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

12 ACTA WASAENSIA2 Statistical Properties of Stock Returns2.1 Unit of MeasurementFrom the viewpoint of an investor, the relevant quantity to measure the performanceof an investment at time t over an investment period τ is its return R t (τ) defined asthe appreciation of its market value V scaled by its original market value: R t (τ) =(V t − V t−τ )/V t−τ .In sufficiently liquid markets we may assume the market price P to be independent ofthe quantity purchased or sold, such that the return of an investment in identical nondividend bearing assets may be written asR t (τ) = P t − P t−τP t−τ= P tP t−τ− 1. (2.1)The return of an investment in stocks may generally not be calculated by (2.1) above,since stocks as a rule pay dividends. Also capital adjustments such as stock splitsand stock dividends imply changes in market prices which do not reflect correspondingchanges in investment value.Returns of dividend paying stocks may thus only be written in the form (2.1) if marketprices are adjusted to neutralize the effects of dividend payments and capital adjustments.Such adjusted prices are nowadays provided by most data vendors and are theappropriate building blocks for the analysis of meaningful investment returns. As iscommon in the empirical finance literature, we will refer with P to the adjusted ratherthan the quoted market prices.Returns depend upon the the investment horizon τ: Multiperiod returns are productsof single period returns. 1 The calculation of multiperiod returns as products of singleperiod returns complicates the analysis of returns over different investment horizons1 More precisely, the multiperiod return R t+τ (τ = τ 1 + τ 2 + ···+ τ n )isrelatedtothesubperiodreturns R t+ ji=1 τ i) (τ j), j =1,...,n by the following product:(1 + R t+τ (τ)) = (1 + R t+τ1 (τ 1 )) · (1 + R t+τ1 +τ 2(τ 2 )) ···(1 + R t+ ni=1 τ i (τ n)).