By default, the standard deviation of a geometric growth type history increases with the square root of time. In some cases, when simulating a stochastic history that increases geometrically, you may not want the standard deviation to grow in this way. Rather, you may want it to stabilize at a constant value.
You can accomplish this by forcing the stochastic history to revert towards its median value at a specified rate. To do so, you must use the Annual Reversion Rate and Initial Value of Median input fields.
Annual Reversion Rate. This is the annual fractional rate at which the history reverts towards its median. Although this is a rate, because it is "hard-wired" to an annual value, this input is dimensionless. That is, it represents the fractional rate per year. It must be non-negative.
Initial Value of Median. In most cases, the initial value of the time history is unlikely to be at the median. Hence, this is the initial value of the median of the time history (the value at time zero). It has the same dimensions as the output. It must be a positive number.
If the Annual Reversion Rate is non-zero, GoldSim generates successive values as follows:
where Vnew is the new value, Vold is the previous value, Δt is the time (in years) between the two values, ε is a random standard normal value (sampled from a distribution with mean 0 and standard deviation 1), r is the reversion rate, and μ' and σ' are the logarithmic growth rate and volatility, respectively. T is the median (a function that grows geometrically from the specified Initial Value of Median using the mean annual growth rate). Told is the previous value of the median.
In practical terms, specifying a non-zero Annual Reversion Rate has two effects:
•The history tracks the median, and if the Initial Value differs from the Initial Value of Median, the history approaches the median history with a half-life of ln(2)/Annual Reversion Rate.
•The standard deviation of the logs initially grows, approaching a constant value with a similar half-life. The "steady-state" standard deviation of the log of the values stabilizes at a value of:
