Continuous compounding

[24fan]

What exactly IS continuous compoudning? Anyone know?

[ecoli]

http://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding

[24fan]

I've seen that. I have no idea what it's talking about.

[ecoli]

oh ok. But it would help to know where you need to start. Do you not understand compounding specifically, or is the concept of interest itself confusing you?

[24fan]

the general concept of the compounding is what is confusing me..

[ecoli]

its not too difficult to understand.

 

Instead of interest compounding quarterly or annually, the time period is infinitely small. So, your continuously adding interest onto the principle.

[24fan]

I think I see what your saying....could you give a brief example?

[BigMoosie]

Suppose a bank provides 12% interest, compounded monthly.

 

In this case 12% is the nominal annual interest rate. The monthly interest rate is 12%/12 = 1%, so over a year this compounds to become 1.01^12-1 = 12.68%. This is called the effective interest rate, because it is how much your money will grow if you leave it for 1 year.

 

Now suppose the bank compounds daily instead of monthly, common sense tells us that you will make more money, but how much? Lets see:

 

Daily interest rate = 12%/365 = 0.03288...% = x. Over a year this compounds to: (1+x)^365 - 1 = 12.74%.

 

So it has risen a little bit, but not substantially more, if you then compound hourly, then every second etc. you will notice that the effective interest rate is approaching a limit (it grows very slowly as you make the time interval smaller but it doesn't go to infinity).

 

It turns out the limit approaches e^i-1, where 'i' is the nominal interest rate. With my example of i=12% this approaches 12.7496852% (not much more than compounding daily).

 

The reason continuous compounding is used in finance is that it is both more fair (arbitrary dates to compound at are just that, arbitrary) and very elegant to work with when calculations become complicated.

 

Note: when using compound interest it is common to state i (the 12% in my example) as the 'force of interest' rather than 'the annual interest rate'.

[ecoli]

I'll just add a bit to that and explain, you're dividing the annual interest rate of the time period which you're compounding it. As the time period gets smaller, the effective interest rate per period also gets really small. Which is why you don't get much more interest by compounding daily and continuously.

 

This is an example of logistic growth (a non linear system so a bit difficult to wrap your head around).