### 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.