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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

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Negative Growth - Constant Rate

Introduction

The table below illustrates exponential population shrinkage at different constant rates of negative annual growth. In the world of finance this is known as fixed rate compound interest. The following table assumes a starting population of 1,000 people (though it could just as easily be your investment of $1,000), and highlights (in yellow) when the population would halve to 500 people at each rate. Population halving leads to exponential shrinkage, and so  - if such growth rates were sustained - each population would proceed through the standard population halving series  - 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

YEAR -1% -2% -3% -4% -5% -6% -7%
0 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00
1 990.00 980.00 970.00 960.00 950.00 940.00 930.00
2 980.10 960.40 940.90 921.60 902.50 883.60 864.90
3 970.30 941.19 912.67 884.74 857.38 830.58 804.36
4 960.60 922.37 885.29 849.35 814.51 780.75 748.05
5 950.99 903.92 858.73 815.37 773.78 733.90 695.69
6 941.48 885.84 832.97 782.76 735.09 689.87 646.99
7 932.07 868.13 807.98 751.45 698.34 648.48 601.70
8 922.74 850.76 783.74 721.39 663.42 609.57 559.58
9 913.52 833.75 760.23 692.53 630.25 572.99 520.41
10 904.38 817.07 737.42 664.83 598.74 538.62 483.98
11 895.34 800.73 715.30 638.24 568.80 506.30
12 886.38 784.72 693.84 612.71 540.36 475.92
13 877.52 769.02 673.03 588.20 513.34
14 868.75 753.64 652.84 564.67 487.67
15 860.06 738.57 633.25 542.09
16 851.46 723.80 614.25 520.40
17 842.94 709.32 595.83 499.59
18 834.51 695.14 577.95
19 826.17 681.23 560.61
20 817.91 667.61 543.79
21 809.73 654.26 527.48
22 801.63 641.17 511.66
23 793.61 628.35 496.31
24 785.68 615.78
25 777.82 603.46
26 770.04 591.40
27 762.34 579.57
28 754.72 567.98
29 747.17 556.62
30 739.70 545.48
31 732.30 534.57
32 724.98 523.88
33 717.73 513.41
34 710.55 503.14
35 703.45 493.07
36 696.41
37 689.45
38 682.55
39 675.73
40 668.97
41 662.28
42 655.66
43 649.10
44 642.61
45 636.19
46 629.82
47 623.53
48 617.29
49 611.12
50 605.01
51 598.96
52 592.97
53 587.04
54 581.17
55 575.35
56 569.60
57 563.91
58 558.27
59 552.68
60 547.16
61 541.69
62 536.27
63 530.91
64 525.60
65 520.34
66 515.14
67 509.99
68 504.89
69 499.84

Note: human population growth is typically measured using an annual growth rate, but other replicators breed at different rates. Simply substitute the appropriate unit of measure (seconds, minutes, hours, days, weeks or months) and you will find the same doubling period (using the new unit of measure) applies.

Many people regard population growth at constant rates as "true" exponential growth, whilst usually pointing out that populations do not normally grow at constant rates. Few people stop to think that negative rates of growth also lead to a form of (negative) exponential growth, otherwise known as exponential shrinkage.

There appear to be two reasons why people regard population shrinkage at a constant rate as true exponential growth:

(99/100)0 (99/100)1 (99/100)2 (99/100)3 (99/100)4 etc

There is no denying that negative population growth at a constant rate is exponential shrinkage. For each constant rate of growth in the table above, each row represents exponential shrinkage at that rate. Each row represents an increment of the exponent by 1.

In real-life, populations do not shrink at constant rates of growth. Modern demographers would claim that because populations do not shrink at constant rates of growth that populations do not experience exponential shrinkage. This claim is disproved in Human Replicators - An Exponentialist View and The Scales Of 70

In a nutshell, it is a simple matter to prove that sustained (negative) variable rates of growth also result in population halving (known in the world of finance as variable rate compound interest). Imagine a population of 1,000 which experiences fluctuations in its growth rate between negative 1% and 7%. It therefore must halve between the start of the 9th year and the end of the 69th year. Sustained population doubling would result in such a population proceeding through the standard population halving series, proving that sustained (negative) variable rates of growth lead to exponential shrinkage.

Logically, given that - for each constant rate of growth listed above - each row represents another year of exponential shrinkage, it must be true that shrinkage for one year at that rate is also exponential. Otherwise, we would be forced to conclude that some minimum number of years must pass before the shrinkage can be called exponential. But then, it would be possible to express the shrinkage over those years as one rate of growth and we would be back in the same situation that we started in. 

So, given that shrinkage at any percentage for any given year can be regarded as exponential growth, it must be true that shrinkage is still exponential when the negative compound interest growth rate varies from year to year.

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Copyright 2001 David A. Coutts
Last modified: 27 October, 2006