Multivariate volatility estimation

• Covariances are relevant for portfolio theory (and therefore asset pricing, capital budgeting, investment decisions)
• Requires estimation of \(\Sigma_t\) instead of just \(\sigma_t^2\)

Problems for estimates \(\hat\Sigma_t\):

1. The curse of dimensionality (too many parameters)
2. Time variation (volatilities and correlations may change over time)
3. Frequency (minute, daily, monthly data?)

Discuss: How would you compute minimum variance portfolio weights for a universe of 1000 assets?

General Framework

• We consider the \(N\)-dimensional return series \[r_t = E\left(r_t|\mathcal{F}_t\right) + \varepsilon_t\] where \(\varepsilon_t\) is a \((N\times 1)\) vector of error terms with (conditional) covariance matrix \(\Sigma_t\)
• Exploiting the symmetry property of \(\Sigma_t\) we can write \[\Sigma_t = D_tQ_tD_t\] where \(Q_t\) is the \((N\times N)\) conditional correlation matrix of \(\varepsilon_t\) and \(D_t = \text{diag}\left({\sigma_{11, t}}, \ldots, {\sigma_{NN, t}}\right)\) is a \((N \times N)\) diagonal matrix
• Because of the symmetry of \(\Sigma_t\), it is sufficient to consider only \(N(N+1)/2\) unique elements

The curse of dimensionality

• Consider the case with a \((T \times N)\) matrix of returns \(R\) where \(T<N\)
• Then the sample variance covariance matrix is \[S = \frac{1}{T-1}R'\left(I_T - \frac{1}{T}\iota\iota'\right)R\]
• The rank of the matrix \(R'R\) is \(\min(T, N) = T\)
• (intuition: \(R'R\) qua linear transformation sends \(\mathbb{R}^N\) into a subspace with dimension at most \(T\))
• This implies that \(R'R\) is singular for \(N\geq T\) and thus \[\exists w\in\mathbb{R}^N: w'Sw = 0\]
• In other words, \(S\) is not invertible and there is no unique efficient portfolio
• More structure is required to overcome this issue (e.g. based on factor structure)

Linear shrinkage (Ledoit Wolf, 2003, 2004)

• The sample covariance matrix is unbiased and the maximum likelihood estimator under normality
• In the context of mean estimation, Stein (1956) showed that a better estimator than the sample mean can be constructed by shrinking the sample mean to a target vector, that is, by using a linear combination of the sample mean and the target vector
• Better in the sense of the mean squared error (MSE) — a little bias in exchange for a large reduction in variance
• Ledoit and Wolf (2003, 2004) apply the same thinking to \(\Sigma\)
• They suggest a linear combination of some target matrix \(F\) (identity, single factor, or equicorrelation matrix) and the sample variance-covariance matrix \(\hat\Sigma\), controlled by the shrinkage intensity \(\alpha\): \[\hat\Sigma^{LW} = \alpha F + (1-\alpha) \hat\Sigma \text{ for } 0\leq\alpha\leq1\]

The shrinkage target (Ledoit Wolf, 2004)

• We already discussed (multi) factor models
• Constant correlation model: all the (pairwise) correlations are assumed identical.
• Estimator of the common constant correlation: average of all the sample correlations

Illustrating shrinkage

• The optimal linear combination depends on unknown population quantities and must, therefore, be thought of as an ideal (or “oracle”) but infeasible estimator
• Ledoit and Wolf derive feasible estimators (in an exercise you are asked to implement the estimator)

The optimal shrinkage intensity (Ledoit-Wolf 2003, 2004)

• To define the MSE for an \((N\times N)\) matrix \(A\), we use the Frobenius norm \[||A||^2_F = {<A, A>} := \frac{1}{N}\sum\limits_{i=1}^N\sum\limits_{j=1}^N A_{i,j}^2 = \frac{1}{N}tr(A'A)\]

• The scaled mean squared error is \[E\left(||\Sigma^{LW} - \Sigma||_F^2\right) \text{ where } \Sigma^{LW} = \alpha F + (1-\alpha) \hat\Sigma\]

• Let \(\pi\) be the sum of asymptotic variances of the entries of the sample covariance matrix scaled by \(\sqrt{T}\), \(\rho\) the scaled asymptotic covariances of the entries of the single-index covariance, and \(\psi\) the bias (misspecification) of the target \[\alpha^* = \frac{1}{T}\frac{\pi - \rho}{\psi}\]

• Interpretation: shrink more when sample noise \(\pi\) is large (high-dimensional, short sample) and less when the target is badly misspecified (\(\psi\) large). The factor \(1/T\) ensures that as the sample grows, the optimal shrinkage vanishes and the estimator collapses to \(\hat\Sigma\)

Dynamic multivariate models

• The generalization of univariate GARCH models to the multivariate domain is conceptually simple
• The literature on the dynamics governing \(\Sigma_t\) may be broadly divided into direct multivariate extensions, factor models, linear combinations of univariate GARCH models (Generalized Orthogonal GARCH), and nonlinear combinations of univariate GARCH models (the broader class of Dynamic Correlation models)
• The need to invert the covariance matrix in many MGARCH parameterizations introduces estimation problems for large systems, as the eigenvalues of the covariance matrix decrease exponentially fast toward zero, even when the covariance is not singular

Constant conditional correlation (CCC)

• Bollerslev (1990)
• In the constant conditional correlation (CCC) model of Bollerslev (1990), the covariance matrix can be decomposed into \[\Sigma_t = D_tRD_t = \rho_{ij}\sqrt{h_{ijt}h_{ijt}}\] where \(D_t = \text{diag}(\sqrt{h_{11t}}, \ldots, \sqrt{h_{NNt}})\) and \(R\) is a positive definite correlation matrix
• Maximum likelihood estimation possible in the multivariate normal case (see Eq. 50 in Boudt et al., 2019)

Obvious extension: DCC

• Covariance Targeting (Engle (2002)) \[\Sigma_t = D_t\mathbf{R_t}D_t = \rho_{ij}\sqrt{h_{ijt}h_{ijt}}\]
• DCC is a two-step estimator: (i) fit a univariate GARCH to each asset to obtain \(D_t\) and the standardized residuals \(z_t = D_t^{-1}\varepsilon_t\); (ii) model the dynamics of the correlation matrix using a proxy process \(Q_t\): \[Q_t = \bar{Q} + \alpha\left(z_tz_t' - \bar{Q}\right) + \beta\left({Q}_{t-1} - \bar{Q} \right)\] where \(\bar Q\) is the sample covariance of the \(z_t\) and \((\alpha, \beta)\) are scalar parameters governing the correlation dynamics
• The correlation matrix \(R_t\) is then obtained as \[R_t = \text{diag}(Q_t)^{-1/2}Q_t\text{diag}(Q_t)^{-1/2}\]

Bridging to large-scale allocation

• At time \(t\), we solve the problem \[ \begin{array}{c} \max_{\omega_{t+1}} \: \omega_{t+1}'\mu_t -\frac{\gamma}{2} \omega_{t+1}'\,\Sigma_{t}\, \omega_{t+1} \quad s.t. \quad \omega'_{t+1}\;\iota= 1, \end{array}\] where \(\Sigma_{t}\) denotes the \(N\times N\) asset return covariance, \(\mu_t\) the mean vector, \(\gamma\) the relative risk aversion, and \(\iota\) a vector of ones

• Optimal portfolio weights: \[\begin{aligned} \omega^*_{t+1} & = \frac{1}{\iota' \Sigma_t^{-1} \iota }\Sigma_t^{-1} \iota+\frac{1}{\gamma}\left(\Sigma_t^{-1} - \frac{1}{\iota' \Sigma_t^{-1}\iota }\Sigma_t^{-1}\iota\iota' \Sigma_t^{-1} \right) \mu_t \end{aligned}\]

… the classical problem

• Noisy estimates of \(\mu_t\) over short horizons
• Solution (?): Fixing \(\mu_t\) (e.g. \(\mu_t=0\)) or using long horizons
• (\(N \times N\)) matrix \(\Sigma_t\) contains \(N(N-1)/2\) distinct elements \(\Rightarrow\) Estimation error
• For \(N=500\): 124,750 elements
• Sample covariance only positive definite if \(T\ge N\)

\(\Rightarrow\) Plugging in sample estimates \(\hat \mu_t\) and \(\hat \Sigma_t\) performs badly!

Solution: Regularizing \(\Sigma\) …

• Shrinking the covariance matrix (Ledoit and Wolf, 2004): \[\begin{aligned} \Sigma_t= \hat{\alpha}F_t+(1-\hat{\alpha})S_t, \end{aligned} \] where \(S_t\) is the sample variance-covariance matrix and \(F_t\) denotes the equi-correlation matrix

Alternative approaches: - Restricting portfolio weights (Jagannathan and Ma, 2003; Fan et al, 2012) - Factor models (MacKinlay Pastor, 2000; Fan et al, 2008) - Parsimonious high-dimensional parametric volatility models (Engle Ledoit Wolf, 2017) - Imposing priors (Barry, 1974; Brown, 1979; Jorion, 1985, 1986; Black Litterman, 1992; Pastor Stambaugh, 2000)

… or using high-frequency data:

• Idea: Constructing a daily covariance estimate using HF data of the same day
• E.g., Liu (2009), Hautsch et al (2012), Hautsch et al (2015), Lunde et al (2015), Bollerslev et al (2016), …

But …

Frictions:

• No consideration of transaction costs!
• Re-balancing is costly!

Estimation:

• Parameters are uncertain!

• Model ambiguity

• Predictive ability of models is time-varying!

• De Miguel et al (2009): Out of sample, the \(1/N\) portfolio cannot be outperformed in high dimensions!

Portfolio optimization under transaction costs

The investor at time \(t\) observes past returns \(R_t=\left(r'_{1},\ldots,r'_{t}\right)'\in\mathbb{R}^{t \times N}\) and assumes an underlying (predictive) model \(\mathcal{M}\).

The (myopic) portfolio optimization problem is \[\begin{aligned} \omega_{t+1} ^* :=&\arg\max_{\omega\in\mathbb{R}^N, \iota'\omega=1} E\left(U_\gamma\left(\omega ' \left(1 + {r}_{t+1}\right)-\nu_t (\omega)\right)|\mathcal{M}, \mathcal{F}_t\right) \\ =&\arg\max_{\omega\in\mathbb{R}^N, \iota'\omega=1} \int U_\gamma\left(\omega '\left(1 + {r}_{t+1}\right)-\nu_t(\omega) \right) p_t ({r}_{t+1}|\mathcal{M})\,\text{d}{r}_{t+1}, \end{aligned}\] where \(p_t({r}_{t+1}|\mathcal{M}):=p_t({r}_{t+1}|\mathcal{M}, \mathcal{F}_t)\) denotes the predictive return distribution and \(\nu_t(\omega)\) the transaction costs.

A closed-form solution for Gaussian returns and quadratic transaction costs

• Initial Markowitz (1952) approach with Gaussian returns: \(p_t({r}_{t+1}|\mathcal{M})=N(\mu,\Sigma)\)
• Can be solved by maximizing the certainty equivalent \[\begin{aligned} \omega_{t+1} ^* & = \arg\max_{\omega \in \mathbb{R}^N, \iota'\omega = 1} \omega'\mu - \nu_t \left(\omega\right) - \frac{\gamma}{2}\omega'\Sigma\omega. \end{aligned}\]
• We assume quadratic transaction costs — mainly because they yield a closed-form solution that is directly comparable to the classical Markowitz problem (Hautsch et al. 2019 show that the \(L_1\) case can be treated similarly): \[\begin{aligned} \nu\left(\omega_{t+1},\omega_{t^+}, \beta\right) = \frac{\beta}{2} \left(\omega_{t+1} - \omega_{t^+}\right)'\left(\omega_{t+1}- \omega_{t^+}\right), \end{aligned}\] with cost parameter \(\beta>0\)
• The pre-rebalancing weights \(\omega_{t^+}\) are not the same as yesterday’s target weights \(\omega_t\): even if the investor does nothing, prices move during the day and drift the portfolio away from \(\omega_t\). Concretely: \[\omega_{t^+} := \frac{\omega_t \circ (1 + r_{t})}{\iota' (\omega_t \circ (1 + r_{t}))}\]

Optimization problem¹

• Proof left for exercise \[\begin{aligned} \omega_{t+1} ^* :=& \arg\max_{\omega \in \mathbb{R}^N, \iota'\omega = 1} \omega'\mu - \nu_t (\omega,\omega_{t^+}, \beta) - \frac{\gamma}{2}\omega'\Sigma\omega \\ =&\arg\max_{\omega\in\mathbb{R}^N,\text{ } \iota'\omega=1} \omega'\mu^* - \frac{\gamma}{2}\omega'\Sigma^* \omega , \end{aligned}\] where the effective mean and covariance absorb the transaction-cost terms: \[\begin{aligned} \mu^*:=\mu+\beta \omega_{t^+} \quad \text{and} \quad \Sigma^*:=\Sigma + \frac{\beta}{\gamma} I_N \end{aligned}\]
• As a result: \[\begin{aligned} \omega^*_{t+1} & = \frac{1}{\gamma}\left(\Sigma^{*-1} - \frac{1}{\iota' \Sigma^{*-1}\iota }\Sigma^{*-1}\iota\iota' \Sigma^{*-1} \right) \mu^* + \frac{1}{\iota' \Sigma^{*-1} \iota }\Sigma^{*-1} \iota\end{aligned}\]

What happens if \(\beta\rightarrow \infty\)?

• Define \(A(\Sigma):=\left({\Sigma}^{-1} - \frac{1}{\iota' {\Sigma}^{-1}\iota }{\Sigma}^{-1}\iota\iota' {\Sigma}^{-1} \right)\).
• Proposition. Let \(\omega_0\) be the initial allocation. Then, \[\begin{aligned} \omega_T^* = \sum\limits_{i=0}^{T-1}\left( \frac{\beta}{\gamma}A(\Sigma^*)\right) ^i \omega(\mu,\Sigma^*) + \left(\frac{\beta}{\gamma}A(\Sigma^*)\right)^T \omega_0, \end{aligned}\] where \(\omega(\mu,\Sigma^*)\) denotes the mean-variance efficient allocation.
• Proposition. \[ \begin{aligned} \lim_{\beta\rightarrow\infty}\omega_{t+1}^* &= \left(I_N - \frac{1}{N}\iota \iota' \right) \omega_{t^+} + \frac{1}{N}\iota = \omega_{t^+} \end{aligned}\]

Efficiency of the portfolio

• Proposition. \(\exists \beta^*> 0 \text{ } \forall \tilde{\beta}\in [0,\beta^*): \left\|\left(\frac{\tilde\beta}{\gamma}A(\Sigma^*)\right)\right\|_F < 1\), where \(\|\cdot\|_F\) denotes the Frobenius norm \(\|A\|_F := \sqrt{\sum\limits_{i=1}^N\sum\limits_{j=1}^N a_{i,j}^2}\)
• For \(\beta < \beta^*\) and \(T \rightarrow \infty\), we have \[\begin{aligned} \sum_{i=0}^{T}\left( \frac{\beta}{\gamma}A(\Sigma^*)\right) ^i \rightarrow \left(I_N - \frac{\beta}{\gamma}A(\Sigma^*)\right)^{-1} \end{aligned}\] and \(\lim_{i\rightarrow\infty}\left( \frac{\beta}{\gamma}A(\Sigma^*)\right) ^i = 0\).
• Proposition. For \(T\rightarrow \infty\) and \(\beta < \beta^{*}\) the series \(\omega_T\) converges to a unique fix-point given by \[\begin{aligned} \omega_\infty & = \left(I_N - \frac{\beta}{\gamma}A(\Sigma^*)\right)^{-1} \omega(\mu,\Sigma^*) = \omega(\mu, \Sigma) \end{aligned}\]

Intuition behind shrinkage

• For asset-specific transaction costs \[\left(\omega_{t+1} - \omega_{t^+}\right)'B\left(\omega_{t+1}- \omega_{t^+}\right)\] where \(B\) is positive definite, the linear shrinkage estimator for \(\Sigma\) of Ledoit and Wolf (2003, 2004) can be reconciled.
• Lemma. Assume a regime of high volatility with \(\Sigma^h = (1 + h)\Sigma\), where \(h > 0\) and \(\Sigma\) is the asset return covariance matrix during calm periods. Assume \(\mu=0\). Then, the optimal weight \(\omega_{t+1}^*\) is equivalent to the optimal portfolio based on \(\Sigma\) and (reduced) transaction costs \(\frac{\beta}{1+h} < \beta.\)

Empirical Illustration

• \(N=308\) assets; 06/2007 - 03/2017 with daily readjustments

• \(\Sigma_t\) estimated by

  1. Sample covariance
  2. Ledoit/Wolf (2004) shrinkage estimator
  3. Rolling window of length \(h = 500\) days

• \(\mu_t\) is fixed to zero. Initial portfolio weights: \(\frac{1}{N}\iota\). \(\gamma=4\).

• Yields realized portfolio returns net of transaction costs \[\begin{aligned} r^\beta_t - \beta \left(\omega^\beta_{t+1} - \omega^\beta_{t^+}\right)^\prime\left(\omega^\beta_{t+1} - \omega^\beta_{t^+}\right) \end{aligned}\] with \(r^\beta_t := r_t ^\prime \omega_t^\beta\).

Realized Sharpe ratios with quadratic transaction costs

• Sample (solid line), LW (dotted line), Equal (dashed)
• X-axis denotes transaction cost parameter \(\beta\) in basis points

\(L_1\) Transaction Costs

• Usual assumption in the literature: \(L_1\) transaction costs \[\begin{aligned} \nu_t\left(\omega, \omega_{t^+}, \beta\right)=\beta ||\omega-\omega_{t^+}||_1 = \beta \sum_{i=1}^N |\omega_{i,t+1}-\omega_{i,t^+}|. \end{aligned}\]

• For \(\mu=0\) and \(\Delta:=\omega_{t+1}-\omega_{t^+}\), we have \[\begin{aligned} \omega_{t+1} ^* = \arg\min_{\Delta\in\mathbb{R}^N,\text{ } \iota'\Delta=0} \gamma\Delta'\Sigma\omega_{t^+} + \frac{\gamma}{2}\Delta'\Sigma \Delta + \beta ||\Delta||_1, \end{aligned}\]

• Solving for \(\omega_{t+1}^*\) yields \[\begin{aligned} \omega_{t+1}^* =\left(1+\frac{\beta}{\gamma}\iota'\Sigma^{-1} \tilde g\right)\omega_\text{mvp} -\frac{\beta}{\gamma}\Sigma^{-1} \tilde g, \end{aligned}\] where \(\tilde g\) is the vector of sub-derivatives of \(||\omega_{t+1} - \omega_{t^+}||_1\) and \(\omega_\text{mvp} := \frac{1}{\iota'\Sigma^{-1}\iota}\Sigma^{-1}\iota\) are the weights of the GMV portfolio.

Recall Jagannathan and Ma?

• Proposition. For \(\mu=0\), the underlying problem is equivalent to \[\begin{aligned} \omega_{t+1} ^* = \arg\min_{\omega\in\mathbb{R}^N,\text{ } \iota'\omega=1} \omega' \Sigma_{\frac{\beta}{\gamma}} \omega, \end{aligned}\] where \(\Sigma_{\frac{\beta}{\gamma}} = \Sigma + \frac{\beta}{\gamma}\left(g^*\iota' + \iota g^{*'}\right)\) and \(g^*\) is the subgradient of \(||\omega^*_{t+1}-\omega_{t^+}||_1\).

What about parameter uncertainty?

• Recall that the optimal portfolio (for now: \(\nu = 0\)) is computed as the solution to \[\begin{aligned}\small \omega_{t+1} ^* :=&\arg\max_{\iota'\omega=1} E\left(U_\gamma\left(\omega ' \left(1 + {r}_{t+1}\right)\right)|\mathcal{M},\mathcal{F}_t\right) \\ =& \arg\max_{\iota'\omega=1} \int U_\gamma\left(\omega '\left(1 + {r}_{t+1}\right) \right) p_t ({r}_{t+1}|\mathcal{M})\,\text{d}{r}_{t+1}\\ =& \arg\max_{\iota'\omega=1} \int\int_\mu\int_\Sigma U_\gamma\left(\omega '\left(1 + {r}_{t+1}\right)\right)p_t ({r}_{t+1}|\mu, \Sigma, \mathcal{M})\,\text{d}\mu\,\text{d}\Sigma\,\text{d}{r}_{t+1}, \end{aligned}\] where \(p_t({r}_{t+1}|\mathcal{M}):=p_t({r}_{t+1}|\mathcal{M},\mathcal{F}_t)\) denotes the predictive return distribution

• Unlike the conditional distribution, the Bayesian predictive distribution accounts for estimation errors by integrating over the unknown parameter space

• The problem above refers to Bayesian Portfolio Allocation

1. Account for estimation risk and model uncertainty
2. Elicit economically meaningful prior beliefs about asset pricing models (distribution of future returns)
3. Highly flexible due to Bayesian computation techniques

Bayesian asset allocation¹

• So far we analysed a two-step procedure

1. Estimate the parameters \(\hat\theta\) (e.g. \(\hat\mu_t\) and \(\hat\Sigma_t\))
2. Maximize the expected utility conditional on the estimated parameters being the true ones \[\max_\omega = E\left(U(\omega)|\theta = \hat\theta\right)\]

• Optimally, however, the investor adjusts for estimation uncertainty
• Thus, portfolio selection based on \(p_t ({r}_{t+1}|\mathcal{M})\) instead of \(p_t ({r}_{t+1}|\hat\theta, \mathcal{M})\)

Example: Bayesian Portfolio Allocation

• Specify the prior distribution of the parameters as diffuse
  \[\pi(\theta) = p(\mu_t|\Sigma_t) \times \pi(\Sigma_t) \propto |\Sigma_t|^{-\frac{N+1}{2}}\]
• Then, assuming that the returns are jointly normally distributed, the posterior distribution is given by \[p(\mu_t|\Sigma_t, \mathcal{D}_t) \propto |\Sigma_t|^{-1/2}\exp\left(-\frac{1}{2}tr\left(T(\mu_t - \hat\mu_t)\right)(\mu_t - \hat\mu_t)'\Sigma_t^{-1}\right)\] and \[\pi(\Sigma_t) \propto |\Sigma_t|^{-(T+N)/2}\exp\left(-\frac{1}{2}tr\Sigma_t^{-1}\left(T\hat\Sigma_t\right)\right)\]
• The predictive distribution corresponds to a multivariate \(t\) distribution with \(T - N\) degrees of freedom
• Under the diffuse prior the mean-variance efficient portfolio is \[\omega^\text{Bayes} = \frac{1}{\gamma}\left(\frac{T - N - 2}{T + 1}\right)\hat\Sigma_t^{-1}\hat\mu_t\]
• Intuition: Invest less in the risky asset and more in the risk-free asset

Extensions and takeaways

• Bayesian computation allows us to sample from (almost) arbitrary posterior predictive distributions
• Optimal allocation can be derived numerically
• Turnover regularization is closely related to parameter shrinkage
• Optimal portfolios are shifted towards current holdings
• This effect is different from statistical approaches: Shrinkage intensity is an exogenous parameter!

Application: Hautsch and Voigt (2019)

Stock characteristics and optimal portfolio choice

• Idea: parametrize weights as a function of the characteristics such that we maximize expected utility
• feasible for large portfolio dimensions (such as the entire CRSP universe)
• proposed by Brandt et al. (2009) in their influential paper Parametric Portfolio Policies: Exploiting Characteristics in the Cross Section of Equity Returns.

Basic idea:

• At each date \(t\) we have \(N_t\) stocks in the investment universe, where each stock \(i\) has return \(r_{i, t+1}\) and is associated with a vector of firm characteristics \(x_{i, t}\) such as time-series momentum or market capitalization
• The investor’s problem is to choose portfolio weights \(w_{i,t}\) to maximize the expected utility of the portfolio return \[\begin{aligned} \max_{w} E_t\left(U(r_{p, t+1})\right) = E_t\left[U\left(\sum\limits_{i=1}^{N_t}w_{i,t}r_{i,t+1}\right)\right] \end{aligned}\]

Where do the stock characteristics show up?

• Parametrize the optimal portfolio weights as a function of \(x_{i,t}\) with the following linear specification for the portfolio weights: \[w_{i,t} = \bar{w}_{i,t} + \frac{1}{N_t}\theta'\hat{x}_{i,t}\]
• here, \(\bar{w}_{i,t}\) is the weight of a benchmark portfolio (value-weighted or naive portfolio), \(\theta\) is a vector of coefficients which we are going to estimate, and \(\hat{x}_{i,t}\) are the characteristics of stock \(i\), cross-sectionally standardized to have zero mean and unit standard deviation. Note on notation: the hat on \(\hat{x}_{i,t}\) denotes standardization, not an estimator — an unusual convention borrowed from Brandt et al. (2009)
• Think of the portfolio strategy as a form of active portfolio management relative to a performance benchmark: Deviations from the benchmark portfolio are derived from the individual stock characteristics
• By construction the weights sum up to one as \(\sum_{i=1}^{N_t}\hat{x}_{i,t} = 0\) due to the standardization. Note that the coefficients are constant across assets and through time
• Implicit assumption is that the characteristics fully capture all aspects of the joint distribution of returns that are relevant for forming optimal portfolios

Optimal Choice of \(\theta\)

• We estimate \(\theta\) by replacing the expectation with its sample analogue and maximizing the sample-average realized utility of the implied portfolio returns: \[\begin{aligned} \hat\theta = \arg\max_\theta \frac{1}{T}\sum\limits_{t=0}^{T-1}U\left(\sum\limits_{i=1}^{N_t}\left(\bar{w}_{i,t} + \frac{1}{N_t}\theta'\hat{x}_{i,t}\right)r_{i,t+1}\right) \end{aligned}\]
• In practice this is a low-dimensional numerical optimization (e.g. via optim in R) — the dimension of \(\theta\) equals the number of characteristics, independent of \(N_t\)
• The allocation strategy is attractive because the number of parameters to estimate is small
• Instead of a tedious specification of the \(N_t\)-dimensional vector of expected returns and the \(N_t(N_t+1)/2\) free elements of the variance-covariance matrix, all we need to focus on in our application is the vector \(\theta\)

What about short-selling constraints?

• Enforcing portfolio constraints via the parametrization is not straightforward, but in our case we simply renormalize the portfolio weights before returning them by computing \[w_{i,t}^+ = \frac{\max(0, w_{i,t})}{\sum\limits_{j=1}^{N_t}\max(0, w_{j,t})}\]