Factor Analysis

kxy.asset_management.factor_analysis.information_adjusted_beta(r, r_m)

Calculates the informmation-adjusted beta.

Note

The standard beta coefficient of an asset or a portfolio is defined by the CAPM model

\[r = \alpha + r_f + \beta (r_m-r_f) + \epsilon\]

where \(r_f\) is a deterministic risk-free rate, \(r_m\) represents market returns, and \(\epsilon\) an idiosyncratic noise term. It follows that

\[\beta = \text{Corr}\left(r, r_m\right) \sqrt{\frac{\mathbb{V}\text{ar}\left(r\right)}{\mathbb{V}\text{ar}\left(r_m\right)}},\]

where \(\text{Corr}\) is Pearson’s correlation coefficient.

The information-adjusted correlation generalizes the foregoing equations and reads

\[\text{IA}\beta = \text{IACorr}\left(\{r\}, \{r_m\}\right) \sqrt{\frac{\mathbb{V}\text{ar}\left(r\right)}{\mathbb{V}\text{ar}\left(r_m\right)}}.\]

While Pearson’s correlation (and therefore beta) only captures linear relationships between portfolio returns and market returns, the information-adjusted correlation fully captures nonlinear and temporal dependencies between portfolio returns and market returns.

Parameters:
  • r ((n,) or (n,d) np.array) – The array of asset(s) or portfolio(s) returns.
  • r_m ((n,) np.array) – The array of market returns.
Returns:

c – The information-adjusted of the asset or portfolio.

Return type:

float

Raises:

AssertionError – If any returns array is not one-dimensional.