The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 325 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 325

Pt – Pt – 1 Pt

Rt = ----------------------- = ------------– 1

Pt – 1 Pt – 1

From this definition it is clear that the compound return Rt(k) over k

periods is:

Pt k –^1 Pti– k –^1

Rt()k = ------------– 1 = ∏------------------– 1 = ∏(Rti– + 1 )– 1

Ptk– i = 0 Pti–+ (^1) i = 0
Consider now the logarithms of prices and returns:
pt = log Pt
rt = log 1 ( + Rt)
rt()k = log 1 [ + Rt()k]
Following standard usage, we denote prices and returns with upper case
letters and their logarithms with lower case letters. As the logarithms of
a product is the sum of the logarithms, we can write:
Pt
rt = log 1 ( + Rt)= log ------------ = pt – pt – 1
Pt – 1
rt()k = log 1 [ + Rt()k] = rt + rt – 1 + ... +rtk– + 1
Note that for real-world price time series, if the time interval is small,
the numerical value of returns will also be small. Therefore, as a first
approximation, we can write
rt = log 1 ( + Rt) ≈Rt
The simplest model of equity prices consists in assuming that loga-
rithmic returns are an IID sequence. Under this assumption we can
write: rt = μ+ εt, where μis a constant and εt is a white noise, that is, a
zero-mean, finite-variance IID sequence. Under this model we can write