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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 multi-billionaire, 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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