The generalized fundamental law of active management

The generalized fundamental law of active management establishes the underlying relationship between expected portfolio performance (IR) and the portfolio characteristics based on bets on quantitative factors such as momentum, value, etc. It extends the work by Grinold and Kahn to the more interesting case where factor information coefficients (ICs) are time-varying.

Let’s say an investor wants to build a long-short dollar neutral portfolio by betting on stocks with high momentum, and shorting stocks with low momentum from a selection universe with N stocks, what will be her/his expected portfolio information ratio (IR)? The answer is the generalized fundamental law of active management:

    \[ \text{IR} = \frac{{\mu}_{\text{IC}}}{\sqrt{(1-{\mu}_{\text{IC}}^2-{\sigma}_{\text{IC}}^2)/N+{\sigma}_{\text{IC}}^2}} \approx \frac{{\mu}_{\text{IC}}}{{\sigma}_{\text{IC}}} \]

where {\mu}_{\text{IC}}  is the mean IC (Information Coefficient, correlation) between risk adjusted stock excess returns and risk adjusted factor exposures, {\sigma}_{\text{IC}}  is the IC standard deviation.

Figure 1. The fundamental law of active management

The generalized fundamental law reveals that portfolio information ratio (IR) is essentially positively related to the mean factor IC and inversely related to the factor IC standard deviation. The effect of selection universe size, or the so-called ‘breadth’, diminishes as it gets larger. This relationship between IR and breadth N is shown in the above above with {\mu}_{\text{IC}}=0.03 and {\sigma}_{\text{IC}}=0.1.

When the portfolio manager is betting on a combination of multiple factors, the generalized multi-factor fundamental law becomes

    \[ \text{IR} = {\sqrt{{\boldsymbol{\mu}_{\text{IC}}^'} ({\sigma}_{\epsilon}^2/N {\mathbf{I}}+{\boldsymbol{\Sigma}}_{\text{IC}})^{-1}{\boldsymbol{\mu}}_{\text{IC}}}} \approx \sqrt{{{\boldsymbol {\mu}}_{\text{IC}}^'}{\boldsymbol{\Sigma}}_{\text{IC}}^{-1}{\boldsymbol{\mu}}_{\text{IC}}}} \]

where {\boldsymbol\mu}_{\text{IC}} is the mean Information Coefficient vector (correlation) between risk adjusted stock excess returns and risk-adjusted factor exposures, {\boldsymbol\Sigma}_{\text{IC}} is the factor IC covariance matrix.

The relation between factor investing and stock selection

The optimal long-short portfolio weight by betting on a combination of multiple factors is given by the formula:

    \[ {\mathbf{w}_{A,t}} = {\frac{1}{N}} {\boldsymbol{\Lambda}}_t^{-1/2} {\mathbf{Z}}_{t-1}{\frac{{{\sigma}_A}{\boldsymbol{\Sigma}}_{\text{IC}}^{-1}{\boldsymbol{\mu}}_{\text{IC}}}{\sqrt{{{\boldsymbol {\mu}}_{\text{IC}}^'}{\boldsymbol{\Sigma}}_{\text{IC}}^{-1}{\boldsymbol{\mu}}_{\text{IC}}}}}} ={\frac{1}{N}} {\boldsymbol{\Lambda}}_t^{-1/2} {\mathbf{Z}}_{t-1}{\mathbf{w}_f}={\left({\frac{1}{N{\sigma}_{r_{it}}}}\sum_{k=1}^{K}{{w_{f,k}}{z_{ik,t-1}}}\right)}_{i=1,\ldots,N} \]

where

    \[ {\mathbf{w}_f} = {\frac{{{\sigma}_A}{\boldsymbol{\Sigma}}_{\text{IC}}^{-1}{\boldsymbol{\mu}}_{\text{IC}}}{\sqrt{{{\boldsymbol {\mu}}_{\text{IC}}^'}{\boldsymbol{\Sigma}}_{\text{IC}}^{-1}{\boldsymbol{\mu}}_{\text{IC}}}}}} \]

is the mean-variance optimal factor portfolio weight when one bets on factors with an expected information coefficient vector {\boldsymbol\mu}_{\text{IC}} and factor covariance {\boldsymbol\Sigma}_{\text{IC}} with target risk  {\sigma}_A. The factor weight times each stock’s corresponding factor exposure,  divided by each stock’s volatility, gives the optimal portfolio’s stock weight.

The formula presented here reveals an underlying connection between factor investing and stock selection. Specifically, it demonstrates that stock selection is equivalent to factor investing when the underlying stock market movement is driven by some factors.

Portfolio Turnover when IC is time varying

We developed new formulas for the turnover and leverage of long-short and long-only constrained mean-variance optimal active portfolios, where active weights are obtained using a factor model conditional mean forecast, and a conditional forecast error covariance matrix that reflects factor risk.

    \[ \text{TO} = \frac{{\sigma}_A \sqrt{1-{\rho}_z}}{\sqrt{\pi} \sqrt{(1-{\mu}_{\text{IC}}^2-{\sigma}_{\text{IC}}^2)/N+{\sigma}_{\text{IC}}^2}} E_{cs}{\left(\frac{1}{{\sigma}_{r_i}}\right)} \]

Our turnover formula shows that portfolio turnover is negatively correlated with factor exposure correlation, {\rho}_z , inversely correlated with factor IC standard deviation and average individual security volatility, and positively related to portfolio tracking error {\sigma}_A.

Figure 2.  Portfolio turnover vs IC volatility and signal autocorrelation

Volatility Forecasting

It is generally agreed that it’s extremely difficult to predict stock’s future returns using its own past returns. However, the absolute returns which is a measure of stock volatility is largely predictable. This phenomenon is commonly referred to as the volatility clustering effect, wherein high volatility tends to be followed by high volatilities. Figure 3 illustrates this effect, with the red line representing the sample autocorrelations for S&P 500 absolute returns.If there is no predictability, this line should be close to zero. The blue line is the theoretical autocorrelation function for a component GARCH model. It can be seen that the component GARCH model does a very good job in capturing the sample autocorrelation in the absolute returns.

Figure 3. S&P 500 Absolute Return Sample Autocorrelations and Component GARCH Fitted