The Time value of money in Continuous Time
Calculating continuous time series in real life scenario means that we might be dealing with infinite frequency and time horizon. The continuous time series is applicable in finance when we deal with continuously changing Foreign Exchange rate, Interest rate (continuously compounding) and price of the perpetual bonds. Normally, to calculate the value of perpetual and high frequency instruments is very difficult, but with the help of mathematics and calculus we can estimate there value within few steps, that also without use of software and spread sheets.Before we began the calculations of perpetual bonds let's go through a important concept that we will use in this section. It is called continuously compounding.
The relationship above illustrates that in continuously compounding formula assuming interest rate is equal to '1%' as value of 'n' (number of times compounding) approaches to infinity the output approaches to 'e' ( Euler's number) which is equal to 2.71828~. The progression is demonstrated by plotting the values in excel and feeding the formula. Further to aid the claim we plot it graphically to illustrate the convergence towards 'Euler's number':
.
This concept is now helpful to understand the continuously compounding interest rate formula. Let us take an example where we have to invest $1 dollar at compounding rate 'r' and compounding 'n' number of times in one (1) year, when 'n' tends to infinity.
The continuous compounding formula looks as follows:
Further in relationship we take 'r' to the denominator and represent (n/r) as N to make it ready for transition into 'Euler's number'
Finally the formula for compounding interest rate at rate 'r'can be written as: 'e'raised to the power 'r'= e^r. Now if we want to calculate for 't' years instead of '1' year we write it as 'e' raised to the power 'rt or e^rt.
For example:
A is the amount that will be received for the principle 'P' with 'r' continuously compounding interest rate in 't' years when number of times compounding 'n' tends to infinity.
Calculating the present value when the time step is very small and we are continuously compounding for 't' year(s).
As mention above section, in the financial market we usually come across situations when the time step is very small and the frequency of compounding gets closer to infinity. In such case inorder to sum the continuously compounding value we take help of Integration formula.
Example :
The interest is 5% per annum. A bond pays $1 coupon in one year (from the beginning of year 0 to the beginning of year 1), but the coupon payment spreads evenly throughout 365 days of the year.
Calculate present value of the bond!
(1/365) means that $1 is divided among 365 days.
Please note that the exponential has negative sign, as we are dealing with present value.
However the challenge is to sum all the them, and if we try to do it in excel we will have to drag it all the way down to 365 days. So in order to sum at time step (1/365) we take integral from 0 to 1 years.
Here the time step is very small (e.g 1/365)
We further solve the integration (sum of many small components) in-order to get:
Please click here to know the process (believe me it is quite simple)
So the Present value of bond with the interest rate 5% per annum which pays $1 coupon in one year (from the beginning of year 0 to the beginning of year 1),whose coupon payment spreads evenly throughout 365 days of the year (very small time step) is $0.9754
Please note that this is just the illustration of the use of integration in continuous time series. In real life small differences in decimal when multiplied with millions of dollar can show huge gap in the result
# Techniques covered in blog posts labelled AED are taught in courses Mathematics for Applied Economics, and Applied Economics Dynamics offered at Crawford School of Public Policy
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