# Continuously Compounded (Future Value)

A very important aspect of compound interest calculations is the re-investment rate that is assumed.

Continuously compounded interest rates are used extensively when pricing derivatives.

In most cases and with all the Excel functions, the re-investment rate is assumed to be the same interest rate for each period.

This re-investment rate is assumed equivalent to the interest rate (or YTM) ??

Most equations in finance theory assume that interest is paid continuously compounded.

### Continuously Compounded Interest

If the 10% interest rate is quoted with continuous compounding it means £100 invested for a year would increase to £??.

### Continuously Compounded Interest Proof (limit)

The entire progression shows that a limit can be defined for continuous compounding (i.e. where m = infinity)

This is repeated at each interest payment date.

Interest paid "t" times a year at a rate of "r/t"

Now lets assume that these interest payments come at an increasingly frequent intervals but an increasingly smaller interest rate.

Lets take the limit t ---> infinity

This will lead to a rate of interest that is paid continuously.

### Continuously Compounded Interest Proof (differential equations)

Another way of deriving this equation is via an ordinary differential equation

Compounding a sum of money at a continuously compounded interest rate R for n years involves multiplying it by earn

Discounting a sum of money at a continuously compounded interest rate R for n years involves multiplying it by e-Rn

The compounding frequency defines the units in which an interest rate is measured.

### Important

Continuously compounded interest is used in quantitative finance.

The future value is also known as the compounded value.

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