Modelling of time
Suppose I lend you a sum of money M for a period of time d, using simple interests, with the rate r (over the whole period d). This means that we agreed that at the end of d, you will owe me
Everyone is familiar with this equation for the calculation of simple interests.
But, suppose now we agreed to compute the repayment due like this: I actually lent you the sum M over two periods of length d/2. And at the end of d you owe me
because at the end of d/2 you owe me M( 1 + r/2 ), which you keep, so at the end of the second period of length d/2, you owe me the sum calculated above.
We may also agree to split the period d into n subperiods of length d/n, in which case the repayment due becomes
A mathematical result establishes that when n tends to + ∞, the coefficient next to M tends to er, which is written with the usual mathematical notations
where e = 2.71828183... It is actually one of the definitions of the number e, which plays such an important role in mathematics (alongside π and a few other constants).
This leads to the formula for "continuously compounded interest" : if r is the simple interest rate for a unitary period d, then if you borrowed a sum M, and we agreed on continuously compounded interests, the repayment after a period t is Mert/d. Taking d as the unit of time (measured by 1) this writes
Notice that r is no longer the interest rate (in the sense of the calculation M( 1 + r) ) of any continously compounded loan. It is just a parameter to which we give the name "simple interest rate" over the time period of length 1.
The modelling with discrete time is for the sake of classroom explanation.
In a first approach, discreteness is simpler to handle than continuity.
C0 will be the initial expenditure, a positive number, appearing with a negative sign in calculations of NPV. In the case of the purchase of a security, it will more usually be called P.
A digression on physics.
Energy was naturally considered a continuous quantity, until 1900 when it became necessary to model it as a physical variable taking only discrete values in order to explain certain phenomena.
The same may happen to time in future physics.
Screens of the video