Risk Model
At its core, the purpose of a risk model is to identify the forces driving the movements in asset prices. The use of risk models in portfolio management is ubiquitous, from the identification of risks to their management via portfolio optimization.
On the Scientific Portfolio platform, the risk model is used to evaluate risk and return decompositions, as well as diversification metrics. It is also used for long-term returns simulations and portfolio allocation tasks. The model we have developed is designed for funds and indices. It differs in some respects from the models developed in the literature that mainly focus on stocks. The next couple of sections will thus be dedicated to providing transparency on its definition, estimation and use throughout the platform.
Factor models have a long history in finance. Their study has been an ongoing focus for financial economists since roughly the publication of the Capital Asset Pricing Model by Sharpe in 1964 [1]. The interest and motivation for the study of factor models arose from early observation that the variations in asset prices over a given period of time shared similarities. The movement of prices from stocks traded on the same market, for instance, were found to exhibit some form of synchrony on any given day. It was also found that across the cross-section of returns, certain groups of stocks shared commonalities in the way their prices changed from one period to the next. These observations hinted at the existence of forces, or factors, driving those variations.
It was the intense interest of academics to understand the origin and nature of these drivers that led to the creation and development of factor models. The development of factor models was pioneered by Sharpe in 1964 [1] and further developed notably by Ross [2] and Fama and French [3] (for an excellent introduction to factor models, see the Nobel lecture by Fama [4] or the guide to factor models edited by the CFA institute [5]).
As we have just mentioned, risk models assume the existence of common forces that drive returns. These forces are often represented by so-called systematic factors, to which financial instruments are exposed. The exposure of an instrument to a factor is often called its factor beta or loading. In their most common form, factor models are linear and written in the form
whereis the return of theinstrument over a period labeled by . The variableis a vector of dimensionthat contains the exposures of the instrument to thesystematic factor. Systematic risk factors are denoted by the variable which is a vector of dimension Note that all variables are associated with individual instruments, while systematic factors are common to all instruments. Finally, the symbol ε corresponds to the residual returns. As their name suggests, residual returns correspond to the part of the returns that are specific to the instrument itself and not due to its exposures to systematic factors. Factor models thus express the relationship between market forces and price movements and allow their decomposition into a systematic and specific part.
Because both factors and exposures are essentially unobservable, researchers have resorted to a wide range of assumptions and techniques to estimate them. Most of these approaches can be categorized broadly into two categories: statistical or fundamental.
Fundamental factor risk model posits that risk factors or loadings are in fact observable and related to fundamental characteristics. Time-series models define risk factors and infer characteristics (see e.g., the Fama and French model [6]), whereas cross-sectional models define characteristics and infer risk factors (see e.g., Rosenberg [7] or Jacob [8]).
Statistical factor risk model makes use of statistical techniques such as principal component analysis (PCA) to simultaneously estimate factors and betas from the instruments' realized returns over a given period of time (see e.g., Chamberlain [9] or Connor [10]) .
Statistical models are used extensively in portfolio optimization due to their enhanced precision and orthogonality. However, they are impractical for understanding risks and rewards due to their black-box nature and complete lack of financial interpretability. On the other hand, fundamental time series models are better suited for interpreting the origins of risks, are intuitive to construct and only require historical prices. However, one of the known issues of time series models is that they are limited to only a few factors due to collinearities that would appear with a large number of time series factors (for instance, when adding industry or country risk factors), impairing the usability of the model for risk analysis. Fundamental cross-sectional models allow for the incorporation of many more factors, notably industry or country factors, and are popular with practitioners who argue that they have greater explanatory and predictive power for individual stock returns [8]. However, they often require large amounts of instrument specific data. Overall, fundamental models are better suited for interpretation but they are less precise than statistical factors. On the other hand, cross-sectional models often suffer from over-parametrization [11], which may cause their associated covariance matrices to be numerically unstable, an issue that makes them unsuitable for portfolio optimization tasks.
Because the guiding philosophy of the Scientific Portfolio platform is to enable users to turn insights into actions, we require a model that is both interpretable in terms of an instrument's fundamentals and amenable for portfolio optimization. Such a model should incorporate the best aspects of the different approaches while avoiding the shortcomings. We believe that the model that we have developed has succeeded in achieving this feat as it incorporates elements of both fundamental and statistical approaches. It is based on the Instrumented Principal Component Analysis (IPCA) model (see e.g., [12] [13] ) and has been built to have the simplicity of time series approach in terms of inputs, the flexibility of a cross-sectional approach and the numerical stability of a statistical approach. The main feature of this model is that it only requires inputs based on a short history of returns, while delivering high precision and granularity for analysis, and numerical stability for portfolio optimization. This allows our model to combine a high level of actionability (based on economically meaningful intuitive factors) and a high level of explanatory power.
To validate our approach, we have compared the precision of our model on a large universe of US equity mutual funds with other commonly used models such as i) to a statistical PCA-based model using the same level of dimensionality, and ii) to a pure time series model using the same seventeen asset pricing and industry factors. One of the most common precision measurement of a risk model is the R-squared, which is defined as:
where the label i refers to a given instrument and the time used is 5 years of weekly returns. As expected, the statistical model obtains a very high average R-squared (i.e., the proportion of risk explained by the model) of 97% across the mutual fund population; however, as we mentioned before, this precision comes at a cost since the statistical model's lack of interpretability undermines its practicality. On the other hand, the pure time series model with seventeen factors, while intuitive, only produces an average R-squared of 88%. Our model produces an average R-squared of 95%, very close to the statistical model in terms of performance, while preserving the actionable and interpretability of the time series model.
In the next section, we provide details on the implementation of the risk model used on our platform.
References
1 William F. Sharpe. (1964). Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. The Journal of Finance. 19(3): 425-442. 10.1111/j.1540-6261.1964.tb02865.x.
2 Stephen A. Ross. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory. 13(3): 341-360. 10.1016/0022-0531(76)90046-6.
3 Eugene F. Fama and Kenneth R. French. (2015). A five-factor asset pricing model. Journal of Financial Economics. 116(1): 1-22. 10.1016/j.jfineco.2014.10.010.
4 Eugene F Fama. (2013). Eugene F. Fama - Prize Lecture: Two Pillars of Asset Pricing. https://www.nobelprize.org/uploads/2018/06/fama-lecture.pdf.
5 Edwin Burmeister and Stephen A. Ross and N. Kahn. (1994). A Practitioner's Guide to Factor Models. CFA Institute. https://rpc.cfainstitute.org/research/foundation/1994/a-practitioners-guide-to-factor-models
6 Eugene F. Fama and Kenneth R. French. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics. 33(1): 3-56. 10.1016/0304-405X(93)90023-5.
7 Barr Rosenberg. (1974). Extra-Market Components of Covariance in Security Returns. The Journal of Financial and Quantitative Analysis. 9(2). 10.2307/2330104.
8 Bruce I. Jacobs and Kenneth N. Levy. (2021). Factor Modeling: The Benefits of Disentangling Cross-Sectionally for Explaining Stock Returns. The Journal of Portfolio Management. 47(6): 33-50. 10.3905/JPM.2021.1.240.
9 Gary Chamberlain and Michael Rothschild. (1983). Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets. Econometrica. 51(5). 10.2307/1912275.
10 Gregory Connor and Robert A. Korajczyk. (1986). Performance measurement with the arbitrage pricing theory. A new framework for analysis. Journal of Financial Economics. 15(3). 10.1016/0304-405X(86)90027-9.
11 Bryan T. Kelly and Dacheng Xiu. (2021). Factor Models, Machine Learning, and Asset Pricing. SSRN Electronic Journal. 10.2139/ssrn.3943284.
12 Bryan T. Kelly and Seth Pruitt and Yinan Su. (2019). Characteristics are covariances: A unified model of risk and return. Journal of Financial Economics. 134(3): 501-524. 10.1016/j.jfineco.2019.05.001.
13 Xi Bai and Katya Scheinberg and Reha Tutuncu. (2016). Least-squares approach to risk parity in portfolio selection. Quantitative Finance. 16(3): 357-376. 10.1080/14697688.2015.1031815.