4 Expected future size of an investment fund
The issues discussed above surrounding the relative merits of geometric and arithmetic measures of expected return also extend to the calculation of projections of the expected size of an investment fund. Consider a stock that has compounding returns over n periods to a value of Sn (with S0=1). The returns in each period are assumed to be random, serially uncorrelated with a constant annual expected value E[rt]. Therefore:
(7)
That is, the expected value of a $1 stock that compounds for n periods at an expected annual arithmetic rate of E[r] is (1+E[r])n.[13] This was illustrated in the numerical example of expected returns in Section Two.[14]
In particular, the expected compounded value of the stock after n periods is not the expected geometric return over that time horizon to the power of n. That would be an understatement of the expected value of the stock (because the expected geometric return is less than the expected annual return). Similarly, the expected geometric return is not the nth root of the expected value of the stock at period n (because that would yield the expected annual arithmetic return).
This result that the expected size of a stock over time is calculated by compounding the expected annual arithmetic return over the time horizon has been applied in the projections of the expected growth of the New Zealand Superannuation Fund over time as illustrated in the Treasury’s spreadsheet model of the New Zealand Superannuation Fund and in McCulloch and Frances (2001). Consistent with the above analysis, the expected Fund size is calculated by compounding the Fund balance (adjusted for capital contributions and withdrawals) by the expected annual arithmetic return.
