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Chapter 8: Problem 55

An account has a nominal rate of \(4.2 \%\). Find the effective annual yield, rounded to the nearest tenth of a percent, with quarterly compounding, monthly compounding, and daily compounding. How does changing the compounding period affect the effective annual yield?

Short Answer

Expert verified
The effective annual yields for quarterly, monthly, and daily compounding are calculated using the given formula. The results indicate that as the number of compounding periods increases, the yield tends to increase, demonstrating the effects of more frequent compounding.

Step by step solution

01

Determine the Nominal Interest Rate

First, note that the nominal interest rate (the interest rate not adjusted for compounding) is 4.2%. This will be represented as 0.042 in decimal form.
02

Calculate Quarterly Compounding

Calculate the effective annual yield with quarterly compounding by using the formula for compound interest, \(A=P(1+\frac{r}{n})^{nt}\) where r is the nominal rate, n is the number of times interest is compounded per unit t, t is the time the money is invested for. Here, \(n = 4\) as interest is compounded quarterly. Substitute \( P = 1, r = 0.042, n = 4, t = 1 \) into the formula to get the annual yield.
03

Calculate Monthly Compounding

Next, calculate the effective annual yield with monthly compounding. Here, \(n = 12\) as interest is compounded 12 times a year. Substitute \(P = 1, r = 0.042, n = 12, t = 1\) into the formula to get the annual yield.
04

Calculate Daily Compounding

Finally, calculate the effective annual yield with daily compounding. Here, \(n = 365\) as interest is compounded 365 times a year. Substitute \(P = 1, r = 0.042, n = 365, t = 1\) into the formula to get the annual yield.
05

Compare yields

Observe the calculated annual yields and note the effect of changing compounding period on the effective annual yield.

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Most popular questions from this chapter

In Exercises 11-14, use the formula $$ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} $$ Round all computations to the nearest dollar. Suppose that you drive 40,000 miles per year and gas averages \(\$ 4\) per gallon. a. What will you save in annual fuel expenses by owning a hybrid car averaging 40 miles per gallon rather than an SUV averaging 16 miles per gallon? b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays \(5.2 \%\) compounded monthly, how much will you have saved at the end of six years?

In Exercises 3-4, find the gross income, the adjusted gross income, and the taxable income. Base the taxable income on the greater of a standard deduction or an itemized deduction. Suppose your neighbor earned wages of \(\$ 86,250\), received \(\$ 1240\) in interest from a savings account, and contributed \(\$ 2200\) to a tax-deferred retirement plan. She is entitled to a personal exemption of \(\$ 3800\) and a standard deduction of \(\$ 5950\). The interest on her home mortgage was \(\$ 8900\), she contributed \(\$ 2400\) to charity, and she paid \(\$ 1725\) in state taxes.

a. Suppose that between the ages of 25 and 37, you contribute \(\$ 3500\) per year to a \(401(\mathrm{k})\) and your employer matches this contribution dollar for dollar on your behalf. The interest rate is \(8.25 \%\) compounded annually. What is the value of the \(401(\mathrm{k})\), rounded to the nearest dollar, after 12 years? b. Suppose that after 12 years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the \(401(\mathrm{k})\). How much money, to the nearest dollar, will you have in the plan when you reach age 65 ? c. What is the difference between the amount of money you will have accumulated in the \(401(\mathrm{k})\) and the amount you contributed to the plan?

Suppose that you are buying a car for \(\$ 56,000\), including taxes and license fees. You saved \(\$ 8000\) for a down payment. The dealer is offering you two incentives: Incentive \(\mathrm{A}\) is \(\$ 10,000\) off the price of the car, followed by a four-year loan at \(12.5 \%\). Incentive \(\mathrm{B}\) does not have a cash rebate, but provides free financing (no interest) over four years. What is the difference in monthly payments between the two offers? Which incentive is the better deal?

In Exercises \(21-34\), express each percent as a decimal. \(62 \frac{1}{2} \%\)
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