BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

ACTA WASAENSIA 13somewhat. For example, if we assume single period returns independently identicallydistributed (iid) under some symmetric distribution, the corresponding multiperiodreturn will be increasingly right-skewed as a function of the investment horizon justdue to the multiplication of single period returns.From a statistical point of view, it is then desirable to transform returns in such a waythat multiperiod returns may be constructed from sums rather than products of singleperiod returns. Such a transformation is given by introducing logreturns r τ (t) asr t (τ) =ln(1+R t (τ)) = ln P t − ln P t−τ . (2.2)Multiperiod returns over long investment horizons become then normally distributedfor iid returns by virtue of the central limit theorem 2 . Logreturns are also calledcontinously compounded returns because they represent the yield of an investmentunder continuous compounding. Their difference from simple returns remains negligiblefor returns in the range of ±15%, implying that logreturns may be cross-sectionallyaggregated with negligible loss of accuracy for investment horizons up to at least oneweek, as long as no extraordinary returns occur.The use of returns (or logreturns) rather than (adjusted) prices in the analysis offinancial time series may also be motivated statistically by the so called unit rootproperty of asset prices and their logs. That is, in autoregressions of the Dickey-Fullertypeln P t = ρ ln P t−1 + u t (2.3)with stationary increments u t one is generally unable to reject the hypothesis ρ =1, 3 implying difference stationarity of the differenced series as is obtained by takinglogreturns. This provides an additional argument for the use of returns beyond thatof reflecting the investors viewpoint, since stationary time series are easier to analysethan those having a unit root.2 provided that single period returns have finite variance. For a discussion of the general case, seesection 3.2.1.3 see, for example, pages 18—21 in Pagan (1996).