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Chapter 5: Problem 16

If Jackson deposits \(\$ 100\) at the end of each month in a savings account earning interest at the rate of \(8 \%\) /year compounded monthly, how much will he have on deposit in his savings account at the end of \(6 \mathrm{yr}\), assuming that he makes no withdrawals during that period?

Short Answer

Expert verified
Jackson will have approximately \(\$8538.33\) in his savings account at the end of 6 years.

Step by step solution

01

Understand the given information

We are given the following information: 1. The regular deposit: \(PMT = \$100\) (at the end of each month) 2. The annual interest rate: \(r = 8\%\) or \(0.08\) 3. The number of years: \(t = 6\) years 4. Interest compounded monthly, so the number of times interest is compounded per year: \(n = 12\)
02

Calculate the monthly interest rate

In order to calculate the monthly interest rate, we will divide the annual interest rate by the number of times interest is compounded per year (12). Monthly interest rate: \(i=\cfrac{r}{n}\) \(i = \cfrac{0.08}{12}\) \(i = 0.0066666667\)
03

Calculate the total number of periods

Since we are given the number of years and the interest is compounded monthly, the total number of periods is as follows: Total number of periods: \(N = n \times t\) \(N = 12\times 6\) \(N = 72\)
04

Use the future value of an ordinary annuity formula

The formula for the future value (FV) of an ordinary annuity is as follows: \(FV = PMT \times \cfrac{(1 + i)^N - 1}{i}\) Let's plug in the values that we have calculated previously: \(FV = 100 \times \cfrac{(1 + 0.0066666667)^{72} - 1}{0.0066666667}\)
05

Calculate the future value

Let's compute the expression inside the parentheses first: \((1 + 0.0066666667)^{72}= 1.568736\) Now, we can compute the rest of the formula: FV equals approximately: \(FV = 100 \times \cfrac{1.568736 - 1}{0.0066666667}\) \(FV = 100 \times 85.383255746\) \(FV 鈮� \$ 8538.33\) At the end of 6 years, Jackson will have approximately \( \$ 8538.33\) on deposit in his savings account, assuming that he makes no withdrawals during that period.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordinary Annuity
An ordinary annuity refers to a series of equal payments or deposits made at regular intervals, with the key characteristic being that these transactions occur at the end of each period. For example, when someone saves money by putting a set amount into a retirement account each month, they're contributing to an ordinary annuity.

Ordinary annuities are a common financial tool used to save for the future or to prepare for retirement. Understanding how ordinary annuities grow over time involves grasping the impact of compounded interest and recognizing the time value of money鈥攖hat future savings are worth more than the same amount saved today due to potential growth.
Compounded Interest
Compounded interest is the process in which earned interest is added back to the principal sum, so that from that moment on, the interest that has been added also earns interest. A key point about compounded interest is that it can significantly increase the growth rate of savings over time, as the interest earns interest.

In the context of Jackson's savings, the question states that the 8% annual interest rate is compounded monthly, which means that each month's accrued interest is added to the principal, affecting the following month's calculations. This concept is fundamental for calculating the future value of an annuity and can make a sizable difference compared to simple interest where interest is not added to the principal.
Savings Account Growth
The growth of a savings account often relies on both the regular contributions made and the compounded interest earned over time. For Jackson, making consistent end-of-month deposits into a savings account with a compounded interest rate, his account reflects the growth of an ordinary annuity.

To visualize how his savings grow over the 6-year period, one can utilize the formula for the future value of an ordinary annuity. With the given interest rate and the power of compounding, the amount not only increases due to his deposits but also accelerates in value due to the interest earned on both the deposits and the accrued interest. It鈥檚 an exemplification of how disciplined, regular saving is a powerful method for increasing wealth.

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Most popular questions from this chapter

The Johnsons have accumulated a nest egg of \(\$ 40,000\) that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of \(\$ 2400 /\) month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed \(\$ 3000\). If local mortgage rates are \(7.5 \% /\) year compounded monthly for a conventional 30 -yr mortgage, what is the price range of houses that they should consider?

Find the effective rate of interest corresponding to a nominal rate of \(9 \% /\) year compounded annually, semiannually, quarterly, and monthly.

The simple interest formula \(A=P(1+r t)\) [Formula (1b)] can be written in the form \(A=P r t+P\), which is the slope-intercept form of a straight line with slope \(P r\) and A-intercept \(P .\) a. Describe the family of straight lines obtained by keeping the value of \(r\) fixed and allowing the value of \(P\) to vary. Interpret your results. b. Describe the family of straight lines obtained by keeping the value of \(P\) fixed and allowing the value of \(r\) to vary. Interpret your results.

Juan is contemplating buying a zero coupon bond that matures in \(10 \mathrm{yr}\) and has a face value of \(\$ 10,000\). If the bond yields a return of \(5.25 \%\) lyear, how much should Juan pay for the bond?

Fleet Street Savings Bank pays interest at the rate of \(4.25 \%\) /year compounded weekly in a savings account, whereas Washington Bank pays interest at the rate of \(4.125 \%\) /year compounded daily (assume a 365day year). Which bank offers a better rate of interest?
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