Multifactor Model Risk And Return

Multifactor models can be useful for return attribution, risk attribution, and portfolio construction.

Return Attribution

Multifactor models can be used to attribute a manager’s active portfolio return to different factors.

Active return equals the differences in returns between a managed portfolio and its benchmark:

active return = RP − RB

We can decompose active return into its two components: (1) factor return (arising from the manager’s decision to take on factor exposures that differ from those of the benchmark) and (2) security selection (arising from the manager choosing a different weight for specific securities compared to the weight of those securities in the benchmark). These two differences also contribute to active risk (discussed later).

active return = factor return + security selection return

where:

The security selection return is then the residual difference between active return and factor return:

security selection return = active return − factor return

Risk Attribution

Active risk (also known as tracking error or tracking risk) is defined as the standard deviation of the active return:

Active risk = tracking error = σ(RP − RB)

The active risk of a portfolio can be separated into two components:

1. Active factor risk: Risk from active factor tilts attributable to deviations of the portfolio’s factor sensitivities from the benchmark’s sensitivities to the same set of factors.
2. Active specific risk: Risk from active asset selection attributable to deviations of the portfolio’s individual asset weightings versus the benchmark’s individual asset weightings, after controlling for differences in factor sensitivities of the portfolio versus the benchmark.

The sum of active factor risk and active specific risk is equal to active risk squared (which is the variance of active returns):

active risk squared = active factor risk + active specific risk

Both components contribute to deviations of the portfolio’s returns from the benchmark’s returns. For example, consider a fundamental factor model that includes industry risk factors. In this case, active risk can be described as follows:

• Active factor risk example: A portfolio manager may decide to under- or overweight particular industries relative to the portfolio’s benchmark. Therefore, the portfolio’s industry factor sensitivities will not coincide with those of the benchmark, and, consequently, the portfolio returns may deviate from the benchmark.
• Active specific risk example: The active portfolio manager may decide to overweight or underweight individual stocks within specific industries. For example, a stock’s market capitalization may comprise 1% of the industry, but the portfolio manager may allocate 2% of industry allocation to the stock, causing the portfolio returns to deviate from the benchmark returns.

Active specific risk can be computed as:

Active factor risk represents the risk explained by deviation of the portfolio’s factor exposures relative to the benchmark and is computed as the residual (plug):

active factor risk = active risk squared − active specific risk

Uses of Multifactor Models

Multifactor models can be useful, for example, to a passive manager who seeks to replicate the factor exposures of a benchmark, or to an active manager who seeks to make directional bets on specific factors. Specific applications of multifactor models include:

1. Passive management. Managers seeking to track a benchmark can construct a tracking portfolio.

Tracking portfolios have a deliberately designed set of factor exposures. That is, a tracking portfolio is intentionally constructed to have the same set of factor exposures to match (track) a predetermined benchmark. 2. Active management.

Active managers use factor models to make specific bets on desired factors while hedging (or remaining neutral) on other factors. A factor portfolio is a portfolio that has been constructed to have sensitivity of one to just one risk factor and sensitivities of zero to the remaining factors. Factor portfolios are particularly useful for speculation or hedging purposes.

For example, suppose that a portfolio manager believes GDP growth will be stronger than expected but wishes to hedge against all other factor risks. The manager can take a long position in the GDP factor portfolio; the factor portfolio is exposed to the GDP risk factor, but has zero sensitivity to all other risk factors. This manager is speculating that GDP will rise beyond market expectations.

Alternatively, consider a manager who wishes to hedge his portfolio against GDP factor risk. Imagine that the portfolio’s GDP factor sensitivity equals 0.8, and the portfolio’s sensitivities to the remaining risk factors are different from zero. Suppose the portfolio manager wishes to hedge against GDP risk but remain exposed to the remaining factors.

The manager can hedge against GDP risk by taking an 80% short position in the GDP factor portfolio. The 0.8 GDP sensitivity of the managed portfolio will be offset by the −0.8 GDP sensitivity from the short position in the GDP factor portfolio.

1. Rules based or algorithmic active management (alternative indices). These strategies use rules to mechanically tilt factor exposures when constructing portfolios.

These strategies introduce biases in the portfolio relative to value weighted benchmark indices.

We will use the Carhart model to illustrate the use of factor portfolios.

Carhart Model

The Carhart four factor model is a multifactor model that extends the Fama and French three factor model to include not only market risk, size, and value as relevant factors, but also momentum.

risk dimensions when modeling asset returns.

Under the CAPM framework, investors choose a combination of the market portfolio and the risk free asset depending on their risk tolerance. By including more risk factors, multifactor models enable investors to zero in on risks that the investor has a comparative advantage in bearing and avoid the risks that the investor is incapable of absorbing. For example, a pension plan invests for long term and, hence, would not be averse to holding a security that bears liquidity risk (and that offers a liquidity risk premium).

Also, if the actual asset returns are better described by multifactor models, then using such models can help investors select more efficient portfolios.

The Information Ratio

Active return alone is insufficient for measuring an investment manager’s performance over a series of measurement periods. For example, imagine that Manager A earned a constant 0.5% (50 bps) active return over each of the last four quarters. Furthermore, suppose Manager B earned active returns of 8%, 5%, −3%, and −8% over the same four quarters.

The average active returns for managers A and B are both 0.5%, but Manager B experienced far more volatility (i.e., less consistency) than Manager A.

To demonstrate a manager’s consistency in generating active return, we use the information ratio, which standardizes average active return by dividing it by its standard deviation. In other words, the historical or ex post information ratio equals the portfolio’s average active return divided by the portfolio’s tracking risk: