Mean-Reverting Random Walks

annual trend rate

annual standard deviation

mean start

months

months to trend

24

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10

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50

60

0

20

40

60

80

100

120

sample ending values

90 th percentile end value	103.422
70 th percentile end value	94.1945
50 th percentile end value	87.2467
30 th percentile end value	81.3451
10 th percentile end value	71.6627

geometric mean of end values	86.8197
arithmetic mean of end values	87.6064
standard deviation of end values	11.7523

Many financial or economic processes can be modeled as mean-reverting random walks. Mean-reverting walks differ from simple diffusion by the addition of a central expectation, usually growing with time, and a restoring force that pulls subsequent values toward that expectation. The random term is lognormally distributed, and the initial value of the mean can be above or below the series start. The strength of the restoring force is given in terms of the time to return to the mean. The plot shows sample paths which result in the

90

th

,

70

th

,

50

th

,

30

th

, and

10

th

percentiles in endpoint value, out of a larger sample.