Exponentials in Finance  A Billionaire's woe
Rahul Bhattacharya August 24, 2006
Exponentials are there in all walks of our social, biological, technological and financial life. Just like in sociology, technology, physics and biology even in finance we encounter exponentials everywhere.
Here is an example of a billionaire's woe. The richest man on earth (of course not Mr. Bill Gates in this case), a multibillionaire, is totally in love with his beautiful, young wife. He asks her what she wants as gift on her birthday  diamonds, a beach house, sports car, whatever she wants. He promises to give her all his wealth. However she asks him for a very modest gift. She tells him that she wants a dollar today (on her birthday), twice that amount (two dollars) the next month, twice that amount (four dollars) on the third month, twice that amount (eight dollars) on the fourth month and so on. She wants to start off with one dollar and keep doubling the amount every consecutive month. The billionaire marveling at his wife's modesty immediately agrees to that gift in writing and gives her a dollar on her birthday. Little does he know that his young and beautiful wife is also an astute mathematician. Let us see what will happen if he keeps giving her what she wants every month.

Billionaire's Gift to his wife in 
Month 
US Dollars 
1 
1 
2 
2 
3 
4 
4 
8 
5 
16 
6 
32 
7 
64 
8 
128 
9 
256 
10 
512 
11 
1,024 
12 
2,048 
13 
4,096 
14 
8,192 
15 
16,384 
16 
32,768 
17 
65,536 
18 
131,072 
19 
262,144 
20 
524,288 
21 
1,048,576 
22 
2,097,152 
23 
4,194,304 
24 
8,388,608 
25 
16,777,216 
26 
33,554,432 
27 
67,108,864 
28 
134,217,728 
29 
268,435,456 
30 
536,870,912 
31 
1,073,741,824 
32 
2,147,483,648 
33 
4,294,967,296 
34 
8,589,934,592 
35 
17,179,869,184 
36 
34,359,738,368 
Actually, at the end of 36 ^{th} month, i.e. three years, the billionaire would have given away more than $34 billion of his wealth to his wife in gift. In fact, if she keeps going then by the end of 41 ^{st} month she would have acquired more than a trillion dollars in wealth which the billionaire does not have and which is more than the known wealth of any man we know today.
If we plot the first 24 months of the above table we will get a graph like this:
In fact it can be easily seen that the above plot very closely resembles the plot of a function of the type given by y = f(x) = Exp(0.68*x):
The above is an exponential function of which in the billionaire's example represents the number of months. Exponentials have a way of blowing. An exponential series starts off with a low value and continues to generate low values for while, and then suddenly it explodes upwards and values increase sharply in a very short span.
Exponentials, or exponential functions, are found everywhere in finance. Let's take another example. Let us look at the monthly closing price history of NASDAQ (the U.S. stock index) between January 3, 1994 and March 1, 2000. If we plot the NASDAQ monthly price against the number of months between the above dates we will get a plot like this:
The above exponential function has a form y = f(x) = 613*Exp (0.021*x)
The above exponential equation explains the movement of NASDAQ monthly closing price between January 3 ^{rd} 1994 and March 1 ^{st} 2000 very well because the goodness of fit given by R squared is 0.94. If you look at daily prices of NASDAQ between 1998 and early 2000 you will find more interesting and better fit exponential functions and the same would be the case if you look at the closing prices of Dow Jones Industrial Average over a 100 year period.
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