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Average Returns

Arithmetic Mean Returns

amr = {1/n} * sum[r(t)]

amr... average mean return
n... number of returns
r(t)... portfolio returns

Geometric Mean Returns

gmr = prod[r(t)] ^ {1/n}

gmr... geometric mean return

As a descriptive measure og historical return, the geometric average provides an annulaized measure of the proportional change in wealth that actually occured over the time horizon being analyzed, as if the wealth grew at a constant rate of return equal to the geometric mean.

The geometric mean return can be approximated with the following expression...

gmr ≈ amr - 0.5*std[r(t)]

gmr... geometric mean return
amr... average mean return
std... Standard deviation
r(t)... portfolio returns

Further, one can deduce from the above expression that the geometric average return will always be smaller than the arithmetic average return, as long as returns are not constant and the time period of measurement is larger than one year

This approximation should not be used in practical applications as an exact formula exists whichh is not too difficult to implement...

Expected (Future) Average Returns

Expected geometric return is often reported as a summary measure of the respective performance of asset classes or investment portfolios. It has intuitive appeal because its historical counterpart, the geometric average, provides a useful descriptive measure of the annualized proportional change in wealth that actually occured over a past time period, as if there had been no volatility in return. However, for applications that involve future projections or other prospective analysis, expected geometric return has limited value and often, the expected annual arithmetic return is a more relevant statistic.

The underlying issue is that the expected future arithmetic average does not depend on the time horizon chosen, while the expected geometric return does: The expected geometric average return declines as the time horizon is increased and therefore is not a particular meaningful measure of the expected growth of wealth over time.

This can be seen when returns are assumed to be lognormal distributed. In this case, the exact relationship between the geometric and arithmetic mean return is...

1 + gmr = {1+amr} * {1 + Var[r]/{1+amr}^2}^{{1-N}/2*N}

Because of the Central Limit Theorem, this result is approximatly true for any return distribution as long as returns are independent over time and have finite variance.

Additionally, the expected variance is known to be..

var[r] = {1+gmr} * { {1 + Var[r]/{1+amr}^2}^{1/N} - 1

Prospective (future-oriented) applications require an unbiased estimate of the expected annual return. The expected geometric return over any period greater than one year will understate the expected annual return, while the expected arithmetic return prvides the appropriate measure for this purpose.

Confidence Intervals

Here is a formula for the 95% confidence interval for the variation of the estimated mean return, denoted here by r, around the unknown true mean return, denoted by r.

r – (t_0.025,n) s/n^1/2 < r < r + (t_0.025,n) s/n^1/2

The notation t_a,n refers to the point x on Student's t distribution with n degrees of freedom, such that P{ t > x } = a. When calculating r from T observations, the degrees of freedom are just n = T–1. When n > 30, the approximation t_a,n = Z_a can be used, where the notation Z_a refers to the point on the standard normal distribution such that P{ Z > x } = a. For the 95% confidence interval, Z_0.025 = 1.96.

It may be argued by some that an observed mean return is not random at all, because it represents reality, and therefore it is inappropriate to calculate a confidence interval. While this is superficially true, it misses the point of carrying out this calculation. We are not actually interested in what the true rate of return might have been for that particular investment, because we already know exactly what it was. Instead, we want the answer to a different question: "How much volatility-induced variation should we expect in point estimates of the mean return, across an ensemble of independent similar investments with the specified mean and standard deviation?" The answer to that question is what the confidence interval provides.

It is important to remember that sampling variation is just one source of the variation that can occur in the estimation of a statistic. Systematic errors such as might be caused by changes in the way the variable is measured, or structural changes in the process itself, can contribute large amounts of additional variability. Even typographical errors in data transcription, whenever they occur, add to the observed variability. Therefore, sampling variation must be seen as a lower bound for the total variability of any statistic.