/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 In Exercises 11-18, a. Determi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 14

In Exercises 11-18, a. Determine the periodic deposit. Round up to the nearest dollar. b. How much of the financial goal comes from deposits and how much comes from interest? \(\$ ?\) at the end of each month \(7.5 \%\) compounded monthly 10 years \(\$ 250,000\)

Short Answer

Expert verified
Given the future value, interest rate and the time period, you can calculate the necessary periodic deposit and then break down the final amount into amounts from deposits and interest. Now, compute the result using the formulas provided in steps 1 and 2 to answer both parts of the problem.

Step by step solution

01

Compute the Periodic Deposit

For an annuity where regular payments are made into an interest-bearing account, the future value \( FV \) can be calculated given a periodic deposit \( PMT \), a periodic interest rate \( r \), and a number of periods \( n \) as: \( FV = PMT \times \left[ \frac{(1 + r)^n - 1}{r} \right] \). Here, we are given \( FV = \$ 250,000 \), \( r = 7.5 \% / 1200 = 0.00625 \) compounded monthly, and \( n = 10 \times 12 = 120 \) months. We need to solve this equation for \( PMT \). Rearranging terms leads to: \( PMT = FV \times \frac{r}{(1 + r)^n - 1} \). Substituting the given values, we compute: \( PMT = \$ 250,000 \times \frac{0.00625}{(1.00625)^{120} - 1} \) it gives the precise value. Round up the result to the nearest dollar to get the final answer.
02

Compute the Total Deposits and Interest

The total amount of deposits over the 10 years is simply the monthly deposit computed in Step 1 multiplied by the number of months, which is \( PMT \times n \). The portion of the final value that comes from interest is then \( FV - PMT \times n \) computed from the given future value and the total deposits.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Compound Interest
Understanding compound interest is crucial when it comes to growing your savings or investments over time. Simply put, compound interest refers to earning or paying interest on the initial principal as well as the accumulated interest from previous periods. In contrast to simple interest, where you only earn interest on the principal, compound interest allows your balance to grow at an accelerating rate.

For the textbook exercise, the future value (\(FV\)) of the periodic deposit is calculated using a formula that factors in compound interest. This formula reflects the effect of earning interest on not only the original deposits but also on the interest that has been previously gained. With a 7.5% interest rate compounded monthly over 10 years, each deposit made will grow more due to the monthly addition of interest to the investment's total.
Future Value of an Annuity
The future value of an annuity is the total amount that will be accumulated at a future point in time, based on regular periodic deposits or payments, plus interest. An annuity is typically a series of equal payments made at regular intervals, such as monthly or annually. In finance, it's important to be able to calculate how much these payments will be worth in the future, considering the effect of compound interest.

As demonstrated in the exercise solution, the formula for computing the future value of an annuity takes into account the periodic deposit (\(PMT\)), the interest rate per period (\(r\)), and the total number of periods (\(n\)). The rearranged formula used to find the periodic deposit required to reach a specific future value goal incorporates these variables and is solved to determine how much needs to be deposited on a regular basis.
Time Value of Money
The time value of money is a foundational concept in finance that suggests money available today is worth more than the same amount in the future, due to its earning potential. Essentially, a dollar today can be invested and earn interest, making it worth more than a dollar in the future that has not yet had the chance to earn interest.

This principle is pivotal in understanding the textbook problem, as it is incorporated into calculating the future value of periodic deposits. By depositing money into an account that yields compound interest, one is effectively leveraging the time value of money. The exercise reveals how much a series of future payments is worth in today's terms and calculates the necessary periodic deposit to achieve a $250,000 goal in 10 years' time. Both the future value and periodic deposit calculations apply the time value of money to compute the amount that needs to be invested consistently over the 10-year period.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

To offer scholarships to children of employees, a company invests \(\$ 15,000\) at the end of every three months in an annuity that pays \(9 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of 10 years? b. Find the interest.

Exercises 19 and 20 refer to the stock tables for Goodyear (the tire company) and Dow Chemical given below. In each exercise, use the stock table to answer the following questions. Where necessary, round dollar amounts to the nearest cent. a. What were the high and low prices for a share for the past 52 weeks? b. If you owned 700 shares of this stock last year, what dividend did you receive? c. What is the annual return for the dividends alone? How does this compare to a bank offering a \(3 \%\) interest rate? d. How many shares of this company's stock were traded yesterday? e. What were the high and low prices for a share yesterday? f. What was the price at which a share last traded when the stock exchange closed yesterday? g. What was the change in price for a share of stock from the market close two days ago to yesterday's market close? h. Compute the company's annual earnings per share using Annual earnings per share $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { 52-Week High } & \text { 52-Week Low } & \text { Stock } & \text { SYM } & \text { Div } & \text { Yld \% } & \text { PE } & \text { Vol 100s } & \text { Hi } & \text { Lo } & \text { Close } & \text { Net Chg } \\ \hline 73.25 & 45.44 & \text { Goodyear } & \text { GT } & 1.20 & 2.2 & 17 & 5915 & 56.38 & 54.38 & 55.50 & +1.25 \\ \hline \end{array} $$

Each group should have a newspaper with current stock quotations. Choose nine stocks that group members think would make good investments. Imagine that you invest \(\$ 10,000\) in each of these nine investments. Check the value of your stock each day over the next five weeks and then sell the nine stocks after five weeks. What is the group's profit or loss over the five-week period? Compare this figure with the profit or loss of other groups in your class for this activity.

S枚lve for \(P\) : $$ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} . $$ What does the resulting formula describe?

In Exercises 1-10, \((n)\) a. Find the value of each annuity. Round to the nearest dollar. b. Find the interest. $$ \begin{array}{|l|l|l|} \hline \begin{array}{l} \$ 150 \text { at the end of } \\ \text { every six months } \end{array} & \begin{array}{l} 6.5 \% \text { compounded } \\ \text { semiannually } \end{array} & 25 \text { years } \\ \hline \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.