Portfolio selection with active strategies: how long only constraints shape convictions

Abstract

In the recent years, an intense effort has been dedicated to the research on equity factors. If the academic literature questions their rewarding nature, the financial industry competes for the best portfolio implementation, either in long-short or in long only vehicles. While both solutions try to exploit the same factors while controlling risk, two kinds of managers co-exist in the meantime: fundamental, discretionary stock-pickers, and quantitative, systematic managers. The contribution of this paper is twofold. The core message is that a quantitative long only portfolio, built with any factor as a future returns’ proxy and a risk control, ends up to be high conviction portfolios: the long only constraint polarizes naturally the portfolio on a concentrated set of nonzero positions. In this respect, actively managed, quantitative long only portfolios share some similarity with discretionary stock pickers. Beyond this message, and backed by numerical experiments, we derive theoretical results and closed-form formulas to show in addition that: (i) selected stocks are those that realize a trade-off between a low $\beta$ and a high loading on the factor; (ii) the thresholds driving this selection are endogenous, leading to a recursive procedure to select the stocks. In other words: the portfolio selection problem may be solved linearly instead of using an optimizer. We highlight in particular the crucial role played by low $\beta$ stocks and by the co-linearity of the risk model with the factor used to forecast returns.

Appendices

Analytics: definition of quantities

We define and describe in this section the quantities and the analytics that we use in the results throughout the paper. We adopt the empirical convention (that holds most of the time among practitioners!) where an active return is the return obtained by subtracting to the original return of the strategy, the return of the Benchmark on the same period; an excess return is the return obtained by subtracting to the original return of the strategy, the return of the Benchmark on the same period times the beta (sensitivity) of the strategy with respect to the Benchmark at the time of the return. In the following, $R_t$ is the time series of daily returns of a strategy and $R_t^B$ the time series of daily returns of the Benchmark.

Risk-related quantities:

• Beta: is the sensitivity of $R_t$ towards $R_t^B$. It is computed as the regression coefficient of $R_t^B$ in the OLS regression of $R_t$ on $R_t^B$ with no intercept. We can estimate this beta as a univariate one that we simply note $\beta$ or as a rolling one $\beta _t$ which is the time series of the same quantity but computed on the subsets of data $(R_{t'}, R_{t'}^B)_{t' \in [t-250;t]}$.

• Volatility: annualized (in percents) standard deviation of weekly returns¹⁹ of the return time series of the strategy. The annualization is done by multiplying with a factor $\sqrt{52}$.

• Active Risk: volatility (see above) of $R_t-R_t^B$.

• Excess Risk: volatility (see above) of $R_t-{\hat{\beta }}_t \cdot R_t^B$.

Gain-related quantities:

• Sharpe: annualized Sharpe ratio of weekly returns of the return time series of the strategy, computed as the ratio of the average return by the standard deviation, annualized by multiplying by $\sqrt{52}$.

• Active Sharpe: Sharpe ratio (see above) of $R_t-R_t^B$.

• Excess Sharpe: Sharpe ratio (see above) of $R_t-{\hat{\beta }}_t \cdot R_t^B$.

• Costs: we estimate costs as the half-spread of consumed by transactions. Our cost of rebalancing is then:

  $$\begin{aligned} \sum _{k=1}^N \frac{\hbox {abs}(w_k(t)-w_k(t-1))}{w({t-1})^T \cdot \mathbf {1}} \times \frac{1}{2} \hbox {spread}_k(t). \end{aligned}$$

  We report the average value of this time series (to provide a univariate quantity) in the tables.

Portfolio-related quantities for a portfolio of positions in cash w:

• Leverage: this quantity is simply $w^T \cdot \mathbf {1}$.

• Turnover: the turnover at date t is the proportion of the portfolio that changes from previous position and equals to:

  $$\begin{aligned} \sum _{k=1}^N \frac{\hbox {abs}(w_k(t)-w_k(t-1))}{w({t-1})^T \cdot \mathbf {1}} . \end{aligned}$$

  We report the median value of this time series in the tables.

• Maximum Drawdown (MaxDD): indicator of maximum value of peak-to-valley cumulated return. Indicates the level of the maximum continuous loss. To be computed, we form the cumulated sum of the returns minus the expanding maximum of the same cumulated return serie. We report the minimum (maximum order of magnitude negative value) of this series in the tables.

• Number of effective positions ($\mathcal{N}_{\!\hbox {\tiny eff}}$): the number of effective positions is the integer part of the ratio

  $$\begin{aligned} \frac{(\sum _{k=1}^N |w_k|)^2}{\sum _{k=1}^N w_k^2}. \end{aligned}$$

  We report the median value of this time series in the tables.

• Number of nonzero positions: to overcome numerical effects, counting strictly positive positions is not convenient, so here is computed the number of realistic positions. A position is defined as realistic if its relative weight in the portfolio is higher than the portion represented by an elementary number of lots²⁰ in a long only fund. So we count as being nonzero a position at time t for an asset k if its relative weight $w_k(t) / w(t)^T \cdot \mathbf {1}$ in the portfolio is greater than:²¹

  $$\begin{aligned} \frac{w_k(t)}{w(t)^T \cdot \mathbf {1}} > \frac{c_k(t) \times 100}{1e9}, \end{aligned}$$

  where $c_k(t)$ is the close price of the asset k. We report the median value of this time series in the tables.

• Diversification Ratio (DR): we take the definition²² of Choueifaty and Coignard (2008) where this quantity is equal to:

  $$\begin{aligned} \frac{w^T \cdot \sigma }{\sqrt{w^T \cdot {\Omega } \cdot w}} \end{aligned}$$

  where $\sigma _t$here is the vector made of the univariate volatilities of the assets at date t. We report the median value of this time series in the tables.

Results and tables

See Table 5.

Table 5 Exhaustive statistics for all portfolios studied in the paper

Weights of a long-short portfolio

We study here the case of a long-short version of our long only portfolio of interest. Therefore, the problem is similar to the one of Sect. 3.2 except that the positions are unconstrained. We control the variance while following a rewarded factor. The portfolio construction following a rewarded factor with expected returns of $\mu$ is then:

$$\begin{aligned} \left. \begin{array}{lll} \max _w &{} &{} w^T \mu \\ \hbox {u.c.} &{} (i) &{} w^T \mathbf {1}= 1\\ &{} (ii) &{} {1\over 2}w^T \Omega w\le {1\over 2}A^2 \end{array}\right. . \end{aligned}$$

Please note that as soon $\mu$ is positive, we expect the risk constraint (ii) to be saturated, i.e. $w^T \Omega w= A^2$.

To find its solution, it is enough to use Lagrange multipliers $\lambda$ and $\gamma$ (given this optimization problem respects the Karush-Kuhn-Tucker conditions since all expressions are smooth and convex). The optimization program reads

$$\begin{aligned} w^T \mu - \gamma (w^T \mathbf {1}- 1) - \frac{\lambda }{2}(w^T \Omega w - A^2). \end{aligned}$$

Deriving, respectively, to w and applying the first order condition gives that we simply have to look for the weights, solving $\mu -\lambda \Omega w -\gamma \mathbf {1}= 0$ as a function of the multipliers, i.e.

$$\begin{aligned} w^{\mu mv} = {1\over \lambda } \Omega ^{-1} (\mu -\gamma \mathbf {1}). \end{aligned}$$

(20)

To find the values of the multipliers, we simply need to inject them in the constraints. In this respect, we introduce the following notation:²³

$$\begin{aligned} \omega _{11}= & {} \mathbf {1}^T\Omega ^{-1}\mathbf {1},\\ \omega _{1\mu }= & {} \mathbf {1}^T\Omega ^{-1}\mu ,\\ \omega _{\mu \mu }= & {} \mu ^T\Omega ^{-1}\mu . \end{aligned}$$

The linear constraint then reads:

$$\begin{aligned} \lambda + \omega _{11}\gamma - \omega _{1\mu } = 0, \end{aligned}$$

(21)

and the quadratic one reads:

$$\begin{aligned} A^2 \lambda ^2 - \omega _{11}\gamma ^2 + 2\omega _{1\mu }\gamma -\omega _{\mu \mu } = 0. \end{aligned}$$

(22)

A simple way to obtain the weights of the portfolio is to express them as

$$\begin{aligned} w^{\mu mv} = a\;\Omega ^{-1}\left( \mu -\frac{\omega _{1\mu }}{\omega _{11}}\mathbf {1}\right) + \frac{1}{\omega _{11}}\Omega ^{-1}\mathbf {1}, \end{aligned}$$

(23)

where $a=1/\lambda$ is chosen to set the risk at the desired level $A^2/2$. Moreover, injecting $\gamma =(\omega _{1\mu } -\lambda )/\omega _{11}$ in Eq. 22 gives the exact and very useful relation between a and A (in other words, this relation links the desired level of risk with the risk aversion in the optimization program):

$$\begin{aligned} a^2 = \frac{\omega _{11}A^2 - 1}{\omega _{\mu \mu } \omega _{11} - \omega _{1\mu }^2}. \end{aligned}$$

(24)

Just note that a is well defined because:

• The numerator of the right-hand side of this equality is positive as far as $A^2>1/\omega _{11}$. It is always the case, and is equal if and only if the constraint on the risk is such that the portfolio is the Minimum Variance one.

• Its denominator is always positive too because of the Cauchy-Schwartz, inequality applied to the Hermitian norm²⁴ associated with $\Omega ^{-1}$. It reaches zero only for the Minimum Variance portfolio, which is a specific case, that we may deal with the specific optimization problem detailed in Sect. 3.4.

Long only minimum variance portfolio

To explore the existence of a subset of stock selected in a Minimum Variance long only portfolio as described in 4.3.1, we will use a continuous framework to show the existence of a threshold on betas such that

$$\begin{aligned} B = \frac{1/{{\bar{\sigma }}}^2 + \sum _{k\in \mathfrak {S}} \beta _k^2/\sigma _k^2}{\sum _{k\in \mathfrak {S}} \beta _k/\sigma _k^2} \end{aligned}$$

where $\mathfrak {S}$ is the set of all $\beta _k$ lower than B. In this continuous framework we will no more have sums but integrals, and the set of stocks $(\beta _k,\sigma _k)_k$ will be defined by a joint density $dm(\beta ,\sigma )$. To reflect the reality of the joint distribution of betas and idiosynchratic volatilities (see Fig. 1), we will make the technical assumption that the support of m is bounded in $\beta$, i.e. there is a maximum beta ${\bar{\beta }}$ and a minimum beta $\underline{\smash {\beta }}$. It allows to rewrite the upper equality as

$$\begin{aligned} B\cdot \int _{\beta<B} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma )= c + \int _{\beta <B} \frac{\beta ^2}{\sigma ^2} dm(\beta ,\sigma ), \end{aligned}$$

where c is a positive constant properly defined.

The existence of a zero for the following function:

$$\begin{aligned} {\mathfrak {B}}(x) = x\cdot \int _{\beta<x} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ) - \int _{\beta <x} \frac{\beta ^2}{\sigma ^2} dm(\beta ,\sigma ) - c \end{aligned}$$

is now enough to proof the existence of B. For reader’s information, Fig. 3 shows the graph of this function on our real dataset (Russel 1000 from 2002-03-29 to 2018-12-27). Start by studying the behaviour of ${\mathfrak {B}}$ when x goes to extreme values:

• When x is very small, equal to the minimum $\underline{\smash {\beta }}$, then it is obvious that

  $$\begin{aligned} {\mathfrak {B}}(\underline{\smash {\beta }})=\underline{\smash {\beta }}\frac{\underline{\smash {\beta }}}{\underline{\smash {\sigma }}^2} m(\underline{\smash {\beta }},\underline{\smash {\sigma }}) - \frac{\underline{\smash {\beta }}^2}{\underline{\smash {\sigma }}^2} m(\underline{\smash {\beta }},\underline{\smash {\sigma }}) - c=-c < 0. \end{aligned}$$

• When x is very large, then ${\mathfrak {B}}(x)$ can be negative. In fact it is easy to see that when

  $$\begin{aligned} c+\int \frac{\beta ^2}{\sigma ^2} dm(\beta ,\sigma ) - \int _{\beta <0} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ) \ge \int _{\beta \ge 0} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ), \end{aligned}$$

  then ${\mathfrak {B}}(x)$ is negative for large values of x. We hence need to ask to $m(\cdot , \cdot )$ (the joint distribution of the betas and the idiosyncratic volatilities of the stocks) to verify the opposite inequality:

  $$\begin{aligned}&c+\int \frac{\beta ^2}{\sigma ^2} dm(\beta ,\sigma ) - \int _{\beta<0} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma )\\&\quad < \int _{\beta \ge 0} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ). \end{aligned}$$

  (P)

  Under such a condition, we observe that as soon as $x>1$,

  $$\begin{aligned} x\cdot \int \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ) - \int \frac{\beta ^2}{\sigma ^2} dm(\beta ,\sigma ) - c \ge \epsilon > 0. \end{aligned}$$

  The only “missing part” between the upper quantity and ${\mathfrak {B}}(x)$ is

  $$\begin{aligned} x\cdot \int _{\beta>x} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ) - \int _{\beta >x} \frac{\beta ^2}{\sigma ^2} dm(\beta ,\sigma ). \end{aligned}$$

  And when x goes towards the maximum value ${\bar{\beta }}$ for the betas, this “missing part” goes to zero by definition of the Lebesgue integral. Thence there is a limit $x_\epsilon$ such that when x is larger than $x_\epsilon$, the missing part is lower than $\epsilon$. As a consequence ${\mathfrak {B}}(x)>0$ as soon as $x>x_\epsilon$.

We now know that ${\mathfrak {B}}$ goes from negative values at the left to positive values at the right. It is enough to know that it has at least one zero. Since the derivative of ${\mathfrak {B}}$ reads

$$\begin{aligned} {\mathfrak {B}}'(x)= & {} \int _{\beta<x} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ) - x \cdot \frac{x}{\sigma ^2} m(x,\sigma ) \\&+ \frac{x^2}{\sigma ^2} m(x,\sigma ) = \int _{\beta <x} \frac{\beta }{\sigma ^2} dm(\beta ,\sigma ). \end{aligned}$$

As underlined in Fig. 3, it is clear that this derivative it monotonous: it starts to be negative (under the realistic assumption that $\underline{\smash {\beta }}<0$) and is positive after a while because of Assumption (P). This is enough to conclude that there is one and only one zero for ${\mathfrak {B}}$ under our realistic assumption. It is equivalent to the existence and unicity of the threshold B needed to have a nonempty long only Minimum Variance portfolio.

Fig. 3

Graph of function ${\mathfrak {B}}(x)+c$ for real data. The shape clearly confirms the theoretical results: going to zero by negative values at the left, towards positive values at the right; moreover the slope starts to decrease and after a while increases, leading to a unique zero for the function, here around 0.22. We know by Table 3 that the zero of ${\mathfrak {B}}(x)$ is around 0.33

Elements of proof for the existence of long only portfolios

Long only Managed Volatility portfolio

The complexity of defining conditions for the existence of a long only Managed Volatility portfolio stems from adding one dimension to the Minimum Variance one: now one has to deal not only with betas but also with factor loadings, and hence one have to consider two variables involving sums over the selected stocks: $\mathbf {B}$ and $\mathbf {C}$. This short appendix simply tries to give an intuition of what is be needed to proof the existence of a long only Managed Volatility portfolio on a given universe.

Going to deep in the details would be out of the scope of this paper, but it is easy to see that provided that $(\mu _k, \beta _k)$ are constrained to lay in the manifold $\mathfrak {M}(a,v,\mathfrak {S})$ defined by

$$\begin{aligned} \mu _k = \frac{\omega _{11}}{1-a\,\omega _{1\mu }}(v-1)+ \frac{\beta _k}{\mathbf {C}(\mu _k,\beta _k,\mathfrak {S})}, \end{aligned}$$

(25)

then the problem is one dimensional and very close to the one of the Minimum Variance, since the condition (13) to keep a beta is now

$$\begin{aligned} \beta _k < v\cdot \mathbf {B}. \end{aligned}$$

Provided that Eq. 25 defines properly a one dimensional manifold, i.e. provided that it can be reduced to an implicit function $\mu _k=f_{\mathfrak {M}(a,v,\mathfrak {S})}(\beta _k)$, then along this line in $\mu _k$ we are facing the long only Minimum Variance problem. This means that we no more face a curve like the one of Fig. 3, but a surface made of such curves, organized in the $(\mu ,\beta )$ plane according to the $\mathfrak {M}(a,v,\mathfrak {S})$.