**Compound Interest**

**Recall:**Interest is the fee we pay to use someone else's money for a specified period of time.

**Recall:**The way simple interest is calculated is expressed by the formula $I=Prt.$

Today we learn about a different kind of interest called

__compound interest.__

But first a thought experiment...

**Thought Experiment: Compound Interest**

By law, the bank is required to maintain a certain level of supply of liquid money. In other words, the bank needs your money and cannot legally do all the shenanigans banks do without it.

In exchange for the privilege of holding on to your sweet, sweet cash, the bank pays you a fee for letting them use it.

Suppose you let the bank keep $\$10000$ worth of your hard-earned greenbacks at an annual interest rate of $2\%.$ How much money will you have at the end of the month?

**Thought Experiment: Compound Interest**

Yes! (You're a clever one, you are!) We have made $$I=Prt=10000 \cdot 0.02 \cdot \frac{1}{12}=16.67$$ after one month for a total future value of $$A=P+I=P+Prt=P(1+rt)=1000(1+0.02 \cdot \frac{1}{12})=10016.67.$$ At the end of the month we could take our money and run, or we could reloan this money to the bank. That is, the $\$10016.67$ of your money sitting in the bank becomes the new principal amount for a new loan you're giving to the bank for yet another month of the privilege of using your money.

And so for the next month... (Cue Jeopardy! music... again...)

**Thought Experiment: Compound Interest**

With the new principal $\$10016.67$ we have $$I=Prt=10016.67 \cdot 0.02 \cdot \frac{1}{12}=16.69,$$ and after another month the total future value at the end of the second month is $$A=P+I=P+Prt=P(1+rt)=10016.67(1+0.02 \cdot \frac{1}{12})=10033.36.$$ At the end of this second month, we again have the option of absconding with our ducats, or we could reloan this money to the bank. That is, the $\$10033.36$ of your money sitting in the bank becomes the new principal amount for a new loan you're giving to the bank for yet another month of getting to use your money.

Do you see the pattern?

**Compound Interest versus Simple Interest**

The above process highlights the difference between compound interest and simple interest.

Compound interest is like a series of short-term simple interest loans where each time the money is re-loaned, the balance owed becomes the new principal amount.

Simple interest is, well... simple interest.

**Thought Experiment: Compound Interest**

Starting Balance | Ending Balance | |

Month 1 | $\$10000$ | $\$10000\cdot(1+0.02 \cdot \frac{1}{12})=\$10016.67$ |

**Burning Question**

Suppose we want to know how much money will be in our account after $10$ months, but we don't want to do $10$ rows of calculations, is there an easier way?

Is there a formula which will allow us to skip to the $10$th line of the table?

**Answer:**Yes!

The formula for the amount of money we'll have by the end of month $m$ is $$A=10000\cdot\left(1+0.02 \cdot \frac{1}{12}\right)^m$$

**The Compound Interest Formula**

The future value $A$ of an investment of principal $P$ at the end of $t$ years at an annual interest rate $r$ is $$A=P\left(1+ \frac{r}{n}\right)^{nt}$$ where interest is compounded $n$ times per year.

**The Compound Interest Formula**

**Example**

Suppose you deposit $\$10000$ into the bank at $2\%$ annual interest compounded monthly. What will be your balance after $15$ months?

Here, we use the compound interest formula with
$$
P=10000,\,\,\,r=0.02,\,\,\,n=12,\,\,\,\mbox{and}\,\,\,t=\frac{15}{12}=1.25
$$
Then
$$
\begin{array}{lll}
\displaystyle A&\displaystyle=P\left(1+ \frac{r}{n}\right)^{nt} &\mbox{}\\
\displaystyle &\displaystyle=10000\left(1+ \frac{0.02}{12}\right)^{12\cdot 1.25} &\mbox{}\\
\displaystyle &\displaystyle=10252.93784&\mbox{}\\
\end{array}
$$
That is, we will have a balance of $\$10252.94$ after $15$ months.

**Another Fun Question**

Suppose that instead of putting our money into the bank, we gave a simple-interest loan to our dear friend Sleazy P. Martini at an annual interest rate of $2\%$ for a $15$-month term.

How much money would Sleazy P. be paying us back at the end of fifteen months?

**Which is the better deal:**simple interest or compound interest?

With the values
$$
P=10000,\,\,\,r=0.02,\,\,\,\mbox{and}\,\,\,t=\frac{15}{12}=1.25
$$
we use the future value of a simple-interest loan formula.
$$
\begin{array}{lll}
\displaystyle A&\displaystyle=P(1+rt) &\mbox{}\\
\displaystyle &\displaystyle=10000(1+0.02\cdot 1.25) &\mbox{}\\
\displaystyle &\displaystyle=10250 &\mbox{}\\
\end{array}
$$
That is, Sleazy P. will pay us back $\$10250.$

Clearly, compound interest is the better deal.

Clearly, compound interest is the better deal.

**Compound Interest versus Simple Interest Redux**

Compound interest assumes that a lender re-loans the balance to the borrower at fixed intervals of time called

__compounding periods__.

The amount the borrower owes then becomes the new principal amount at the beginning of each compounding period.

The effect is that future interest is made from the principal

__AND past interest.__

Thus, compound interest

__accrues__faster than simple interest.

Over the long term, this can have a substantial effect on how much interest is paid.

**Which Investment is Better?**

Mr. Holt is saving for retirement and is considering investing a sum of $\$7880$ into one of two investments:

Investment 1 which offers $5.2\%$ compounded daily, and

Investment 2 which pays offers $5.3\%$ compounded quarterly.

Which investment should Mr. Holt choose?

Before we answer this question, we a new idea will help us...

**Effective Interest Rate**

The

__effective interest rate__of an investment is the simple interest rate which would pay the same amount of interest over one year.

The effective interest rate is a good way to compare investments involving simple and compound interest.

The effective interest rate $E$ is given by the formula $$E=\left(1+\frac{r}{n}\right)^n-1$$ where $r$ is the annual interest rate and $n$ is the number of times interest is compounded in a year.

**Note #1:**The effective interest rate is sometimes referred to as the

__annual yield.__

**Note #2:**The rate advertised by the bank is called the

__nominal rate.__

**Example: Using Effective Interest Rate to Compare Investments**

Mr. Holt is saving for retirement and is considering investing a sum of $\$7880$ into one of two investments:

Investment $1$ which offers $5.2\%$ compounded daily, and

Investment $2$ which pays offers $5.3\%$ compounded quarterly.

Which investment should Mr. Holt choose?

For Investment $1,$
$$
r=0.052\,\,\,\mbox{and}\,\,\,n=365
$$
So the effective interest rate of Investment $1$ is
$$
\begin{array}{lll}
\displaystyle E_1&\displaystyle=\left(1+\frac{r}{n}\right)^n-1 &\mbox{}\\
\displaystyle &\displaystyle=\left(1+\frac{0.052}{365}\right)^{365}-1 &\mbox{}\\
\displaystyle &\displaystyle=0.05337184107 &\mbox{}\\
\end{array}
$$
That is, the effective interest rate of Investment $1$ is about $5.3\%.$

For Investment $2,$ $$ r=0.053\,\,\,\mbox{and}\,\,\,n=4 $$ So the effective interest rate of Investment $2$ is $$ \begin{array}{lll} \displaystyle E_2&\displaystyle=\left(1+\frac{r}{n}\right)^n-1 &\mbox{}\\ \displaystyle &\displaystyle=\left(1+\frac{0.053}{4}\right)^{4}-1 &\mbox{}\\ \displaystyle &\displaystyle=0.05406271063 &\mbox{}\\ \end{array} $$ That is, the effective interest rate of Investment $2$ is about $5.4\%.$

We conclude that Investment $2$ is the better investment.

For Investment $2,$ $$ r=0.053\,\,\,\mbox{and}\,\,\,n=4 $$ So the effective interest rate of Investment $2$ is $$ \begin{array}{lll} \displaystyle E_2&\displaystyle=\left(1+\frac{r}{n}\right)^n-1 &\mbox{}\\ \displaystyle &\displaystyle=\left(1+\frac{0.053}{4}\right)^{4}-1 &\mbox{}\\ \displaystyle &\displaystyle=0.05406271063 &\mbox{}\\ \end{array} $$ That is, the effective interest rate of Investment $2$ is about $5.4\%.$

We conclude that Investment $2$ is the better investment.

**Reaching an Investment Goal: Investing the Right Amount**

Sleazy P. Martini is saving for grad school and needs to have a sum of $\$18500$ saved in $6$ years from a certificate of deposit which pays $3.7\%$ interest compounded quarterly.

How much money will Sleazy P. need to invest now to reach the investment goal?

From the problem we know that
$$
A=18500,\,\,\,r=0.037,\,\,\,n=4,\,\,\,\mbox{and}\,\,\,t=6
$$
with $P$ unknown.

Plugging what we know into the compound interest formula, $$ \begin{array}{lll} &\displaystyle A=P\left(1+\frac{r}{n}\right)^{nt}&\mbox{}\\ \implies &\displaystyle 18500=P\left(1+\frac{0.037}{4}\right)^{4\cdot 6}&\mbox{$P$ is unknown}\\ \implies &\displaystyle 18500=P\cdot 1.247297908&\mbox{}\\ \implies &\displaystyle \frac{18500}{\color{magenta}{1.247297908}}=\frac{P\cdot 1.247297908}{\color{magenta}{1.247297908}}&\mbox{}\\ \implies &\displaystyle \frac{18500}{1.247297908}=P&\mbox{}\\ \implies &\displaystyle 14832.06208=P&\mbox{}\\ \end{array} $$ That is, Sleazy P. will need to deposit $\$14832.6$ to get back $\$18500$ back in $6$ years.

Plugging what we know into the compound interest formula, $$ \begin{array}{lll} &\displaystyle A=P\left(1+\frac{r}{n}\right)^{nt}&\mbox{}\\ \implies &\displaystyle 18500=P\left(1+\frac{0.037}{4}\right)^{4\cdot 6}&\mbox{$P$ is unknown}\\ \implies &\displaystyle 18500=P\cdot 1.247297908&\mbox{}\\ \implies &\displaystyle \frac{18500}{\color{magenta}{1.247297908}}=\frac{P\cdot 1.247297908}{\color{magenta}{1.247297908}}&\mbox{}\\ \implies &\displaystyle \frac{18500}{1.247297908}=P&\mbox{}\\ \implies &\displaystyle 14832.06208=P&\mbox{}\\ \end{array} $$ That is, Sleazy P. will need to deposit $\$14832.6$ to get back $\$18500$ back in $6$ years.

**Reaching an Investment Goal: How Much Time?**

We could simply put a sum of money into an investment and wait for our money to grow to a desired amount. The formula for the amount of time $t$ it will take a principal amount $P$ to grow to a future value $A$ at an annual interest rate $r$ compounded $n$ times per year is $$t=\frac{\ln(A/P)}{n\ln(1+r/n)}=\frac{\log(A/P)}{n\log(1+r/n)}$$ where $\ln$ and $\log$ are logarithmic functions which you can find on your scientific calculator. Either $\ln$ or $\log$ will work fine.

**Example:**Recall that Sleazy P. Martini wants to save for grad school and needs to have a sum of $\$18500$ saved in $6$ years. If Sleazy P. is willing to wait a bit longer, and deposits $\$10000$ into an account and which pays $3.7\%$ interest compounded quarterly, how long will Sleazy P. have to wait before reaching his investment goal?

$$
A=18500,\,\,\,P=10000\,\,\,r=0.037,\,\,\,\mbox{and}\,\,\,n=4
$$
the time Sleazy P. would need to wait is
$$
\begin{array}{lll}
\displaystyle t&\displaystyle=\frac{\ln(A/P)}{n\ln(1+r/n)} &\mbox{}\\
\displaystyle &\displaystyle=\frac{\ln(18500/10000)}{4\ln(1+0.037/4)} &\mbox{}\\
\displaystyle &\displaystyle=16.70341909 &\mbox{}\\
\end{array}
$$
That is, Sleazy P. would need to wait about $16.7$ years before attaining a sum of $\$18500.$

On your calculator, you will need to type something like the following: $$ \mbox{ln(18500/10000)/(4*ln(1+0.037/4))} $$

On your calculator, you will need to type something like the following: $$ \mbox{ln(18500/10000)/(4*ln(1+0.037/4))} $$

**Yet Another Way to Reach an Investment Goal: Annuities**

When we make regular payments into an account which draws compound interest, we are paying into an

__annuity.__

People will often use annuities to reach an investment goal when they don't have all the money for the principal up front.

**The Big Question:**Suppose we make regular payments of $R$ dollars $n$ times per year into an account with annual interest rate $r$ over the course of $t$ years, what is the future value $A$ of this investment?

**The Future Value of an Annuity**

Suppose we make regular payments of $R$ dollars $n$ times per year into an annuity with annual interest rate $r$ over the course of $t$ years.

Then the future value $A$ of the annuity is $$A=\frac{nR\left[\displaystyle \left(1+\frac{r}{n}\right)^{nt}-1\right]}{r}$$

**Reaching an Investment Goal: Annuities**

Mr. Holt is planning on retiring in $17.5$ years and has decided to save money purchasing an annuity which earns $6.2\%$ interest compounded semi-annually.

If Mr. Holt makes semi-annual payments of $\$500,$ find the future value of the annuity after $17.5$ years.

Using
$$
R=500,\,\,\,r=0.062,\,\,\,\mbox{and}\,\,\,n=2
$$
we compute the future value of the annuity.
$$
\begin{array}{lll}
\displaystyle A&\displaystyle=\frac{nR\left[\displaystyle \left(1+\frac{r}{n}\right)^{nt}-1\right]}{r} &\mbox{}\\
\displaystyle &\displaystyle=\frac{2\cdot 500 \left[\displaystyle \left(1+\frac{0.062}{2}\right)^{2\cdot 17.5}-1\right]}{0.062} &\mbox{}\\
\displaystyle &\displaystyle=30823.77846 &\mbox{}\\
\end{array}
$$
The future value of the annuity is $\$30823.78.$

The calculator key strokes are $$ \mbox{(2*500*((1+0.062/2)^(2*17.5)-1))/0.062} $$

The calculator key strokes are $$ \mbox{(2*500*((1+0.062/2)^(2*17.5)-1))/0.062} $$

**Monthly Payment to Reaching an Investment Goal**

Suppose we want to reach an investment goal of $A$ dollars by paying an unknown amount of $R$ dollars into an annuity $n$ times per year with annual interest rate $r$ over the course of $t$ years.

Then the regular payment $R$ is given by $$\displaystyle R=\frac{rA}{\displaystyle n\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}$$

**Reaching an Investment Goal: Annuities**

Sleazy P. Martini is saving for grad school and needs to have a sum of $\$18500$ saved in $6$ years. Instead of investing more than he actually has, or waiting longer than he had hoped, Sleazy P. has decided pay into an annuity which pays $3.7\%$ interest compounded quarterly.

How much should the quarterly payment be in order for Sleazy P. to reach his investment goal?

Using
$$
A=18500,\,\,\,r=0.037,\,\,\,\mbox{and}\,\,\,n=4
$$
we compute the quarterly payment.
$$
\begin{array}{lll}
\displaystyle R&\displaystyle=\frac{rA}{\displaystyle n\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]} &\mbox{}\\
\displaystyle &\displaystyle=\frac{0.037\cdot 18500}{\displaystyle 4\left[\left(1+\frac{0.037}{4}\right)^{4\cdot 6}-1\right]} &\mbox{}\\
\displaystyle &\displaystyle=691.9791651 &\mbox{}\\
\end{array}
$$
That is, Sleazy P. will need to make quarterly payments of about $\$691.98$ in order to reach his investment goal
of $\$18500$ in $6$ years.

The calculator key strokes are $$ \mbox{0.037*18500/(4*((1+0.037/4)^(4*6)-1))} $$

The calculator key strokes are $$ \mbox{0.037*18500/(4*((1+0.037/4)^(4*6)-1))} $$