Active Weight Optimization

You can choose to run an active weight optimization in order to optimize your portfolio's performance relative to a benchmark portfolio.  If you enable active weight optimization through the Options Menu, optimality is determined based on risk-return characteristics of the active weights of the portfolio, and the Optimizer displays portfolio risk and return relative to the benchmark.  To illustrate this, the benchmark portfolio always has active weights of zero for all assets.  In general, if the vector of absolute portfolio weights is given by p, the vector of benchmark weights is given by b, the vector expected absolute returns of assets by μ, and the covariance matrix of absolute returns by Σ; then the active weights of a portfolio are given by p-b, the active return of the portfolio is μ' * (p-b), and the active variance is (p-b)' * Σ * (p-b).  Active standard deviation, more commonly called tracking error, is the square root of the active variance.

For example, an initial portfolio with a 7% absolute return changes to a 0% active return when active weight optimization is employed, given that the benchmark portfolio has a return of 7%. However, the risk transformation is not so straightforward. If a portfolio and the benchmark both have a 10% standard deviation, the active risk of the portfolio might be anywhere from 0 to 20% depending on the cross-correlation between initial and benchmark portfolios. Furthermore, very low risk assets in absolute terms can become highly risky relative to the benchmark.

When active weight optimization is enabled, the following conditions occur:

• The benchmark portfolio's active risk and return are always 0%.  

• The optimal portfolio with the lowest risk mirrors the benchmark portfolio exactly in the absence of constraints which would exclude this portfolio.  

• Though the absolute asset weights (active weight plus benchmark) of the optimal portfolios appear on the Results Worksheet, the optimization is constrained to ensure that the active weights, the benchmark-relative asset weights, sum to one minus the total benchmark weight (i.e. usually sum to zero).  

• The returns and standard deviations of the initial, optimal, reference, and investable portfolios are all displayed relative to the benchmark portfolio.  

• Active weight optimization also affects the constraints. All constraints become benchmark relative, meaning that the quadratic constraints penalize the optimization for straying from the benchmark portfolio, etc. You can also set benchmark-relative or absolute asset bounds independently from active weight optimization.

• A multi-period horizon cannot be calculated.

• The efficient frontier may dip because of the presence of negative-return assets relative to the benchmark. Additional constraints may solve this problem.

• Standard deviation is renamed tracking error in all columns and charts.