4.4: Simple Interest - Today’s Dollar vs. Tomorrow’s Dollar
- Page ID
- 132022
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Describe interest as compensation for deferring the use of money.
- Differentiate between simple and compound interest.
- Apply interest calculations to measure future or present value.
A Graduation Gift
Let’s say your grandparents want to gift you $1,000 for graduation next year. They face two options:
- Set aside $1,000 today and let it grow with simple interest for one year.
- Work backward to decide how much they need to set aside today to ensure you receive exactly $1,000 next year.
By exploring both options, we’ll see why today’s dollar isn’t quite the same as tomorrow’s dollar.
Future Value: How Much Will You Have?
Your grandparents decide to set aside $1,000 today in a simple interest account that earns 5 percent per year. How much will it be worth when you graduate next year?
Here’s the idea:
At the end of the year, the interest you earn is calculated based on the amount you have deposited.
The formula for simple interest is:
\(\displaystyle{FV = PV \times (1+r)}\)
Where:
PV = Present value (amount today) = $1,000
r = Interest rate = 5% = 0.05
Plugging in the numbers:
\(\displaystyle{FV = 1,000 \times (1+0.05)}\)
\(\displaystyle{FV = 1,000 \times (1.05)}\)
\(\displaystyle{FV = 1,050}\)
Result: At graduation, you’ll receive $1,050. The additional $50 represents the interest earned over one year.
Present Value Example: How Much Should They Set Aside Today?
Now, suppose your grandparent changes their mind and decides they want to set aside an amount today that will grow to exactly $1,000 in one year. How much would they need to deposit now?
Here’s the idea: We’re bringing tomorrow’s dollar back to today’s value - a process called “discounting.”
The formula for present value (PV) is:
\(\displaystyle{PV = \frac{FV}{(1+r)}}\)
Where:
FV = Future value = $1,000
r = Interest rate = 5% = 0.05
Plugging in the numbers:
PV = 1,000/(1+0.05)
PV = 1,000/(1.05)
PV = 952.38
Result: To ensure you receive $1,000 at graduation, your grandparents would need to set aside approximately $952.38 today.
You don’t need to do these calculations by hand - online tools and Excel spreadsheets can handle them for you! Here’s how:
Online Calculators:
Search for “Future Value Calculator” or “Present Value Calculator” online.
Enter the values: PV, FV, interest rate (r), and the number of periods (1 year in this case).
Excel Formulas:
Future Value: Use the formula =FV(rate, nper, pmt, pv)
Example: =FV(5%, 1, 0, -1000) → Returns $1,050
Present Value: Use the formula =PV(rate, nper, pmt, fv)
Example: =PV(5%, 1, 0, -1000) → Returns $952.38
Tip: In Excel, enter interest rates as decimals (5% = 0.05) and be mindful of positive and negative signs when entering amounts.
Today’s Dollar vs. Tomorrow’s Dollar
What does this tell us?
- The future value shows how money today grows into tomorrow’s dollars with interest.
- The present value indicates that tomorrow’s dollars are worth less today because they have not yet earned any interest.
This simple relationship between today’s value and tomorrow’s value forms the foundation for everything that follows.
What if You Had to Wait Longer?
So far, we’ve looked at lump sums and annuities with simple, single-period interest. But what happens when the interest we earned last year starts to earn interest for us this year? That’s where things get interesting - and it’s what we’ll explore next.
Interest is the reward (or cost) for letting time pass without spending money. When you lend money or delay your own spending, interest measures the value of that wait. This section introduces how interest works, starting with simple examples and gradually introducing formulas.
- Interest is what bridges today’s dollar and tomorrow’s.
- Simple interest grows in a straight line—only the principal earns interest.
- Interest lets us assign comparable value to different moments in time.
This section prepares students to interpret and calculate future and present values using interest, while still grounding the concepts in real-world decisions.
- Have you ever loaned money to someone (or borrowed it)? How did interest factor into the decision?
- Why might someone prefer to accept less money today than more money in the future?
- You invest $1,000 at 5 percent simple interest for 3 years. How much interest do you earn, and what’s the total future value?