6 Using estimation risk reduction techniques
Under the EW and MW portfolios, the portfolio weights do not require estimation whereas the GMV and ERC portfolios do. For each of these portfolios, there are six given ways of performing the estimation in this research. The first is by using the standard sample covariance matrix, the next four are to use techniques outlined in chapter 5, and the final one is to use a combination of quantile factor modelling and regime switching. A summary of all of the portfolio technique pairs is given in table 6.1 below.
| Portfolio | Estimation technique | Abbreviation |
|---|---|---|
| Equal weight | - | EW |
| Market weight | - | MW |
| Global minimum variance | Sample covariance matrix | GMV-SCM |
| Global minimum variance | Quantile factor modelling | GMV-QFM |
| Global minimum variance | Regime switching | GMV-RS |
| Global minimum variance | Quantile factor modelling with regime switching | GMV-QRS |
| Global minimum variance | Ridge regression | GMV-RR |
| Global minimum variance | Subset re-sampling | GMV-SRS |
| Equal risk contribution | Sample covariance matrix | ERC-SCM |
| Equal risk contribution | Quantile factor modelling | ERC-QFM |
| Equal risk contribution | Regime switching | ERC-RS |
| Equal risk contribution | Quantile factor modelling with regime switching | ERC-QRS |
| Equal risk contribution | Ridge regression | ERC-RR |
| Equal risk contribution | Subset re-sampling | ERC-SRS |
6.1 Quantile factor modelling and regime switching
QFM and RS improve the estimation of the inputs into the optimisation and therefore have the same implementation for the GMV and ERC portfolios. To use the RS model for covariance prediction, the probability of being in either a quiet or a turbulent state needs to be estimated. If \(s_t\) is the random variable that takes on the value of the state at time \(t\) then \(s_t \in \{Q, T\}\). The true parameters are defined as: \[\begin{align*} \pi_{Q, t} &:= \mathbb{P}\{s_t = Q\}, \\ \pi_{T, t} &:= \mathbb{P}\{s_t = T\} = 1 - \pi_{Q, t}. \end{align*}\] These can be estimated using the emission quantities from the HMM, \(\hat{d}_t\), and a maximum likelihood approach - the Baum-Welch algorithm3. Kritzman, Page, and Turkington (2012) determine a turbulent-quiet data split through the parameter \(\zeta\), which is the proportion of the data allocated to the quiet regime for covariance estimation. They suggest a value of \(0.7\) or \(0.8\), but due to constraints on input data length we use \(0.7\) in this research. The investor has nearly estimated enough parameters to blend the variance-covariance matrix using the method of Flint and Du Plooy (2018) outlined below in equation (6.1). \[\begin{align} \hat{\Sigma}_{\text{blend}} & = \hat{\pi}_{Q, t + 1} \hat{\eta}_Q \hat{\Sigma}_Q + \hat{\pi}_{T, t + 1} \hat{\eta}_T \hat{\Sigma}_T. \tag{6.1} \end{align}\] The estimated probabilities, \(\hat{\pi}_{Q, t + 1}\) and \(\hat{\pi}_{T, t + 1}\), are forward-looking. Predictions can be made empirically using the current state because the transition probabilities are generally near \(1\) or \(0\) in the data. The investor still has to estimate their normalised aversion to each regime \(\hat{\eta}_{s_{t+1}}\), which they can do using the estimation procedure from Bodnar et al. (2018). For the experiment, the investor was indifferent between regimes.
The intention of this blending procedure is for money managers to estimate the state probabilities themselves, thus incorporating their future beliefs. In reality, when implementing a quantitative model, it is common to use a rolling estimation window of data. Asset return data is often not long enough to accommodate sophisticated portfolio construction methods. Therefore, assigning \(\hat{\pi}_{T, t + 1}\) values near \(1\) results in the estimation and application of large covariance matrices with potentially only \(30 \%\) of the data, without the requisite weighting of the other covariance matrix. The inaccurate covariance matrix leads to the very same input sensitivity problems that the RS model attempts to avoid. It is worth clarifying that the sensitivity is due to an incorrectly estimated \(\hat{\pi}_{T, t + 1}\), not the absence of underlying regimes. To correct this sensitivity concern, a discretised simplification is used. If the state is estimated to be quiet, nothing is done to the weights implied by the volume of data: \[\begin{align*} (\hat{\pi}_{Q, t + 1}, \hat{\pi}_{T, t + 1}) = (\zeta, 1 - \zeta). \end{align*}\] If the state is estimated to be turbulent, then the weights implied by the volume of data are adjusted to overweight the turbulent regime by a proportion of \(\gamma\):
\[\begin{align*} (\hat{\pi}_{Q, t + 1}, \hat{\pi}_{T, t + 1}) = (\zeta - \frac{\gamma}{2}, 1 - \zeta + \frac{\gamma}{2}). \end{align*}\] In the experiment, we set \(\gamma\) to 0.1. The effect is that in the quiet regime, the covariance matrix is the same as without the RS model. While during the turbulent regime, the turbulent covariance matrix is given a weighting of \(\gamma\) more than what is implied by the data split.
The QFM technique is examined next. Chen, Dolado, and Gonzalo (2019) use the QFM technique from equation (??) for prediction. They select risk factors and estimate factor loadings using a simultaneous procedure. Factor-based modelling is not the focus of this research, although it can be used in conjunction with the general framework (??). In this experiment, mainly for pedagogical purposes, simple factors are used for the QFM portfolios; namely, a market factor and a squared market factor. This choice is consistent with the findings of Treynor and Mazuy (1966), with more detail given in appendix insert appendix reference here. Additionally, the QFM user has to choose how to partition the interval \([0, 1]\), i.e. decide which quantiles to use. The partition is an important input which could be estimated. We use the same split as Ma and Pohlman (2008)}, where: \[[\tau_0 = 0, \tau_1 = 0.1]\cup(\tau_1 = 0.1, \tau_2 = 0.9] \cup (\tau_2 = 0.9, \tau_3 = 1].\] In the notation, the intervals are referred to by their right endpoint. Flint and Du Plooy (2018) note that the QFM, as stated, does not provide variation between quantiles, as the idiosyncratic error term will always adjust so that the total estimated covariance matrix is always equal to the sample covariance matrix. Therefore, to estimate the quantile specific covariance matrices, they propose fixing the error term to the median error such that \(\hat{\epsilon}(\tau) = \hat{\epsilon}(0.5)\). Then each set of quantile returns can be used to construct a sample covariance matrix denoted \(\hat{\Sigma}^{(\tau)}\). Ma and Pohlman (2008) propose a strategy for portfolio forecasting called quantile regression portfolio distribution (QRPD). It involves interval-weighting the portfolio allocations \(\hat{w}^{(\tau)}\) for each quantile covariance matrix. The QRPD portfolio is: \[\begin{align} \hat{w}_{\text{QRPD}} & = \sum_{i = 1}^{l} p_i\hat{w}^{(\tau_i)},\;\; \end{align}\] where \(p_i = \tau_i - \tau_{i - 1}\), and \(l\) is the number of intervals. Like the SRS method, the QRPD method is comparable to ensemble methods because different models are being aggregated in the hope of reducing total estimation risk.
The QRS technique from table 6.1 is laid out by Flint and Du Plooy (2018). They first separate the data history into regimes. Within each regime, they apply the QFM approach. Once they find \(\hat{\Sigma}_{s_t}^{(\tau_i)}\) for every quantile and state, they blend the covariance matrices between regimes. For every possible pair of quantiles \(\tau_i, \tau_j \in \{\tau_1, ..., \tau_{|\tau|}\}\), a blended SCM can be constructed: \[\begin{align} \hat{\Sigma}^{(\tau_i, \tau_j)}_{\text{blend}} & = (\hat{\pi}_{Q, t + 1}\hat{\eta}_Q\hat{\Sigma}^{(\tau_i)}_Q) + (\hat{\pi}_{T, t + 1}\hat{\eta}_T\hat{\Sigma}^{(\tau_j)}_T). \end{align}\] Each of the SCMs can be used to find a portfolio \(\hat{w}^{(\tau_i, \tau_j)}\). Once the \(l \times l\) portfolios have been found they can also be blended with the QRPD method: \[\begin{align} \hat{w}_{\text{QRPD}} & = \sum_{i = 1}^{|\tau|} \sum_{j = 1}^{|\tau|} p_i p_j \hat{w}^{(\tau_i, \tau_j)}. \end{align}\]
6.2 Ridge regression
As stated previously, the QFM and RS techniques do not require separate workarounds for the ERC and GMV portfolios. This is not the case for the RR method. First, we consider the GMV portfolio estimation and then the ERC portfolio estimation. The Lagrangian multiplier, \(\lambda\), for the penalty in framework (??), should be chosen to minimise the estimation risk. However, the estimation risk is unobservable. Kinn (2018) suggests the k-folds cross-validation estimation procedure to select this parameter4. The broad idea of this procedure is to approximate the estimation risk and choose a value of lambda, \(\lambda^*\), that minimises the approximated estimation risk. To perform k-folds cross-validation, we initially arrange the observations into \(K\) subsets without replacement. The subsets are taken over time and not assets as with SRS technique. The set of observations for the \(k^{\text{th}}\) subset is denoted \(\mathcal{I}_k\). All of the observations in the \(K - 1\) remaining subsets (i.e. excluding those in the \(k^{\text{th}}\) subset) are stored in a set denoted \(\mathcal{I}_{-k}\). This yields the optimisation: \[\begin{align} \lambda^* & = \underset{\lambda}{\text{argmin}} \Big \{ \frac{1}{K} \sum_{k =1}^K \hat{h}_k(\lambda, \mathcal{I}_k, \mathcal{I}_{-k})\Big \}, \tag{6.2} \end{align}\] where \(\hat{h}_k(\lambda)\) is a function that approximates estimation risk given a value of \(\lambda\). One choice for \(h\) is the mean squared error penalty for a portfolio found using the data from \(\mathcal{I}_{-k}\) and then tested on \(\mathcal{I}_k\). This procedure is described below:
For each subset \(k = \{1, 2, ..., K\}\), find the optimal portfolio weights using the training data \(\mathcal{I}_{-k}\) and apply the penalty scaled by \(\lambda\).
Evaluate the performance of this portfolio on the unused data \(\mathcal{I}_k\) with the mean squared error loss function and retain the score \(\hat{h}_k\): \[\begin{align} \hat{h}_k (\lambda, \mathcal{I}_k, \mathcal{I}_{-k}) = \frac{1}{|\mathcal{I}_k|} \sum_{i \in \mathcal{I}_k} (\bar{r}_{gmv} - \textbf{X}_i^\intercal \hat{w}_{\mathcal{I}_{-k}}(\lambda))^2, \tag{6.3} \end{align}\] where \(\bar{r}_{gmv}\) is determined as in chapter 5, and the function \(|\cdot|\) counts the number of observations in a set.
The range of possible \(\lambda\) values can be discretised to find a solution that minimises the objective function in (6.2)5.
For the ERC portfolio implementation of the RR method, there is an additional hyperparameter, \(\eta_{erc}\), that needs to be estimated. This hyperparameter should be found before the penalty hyperparameter \(\lambda^*\). We also use the k-folds cross-validation technique to find this parameter. However, the mean squared error loss to the desired global minimum variance portfolio from equation (6.3) is not appropriate for optimising an ERC portfolio. Ideally, \(\eta_{erc}^*\) should minimise the distance between all of the total risk contributions so that they are as equal as possible. Therefore, we make use of the following hyperparameter selection function (see appendix insert reference to appendix here for motivation): \[\begin{align} \hat{h}_k^{erc} (\eta, \mathcal{I}_k, \mathcal{I}_{-k}) & = \sum_{i = 1}^N \sum_{j \geq i}^N(\hat{w}_{i, \mathcal{I}_{-k}}(\textbf{S}_{\mathcal{I}_k} \hat{w}_{\mathcal{I}_{-k}})_i - \hat{w}_{j, \mathcal{I}_{-k}}(\textbf{S}_{\mathcal{I}_k} \hat{w}_{\mathcal{I}_{-k}})_j )^2. \end{align}\]
References
Bodnar, Taras, Yarema Okhrin, Valdemar Vitlinskyy, and Taras Zabolotskyy. 2018. “Determination and Estimation of Risk Aversion Coefficients.” Computational Management Science 15 (2): 297–317.
Chen, Liang, J. Juan Dolado, and Jesus Gonzalo. 2019. “Quantile Factor Models.”
Flint, Emlyn James, and Simon Du Plooy. 2018. “Extending Risk Budgeting for Market Regimes and Quantile Factor Models.” Available at SSRN 3141739.
Kinn, Daniel. 2018. “Reducing Estimation Risk in Mean-Variance Portfolios with Machine Learning.” arXiv Preprint arXiv:1804.01764.
Kritzman, Mark, Sebastien Page, and David Turkington. 2012. “Regime Shifts: Implications for Dynamic Strategies (Corrected).” Financial Analysts Journal 68 (3): 22–39.
Ma, Lingjie, and Larry Pohlman. 2008. “Return Forecasts and Optimal Portfolio Construction: A Quantile Regression Approach.” The European Journal of Finance 14 (5): 409–25.
Treynor, Jack, and Kay Mazuy. 1966. “Can Mutual Funds Outguess the Market.” Harvard Business Review 44 (4): 131–36.