Abstract
This chapter initially underlines the distinguishing features of the so called risk-based strategies for asset allocation in comparison with the Mean-Variance Analysis and proposes two criteria which highlight possible discrepancies among risk-based strategies. This is followed by a detailed illustration and interpretation of the theoretical concepts and tools of risk budgeting literature used to set up or analyse risk-based portfolios. We take into consideration the marginal risk, the total risk contribution and the percentage total risk contribution. The rest of the chapter offers an in depth discussion of the risk parity strategy, starting from the rudimental version, also known as naïve risk parity or inverse volatility strategy, to the optimal one which can really build a portfolio such that risk contributions from the different asset classes to the portfolio overall risk are equalized. We explore the use of leverage combined with a risk parity strategy. The chapter concludes giving attention to the potential evolutions of the risk parity strategy. In particular, we look at points of attractiveness and shortcomings of a potential new version of risk parity strategy that considers “risk factors” rather than asset classes as the building blocks of a portfolio construction approach.
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Notes
- 1.
The assumption of cash funding in calculating the marginal risk seems strong. However, we have to admit it is more acceptable in the context of finding a policy portfolio that is a strategic asset allocation solution than it would be in the context of tactical asset allocation or active portfolio management. On the issue, see Scherer (2015).
- 2.
The concepts and measurements of marginal risks and total risk contributions can also be referred to different measures for portfolio risk, for example Value at Risk (VaR) and Expected Shortfall (ES). Assuming a Gaussian context and a parametric estimation approach, such risk measures are strongly related to volatility and so the extension is simple. For clarity on this point, we note that marginal risks for position i in case VaR or ES measures are used with a confidence level α (for example 95 % or 99 %) can be written under the zero-mean hypothesis as follows, respectively:
$$ MR_{VaR\_i} = N_{1 - \alpha }^{ - 1} \cdot \frac{{\sum\nolimits_{j = 1}^{N} {w_{j} \sigma_{ij} } }}{{\sigma_{P} }} $$$$ MR_{ES\_i} = \frac{1}{1 - \alpha } \cdot \frac{{\sum\nolimits_{j = 1}^{N} {w_{j} \sigma_{ij} } }}{{\sigma_{P} }} \cdot \phi \left( {N_{1 - \alpha }^{ - 1} } \right) $$where \( N_{\alpha }^{ - 1} \) is the inverse of the standard normal cumulative distribution function and ϕ indicates the probability density function of a Normal distribution. Moving from marginal risks to risk contributions, as usual, just requires the product with w i . We observe that both marginal risks and risk contributions are different from the ones obtained in the decomposition of portfolio volatility, but what remains unchanged are the percentage total risk contributions. Considering our notation, we assume VaR and ES are expressed as losses keeping their natural sign.
- 3.
We note that in the classic Mean-Variance Optimization the portfolio weights are the main outputs and risk contributions are a secondary output determined by the first.
- 4.
Risk budgets (BD i ) are indicated in relative terms (that is they are expressed in %), not as nominal values.
- 5.
From Sect. 3.2, we remember that a general TRC i corresponds to \( w_{i}\,\cdot\,\frac{\partial RM}{{\partial w_{i} }} \).
- 6.
In spite of the general form of the mathematical system we have adopted, we observe that with reference to the constraint on the risk budgets, Roncalli (2014) considers not very reasonable to set one risk budget to zero, it would be better to reduce the investment universe excluding the corresponding asset in that case. For this reason, he suggests a strict positivity constraint on the BD i .
- 7.
Explicit solutions are admissible just in the bivariate case and in the case of constant correlation. See Sect. 3.3.
- 8.
It is useful to keep in mind that try to model a constrained nonlinear optimization problem as a quadratic programming subproblem means write the second-order Taylor expansion of f(w). Since it is an N-dimensional function, the expansion involves the gradient and the hessian of the function. In particular, the quadratic subproblem at the current approximation w s has the form:
$$ \begin{aligned} & \mathop {Min}\limits_{{{\mathbf{d}}_{w}^{*} }} \nabla f\left( {{\mathbf{w}}_{{\mathbf{s}}} } \right) {\mathbf{d}}_{{\mathbf{w}}} + \frac{1}{2}{\mathbf{d}}_{{{\mathbf{w}}_{{}} }}^{{\mathbf{'}}} \nabla_{{}}^{2} f\left( {{\mathbf{w}}_{{\mathbf{s}}} } \right)^{{}} {\mathbf{d}}_{{\mathbf{w}}} \\ & s.t. \\ & \nabla h_{f} \left( {{\mathbf{w}}_{{\mathbf{s}}} } \right){\mathbf{d}}_{{\mathbf{w}}} + h_{f} \left( {{\mathbf{w}}_{{\mathbf{s}}} } \right) = 0\,\,with\,\,f = 1, \ldots ,F \\ & \nabla y_{g} \left( {{\mathbf{w}}_{{\mathbf{s}}} } \right){\mathbf{d}}_{{\mathbf{w}}} + y_{g} \left( {{\mathbf{w}}_{{\mathbf{s}}} } \right) \ge 0\,\,with\,\,g = 1, \ldots ,G \\ \end{aligned} $$where d w = w – w s .
The solution of the subproblem is used as a search direction to determine the next iterate. The final solution must satisfy the so called Karush-Kuhn–Tucker conditions that, in this case, are referred to a Lagrangian function that takes into account the constraints.
- 9.
See Anderson et al. (2014).
- 10.
Similarly to the standard CAPM, Fig. 3.1 accepts the assumption that investors can lend and borrow money at the same rate, that is the risk-free rate.
- 11.
Evidently, this real experience contrasts with the predictions of Modern Portfolio Theory that would expect the higher risk-adjusted performance from the market and not from a portfolio that is largely exposed to “safer” asset classes. However, some years ago, Frazzini and Pedersen (2013) and Asness et al. (2012) have proposed the leverage aversion argument as a theoretical explanation for this real case. According to this argument, the significant pressure in the market coming from leverage averse investors (by choice or by regulation) implies that they prefer to select a portfolio with risky assets rather than levering low-risk positions in the search for more interesting return targets. As a consequence of their pressure on the demand for riskier assets, the expected return of such assets is reduced while disregarded low-risk assets can trade at lower prices thus offering higher returns with a consequent improvement of their risk-adjusted performance. Those investors who are less leverage averse and apply leverage can benefit from the overweighting of “safer” assets in their portfolio.
- 12.
If borrowing is added exclusively to the low-risk component, the amount of leverage (LEV) that is necessary can be written, according to Ruban and Meles (2011), as follows:
$$ LEV = \frac{{\sigma_{Equity} \cdot w_{Equity} }}{{\sigma_{Bond} \cdot \left( {1 - w_{Equity} } \right)}} $$In our example, where the equity component is 3.10 times more volatile than the fixed income component and has a 40 % original allocation, the fixed income component must accordingly be levered 2.07 times. The resulting portfolio would reach the equalization between risk contributions and would have a higher volatility. It would include a 124.08 % exposure in the fixed income component and a 40 % exposure in equity.
- 13.
The reason is that we are ignoring specific risks.
- 14.
- 15.
The author who established the original concept of risk parity as a portfolio construction process, Qian, declared a preference for macroeconomic risk factors. He has argued that “…a portfolio with true risk parity should have balanced risk exposure to the economic risks of growth and inflation. As a consequence, it should have balanced (but not necessarily equal) risk contribution from three risk sources: equity, interest rates, and inflation”. See Qian (2013).
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Braga, M.D. (2016). Risk-Based Approaches to Asset Allocation: The Case for Risk Parity. In: Risk-Based Approaches to Asset Allocation. SpringerBriefs in Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-24382-5_3
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