the fraction inside the parentheses: \(1 + \dfrac{0.08}{12} = 1 + 0.0066667 = 1.0066667\) Now, raise this value to the power of the number of compounding periods: \((1.0066667)^{12} = 1.0830192\) Finally, subtract 1 to find the effective interest rate: \(1.0830192 - 1 = 0.0830192\)
04

Convert the Effective Interest Rate to Percentage

To convert the effective interest rate from decimal to percentage, multiply by 100: \(0.0830192 * 100 = 8.30192 \% \) Therefore, the effective interest rate corresponding to the given nominal rate of 8% compounded monthly is approximately \(8.30\%\) per year.

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Most popular questions from this chapter

Use logarithms to solve each problem. How long will it take \(\$ 5000\) to grow to \(\$ 6500\) if the investment earns interest at the rate of \(12 \% /\) year compounded monthly?

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