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(Video) Compound Interest

by ThinkwellVids

Here's another great video by ThinkwellVids showing you how to solve compound interest problems.

$\large{\text{Summary}}$

The formula for this is given by: $$A=P\bigg(1+\frac{r}{m}\bigg)^{tm}$$ Where $A$ is the future value, $P$ is the present value, $r$ is the interest rate, $t$ is the time, and $m$ is the number compounding per year.

Compound Interest vs Simple Interest

Recall that, when calculating simple interest, you are multiplying the interest rate by the principal amount (what you initially invested).

On the other hand, compound interest is basically interest on interest. Instead of just earning interest on the principal amount you invested, you earn interest on the principal and on interest you've already accumulated on it. - Hannah Bonville

Below is a story to help illustrate the difference in growth between simple and compound interest.

Bob's Bank

COMPOUND INTEREST

Bob starts with $\$5000$ in his bank account and receives $5\%$ interest compounded daily for $8$ days.

$$\begin{array}{|l|l|l|} \hline \text{Day}&\text{Amount in Interest Gained}&\text{Amount in Bob's Account}\quad\quad\\ &(0.05\times\text{balance})&(\text{balance}+\text{interest})\\ \hline 0&0&\$\color{blue}{5000}\\ \hline 1&0.05\times\color{blue}{5000}=250&5000+250=\$\color{Violet}{5250}\\ \hline 2&0.05\times\color{Violet}{5250}=262.50&5250+262.50=\$\color{RedOrange}{5512.50}\\ \hline 3&0.05\times\color{RedOrange}{5512.50}=275.63&5512.50+275.63=\$\color{Red}{5788.13}\\ \hline 4&0.05\times\color{Red}{5788.13}=289.41&5788.13+289.41=\$\color{Green}{6077.54}\\ \hline 5&0.05\times\color{Green}{6077.54}=303.88&6077.54+303.88=\$\color{Magenta}{6381.42}\\ \hline 6&0.05\times\color{Magenta}{6381.42}=319.07&6381.42+319.07=\$\color{RawSienna}{6700.49}\\ \hline 7&0.05\times\color{RawSienna}{6700.49}=335.02&6700.49+335.02=\$\color{TealBlue}{7035.51}\\ \hline 8&0.05\times\color{TealBlue}{7035.51}=351.78&7035.51+351.78=\$7387.29\\ \hline \end{array}$$

So, Bob removes $\$7,387.29$ from his bank on the eighth day.

We can also use the compound interest formula to find the balance on the eighth day.

$A = \text{ final amount}$

$P = \text{ principal (starting) amount} = 5000$

$r = \text{ interest rate (expressed in same units as } n \text{ and } t) = 5\% \text{ per day }= 0.05$

$n = \text{ compounding frequency} = 1 \text{ time per day}$

$t = \text{ number of time periods} = 8 \text{ days}$

$$\begin{align} A &= P(1 + \frac{r}{n})^{nt}\\ &= 5000(1+ \frac{0.05}{1})^{1*8}\\ &= 5000(1+0.05)^{8}\\ &= 5000(1.05)^{8}\\ &= 7387.28 \end{align}$$

As you can see, if you use the formula, you get pretty much the same amount! (The difference of one cent is due to rounding in the table.)

SIMPLE INTEREST

Bob starts with 5000 dollars in his bank account and receives 5% (simple interest) daily for 8 days.

$$\begin{array}{|l|l|l|} \hline \text{Day}&\text{Amount in Interest Gained}&\text{Amount in Bob's Account}\quad\quad\\ &(0.05\times\color{blue}{\text{principle balance}})&(\text{balance}+\text{interest})\\ \hline 0&0&\$\color{blue}{5000}\,(\color{blue}{\text{principle balance}})\\ \hline 1&0.05\times\color{blue}{5000}=250&5000+250=\$\color{Violet}{5250}\\ \hline 2&0.05\times\color{blue}{5000}=250&5250+250=\$\color{RedOrange}{5500}\\ \hline 3&0.05\times\color{blue}{5000}=250&5500+250=\$\color{Red}{5750}\\ \hline 4&0.05\times\color{blue}{5000}=250&5750+250=\$\color{Green}{6000}\\ \hline 5&0.05\times\color{blue}{5000}=250&6000+250=\$\color{Magenta}{6250}\\ \hline 6&0.05\times\color{blue}{5000}=250&6250+250=\$\color{RawSienna}{6500}\\ \hline 7&0.05\times\color{blue}{5000}=250&6500+250=\$\color{TealBlue}{6750}\\ \hline 8&0.05\times\color{blue}{5000}=250&6750+250=\$7000\\ \hline \end{array}$$

Bob removes $\$7,000$ from his bank on the 8th day.

We can also use the simple interest formula to find the balance on the eighth day.

$A = \text{ final amount}$

$P = \text{ principal (starting) amount} = 5000$

$r = \text{ interest rate (expressed in same units as } n \text{ and } t) = 5\% \text{ per day }= 0.05$

$t = \text{ number of time periods} = 8 \text{ days}$

$$\begin{align} A &= P(1 + rt)\\ &= 5000(1+ 0.05\cdot 8)\\ &= 5000(1+0.4)\\ &= 5000(1.4)\\ &=7000 \end{align}$$

It is clear that over time, compound interest makes much more interest than its linear counterpart, simple interest.

(Video) Compound Interest

by HEGARTYMATHS

Here's a video by HEGARTYMATHS where he goes over some word problems involving compound interest.

$\large{\text{Summary}}$

Compound interest is the addition of interest to the principle sum of a loan or deposit and we use the this formula $\text{Future Value}=\text{Present Value}(1+\text{Interest Rate})^{\text{Time}}$ $$FV=PV(1+r)^t$$

(Video) Continuous Compound Interest

by mathman1024

Here's a video by mathman1024 explaining continuous compound interest.

$\large{\text{Summary}}$

To evaluate compound interest we used the formula: $A=P(1+\frac{r}n)^{nt}$, where $n$ was the number of times we compounded it. Now we're looking at continuous compounded interest or when number of compoundings approaches infinity: $n\rightarrow\infty$. This changes the formula and we get a new one for continuous interest: $$A=Pe^{rt}$$ Where,

• $A=$ the final amount
• $P=$ the principal (or starting) amount
• $e\approx2.71828...$ a constant irrational number
• $r=$ the annual rate
• $t=$ time (in years)

Let's take a look at an example. In the previous video on compound interest we were told that: $P=12000$ $r=0.08$ $t=5$. So let's see what happens with our new formula. $$\begin{align} &A=Pe^{rt}\\ &A=\color{red}{12000}e^{\color{red}{0.08\cdot5}}\\ &A=12000e^{0.4}\\ &A=17901.90 \end{align}$$ So as you see it gives an even larger amount of money back. $$\begin{array}{|c|c|} \hline n&A\\ \hline \text{Yearly}&17631.94\\ \hline \text{Quarterly}&17831.37\\ \hline \text{Monthly}&17878.15\\ \hline \text{Continuously}&17901.90\\ \hline \end{array}$$