This chapter introduces stochastic processes (time series) and explains how they are related to decisions as well as risks. First, let us revisit the traditional finance theory: the capital asset pricing model (CAPM), which is a single-factor model. As discussed previously in this book, the CAPM can be used to determine the necessary rate of return on an asset. The private risks of the asset can be diversified away and will not be priced by the market, so the only relevant risk is the systematic risk that is represented by β. Mathematically,

E R R E R Ra f am m f( ) ( ( ) )= + −β

E Ra( ) = Expected return on asset a

E Rm( ) = Expected return on market

Rf = Risk-free rate

βam = β of asset a against market

βam a m mCov R R Var R= ( , )/ ( )

Cov R Ra m( , ) = Covariance between asset returns and market returns

Var Rm( ) = Variance of market returns

When we utilize the CAPM to price an asset, we need to calculate a β first. To do this, we need the covariance of the asset’s return against the market’s return as well as the variance of market returns. Note that the “market” is the summation of all investable assets and by definition nearly impossible to observe. So, a strict application of CAPM is impossible, but in most cases we select imperfect proxies for the market such as the Wilshire 5000 stock index in the United States.