Details on the Hierarchical Model

factor driven performance attribution decomposing excess return the hierarchical factor attribution model is an advanced methodology designed to explain the excess return r {\text{excess}} —the difference between the portfolio's return r p and the benchmark's return r b this model moves beyond traditional asset level attribution by linking performance directly to systematic risk factors this method adheres to the principles of balanced portfolio attribution (campisi) , where the total excess return is perfectly and additively decomposed into two essential components \mathbf{r {\text{excess}}} = \mathbf{\text{allocation effect}} + \mathbf{\text{selection effect}} 1\ the factor return model and exposure the attribution relies on the assumption that every instrument's return is driven by its sensitivity to broad market factors r i = \sum {k=1}^{k} (\beta {i,k} \times f k) + \epsilon i where \beta {i,k} is the instrument's exposure to factor k , and \epsilon i is its idiosyncratic return (or specific alpha) the manager's active bets are defined by the portfolio's aggregated factor exposure \mathbf{\beta {p,k}}, which is the weighted average of the individual instrument exposures \mathbf{\beta {p,k}} = \sum {i} w {p,i} \times \beta {i,k} 2\ arithmetic excess returns decomposition when using arithmetic returns, the decomposition provides contribution values that are strictly additive, allowing for the easiest interpretation and reporting a allocation effect (systematic bet) the allocation effect measures the return generated by the manager's decision to actively differ from the benchmark's systematic risk profile \mathbf{\text{allocation effect}} k = (\mathbf{\beta {p,k}} \mathbf{\beta {b,k}}) \times (\mathbf{r {\text{factor}, k}} \mathbf{r {\text{total\\,b}}}) interpretation this component rewards the manager for making an active bet on a factor (\mathbf{\beta {p,k}} \mathbf{\beta {b,k}}) whose return \mathbf{r {\text{factor}, k}} was superior to the total benchmark return \mathbf{r {\text{total\\,b}}} while structurally derived from the brinson model, this calculation is necessary to ensure the total contribution of all factors perfectly reconciles with the final \mathbf{r {\text{excess}}} b selection effect (factor adjusted alpha) the selection effect measures the manager's ability to generate specific return ($\epsilon$) superior to the benchmark's this term successfully isolates pure, factor adjusted alpha \mathbf{\text{selection effect}} k = \mathbf{\beta {p,k}} \times \left\[ (\mathbf{r {p, \text{residual}}}) (\mathbf{r {b, \text{residual}}}) \right] calculation principle the calculation first determines the residual return for both the portfolio and the benchmark the selection effect then weights the difference between the two residuals by the portfolio's factor exposure \mathbf{\beta {p,k}} interpretation a positive selection effect means the securities chosen by the manager within that factor grouping (or country/sector) produced returns greater than what was predicted by the factor model 3\ geometric excess returns decomposition geometric decomposition presents a challenge because the final result \frac{1 + r p}{1 + r b} is multiplicative, while the underlying factor model is additive to achieve a per row decomposition that still reconciles the total, the model uses the standard geometric two factor decomposition applied to factor exposure data as a necessary structural approximation \frac{1 + r p}{1 + r b} \approx \mathbf{\text{allocation factor}} \times \mathbf{\text{selection factor}} allocation factor measures the ratio of the portfolio's factor explained return to the benchmark's factor explained return, adjusted for the total benchmark gross return selection factor measures the ratio of the portfolio's relative return to the benchmark's relative return, adjusted by the reconciliation factor to ensure the product of all row components equals the total geometric excess return 4\ the idiosyncratic component the idiosyncratic row represents the portfolio's total residual return \mathbf{r {\text{residual}}} and is not a tradable asset but a reporting component function it is calculated as the final difference between the actual total return and the total factor explained return \mathbf{r {\text{actual total}}} \mathbf{r {\text{factor explained total}}} significance by including this row in the final attribution table, the model ensures that the sum of the allocation and selection effects across all factors and the idiosyncratic row perfectly equals the total excess return this preserves the essential mathematical integrity (additivity) of the model, even in the presence of systematic leverage