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Linear Growth Versus Exponential Growth
Logistic Growth Versus Exponential Growth
Compound Growth Versus Exponential Growth
Exponential Brownian Motion
The Mechanism Of Population Doubling
Generations and Population Doublings
What Is Exponential?

The Rule of 70 and The Rule of 72 Compared
The Scales of e

External Links:

Geometric Brownian Motion - SITMO

Brownian Motion - Wikipedia

Google Shares (Historical Prices) - Yahoo! Finance


Exponential Brownian Motion (or Geometric Brownian Motion)

(The shockingly simple truth behind modelling share price fluctuations)

Introduction

Many websites devoted to loans, investments and compound interest make mention of the Rule of 72. Most of those only tell the positive side of things (how your money doubles), and fail to adequately explain the dark side of the Rule of 72. In this article I will explore both aspects of the Rule of 72 - how your money doubles as well as how your money halves.

Dig deeper into financial modelling and you will come across a type of growth known as Exponential Brownian Motion (or Geometric Brownian Motion), typically used in explanations of the chaotic nature of share price fluctuations on the stock market. As far as I can tell, nobody has proposed any simple growth model that can explain how share prices rise and fall. Not why, but how. In this article I will extend the Rule of 72 to the Scales of 72, which can be used as a simple rule of thumb to predict both wealth doubling and wealth halving.

The Rule of 72

The first thing you should know about the Rule of 72 is that it has rival "rules". The most commonly used rival is the Rule of 70. For a comparison of the two read my Exponentialist article The Rule of 70 and The Rule of 72 Compared. In essence they are the same rule. However, because 72 has more whole number divisors than 70 it is the "rule" of choice in the world of finance. This makes it easier to explain to clients making investments or taking out loans. 

Also worth noting is that both the Rule of 70 and the Rule of 72 are increasingly inaccurate as the percentage growth rate increases (values between positive and negative 10% work quite well).

The majority of the time, the Rule of 72 is used to explain how to "double your money". It's quite simple, all you do is divide the interest rate into 72 to get the doubling period. Thus, a 1% annual growth rate yields a 72-year doubling period and a 2% annual growth rate yields a 36-year doubling period. 

For share prices, the doubling period might be measured in days, weeks or months. Thus, a share price that experiences a sustained 1% growth rate per day for 72 days will double its share price around the 72nd day. Or, a share price that experiences a 2% growth rate per week will double its share price around the 35th week.

However, to understand the shockingly simple truth behind share price fluctuations, it is necessary to delve into Exponential Brownian Motion (also known as Geometric Brownian Motion). This poorly named mathematical model has nothing to do with actual Brownian Motion, and everything to do with growth modelling the apparently random (or stochastic) process of share price fluctuations. 

Exponential Brownian Motion

Existing explanations of  Exponential Brownian Motion are anything but simple, and rely heavily on complicated mathematics (see external link).

When considering the apparently random nature in the rise and fall of share prices wouldn't it be nice instead if we could apply a simple model such as the Rule of 72 to calculate the doubling period for a given value or the halving period for a given value? 

The problem appears to be that the Rule of 72 relies on a fixed rate of compound interest, and there is nothing fixed about the value of share prices. Also, conventional explanations of the Rule of 72 rely solely on positive growth rates.

Yet it should be entirely logical to suppose that any mixture of positive growth rates which add up to 72 will result in wealth doubling, and any mixture of negative growth rates which add up to negative 72 will always result in a wealth halving. Of course, using the Rule Of 72, these are approximate calculations. 

Before we explore a mixture of positive and negative variable growth rates, let's examine an example of variable positive growth rates on their own:

Example 1: Variable positive growth rates:

1 + 2 + 1 + 3 + 4 + 2 + 1 + 3 + 4 + 5 + 3 + 2 + 1 + 2 + 3 + 4 + 2 + 1 + 2 + 4 + 2 + 3 + 3 + 2 + 5 + 5  +2 = 72

How many years did it take the wealth to double? Rather than adding the rates together, simply count how many rates there are - the answer is 27. In fact, it doesn't even matter which order these growth rates appear in, the result is always the same for the same set of growth rates.

The same thing works in reverse, with only a minor adjustment! This is because, for a given growth rate, halving times are always slightly shorter than doubling times. This imbalance between doubling and halving is explored in more detail in my article The Rule of 70 and The Rule of 72 Compared.

However, as an approximation, it is reasonably accurate to simply ignore this minor adjustment. Thus, for variable negative growth rates, whenever they reach 72 then the value of the shares (or loan, or investment) will halve. 

The Scales of 72

If there is a mixture of positive and negative rates, together with variable rates for both, then this is as complex as it can possibly get. 

The solution is the Scales Of 72.  Imagine a weighing scale with negatives on one side and positives on the other. As soon as one side "weighs" 72 more than the other side then that side "wins". If the winning side is negative, then the value  in question is halved.  If the winning side is positive, then the value in question is doubled.

Rates for years of negative growth Rates for years of positive growth
1, 3, 2, 5, 1 and 1 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 5, 5, 4, 3, 3, 5, 3, 5, 5 and 3
Red Time : 6 years @ variable rates 

Red Rate Total : 13

Black Time : 23 years @ variable rates  

Black Rate Total : 85

Example 2: This share price (or investment / loan value)  will double. Using the Scales Of 72 measure the doubling time for mixed positive and negative variable growth rates. The sum of the growth rates can be calculated as 85 - 13 = 72 (Black Rate Total - Red Rate Total = 72).  The elapsed time can be calculated as 23 + 6 = 29 (Black Time + Red Time = 29 years). 

A zero growth rate for a given year simply adds 1 to the elapsed time.

Note that this model is an approximation, though a very simple and useful approximation at that. It is close enough to prove that there is a predictable order to the apparently chaotic nature of variable rate compound interest (as seen in the share price fluctuations, or a variable rate mortgage). Doublings and halvings are not occurring randomly! 

I explore the same concept for the Rule of 70 in my article The Scales of 70. To differentiate my growth model (based on variable rate compound interest) from the various Exponential Brownian Motion growth models and the simple Malthusian Growth Model (constant rate exponential), I refer to growth based on variable rate compound interest as Couttsian Growth and the model as the Couttsian Growth Model.

Let's try a real example  - in this case Google - from Yahoo! Finance. As some of the monthly growth figures exceed 10%, I will now swap from the approximate Scales of 72 (useful for percentages up to + / - 10%) to a totally accurate growth model called the Scales of e.

Google Share Price (Historical)

Why choose the Google share price in this analysis? Because in 2004 the chief executive of Google - Eric Schmidt - announced he intended to raise US $ 2,718,281,828 in the initial sale of Google shares (Elwes, 2007). This figure is derived from the value of e (~2.718281828).

In this example I will use the Scales of e to analyse historical Google share price fluctuations to "predict" historical share price doubling. To do so I use Natural Logarithms, denoted as LN. Note that the LN of 2 is 0.693147181, and the LN of 4 is 1.386294361 (which is twice 0.693147181).

The original Google share price is 100. Double 100 is 200, and quadruple 100 is 400. 

Date Open Adjusted Close Monthly
Growth Amount
Variable
Compound Interest Rate (monthly)
Natural Logarithm (LN) Running total
(LN)
Scales of e
Predictions
19-Aug-04 100 102.37 2.37 2.37% 0.02342351 0.02342351  
1-Sep-04 102.7 129.6 26.9 26.19% 0.23264067 0.25606418  
1-Oct-04 130.8 190.64 59.84 45.75% 0.37671739 0.63278158  
1-Nov-04 193.55 181.98 -11.57 -5.98% -0.06163909 0.57114249  
1-Dec-04 181.95 192.79 10.84 5.96% 0.05786959 0.62901208  
3-Jan-05 197.4 195.62 -1.78 -0.90% -0.00905813 0.61995395  
1-Feb-05 194.38 187.99 -6.39 -3.29% -0.03342624 0.58652772  
1-Mar-05 189.29 180.51 -8.78 -4.64% -0.04749405 0.53903367  
1-Apr-05 181.76 220 38.24 21.04% 0.19094041 0.72997408 Doubled
2-May-05 222.05 277.27 55.22 24.87% 0.22208918 0.95206325  
1-Jun-05 283.2 294.15 10.95 3.87% 0.03793648 0.98999973  
1-Jul-05 295.04 287.76 -7.28 -2.47% -0.02498414 0.96501559  
1-Aug-05 288.12 286 -2.12 -0.74% -0.00738525 0.95763034  
1-Sep-05 285.91 316.46 30.55 10.69% 0.10151978 1.05915012  
3-Oct-05 313.63 372.14 58.51 18.66% 0.17105618 1.23020630  
1-Nov-05 371.86 404.91 33.05 8.89% 0.08514738 1.31535368  
1-Dec-05 409.2 414.86 5.66 1.38% 0.01373708 1.32909076  
3-Jan-06 422.52 432.66 10.14 2.40% 0.02371542 1.35280618  
1-Feb-06 389.03 362.62 -26.41 -6.79% -0.07030101 1.28250517  
1-Mar-06 368.56 390 21.44 5.82% 0.05654322 1.33904839  
3-Apr-06 389.53 417.94 28.41 7.29% 0.07039700 1.40944539 Quadrupled
1-May-06 418.47 371.82 -46.65 -11.15% -0.11819534 1.29125005  
1-Jun-06 373.54 419.33 45.79 12.26% 0.11563311 1.40688316  
3-Jul-06 420.04 386.6 -33.44 -7.96% -0.08295938 1.32392378  
1-Aug-06 385.11 378.53 -6.58 -1.71% -0.01723368 1.30669010  
1-Sep-06 380.99 401.9 20.91 5.49% 0.05343017 1.36012027  
2-Oct-06 401.9 476.39 74.49 18.53% 0.17003355 1.53015382  
1-Nov-06 478.76 484.81 6.05 1.26% 0.01255763 1.54271145  
1-Dec-06 485.98 455.58 -30.4 -6.26% -0.06459614 1.47811531  

Table 1 - Scales of e analysis of historical Google share price fluctuations. Open and Adjusted Close prices sourced from Yahoo! Finance (on 27th December, 2006). LNs calculated using MS Excel LN function.

Each row represents one month of growth at variable rates of compound interest. These can be positive or negative. For each row the LN is indicated. When the LN running total equals (or first exceeds) the LN of 2, then the Google share price has doubled (during the month starting 1st April 2005). When the LN running total equals (or first exceeds) the LN of 4 then the Google share price has doubled for the second time (or quadrupled). This is during the month starting 3rd April 2006

Note that the Scales of e has coped with occasional months of negative growth, and the more prevalent months of positive growth. If negative growth is prevalent, then the value will halve when the LN running total equals (or first exceeds) the LN of 1/2 which is -0.693147181.  

The Scales of e can be used to analyse historical growth for the share price for any company's shares, for any growth period. It works just as well for predicted future growth just as well. 

It is also worth repeating that the Scales of e can be used for any growth factor, and is not restricted to doublings and halvings. However, the ability to explain growth in terms of doublings and halvings is both surprisingly powerful and pleasantly simple. 

Exponential Method

The Scales of e also explains the Exponential Method, universally used but little understood. For more read my articles The Exponential Method and US Census Bureau - Incorrect Use of the Exponential Method

Conclusion

For small percentage growth rates the Scales of 72 (or the Scales of 70, depending upon personal preference) provides a useful insight into the predictable nature of wealth doubling and wealth halving, even in the face of apparent chaos caused by variable rate compound interest at both positive and negative rates.

For an accurate model of wealth doubling and halving (or growth by any other factor) based on Natural Logarithms, allowing for positive and negative growth rates of any size, use the Scales of e

To those that argue that such processes are stochastic (or random) I would argue that there is a predictable order and shape to such growth. Such growth is, in fact, comparable to exponential growth in power. Thus, in terms of time frames and predictability, there is a very close bond between variable rate compound interest and exponential growth (which is, in essence, fixed rate compound interest).

Through the Scales of e, variable rate compound interest becomes just as predictable as exponential growth.

In summary the Scales of e works for any combination of

In short, variable rate compound interest (Couttsian Growth) is a universal form of growth. All loans, and all investments (including share price fluctuations), adhere to the Scales of e (which describes the Couttsian Growth Model in mathematical terms). So do all populations of all species, all of the time. It also applies to cell growth.

Thus the same growth model that applies to share price fluctuations (and wealth in general) also applies to all life. 

References

e: the mystery number - Richard Elwes, 18th July 2007 - from NewScientist.com (accessed 27/07/2007)

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Copyright 2006 David A. Coutts
Last modified: 18 November, 2008