Question

What is compounding of interest. Explain with examples.

Answer 1

Interest generates interest--that is the simple meaning of compounding of interest.

In simple interest , we receive only periodic interest on the principal amount alone.For example,if we invest $ 1000 in an account yielding 8% simple interest per year, we will receive(or the a/c increases by) the same $1000*8%=$ 80 in simple interest , for all the years we maintain the account.

whereas,

if the same account gives compound interest of 8%, at end of yr. 1, we will receive(or the a/c grows by) $ 1000*8%=$ 80 & in the 2nd year , it increases by (1000+80)*8%= $ 86.4---ie. $ 80 interest received in the 1st year also earns interest at the rate of 80*8%=$ 6.4.

Thus, the interest of the previous year gets added to the principal , on which it was calculated & the resultant sum becomes the new principal for this year.

That is the compounding effect of interest .

The simple formula for compound interest is

A=P*(1+r)^n

where,

A= The amount or the Future Value

P=the initial amount invested

r= the rate of interest per period

n= no.of periods

Continuing the above example,

we can find the future value of $ 1000 at the end of say, 6 yrs at 8% interest, compounded annually,

here, P= $ 1000; r=8% or 0.08 ; n=6 & we need to find the A or the FV

ie. FV=1000*(1+0.08)^6= $ 1586.87

Thus, we can see that ,

it is solely due to the effect of compounding , that money gains value over time.So, the concept of time value of money , is born due to this very compounding effect or earning capacity of the interest amount.

Interest may be compounded yearly(only once in a year), semi-annually(twice in a year), quarterly(once in 4 months in a year), monthly(12 times in a year) or even daily(365 times in a year) or even more than that continuously compounded.

the more times ,interest is compounded, the more the future value of the initial amount invested.

We will see this effect with the above same numbers for all parameters & the results will be interesting to note as well as easy to understand.

we have seen interest compounded annually

now , we will see, one by one, the rest of the compounding frequencies:

Interest compounded semi-annually(twice in a year)

So, now, the principal is the same $ 1000

r= the rate of interest per period, ie. 8%/2=4% &

n= no.of compounding periods, ie .6 yrs.*2=12

Now applying the formula,

FV=1000*(1+0.04)^12=1601.03

Interest compounded quarterly(once in 4 months in a year)

now, the principal is the same $ 1000

r= the rate of interest per period, ie. 8%/4=2% &

n= no.of compounding periods, ie .6 yrs.*4=24

Now applying the formula,

FV=1000*(1+0.02)^24=1608.44

Interest compounded monthly (ie. 12 times in a year)

now, the principal is the same $ 1000

r= the rate of interest per period, ie. 8%/12= 0.67% or 0.0067 p.m. &

n= no.of compounding periods, ie .6 yrs.*12=72

Now applying the formula,

FV=1000*(1+0.0067)^72=1613.50

Interest compounded daily (ie. 365 times in a year)

now, the principal is the same $ 1000

r= the rate of interest per period, ie. 8%/365= 0.0219% or 0.000219 per period &

n= no.of compounding periods, ie .6 yrs.*365= 2190

Now applying the formula,

FV=1000*(1+0.000219)^2190=1615.99

Interest compounded continuously(ie. Constant compounding for Infinite times)

now, the principal is the same $ 1000

r= the rate of interest per period, ie. 8%

n= no.of compounding periods, ie .6 yrs.

Now applying adifferent formula foe continuous compounding,

A=PV*e^(r*n)

e is the mathematical constant, 2.7183

So, A or FV=1000*2.7183^(0.08*6)= 1616. 08

Thus we can see that no.of times of compounding of interest increases the future value of the principal amount.

The r,ie. The rate of interest which is normally given as % per annum, is converted to rate per period(semi-annual, quarterly , monthly or daily)   &

the no.of compounding periods is arrived at by multiplying with the no.of years given.

ie. A/FV= P*(1+(r/n))^(r*n)

It is interesting to summarise the effect of above different compoundings:

Compounding periods	Future value	Increase in FV
Annual	1587
Semi-annual	1601	14
Quarterly	1608	7
Monthly	1614	6
Daily	1615.99	1.99
Continuous	1616.08	0.09

We can see that the initial money does increase , but by decreasing amounts ,as the compounding frequency increases.

The increase is the least under constant compounding.

Mostly, only semi-annual, quarterly or monthly compounding are more prevalent.