Page 4 - Rhea

Financial use of e

As discussed on the page Defining e, Jacob Bernoulli discovered the value of e in 1683 while studying compound interest. However, <in progress>

Bernoulli discovered this constant in 1683 by studying a question about compound interest, but people in far more ancient time already came up with similar question. For example, a clay tablet from Mesopotamia, dated to about 1700 B.C. and now in the Louvre, poses the following problem: How long will it take for a sum of money to double if invested at 20 percent interest rate compounded annually? We now know this could be solved by using logarithms which the Babylonians did not have. Suppose we have $100 (the "principal") in bank that pays 5% interest compounded annually. At the end of one year, our balance will be 100 x 1.05 = $105. At the end of the second year the balance will therefore be 105 x 1.05 = $110.25, at the end of the third year 110.25 x 1.05 = $115.76, and so on. It is easily to get the formula: Cumulated money(S) Principal (P) Interest rate(r) Time(t)

                $ \begin{align} S = P\left(1+ r\right)^t \end{align} $

This formula is the basis of financial math,and it will apply to bank accounts, loans, mortgages, or annuities. Since time could be other than year more precise formula would be;

                $ \begin{align} S = P\left(1+\frac rn\right)^nt \end{align} $

Let the annual interest rate be 100% then we have

                $ \begin{align} S = P\left(1+\frac 1n\right)^n \end{align} $

As the n increases we have:

When n=50 with 5% interest rate,the growth rate would almost be $ e^ {\frac{1}{20}} $. So we rewrite:

                $ \begin{align} growth = e^{r} \end{align} $

If the time is t , the growth rate would be:

                $ \begin{align} growth = e^{rt} \end{align} $

This means e could be used to calculate the growth rate for continuous same interest rate with the above general formula. So if we want to know when will $100 double with 5% interest rate we have:

                $ \begin{align} 100e^{5%t}=200 \end{align} $

After calculation t will be 13.86 years:

                $ \begin{align} t=\frac{ln2}{5%}=\frac{0.693}{5%}=\frac{69.3}{5}≈\frac{72}{5} \end{align} $

Easily we can tell the time is roughly about 72 divided by the interest rate.This is well know as "Rule of 72".

Further more as n increases to infinity we could define is continuous compounding. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P as

                $ \begin{align} S = Pe^rt \end{align} $

As the number of compounding periods n reaches infinity in continuous compounding, the continuous compound interest is referred to as the force of interest ẟ. In financial math ,let the amount of money at time T0 be A the amount of money at T be B define accumulation function a(t)is a function defined in terms of time t expressing the ratio of the value at time t (future value A) and the initial investment(present value I)

which is

                $ \begin{align} A=a(t)I \end{align} $

The accumulation functions are often expressed in terms of e.

So we could define the force of interest as:

                $ \begin{align} ẟ = \frac {a'(t)}{a(t)} \end{align} $

Conversely:

                $ a(t)=e^{\int_0^t ẟ\ du} $ .

easily we could get a(t):

                $ \begin{align} a(t) = e^{tẟ} \end{align} $

The force of interest is widely used in financial math to calculate loans mortgages treasure bill and annuities with ẟ being a function.

References:
O'Connor, J J; Robertson, E F. "The number e". MacTutor History of Mathematics.

Howard Eves, An Introduction to the History of Mathematics (1964; rpt. Philadelphia: Saunders College Publishing, 1983), p. 36.