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Chapter 5: Problem 21

Find the effective rate corresponding to the given nominal rate. \(10 \% /\) year compounded semiannually

Short Answer

Expert verified
The effective interest rate corresponding to a nominal rate of 10% per year compounded semiannually is 10.25% per year.

Step by step solution

01

Identify the variables

The nominal interest rate (i) is given as 10% per year, and the number of compounding periods (n) per year is given as semiannually, which means it's compounded 2 times a year.
02

Convert the nominal rate into a decimal

Convert the nominal interest rate percentage to its decimal form. \(i = 10\% = 0.10\)
03

Use the formula to compute the effective interest rate

The formula to calculate the effective interest rate (r) is given as follows: \[r = (1 + \frac{i}{n})^n - 1\] Plug in the decimal form of the nominal interest rate (i) and the number of compounding periods (n): \(r = (1 + \frac{0.10}{2})^2 - 1\)
04

Calculate the effective interest rate

Perform the calculations: \(r = (1 + 0.05)^2 - 1\) \(r = (1.05)^2 - 1\) \(r = 1.1025 - 1\) \(r = 0.1025\)
05

Convert the effective interest rate to percentage

To convert the decimal form of the effective interest rate to a percentage, multiply by 100: \(r = 0.1025 * 100\) \(r = 10.25\% \) Thus, the effective interest rate corresponding to a nominal rate of 10% per year compounded semiannually is 10.25% per year.

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Most popular questions from this chapter

FINANCING A HomE Eight years ago, Kim secured a bank loan of \(\$ 180,000\) to help finance the purchase of a house. The mortgage was for a term of \(30 \mathrm{yr}\), with an interest rate of \(9.5 \% /\) year compounded monthly on the unpaid balance to be amortized through monthly payments. What is the outstanding principal on Kim's house now?

FINANCING A CAR Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of \(\$ 16,000\) at a rate of \(7.9 \% /\) year compounded monthly. Her bank is now charging \(11.5 \%\) year compounded monthly for new car loans. Assuming that each loan would be amortized by 36 equal monthly installments, determine the amount of interest she would have paid at the end of \(3 \mathrm{yr}\) for each loan. How much less will she have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank?

Suppose payments were made at the end of each quarter into an ordinary annuity earning interest at the rate of \(10 \% /\) year compounded quarterly. If the future value of the annuity after 5 yr is \(\$ 50,000\), what was the size of each payment?

The inflation rates in the U.S. economy for 2003 through 2006 are \(1.6 \%, 2.3 \%, 2.7 \%\), and \(3.4 \%\), respectively. What was the purchasing power of a dollar at the beginning of 2007 compared to that at the beginning of 2003 ? Source: U.S. Census Bureau

Find how much money should be deposited in a bank paying interest at the rate of \(8.5 \% /\) year compounded quarterly so that, at the end of \(5 \mathrm{yr}\), the accumulated amount will be \(\$ 40,000\).
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