Compound Interest

“Compound interest is the 8^th wonder of the world. He who understands it, earns it; he who doesn't, pays it.”
       - Albert Einstein 

Introduction

In this topic, we will first see the impact of compounding, then we will understand what compound interest means, and how to calculate it next we will understand compounding frequency and how to convert one compounding frequency into another. In the end, we will understand the effective annual interest rate.

Impact of compound interest 

  

Compounding creates an exponential effect on the interest earned, and the best way to understand its impact is visually.

From the above figure we see how over the time compound interest grows exponentially, and this is the kind of impact compounding has in the long run.

Compound Interest

Compound interest can be understood as interest earned on interest. Let say we have INR 1000 and we invested it to earn 10% per annum compounded annually. Now using the below formula we calculated the total amount after 2 years.

             A = p*(1 +r/100)^n
  
             A = amount after 2-year
             p = initial amount invested
             n = number of year
             r = interest rate
So, A = 1000*(1 + 10/100)^2
          =  1210
      C.I( compound interest) = A-p
                                             = 1210-1000
                                             =  210
Now if we expand the equation a little
        A = 1000*(1 + 10/100)*(1 + 10/100) 
the underlined portion shows interest earned in the first year, and the bold portion shows interest earned in the second year. As can be seen from the bold part of the equation, second-year interest is not only earned on the principal but also interest earned in the first year. So, let's further simplify it.

CI = interest earned on the principal in the first year +  interest earned on (principal + first-year        interest) in the second year
     = 100 + (1000 + 100)*10/100
     = 100 + 110
     = 210
The same logic can be extended to n>2 also.

Compounding Frequency 

In the financial world, we encounter statements like x % per annum compounded semiannually. So, what is compounded semiannually means, it is what we called compounding frequency. It basically means how many times we pay interest in a year. Here compounding semiannually means interest will be paid 2 times a year. So, here compounding frequency is 2. Similarly, it could be quarterly, monthly or daily. Now, how we incorporate this into our earlier formula.

        

          A = p*( 1 + r/100*m)^n*m

           

          r = rate of interest given per annum

          n = no of year 

          m = compounding frequency

So, if it is given that INR 1000 is invested at 10% per annum compounded quarterly for 3 years, then

          A =1000*( 1 + 10/100*4)^4*3

              = 1344.88

Continuous compounding

 When compounding frequency becomes very small i.e compounding takes place every moment then it is called continuous compounding. For continuous compounding we use formula.

         A = p*e^rn

         

         r = rate of interest given continuously compounded

         n = number of year

Converting on compounding frequency to another 

Let say we are given interest rate as 10% per annum compounded semiannually, and we need to find an equivalent rate for quarterly compounding i.e a quarterly compounded interest rate which will give us equivalent output for the same initial investment after

So if we desire the same output assuming an initial investment of INR 1000 for 2 years.

         1000*(1 + 0.10/2)^2*2  =  1000*( 1 + r/100*4)^4*2

solving for r we get 9.87%

We could generalise the above formula

         A*( 1 + rp/100*p)^p*n = A*( 1 + rq/100*q)^q*n

         n = number of year

         p,q = componding frequencies 

         rp = interest rate with compounding frequency p

         rq = interest rate with compounding frequency q

  

Important point: one thing to observe above is the equivalent r for quarterly compounded is lower than semiannually compounded rate. So, as we increase the compounding frequency equivalent compounding rate will decrease.

Effective annual interest rate

When we convert a rate with a given compounding frequency to the equivalent rate for compounding frequency of 1, the equivalent rate is called an effective annual interest rate. Effective annual interest rate basically tells the actual interest earned as a result of compounding over the period. To find an effective annual interest rate we use the above formula and put p = 2 and q = 1

      

          1000*(1 + 0.10/2)^2*2 = 1000*(1 + r/100)^2

EAIR = 10.25%

Important point: From the above image we can note that as the compounding frequency increases for the given interest rate EAIR increases. Similarly, we can say that for a given EAIR equivalent compounded rate decrease as compounding frequency increases.

This ends our discussion on compound interest. For any doubt please comment.