Over the next 30 days, economists forecast that the pound may weaken relative to the dollar by as much as 6%, or it may strengthen by as much as 7%
The possible values for the rate of change of the dollar–pound spot exchange rate are –7%, –5%, –3%, –1%, 0%, 2%, 4%, and 6%. If these values are equally likely, what are the mean and standard deviation of the future spot exchange rate if the current rate is $1.5845/£?
The mean is the probability weighted average of the future possibilities. Because the events are equally likely and there are 8 events, each gets weight 1/8 in the average. Thus the mean is
(1/8) [(-7%) + (-5%) + (-3%) + (-1%) + 0% + 2% + 4% + 6%] = -0.5%
Consequently, the expected future spot rate is (1 – 0.005) $1.5845/£ = $1.5766/£. The standard deviation is the square root of the variance. The variance is the probability weighted average of the squared deviations of the possible realizations from their mean. Upon taking the average of the squared deviations of the realizations from their means, we find the standard deviation equal to 4.1533%. Because the standard deviation of the future exchange rate is the current exchange rate times the volatility of the rate of change, we find the volatility of the future exchange rate to be 4.1533% $1.5845/£ = $0.0658/£.
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