What is Compound Interest? 

Compound interest is a method that is used to estimate the interest paid on the principal and the accrued interest as opposed to the simple interest that is estimated only on the principal amount.  

Real-Time Application 

Let us take an image posted by you on social media. If you have 10 friends viewing the image, your views become 10. Now, if any out of the 10 further share the image to his/her friends, views of the image increase by a lot more. For a post to become viral, this sharing is multifold, increasing exponentially with each successive person who shares the image. This can be related to compound interest where interest earned on an existing amount is added to the original amount so that for the next time period that the interest is calculated, there is an increase in the initial amount used for calculating interest. Just like any additional share of your post is adding to the number of views by a large amount, the addition of interest to the principal amount continuously increases the principal amount over each iteration of the time period.  

Another common analogy is the snowball effect. Imagine standing on top of the hill and making a snowball roll down the hill. Over each revolution, the snowball accumulates more snow and can become a really big one by the time it reaches the bottom of the hill. Well, if the snow was money, you would be a very rich person by the time you came down the hill. 

Deriving Compound Interest 

Compound interest, or 'interest on interest', is determined using the formula —$CI=P{\left(1+\frac{r}{100k}\right)}^{kn}-P$,

where P is principal, r is the annual rate of interest (in percentage), k is the number of compounding periods per year and n is the number of years.  

It's deserving to remark that this method provides you the ultimate value of an expense or investment, where the interest keeps adding to the principal. If you hold a developing condition, which generates more growing circumstances, which in turn produces more developing things for your income to build up quickly. To learn compound interest, primarily begin with the idea of simple interest, that you invest money, and the bank gives you credit on your deposit.

Famous Quotes: 

Einstein described it as a common strong force in the creation and is rumored to have said this about compound interest — “It is the 8th wonder of the world. He who understands it, earns it; he who doesn't, pays it.” 

Benjamin Franklin’s quote takes the cake though — “Money makes money. And the money that money makes, makes money.”

Formulas 

The Formula for Future Value (FV) 

P is principal, r is the annual rate of interest (in percentage), k is the number of compounding periods per year and n is the number of years. 

$FV=PV{\left(1+\frac{r}{100k}\right)}^{kn}$

The Formula for Interest Rate (r) 

You should want to operate from the common annual interest percentage you're taking on your profits, finance, private loan, or vehicle loan, this method can assist. Remark that you must multiply your returns by 100 to generate a percentage figure (%).

$r=k\left[{\left(\frac{FV}{PV}\right)}^{1/nk}-1\right]$

The Formula for Time (n) 

This modification of the method is used for estimating time (n), by applying natural logarithms. 

$n=\frac{\ln \left(\frac{FV}{PV}\right)}{k\ln \left(1+\frac{r}{100k}\right)}$

Practice Problems

Example 1 

Q. Harry decided to invest $55000 at a nominal annual interest rate of 6%, compound quarterly. How much will his investment be worth in 6 years? 

A. Here PV = 55000, r = 6, k = 4, n = 6 

$\begin{array}{l}FV=PV{\left(1+\frac{r}{100k}\right)}^{kn}\\ =55000{\left(1+\frac{6}{100\left(4\right)}\right)}^{4\left(6\right)}\\ =55000{\left(1+\frac{6}{400}\right)}^{24}\\ =55000{\left(1+0.015\right)}^{24}\\ =55000{\left(1.015\right)}^{24}\\ =75622.65\end{array}$

The investment will be worth $75,622.65 in 6 years. 

Example 2 

Q. Lisa invests $ 15,000 into a saving account that pays 4.25% interest, compounded monthly. Estimate the interest earned in the account after 5 years. 

A. Here P = 15000, r = 4.25, k = 12, n = 5 

$\begin{array}{l}CI=P{\left(1+\frac{r}{100k}\right)}^{kn}-P\\ =15000{\left(1+\frac{4.25}{100\left(12\right)}\right)}^{12\left(5\right)}-15000\\ =15000{\left(1+\frac{4.25}{1200}\right)}^{60}-15000\\ =3544.53\end{array}$

The interest compounded is $3,544.53. 

Example 3

Q. If an investment gives you 8% interest compounded annually, how long will it take to double your money, invested in the present time?

A. Here FV = 2PV, k = 1, r = 8

n = ln (FV/PV) / k [ln (1 + r/100k)]

n = ln (2PV/PV) / [ln (1 + 8/100)]

n = ln 2 / ln 1.08

n = 9 years

Key Variables of Compound Interest

There are a few key factors that play in the final product and that impact the returns. Let us look at some of them:

Annual Interest Rate: This is the rate that one earns or gets charged. You earn or owe more money when the interest rate is high.

Initial Deposit/Initial Balance: The initial amount that one borrows or deposits adds up over time, by compound interest formula.

Compounded Frequency: How often the interest is compounded while taking a loan or depositing: daily, monthly, or yearly determines the balanced growth.

Duration: The duration or period that one leaves money in the account or longer a debt is held, the compounded earning or debt is higher.

Withdrawals and Deposits: Over the long run, the pace at which one grows the initial investment balance makes a big difference.

Simple Interest Vs Compound Interest 

$CI=P{\left(1+\frac{r}{100k}\right)}^{kn}-P$.

Compound Interest Basic Applications 

The basic application of compound interest is the fields of finance, specifically as related to savings and loans. It is worth noting that a CI on loans is very bad, however, a CI on savings is always welcome. Although students might not be the target users of this formula, people with a good grasp of this topic can generally be assumed to have a better financial future. Following this formula, the interest is credited on the principal in addition to any accrued interest. The measure of interest for a time is attached to that measure of principal to estimate the interest for the following years. In different terms, the interest is reinvested to get more interest. The interest may be increased daily, periodically, semiannually, or yearly.  

Common Mistakes 

• One of the often-repeated errors is assuming that the period of compounding interest is annual, when in fact it could be semi-annual, quarterly, monthly, or even daily. Always double-check the time period to avoid mistakes in calculation. 
• The interest rate in percentage will need a divisor of 100 in the formula to be correct and most students often overlook this simple thing. 
• The formula is for future value and not for the interest itself. To calculate only the interest, the principal amount needs to be subtracted from the final amount. 
• Last but not least; make sure to read the question correctly to find out what is being asked. 

Context and Application 

The topic is relevant for the following exams and courses: 

• Finance and Accountancy (Bachelors and Masters) 
• Chartered Accountancy 
• ICWA 
• CPA 
• Bachelor in Commerce 
• Masters in Commerce 
• MBA Finance