Factor Models in Return Attribution

What are the sources of a manager's returns relative to a benchmark? Fundamental multifactor models can be used to decompose the sources of returns in a meaningful way. Active return is the return in excess of the return on the benchmark. It can be either positive or negative.

Active return = R_P - R_B We can also analyze a portfolio manager's active return as the sum of two components:

• The return from factor tilts. This is the product of the portfolio manager's factor tilts (active factor sensitivities) and the factor returns.
• Asset selection. This is the part of active return reflecting the manager's skill in individual asset selection.

Factor Models in Risk Attribution

Active risk, also known as tracking risk (TR) or tracking error (TE), is the standard deviation of active returns.

Active risk = s(R_P - R_B) Active risk squared can be decomposed as the sum of active factor risk and active specific risk. Both components contribute to the risk of active returns.

Active risk squared = Active factor risk + Active specific risk

• Active factor risk is the contribution to active risk squared resulting from the portfolio's different-than-benchmark exposures relative to factors specified in the risk model. For example, a portfolio manager may under- or overweight some industries in the portfolio relative to its benchmark. The portfolio's industry factor sensitivities will not be the same as those of the benchmark.

• Active specific risk, also known as asset selection risk, is the contribution to active risk squared resulting from the portfolio's active weights on individual assets as those weights interact with assets' residual risk. For example, a portfolio manager may under- or overweight some stocks within a particular industry, and consequently the portfolio returns may deviate from the benchmark.

Information Ratio

The information ratio (IR) is mean active return divided by active risk. It measures the increment in mean active return per unit of active risk.

• Since this ratio considers the annualized standard deviation of both series (as measures of risks inherent in owning either the portfolio or the benchmark), the ratio shows the risk-adjusted excess return of the portfolio over the benchmark. The higher the ratio, the higher the excess return of the portfolio, given the amount of risk involved, and the better the portfolio manager.

• The information ratio is similar to the Sharpe ratio, but there is a major difference. The Sharpe ratio compares the return of an asset against the return of Treasury bills; the information ratio compares excess return to the most relevant equity (or debt) benchmark index.

• A high IR can be achieved by having a high return in the portfolio, a low return in the index, and a low tracking error.

Factor Models in Portfolio Construction

A factor portfolio is a well-diversified portfolio constructed to have a beta of one on one factor and a beta of zero on any other factors. It represents the risk of that factor only. A portfolio manager can use a factor portfolio to hedge that risk or speculate on it.

Strategic Portfolio Decisions

Traditionally, the CAPM approach would allocate assets between the risk-free asset and a broadly diversified index fund. Considering multiple sources of systematic risk may allow investors to improve on that result by tilting away from the market portfolio. Generally, investors would gain from accepting above average (below average) exposures to risks that they have a comparative advantage (comparative disadvantage) in bearing.

Different individual investors will have different individual degrees of ability to bear risk. Therefore, not all investors should hold the same portfolio. A multifactor model can be used to design the optimal portfolio for the individual investor by overweighting and underweighting exposure in different asset classes in the portfolio.