# Portfolio turnover when IC is time-varying

## Abstract

We develop new formulas for the turnover and leverage of mean–variance optimal long–short market neutral portfolios, where active weights are obtained using a factor model conditional mean forecast and a conditional forecast error covariance matrix that reflects strategy risk. We show that for eight commonly used quantitative factors, the turnovers and leverages derived using our long–short formulas are quite close to what the practitioners actually implement. We further carry out extensive simulations for long-only active portfolios and develop a highly accurate empirical formula that relates long-only turnover to long–short turnover, a transfer coefficient, portfolio target tracking error, strategy risk and a benchmark choice coefficient. Our result shows that when the proper risk model is used in factor investing, the optimal portfolio’s turnover and leverage are well within reasonable practically implementable ranges even if no additional constraints are imposed.

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1. 1.

This is to the assumption that the two terms on the right-hand side of (1) are uncorrelated, and the fact that the left-hand side has unconditional variance one. For then, it follows that

$$\text{var} ({\text{IC}}_{t} {\kern 1pt} z_{i,t - 1} ) + \text{var} (\varepsilon_{it} ) = E(({\text{IC}}_{t} {\kern 1pt} - \mu_{\text{IC}} ){\kern 1pt} z_{i,t - 1} + \mu_{\text{IC}} z_{i,t - 1} )^{2} + \sigma_{\varepsilon }^{2} = \sigma_{\text{IC}}^{2} + \mu_{{\text{IC}}}^{2} + \sigma_{\varepsilon }^{2} = 1.$$
2. 2.

This is easily checked by notating that the pairwise conditional forecast error covariances are given by \begin{aligned} E\left( {(r_{it} - \alpha_{it} )(r_{jt} - \alpha_{jt} )\left| {\,{\text{I}}_{t - 1} } \right.} \right) & = E\left( {\sigma_{{r_{i} }} ((\text{IC}_{t} - \mu_{{\text{IC}}} )z_{i,t - 1} + \varepsilon_{it} )\sigma_{{r_{j} }} ((\text{IC}_{t} - \mu_{{\text{IC}}} )z_{j,t - 1} + \varepsilon_{jt} )\left| {\,{\text{I}}_{t - 1} } \right.} \right) \\ & = \sigma_{{r_{i} }} \left( {E(\text{IC}_{t} - \mu_{{\text{IC}}} )^{2} z_{i,t - 1} z_{j,t - 1} + E(\varepsilon_{it} \varepsilon_{jt} \left| {\,{\text{I}}_{t - 1} } \right.)} \right)\sigma_{{r_{j} }} \,. \\ \end{aligned}

3. 3.

These ratios are B/P = book to price, C/P = cash flow to price, D/P = dividend yield to price, E/P = earnings to price, FE/P = forward earnings to price, S/P = sales to price, MOM = cumulative 11-month return from t-12 to t-2, SHORT = short as a percent of total shares float.

4. 4.

We have a total of 845 realized portfolio turnovers of which 88 were for averaging 10 simulations for each of the $$11 \times 8 = 88$$ combinations of sample size N and tracking error $$\sigma_{A}$$ in Exhibit 6, and 64 were for averaging 10 simulations for each of the $$8 \times 8 = 64$$ combinations of signal autocorrelation $$\rho_{z}$$ and tracking error $$\sigma_{A}$$. The remaining 693 simulations are obtained from various combinations of model and portfolio parameters within the ranges discussed above.

5. 5.

Other model parameter values are: $$\mu_{\text{IC}} = 0.05$$, $$\sigma_{\text{IC}}$$ varies from 0.1 to 0.5 with a majority of them being 0.2, $$\rho_{z}$$ has values 0.85, 0.9 and 0.95, and $$\sigma_{{r_{i} }}$$ are drawn from a fixed uniform distribution with a minimum value of 0.05 and a maximum value of 0.4.

## References

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Correspondence to Zhuanxin Ding.

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## Appendix

### Appendix

The derivation of our formulas for the expected turnover and leverage of the optimal long–short portfolio is quite similar to that in Appendix of QSH, the difference being that our optimal weights formula (9) is different. The one-way turnover from time $$t - 1$$ to time t is

\begin{aligned} {\text{TO}}_{t} & = \frac{1}{2}\sum\limits_{i = 1}^{N} {|w_{A,it} - w_{A,i,t - 1} |} \\ & = \frac{{\sigma_{A} }}{{2\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{\text{IC}}^{2} } }}\frac{1}{N}\sum\limits_{i = 1}^{N} {\left| {\frac{{z_{i,t - 1} }}{{\sigma_{{r_{i} }} }} - \frac{{z_{i,t - 2} }}{{\sigma_{{r_{i} }} }}} \right|} . \\ \end{aligned}
(20)

The expected turnover is then

\begin{aligned} {\text{TO}} & = \frac{{\sigma_{A} }}{{2\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{\text{IC}}^{2} } }}\frac{1}{N}\sum\limits_{i = 1}^{N} {E\left| {\frac{{z_{i,t - 1} }}{{\sigma_{{r_{i} }} }} - \frac{{z_{i,t - 2} }}{{\sigma_{{r_{i} }} }}} \right|} \\ & = \frac{{\sigma_{A} }}{{2\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{\text{IC}}^{2} } }}\frac{1}{N}\sum\limits_{i = 1}^{N} {E\left| {z_{i,t - 1} - z_{i,t - 2} } \right|E\left( {\frac{1}{{\sigma_{{r_{i} }} }}} \right)} \\ & = \frac{{\sigma_{A} }}{{2\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{\text{IC}}^{2} } }}\frac{1}{N}\sum\limits_{i = 1}^{N} {\frac{{2\sqrt {1 - \rho_{z} } }}{\sqrt \pi }\frac{1}{{\sigma_{{r_{i} }} }}} \\ & = \frac{{\sigma_{A} }}{{2\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{\text{IC}}^{2} } }}\frac{{2\sqrt {1 - \rho_{z} } }}{\sqrt \pi }\frac{1}{N}\sum\limits_{i = 1}^{N} {\frac{1}{{\sigma_{{r_{i} }} }}} \\ & = \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) \, .\\ \end{aligned}
(21)

In the above derivation, we assumed that the $$z_{i}$$ are standard normal random variables and $$\sigma_{{r_{i} }}$$ are fixed parameters for each i, and used the fact that the difference $$z_{i,t - 1} - z_{i,t - 2}$$ of two correlated standard normal random variables with correlation coefficient $$\rho_{z}$$ is normally distributed with variance $$2(1 - \rho_{z} )$$, along with the fact that the expected absolute value of a standard normal random variable is $$\sqrt {2/\pi } .$$

It should be noted that we ignored the weight drift due to price movement in the above derivation and calculated turnover solely due to forecast changes. As pointed out in QSH, this will be an excellent approximation because the majority of the turnover is created by changes in the forecasts.

Furthermore, the leverage of the optimal portfolio at time t is

\begin{aligned} L_{t} & = \sum\limits_{i = 1}^{N} {|w_{A,it} |} \\ & = \sum\limits_{i = 1}^{N} {\frac{{\sigma_{A} }}{{N\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{\text{IC}}^{2} } }}\left| {\frac{{z_{i,t - 1} }}{{\sigma_{{r_{i} }} }}} \right|} \\ \end{aligned}
(22)

and its expected value is

\begin{aligned} L & = \sum\limits_{i = 1}^{N} {\frac{{\sigma_{A} }}{{N\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{IC}^{2} } }}E\left| {\frac{{z_{i,t - 1} }}{{\sigma_{{r_{i} }} }}} \right|} \\ & = \frac{{\sigma_{A} }}{{\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{IC}^{2} } }}\frac{1}{N}\sum\limits_{i = 1}^{N} {E\left| {z_{i,t - 1} } \right|E\left( {\frac{1}{{\sigma_{{r_{i} }} }}} \right)} \\ & = \frac{{\sigma_{A} }}{{\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{IC}^{2} } }}\frac{1}{N}\sum\limits_{i = 1}^{N} {\sqrt {\frac{2}{\pi }} \cdot \frac{1}{{\sigma_{{r_{i} }} }}} \\ & = \sqrt {\frac{2}{\pi }} \frac{{\sigma_{A} }}{{\sqrt {\sigma_{\varepsilon }^{2} /N + \sigma_{IC}^{2} } }}\frac{1}{N}\sum\limits_{i = 1}^{N} {\frac{1}{{\sigma_{{r_{i} }} }}} \\ & = \sqrt {\frac{2}{\pi }} \frac{{\sigma_{A} }}{{\sqrt {(1 - \mu_{IC}^{2} - \sigma_{IC}^{2} )/N + \sigma_{IC}^{2} } }}E_{cs} \left( {\frac{1}{{\sigma_{{r_{i} }} }}} \right). \\ \end{aligned}
(23)

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