6.4.0: Exercises
- Page ID
- 171716
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)What is the difference between simple interest and compound interest?
What is a direct way to compare accounts with different interest rates and number of compounding periods?
Which type of account grows in value faster, one with simple interest or one with compound interest?
How many periods are there if interest is compounded?
Daily
Weekly
Monthly
Quarterly
Semi-annually
In the following exercises, compute the future value of the investment with the given conditions.
Principal = $15,000, annual interest rate = 4.25%, compounded annually, for 5 years
Principal = $27,500, annual interest rate = 3.75%, compounded annually, for 10 years
Principal = $13,800, annual interest rate = 2.55%, compounded quarterly, for 18 years
Principal = $150,000, annual interest rate = 2.95%, compounded quarterly, for 30 years
Principal = $3,500, annual interest rate = 2.9%, compounded monthly, for 7 years
Principal = $1,500, annual interest rate = 3.23%, compounded monthly, for 30 years
Principal = $16,000, annual interest rate = 3.64%, compounded daily, for 13 years
Principal = $9,450, annual interest rate = 3.99%, compounded daily, for 25 years
In the following exercises, compute the present value of the accounts with the given conditions.
Future value = $250,000, annual interest rate = 3.45%, compounded annually, for 25 years
Future value = $300,000, annual interest rate = 3.99%, compounded annually, for 15 years
Future value = $1,500,000, annual interest rate = 4.81%, compounded quarterly, for 35 years
Future value = $750,000, annual interest rate = 3.95%, compounded quarterly, for 10 years
Future value = $600,000, annual interest rate = 3.79%, compounded monthly, for 17 years
Future value = $800,000, annual interest rate = 4.23%, compounded monthly, for 35 years
Future value = $890,000, annual interest rate = 2.77%, compounded daily, for 25 years
Future value = $345,000, annual interest rate = 2.99%, compounded daily, for 19 years
In the following exercises, compute the effective annual yield for accounts with the given interest rate and number of compounding periods. Round to three decimal places.
Annual interest rate = 2.75%, compounded monthly
Annual interest rate = 3.44%, compounded monthly
Annual interest rate = 5.18%, compounded quarterly
Annual interest rate = 2.56%, compounded quarterly
Annual interest rate = 4.11%, compounded daily
Annual interest rate = 6.5%, compounded daily
The following exercises explore what happens when a person deposits money in an account earning compound interest.
Find the present value of $500,000 in an account that earns 3.85% compounded quarterly for the indicated number of years.
- 40 years
- 35 years
- 30 years
- 25 years
- 20 years
- 15 years
Find the present value of $1,000,000 in an account that earns 6.15% compounded monthly for the indicated number of years.
- 40 years
- 35 years
- 30 years
- 25 years
- 20 years
- 15 years
In the following exercises, the number of years can reflect delaying depositing money. 40 years would be depositing money at the start of a 40-year career. 35 years would be waiting 5 years before depositing the money. Thirty years would be waiting 10 years before depositing the money, and so on. What do you notice happens if you delay depositing money?
For each 5-year gap for exercise 32, compute the difference between the present values. Do these differences remain the same for each of the 5-year gaps, or do they differ? How do they differ? What conclusion can you draw?
Daria invests $2,500 in a CD that yields 3.5% compounded quarterly for 5 years. How much is the CD worth after those 5 years?
Maurice deposits $4,200 in a CD that yields 3.8% compounded annually for 3 years. How much is the CD worth after those 3 years?
Georgita is shopping for an account to invest her money in. She wants the account to grow to $400,000 in 30 years. She finds an account that earns 4.75% compounded monthly. How much does she need to deposit to reach her goal?
Zak wants to create a nest egg for himself. He wants the account to be valued at $600,000 in 25 years. He finds an account that earns 4.05% interest compounded quarterly. How much does Zak need to deposit in the account to reach his goal of $600,000?
Eli wants to compare two accounts for their money. They find one account that earns 4.26% interest compounded monthly. They find another account that earns 4.31% interest compounded quarterly. Which account will grow to Eli’s goal the fastest?
Heath is planning to retire in 40 years. He’d like his account to be worth $250,000 when he does retire. He wants to deposit money now. How much does he need to deposit in an account yielding 5.71% interest compounded semi-annually to reach his goal?
Jo and Kim want to set aside some money for a down payment on a new car. They have 6 years to let the money grow. If they want to make a $15,000 down payment on the car, how much should they deposit now in an account that earns 4.36% interest compounded monthly?
A newspaper’s business section runs an article about savings at various banks in the city. They find six that offer accounts that offer compound interest.
Bank A offers 3.76% compounded daily.
Bank B offers 3.85% compounded annually.
Bank C offers 3.77% compounded weekly.
Bank D offers 3.74% compounded daily.
Bank E offers 3.81% compounded semi-annually.
To earn the most interest on a deposit, which bank should a person choose?
Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a principal of $100,000 that are allowed to grow for 20 years. What is the difference in the future values of the account with the highest effective annual yield, and the account with the second highest effective annual yield?
Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a principal of $100,000 that are allowed to grow for 20 years. What is the difference in the future values of the account with the highest effective annual yield, and the account with the lowest effective annual yield?
Jesse and Lila need to decide if they want to deposit money this year. If they do, they can deposit $17,400 and allow the money to grow for 35 years. However, they could wait 12 years before making the deposit. At that time, they’d be able to collect $31,700 but the money would only grow for 23 years. Their account earns 4.63% interest compounded monthly. Which plan will result in the most money, depositing $17,400 now or depositing $31,700 in 12 years?
Veronica and Jose are debating if they should deposit $15,000 now in an account or if they should wait 10 years and deposit $25,000. If they deposit money now, the money will grow for 35 years. If they wait 10 years, it will grow for 25 years. Their account earns 5.25% interest compounded weekly. Which plan will result in the most money, depositing $15,000 now or depositing $25,000 in 10 years?