3 A framework for constructing risk-based portfolios
3.1 An overview of modern portfolio theory
Every investor has a universe of \(N\) assets to which they can allocate capital. The proportion of their allocation to the i\(^{\text{th}}\) asset, \(w_i\), depends on the investor’s risk and return preferences. They could prefer riskier asset combinations because they require high capital growth, or they could prefer more stable asset combinations that prioritise capital preservation. The column vector \(w = [w_1,w_2,\text{…},w_N]^\intercal\) is an \(N \times 1\) vector of allocations that define an investor’s portfolio. This portfolio is constrained by the investor’s capital budget, which may be articulated through the notion of weights. Therefore, all considered portfolios should adhere to the budget constraint: \(\sum_i w_i = 1\).
In his seminal paper, Markowitz (1952) attempts to quantify the asset allocation process. Herein he posits that the investor has to make a risk-reward trade-off when considering their portfolio returns, \(R_\mathrm{p}\), over a pre-determined time horizon \([0, T]\). In his framework, he measures risk with the variance of portfolio returns \(\mathbb{V}\mathrm{ar}[R_\mathrm{p}] = w^\intercal \Sigma w\), where \(\Sigma\) is the \(N \times N\) asset return covariance matrix. Markowitz measures reward by the expectation of portfolio returns \(\mathbb{E}[R_\mathrm{p}] = w^\intercal \mu\), where \(\mu\) is the \(N \times 1\) vector of expected asset returns. If the investor fixes expected returns to a constant, \(\mathbb{E}[R_\mathrm{p}] = c\), they encounter the problem of minimising portfolio risk, \(\mathbb{V}\mathrm{ar}[R_\mathrm{p}]\). The mean-variance optimal (MVO) portfolio \(w_\mathrm{mvo}\) achieves this goal while adhering to the budget constraint and the fixed expected return constraint. Written mathematically, \(w_\mathrm{mvo}\) is the solution to the following Markowitz optimisation: \[\begin{align} w_\mathrm{mvo} &= \underset{w}{\text{argmin}} \Big \{ w^\intercal \Sigma w \Big \}, \tag{3.1} \end{align}\] subject to the constraints: \[\begin{align*} \mathcal{C}(w) &= \begin{cases} w^\intercal \mu &= c \;\;\; \\ w^\intercal \underline{1} &= 1 \;\; . \end{cases} \end{align*}\]
There is a complication when using this framework in practice because \(\Sigma\) and \(\mu\) are unknown. The investor can only observe a small set of sample returns from the underlying stock processes that use these quantities as inputs. The sample returns, represented by a \(N \times T\) matrix \(\textbf{X}\), can be used to estimate \(\Sigma\) and \(\mu\). The sample estimates are:
\[\begin{align} \textbf{S}&= \frac{1}{T - 1}(\textbf{X} - \mathbf{\bar{x}} \underline{1}^\intercal_T)(\textbf{X} - \mathbf{\bar{x}} \underline{1}^\intercal_T)^\intercal, \tag{3.2} \\ \mathbf{\hat{\mu}} &= \mathbf{\bar{x}} , \tag{3.3} \end{align}\] where \(\mathbf{\bar{x}}\) is a \(N \times 1\) vector of mean returns. A total of \(\frac{N(N+1)}{2}\) distinct parameters are being estimated for the sample covariance matrix (SCM) while \(N\) distinct sample expected returns are estimated. \(\textbf{S}\) and \(\hat{\mu}\) can be substituted into equation (3.1) to infer an MVO portfolio, \(\hat{w}_\mathrm{mvo}\). For each level of portfolio return \(c = \hat{w}^\intercal \hat{\mu}\), an estimated portfolio volatility level \(\mathbb{SD}[\hat{R}_\mathrm{p}] = \sqrt{\hat{w}^\intercal\textbf{S}\hat{w}}\) is realised. These estimated expected return-volatility pairs induce a frontier in the expected return-volatility plane, which may or may not be close to the frontier implied by the actual inputs \(\Sigma\) and \(\mu\). The actual frontier is optimal for the Markowitz framework, and he terms it the `efficient frontier’.
The above framework and estimation procedure yields a solution to the asset allocation dilemma, but there is still a sensitivity predicament. Michaud (1989) shows that the MVO procedure, as described above, overweights assets with substantial estimated returns \(\mathbf{\bar{x}}\). However, these are the same assets that are likely to have been misestimated. Thus, any potential estimation errors are `maximised’ - an undesirable property that makes \(\hat{w}_\mathrm{mvo}\) a potential liability for the investor to hold.
The MVO procedure is commonly adjusted to reduce sensitivity to the input \(\mathbf{\bar{x}}\) in one of two ways. The first is to estimate expected returns in a manner that targets return drivers or factors, which has motivated the rise of factor-based investing (Ang 2014). The second adjustment is to remove the dependency on expected return estimates altogether and only construct portfolios based on their risk properties. The latter has inspired the field of risk-based investing and is the focus of this dissertation.
3.2 Introducing risk-based investing
A general risk-based portfolio optimisation programme is given below in equation (3.4). Removing the expected return constraint and altering the previous MVO optimisation problem for the consideration of a generalised objective risk function \(f(\cdot| \textbf{X})\) yields: \[\begin{align} w^* = \underset{w}{\text{argmin}} \Big \{ f(w, \Sigma| \textbf{X}) \Big \}, \tag{3.4} \end{align}\] subject to the constraint: \[\begin{align*} w^\intercal \underline{1} &= 1 \;\; , \end{align*}\]
where \(f(\cdot | \textbf{X})\) is a risk metric to be minimised. The choice of \(f(\cdot | \textbf{X})\) determines which risk-based portfolio is the optimal solution. Chapter make chapeter expounds on the risk-based portfolio types relevant to this research.
The range of feasible portfolios given by equation (3.4) is practically too general because unlikely single asset weights are still possible. To this end, an investor should apply a weight constraint to limit feasible allocations, ensuring comparability with practical investing. Jagannathan and Ma (2003) show that risk-based long-short portfolios can have extreme weights in practice, which are unlikely to be accepted by an investor.
Additionally, Jagannathan and Ma (2003) also conclude that imposing the long-only investment constraint on US equities leads to improved efficiency for optimal portfolios constructed with the first two sample moments \(\hat{\mu}\) and \(\textbf{S}\). Hence, applying the long-only constraint is both statistically appropriate and practically relevant for most investors.
Equally important is that an investor will be reluctant to concentrate their portfolio in a small number of assets. Limiting the maximum single asset allocation to a selected weight \(\alpha\) avoids such concentration. These constraints are concurrently expressed as \(\{w :0 \leq w_i \leq \alpha, \; \forall i \}\) and can be added to optimisation (3.4). In its current form, the developed framework is still somewhat abstract, so it is not obvious how to improve it. Even so, one can always define specific properties that the framework ought to have for there to be a good chance of it operating as intended.
3.3 Improving risk-based portfolios
Hadamard (1923) defines a mathematical problem as ‘well-posed’ if:
- the solution exists,
- the solution is unique,
- the solution is not overly sensitive to small perturbations in inputs.
Well-posed problems are easier to work with and are more stable than ill-posed problems - ones that fall short of the definition. The expected return constraint was previously disregarded for MVO portfolios because the MVO framework often does not meet requirement 3. when the sample mean returns estimate \(\mu\). Risk-based portfolios are therefore more `well-posed’ than MVO portfolios.
However, there are two common ways in which risk-based portfolios are also ill-posed. The first is if there are very few sample observations; namely, if \(T < N\). In this scenario, the covariance matrix is not of full rank and is therefore not invertible, causing non-unique solutions to \(w^*\). The framework is then ill-posed by 2. The second is if small changes in \(\textbf{X}\) result in large deviations of \(w^*\). The framework is then ill-posed by 3. Many researchers such as Jobson and Korkie (1981) and Best and Grauer (1991) have shown that the latter phenomenon is observed in practice, and often persists even if \(T\) is much larger than \(N\). To address point (iii) and make the problem well-posed, one needs a measure of sensitivity. We outline a pseudo-derivation of a sensitivity measure below.
Kinn (2018) views the portfolio optimisation problem from a modelling perspective. The returns on the portfolio are modelled directly by an unknown function \(g(\cdot)\): \[\begin{align} R_p & = g(\textbf{X}) + \epsilon, \tag{3.5} \end{align}\] where \(\epsilon\) has the normal distribution with zero mean and variance \(\phi^2\). Estimating \(g(\cdot)\) is the aim of using the framework, and it is a real-valued function that is not necessarily differentiable or continuous. Each algorithm \(q\) refers to a combination of estimation procedure for the inputs, and a computation of a risk-based portfolio using equation (3.5). Each \(q\) specifies an estimate of the function \(g\), denoted \(\hat{g}_q\). In the same way that \(\hat{w}^*\) (\(w^*\) calculated with sample inputs \(\textbf{S}\) and \(\hat{\mu}\)) can be used to estimate the out-of-sample risk-based portfolio, \(\hat{g}_q\) can predict out-of-sample portfolio returns, which are forecasted most accurately by the unobservable function \(g\).
It is necessary to distinguish between the two types of data that are available. The first is historical data comprising of the matrix \(\textbf{X}_0\) and in-sample returns \(R_{\mathrm{p}, 0}\), which combine to form the set \(\mathcal{H}_0\). The second is out-of-sample data \(\textbf{X}_1\) and \(R_\mathrm{p, 1}\), which combine to form the set \(\mathcal{H}_1\). Algorithm \(q\) does not utilise the data contained in \(\mathcal{H}_1\), which are chronologically realised after the most recent points in \(\mathcal{H}_0\). A mean squared error penalty is appropriate to measure the accuracy with which \(\hat{g}_q (\textbf{X}_1)\) (estimated using \(\mathcal{H}_0\)) predicts the out of sample returns \(R_\mathrm{p, 1}\). Kinn (2018) terms the expectation of the out-of-sample mean squared error “generalisation error” (GE), shown mathematically as: \[\begin{align} GE(\hat{g}_q) & = \mathbb{E}[(R_\mathrm{p, 1} - \hat{g}_q (\textbf{X}_1))^2|{\mathcal{H}_1}], \end{align}\] where GE is specific to a sample. The actual quantity of interest is the expected performance of \(q\) for many potential sample sets, as we are evaluating \(q\)’s efficacy holistically. This quantity is called the expected generalisation error across all samples, denoted \(G_q\). Furthermore, \(G_q\) may be decomposed3 to reflect a common modelling trade-off between bias and variance: \[\begin{align} G_q &:= \mathbb{E} \Big [ \mathbb{E}[(R_\mathrm{p, 1} - \hat{g}_q (\textbf{X}_1))^2|{\mathcal{H}_1}] \Big | \mathcal{H}_0\Big ] , \\ & = \underbrace{ \Big (g(\textbf{X}_1) - \mathbb{E}[\hat{g}_q (\textbf{X}_1)|\mathcal{H}_0] \Big )^2}_{\text{squared bias}} + \underbrace{\mathbb{V}\mathrm{ar} [\hat{g}_q(\textbf{X}_1)|\mathcal{H}_0]}_{\text{variance}} + \underbrace{\phi^2}_{\text{irreducible error}}.\tag{3.6} \end{align}\] The squared bias is the extent to which the expectation of predicted returns differs from the best possible predictor of returns, the correct function \(g(\cdot)\). The variance measures the magnitude by which the predicted returns will vary under repeated sampling. By setting \(\hat{g}_q(\cdot) = g(\cdot)\) in equation (3.6) only statistical noise remains; hence, the noise is irreducible. The risk of misestimating \(g(\cdot)\) should not include the risk that is retained by even the best estimator. Therefore, estimation risk is considered as the sum of the first two terms only, the squared bias and the variance. An over-fitted algorithm will have high variation for repeated samples. An under-fitted algorithm will have high bias and be consistently poor for repeated samples. The over- and under-fitting trade-off is an example of how the bias-variance trade-off works in practice.
Until now, we have assumed that the estimated portfolio \(\hat{w}^*\) from equation (3.4) is an unbiased estimate of the actual risk metric-minimising portfolio \(w^*\) because the choice of \(f(\cdot | \textbf{X})\) determines precisely the type of risk-based portfolio. However, \(f(\cdot|\textbf{X})\) does not precisely determine the estimation risk. Employing a penalty on the objective function introduces bias to reduce estimation risk, i.e. hopefully, the squared bias increase does not outweigh the variance decrease. The introduction of the penalty yields an estimated portfolio \(\hat{w}^*\) that is consistently closer to \(w^*\) than an unbiased portfolio would be. The penalty constraint can be stated as \(P(w) \leq s\) and reduces the set of all possible portfolios. If done correctly allocations that are misestimated by the heftiest margins are excluded by this constraint, and allocations that are consistently closer to the actual portfolio remain. The general risk-based framework can now be restated as below to accommodate the penalty using a Lagrangian multiplier approach: \[\begin{align} w^* = \underset{w}{\text{argmin}} \Big \{ f(w, \Sigma| \textbf{X}) + \lambda P(w) \Big \}, \tag{3.7} \end{align}\] subject to the constraints: \[\begin{align*} \mathcal{C}(w) &= \begin{cases} w^\intercal \underline{1} = 1 \;\; \\ 0 \leq w_i \leq \alpha, \; \forall i \;\;\; , \end{cases} \end{align*}\]
where \(P(\cdot)\) is the penalty function, and \(\lambda\) is the Lagrangian multiplier. Kinn proposes estimating \(\lambda\) in a way that is consistent with the rest of the optimisation. Therefore, we apply the \(\lambda\) estimation that minimises the portfolio specific risk metric using in-sample data. Additionally, by setting \(\lambda = 0\), the unpenalised portfolio can still be recovered. Optimisation (3.7) will be the general risk-based portfolio optimisation going forward.
Improving on the vanilla risk-based optimisation is done in three ways in this research, and these improvements are also common in the literature. Approach one deals with the assumption in equation (3.5) that \(\epsilon\) has a normal distribution, and that the errors through time are independent and identically distributed. If \(\hat{g}_q\) fails to approximate \(g\) correctly, then the assumption is violated. There could be natural heterogeneity in the data accounted for by \(g\). Flint and Du Plooy (2018) attempt to deal with heterogeneity by changing the estimation procedure of \({g}\) so that it includes and accounts for potential differences in each observation of sample data. They do this through the application of regimes and quantiles, grouping the input data to increase the accuracy of the estimated model. The second approach adapted from Kinn (2018) has already been shown above and involves penalising the optimisation and setting \(\lambda > 0\) in equation (3.7). The third and final approach requires resampling from the observations in \(\textbf{X}\) for different subsets of assets and hinges on implementation adjustments. Shen and Wang (2017) suggest finding \(w^*\) multiple times for different resampled subsets and blending the result afterwards to find an aggregate weight, that hopefully reduces estimation risk. Chapter insert chapter ref contains a more detailed exploration of these techniques. Before that, further investigation of desirable risk properties and the portfolios that have them is required to specify \(f(\cdot|\textbf{X})\).
References
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