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Calculating the Annual Pecentage Yield (APY) And Continuous Compounding
Negative Growth - Constant Rate
Positive Growth - Constant Rate
Positive and Negative Growth Compared

The Rule of 70 and the Rule of 72 Compared

External Links:
Percentage - Wikipedia article

Fraction - Wikipedia article

Natural Logarithm - Wikipedia article



Positive and Negative Growth Compared

Introduction

Percentages are nothing more than fractions of the form x /100, or so many hundredths. However, the modern usage of percentages has all but eclipsed the usage of other fractions. Does this matter? To be honest, percentages are easily understood and universally applicable. However, when it comes to the comparison of positive and negative growth for a given rate, percentages can produce some counter-intuitive results.

In this article I examine the cause of that imbalance. I then show how fractions other than hundredths can be used to explain the cause in a different way, and can demonstrate that positive and negative growth are not necessarily imbalanced at all. 

Finally, I reveal a unique new way of calculating the net result of applying a growth rate once for negative growth, and once for positive growth.

Imbalance Using Percentages

Common sense might cause you to believe that for a given rate, 1 year of positive growth (at that rate) is cancelled by 1 year of negative growth at that year. So, for example, if some grows at 1% for a year and then shrinks at 1% for the next year it will return to its original value. 

Another way of putting it is to say that -1% is the inverse of +1%. Except that it isn't. 

This is easily tested on your calculator, or an MS Excel spreadsheet. Assuming a starting value of 100, something that grows at 1% increases by 1 and will become 101. However, 1% of 101 is not 1 but 1.01. Thus, if a value of 101 grows at minus 1% it will become 101 - 1.01 = 99.99.

Imbalance Using Natural logarithms

In my article The Rule of 70 and the Rule of 72 Compared I used Natural Logarithms (LN) to reveal how, for a given rate of growth (expressed as a positive or negative rate), there is an imbalance between the respective accurate doubling and halving periods.

So, for 1%:

Applied to 100, we again obtain a result of 99.99

Balance Using Natural logarithms

Worth noting, if using LN, is that the inverse of any positive LN is always the same value as the positive LN but with a minus sign. Thus, added together, the net result LN = 0. LN(0) = 1. Hence, anything that grows by LN(X) one year and shrinks by LN(1/X) the next will return to its original value. 

Thus, the inverse LN of 0.0099503309 (LN of 1% for positive growth) is - 0.0099503309 (for negative growth). 

to convert back to a real number, this is:

e ^ 0.0099503309 = 0.99 (using MS Excel that is EXP(1) ^^ 0.0099503309)

Imbalance Using Other Fractions

Hence positive 1% is clearly 101/100, but negative 1% is not -101/100. It is 99/100.

Thus, the calculation becomes:

(101/100) * (99/100) = (101 * 99) / (100 * 100) = 100 * 9999/10000 = 0.9999

Applied to 100 that is 99.99

Balance Using Other Fractions

Without the need to resort the LN, it is possible to prove that that the inverse growth of 101/100 is not 99/100 but 100/101. 

Thus, if something grows by 101/100 one year, and shrinks by 100/101 the next, we have:

101/100 * 100/101 = (101 * 100) / (100 * 101) = 10100 / 10100 = 1

Alternately, if something shrinks by 99/100 one year then the inverse growth is 100/99. Thus we have:

99/100 * 100/99 = (99 * 100) / (100 * 99) = 9900 / 9900 = 1

The Inverse Square Law of Growth

During my investigations I discovered an unusual, and possibly original, way of calculating the unbalanced growth for a given percentage rate of growth (using 1 period of positive growth, and 1 period of negative growth, at the same rate) is as follows:

The Growth Factor = 1 - (r/100)2

where r = the rate

Thus, if the rate is 1%, this becomes:

1 - (1/100)2  = 1 - (1/100 * 1/100) = 1 - (1/10000) = 0.9999.

For a 5% growth rate this becomes:

1 - (5/100)2  = 1 - (5/100 * 5/100) = 1 - (25/10000) = 0.9975. 

This pattern was revealed when I laid out the following table:

Rate Negative Rate
(NR)
Positive Rate
(PR)
ln(NR) ln(PR) ln(NR) + ln(PR) e ^ (ln(NR) + ln(PR)) 1 -  (rate/100) ^ 2
1 0.99 1.01 -0.0100503359 0.0099503309 -0.0001000050 0.99990 0.99990
2 0.98 1.02 -0.0202027073 0.0198026273 -0.0004000800 0.99960 0.99980
3 0.97 1.03 -0.0304592075 0.0295588022 -0.0009004052 0.99910 0.99970
4 0.96 1.04 -0.0408219945 0.0392207132 -0.0016012814 0.99840 0.99960
5 0.95 1.05 -0.0512932944 0.0487901642 -0.0025031302 0.99750 0.99950
6 0.94 1.06 -0.0618754037 0.0582689081 -0.0036064956 0.99640 0.99940
7 0.93 1.07 -0.0725706928 0.0676586485 -0.0049120444 0.99510 0.99930
8 0.92 1.08 -0.0833816089 0.0769610411 -0.0064205678 0.99360 0.99920
9 0.91 1.09 -0.0943106795 0.0861776962 -0.0081329832 0.99190 0.99910
10 0.90 1.10 -0.1053605157 0.0953101798 -0.0100503359 0.99000 0.99900

 

Conclusion

I hope that you agree that it has proven useful to examine positive and negative growth for a given rate using percentages, Natural Logarithms, and other fractions. It is worth remembering that, if we use percentages, then nature is inclined to favour negative growth over positive growth. However, using Natural Logarithms or other fractions, it is a relatively simple matter to demonstrate inverse negative growth.

Finally, in examining the nature of positive and negative growth, I happened to stumble upon the Inverse Square Law of Growth. Possibly this has been discovered already, I don't know, but it is always satisfying to discover mathematical patterns in natural processes such as growth. 

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Copyright 2007 David A. Coutts
Last modified: 18 November, 2008