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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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Positive Growth - Constant Rate

Introduction

The table below illustrates exponential population growth at different constant rates of positive 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 $1,000 investment), and highlights (in yellow) when the population would double to 2,000 people at each rate. Population doubling leads to exponential growth, and so  - if such growth rates were sustained - each population would proceed through the standard population doubling series (in thousands) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 etc.

YEAR 1% 2% 3% 4% 5% 6% 7%
0 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00
1 1010.00 1020.00 1030.00 1040.00 1050.00 1060.00 1070.00
2 1020.10 1040.40 1060.90 1081.60 1102.50 1123.60 1144.90
3 1030.30 1061.21 1092.73 1124.86 1157.63 1191.02 1225.04
4 1040.60 1082.43 1125.51 1169.86 1215.51 1262.48 1310.80
5 1051.01 1104.08 1159.27 1216.65 1276.28 1338.23 1402.55
6 1061.52 1126.16 1194.05 1265.32 1340.10 1418.52 1500.73
7 1072.14 1148.69 1229.87 1315.93 1407.10 1503.63 1605.78
8 1082.86 1171.66 1266.77 1368.57 1477.46 1593.85 1718.19
9 1093.69 1195.09 1304.77 1423.31 1551.33 1689.48 1838.46
10 1104.62 1218.99 1343.92 1480.24 1628.89 1790.85 1967.15
11 1115.67 1243.37 1384.23 1539.45 1710.34 1898.30 2104.85
12 1126.83 1268.24 1425.76 1601.03 1795.86 2012.20
13 1138.09 1293.61 1468.53 1665.07 1885.65
14 1149.47 1319.48 1512.59 1731.68 1979.93
15 1160.97 1345.87 1557.97 1800.94 2078.93
16 1172.58 1372.79 1604.71 1872.98
17 1184.30 1400.24 1652.85 1947.90
18 1196.15 1428.25 1702.43 2025.82
19 1208.11 1456.81 1753.51
20 1220.19 1485.95 1806.11
21 1232.39 1515.67 1860.29
22 1244.72 1545.98 1916.10
23 1257.16 1576.90 1973.59
24 1269.73 1608.44 2032.79
25 1282.43 1640.61
26 1295.26 1673.42
27 1308.21 1706.89
28 1321.29 1741.02
29 1334.50 1775.84
30 1347.85 1811.36
31 1361.33 1847.59
32 1374.94 1884.54
33 1388.69 1922.23
34 1402.58 1960.68
35 1416.60 1999.89
36 1430.77 2039.89
37 1445.08
38 1459.53
39 1474.12
40 1488.86
41 1503.75
42 1518.79
43 1533.98
44 1549.32
45 1564.81
46 1580.46
47 1596.26
48 1612.23
49 1628.35
50 1644.63
51 1661.08
52 1677.69
53 1694.47
54 1711.41
55 1728.52
56 1745.81
57 1763.27
58 1780.90
59 1798.71
60 1816.70
61 1834.86
62 1853.21
63 1871.74
64 1890.46
65 1909.37
66 1928.46
67 1947.74
68 1967.22
69 1986.89
70 2006.76

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

(101/100)0 (101/100)1 (101/100)2 (101/100)3 (101/100)4 etc

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

In real-life, populations do not grow at constant rates of growth. Modern demographers claim that because populations do not grow at constant rates of growth that therefore populations do not grow exponentially. 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 (positive) variable rates of growth also result in population doubling (known in the world of finance as variable compound interest). Imagine a population of 1,000 which experiences fluctuations in its growth rate between 1% and 7%. It therefore must double between the start of the 10th year and the end of the 70th year. Sustained population doubling would result in such a population proceeding through the standard population doubling series, proving that sustained (positive) variable rates of growth lead to exponential growth.

Logically, given that - for each constant rate of growth listed above - each row represents another year of exponential growth, it must be true that growth 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 growth can be called exponential. But then, it would be possible to express the growth over those years as one rate of growth and we would be back in the same situation that we started in. 

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

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