Active Return

\[ R_A = \sum_{i=1}^{N}\Delta W_i R_i \]

or in terms of exposure to factor k

\[ R_A = \sum_{}^{}(\beta _{pk} - \beta _{bk}) \times F_k + (\alpha + \varepsilon ) \]

The expected active return is \[ E(R_A) = IC\ \sqrt[]{BR}\ \sigma _{R_A}\ TC \] | where: | BR is the number of truly independent decisions made each year, and | TC is the transfer coefficient; the ability to translate insights into investments

Construction Approaches

Factor exposures
Timing
Position sizing
Breadth or depth
Systematic or discretionary
Bottom-up or top-down

Active share is the percentage of assets that are not part of the benchmark.

Measures of Relative Risk

\[ \text{Active Share} = \frac{1}{2} \sum_{i=1}^{n}|\text{Weight}_{\text{portfolio},\ i} - \text{Weight}_{\text{benchmark},\ i}| \]

I’m not sure where the 1/2 comes from.

Active risk is affected by correlations between holdings.

Measures of Performance

Performance can be measured absolutely (Sharpe ratio) or relatively (information ratio).

Replacing or adding an asset with high covariance to the existing portfolio will increase total portfolio risk.

The contribution of an asset to a portfolio variance is the sum of the product of the weight of the asset and its covariance with each constituent of the portfolio.

Determining the Appropriate Level of Risk

High leverage can reduce compound return if the increase in variance is greater than the increase in arithmetic return.

Implicit Costs

Smaller funds have higher explicit costs but lower implicit costs.

The Well Constructed Portfolio

A well structured portfolio delivers the promised characteristics in a cost and risk efficient way.