Continuous Compounding

From our examples of Finance, we have see that the rate of return on an investment, invested at a rate r, for a period m can be expressed as follows

where

• r is the rate of interest

• m is the frequency of interest payment

• t is the term for the deposit

• FV is the Future Value

• PV is the Present Value

For example, if we deposit 1000 today (PV) at a rate (r) 5% for a one year term (t), when the interest is paid annually (m), the the future value (FV) is

The Future value, or the value of the deposit in one years time works out to 1050.

Now assume that the interest was paid on a six monthly basis. This will result in two payments in one year, so m becomes 2. Plugging this back into equation (1) will give you a number sightly higher than 1050.

The future value rises as the compounding frequency increases, as shown in the table. Frequency of 4 implies interest is paid every three months. Clearly this frequency can rise to every day, every hour in the day, every minute and so on. As the frequency rises, we get closer to what is known as continuous compounding.

In mathematics, there is a mathematical constant called e.

The expression for e is as follows

So, in other words, as the frequency m approaches infinity which means that the rate of interest accrual is continuous the expression becomes e.

We can therefore rewrite equation (1) as

Going back to the table we showed before, what would be the future value were you to plug the value of r=0.05 and t=1 into the above formula?

Continuous compounding is rarely used in practical everyday products such as bank deposits or mortgages. Most retail banking systems are not really designed for such complex calculations. So why did we introduce this?

Continuous compounding is used in Quantitative Finance where models are built to price instruments such as options (a derivative). They are also used in modelling market behaviour such as the behaviour of interest rates in the Investment Banking world.

We can now develop the model for zero coupon bonds using the principles discussed above.

Let us define the terms for the model.

We define R(t,T) to be the spot interest rate prevailing at time t for an investment P maturing at time T, denoted by P(t,T).