Learning Risk Preferences from Investment Portfolios Using Inverse Optimization
aa r X i v : . [ q -f i n . P M ] O c t Learning Time Varying Risk Preferences fromInvestment Portfolios using Inverse Optimizationwith Applications on Mutual Funds
Shi Yu
The Vanguard Group, Malvern, PA, USA, [email protected]
Yuxin Chen
The Vanguard Group, Malvern, PA, USA
Chaosheng Dong*
Amazon, Seattle, WA, USA
The fundamental principle in Modern Portfolio Theory (MPT) is based on the quantification of the portfolio’srisk related to performance. Although MPT has made huge impacts on the investment world and promptedthe success and prevalence of passive investing, it still has shortcomings in real-world applications. One ofthe main challenges is that the level of risk an investor can endure, known as risk-preference , is a subjectivechoice that is tightly related to psychology and behavioral science in decision making. This paper presentsa novel approach of measuring risk preference from existing portfolios using inverse optimization on themean-variance portfolio allocation framework. Our approach allows the learner to continuously estimatereal-time risk preferences using concurrent observed portfolios and market price data. We demonstrateour methods on real market data that consists of 20 years of asset pricing and 10 years of mutual fundportfolio holdings. Moreover, the quantified risk preference parameters are validated with two well-knownrisk measurements currently applied in the field. The proposed methods could lead to practical and fruitfulinnovations in automated/personalized portfolio management, such as Robo-advising, to augment financialadvisors’ decision intelligence in a long-term investment horizon.
1. Introduction
Risk preference (risk tolerance or risk aversion) is a fundamental concept modeling individualpreference for certainty under uncertainty. In portfolio allocation, one primary goal has been to rec-oncile empirical information about securities prices with theoretical models of asset pricing underconditions of inter-temporal uncertainty (Cohn 1975). The notion of risk preference has been anessential assumption underlying almost all such models (Cohn 1975). However, measurement ofrisk preference has been treated in separate paths. In finance domain, quantification of risk can beroughly summarized as ratio comparisons between wealth and risky assets under market volatility. * Work done prior to joining Amazon hi, Yuxin, and Chaosheng: Learning Time Varying Risk Preferences from Investment Using Inverse Optimization However, the underlying biological, behavioral, and social factors behind risk appetite are com-monly studied in many other disciplines, i.e., social science (Payne et al. 2017, Guiso and Paiella2008), behavior science (Brennan and Lo 2011), mathematics (von Neumann and Morgenstern1947), psychology (Sokol-Hessner et al. 2009, Mcgraw et al. 2010), and genetics (Linn´er 2019). Formore than half of a century, many measures of risk preference have been developed in variousfields, including curvature measures of utility functions (Arrow 1971, Pratt 1964), human subjectexperiments and surveys (Rabin and Thaler 2001, Holt and Laury 2002), portfolio choice for finan-cial investors (Guiso and Paiella 2008), labor-supply behavior (Chetty 2006), deductible choicesin insurance contracts (Cohen and Einav 2007, Szpiro 1986), contestant behavior on game shows(Post et al. 2008), option prices (A¨ıt-Sahalia and Lo 2000) and auction behavior (Lu and Perrigne2008). Nowadays, investor-consumers’ risk preferences are mainly investigated through one or com-bination of three ways. The first one is assessing actual behavior. Two examples are inferring house-holds’ risk attitudes using regression analysis on historical financial data (Schooley and Worden1996) and inferring investors’ risk preferences from their trading decision using reinforcementlearning (Alsabah et al. 2020, Wang et al. 2020). The second one is assessing responses to hypo-thetical scenarios about investment choices (see Barsky et al. (1997) and (Hey 1999)). In practice,online questionnaires are widely adopted by Robo-advising firms to evaluate investors’ risk profiles(Alsabah et al. 2020). The third one is subjective questions (see Hanna et al. (1998) for a surveyof these different techniques).Despite its profound importance in economics, there remain some limitations with respect tolearning the risk preference using classic approaches. First, many existing methods assume inher-ent small deviations of the risk preference, which is “ valid only for potential losses that are rel-atively unthreatening to the individuals’ wealth ” (Thomas 2016). Second, most practices in placeare often insufficient to deal with scenarios where investment decisions are managed by machinelearning processes (Robo-advising), and risk preferences are expected as input parameters thatcan generate new decisions directly (auto-rebalancing). Currently, Robo-advisors first communi-cate and categorize clients’ risk preferences based on human interpretation, and later map themto the nearest values of a finite set of representative risk preference levels (Capponi et al. 2019).Those limitations make existing theories and approaches challenging to deal with prominent sit-uations in which risk preference changes dramatically in an inter-temporal dimension, such assavings, investment, consumption problems, dynamic labor supply decisions, and health decisions(O’Donoghue and Somerville 2018). Real-world risk preference is clearly not as straightforward asmany theories have assumed, and perhaps individuals even do not exhibit risk appetite consistentlyin their behaviors across domains (O’Donoghue and Somerville 2018). Namely, risk preference is hi, Yuxin, and Chaosheng:
Learning Time Varying Risk Preferences from Investment Using Inverse Optimization time varying in reality. Nowadays, due to the technological advances, we already have overwhelm-ing behavioral data across all domains, providing a myriad of additional sources to help us decipherthe perplexity of risk preference from different angles. Although these new approaches might notultimately prove to be the best models for studying risk preference, they can be used in conjunctionwith traditional methods with additional, more nuanced implications that are borne out by data(O’Donoghue and Somerville 2018).To tackle aforementioned limitations, we present a novel inverse optimization approach to mea-sure risk preference directly from market signals and portfolios. In a volatile market, the sameportfolio created by buy and hold investors could reflect very different risk levels when market con-ditions change, regardless most of the time, investor’s subjective risk preference remains unchanged.Our approach’s primary motivation is to complement traditional approaches through continuousmonitoring and evaluation of embedded portfolio risks and to facilitate the automated decisionprocess of portfolio adjustment when necessary to ensure that real portfolio risk is aligned withinvestor’s true risk preference. Our method is developed based on two fundamental methodologies:convex optimization based Modern Portfolio Theory (MPT) and learning decision-making schemethrough inverse optimization. We assume investors are rational and their portfolio decisions arenear-optimal. Consequently, their decisions are affected by the risk preference factors through port-folio allocation model. We propose an inverse optimization framework to infer the risk preferencefactor that must have been in place for those decisions to have been made. Furthermore, we assumerisk preference stays constant at the point of decision, but varies across multiple decisions overtime, and can be inferred from joint observations of time-series market price and asset allocations.Our inverse optimization approach represents an integral component of a general view of usingmachine learning to learn individual investor’s risk preference, which embodies data and modelsfrom three aspects. The first aspect is leveraging demographic features, such as education, financialstatus, gender, age, to learn risk preference. Such type of information is collected once, and itsimpact on risk preference can be learned by applying statistical models on a large amount of clientdata. The second aspect reflects influences on risk preference from lifetime events, such as mortgagepayment, disaster loss, accidental medical conditions and expense. These significant events typi-cally have huge impacts on investment goals or contribution plans, leading to subsequent changesin risk preferences. The third aspect is extracting insights from consumer financial behavior, whichinvolves understanding how investor-consumers make financial decisions, and how these decisionsare reflected in the interactions of financial products. The data that captures these behaviors isprobably the best source to investigate the underlying risk preferences behind decisions. In an auto-mated financial advice system, risk evaluation coming from the first and second aspects are oftensporadic, whereas the estimation from the third aspect is real-time insights based on continuous hi, Yuxin, and Chaosheng: Learning Time Varying Risk Preferences from Investment Using Inverse Optimization monitoring and measurement. Therefore, a successful investment management system shall alwayscompare and validate results from all three components and ensure their consistency throughoutthe entire investment horizon. Unfortunately, the third aspect was considered intractable until therecent advances of big data and machine learning offer new approaches to solve those challengingproblems. Our approach offers a tractable way to monitor and measure investors’ risk preferencesby observing their portfolios, since portfolios are considered as outcomes of an investor’s investmentdecisions.We demonstrate our approach using 20 years of market data and 10 years of mutual fundquarterly holding data. The reasons of using mutual fund portfolios are threefold. First, mutualfund holdings are freely accessible public data and it is easier to discuss and interpret results usingpublic data than clients’ private data. Second, mutual funds are usually constructed by trackingindustry capitalization indices, or managed by fund managers. Thus, they can be considered asoptimal portfolios constructed through rational decision making. Moreover, risk measurements ofmutual funds are common knowledge reported by several well-known metrics and estimations withour methods can be directly validated by these metrics. Third, mutual funds usually are diversifiedon large numbers of assets and they fit the underlying MPT used in our model. Moreover, high-dimensional portfolios are challenging to optimize in the inverse optimization problem and theyreally test the efficiency of our approach. Numerical experiments show that our approach candirectly tackle learning tasks involving hundreds of assets. For tasks involving more than onethousand assets, we propose two different approaches, sector aggregation and factor projection, totransform the problems into lower dimensional space. Our contributions
We summarize the major contributions of our paper as follows: • To the best of our knowledge, we propose the first inverse optimization approach for learningtime varying risk preference parameters of the mean-variance portfolio allocation model, based ona set of observed mutual fund portfolios and underlying asset price data. The flexibility of ourapproach also enables us to move beyond mean-variance and adopt more general risk metrics. • The proposed method provides an effective solution in Robo-advising where risk-return trade-off needs to be dynamically updated based on the risk profile communicated by the client. Riskpreference values learned from our approach can be used directly as input parameters, or as refer-ences of market risk preference, in Robo-advising portfolio construction. • Our inverse optimization approach is able to handle learning task that consists of hundredsof assets in portfolio, and efficiently learn from long sequences of time-series data composed bythousands of observations. For portfolios composed by more than one thousand assets, we proposeSector-based and Factor-base aggregation to improve the computational efficiency. In particular,to our knowledge, it is the first time factor analysis is introduced in inverse optimization approachon portfolio allocation. hi, Yuxin, and Chaosheng:
Learning Time Varying Risk Preferences from Investment Using Inverse Optimization • We collect and process 10 years of mutual fund portfolio holding and 20 years of marketprice data to demonstrate the proposed algorithms. Our data collection and engineering processis scalable to gather all available mutual fund historical holdings. To the best of our knowledge,historical fund portfolio holding data has been very difficult to find, and we aim to share this datato the public to facilitate related researches.
2. Background
The fundamental mean-variance portfolio optimization model developed by Markowitz (1952) andits variants assume that investors estimate the risk of the portfolio according to the variabilityof the expected return. Moreover, Markowitz (1952) assumes that investors make decisions solelybased on the preferences of two objectives: the expected return and the risk. The trade-off betweenthe two objectives is typically denoted by a positive coefficient and referred to as risk tolerance (or risk aversion ). Later, Black and Litterman (1992) extends the framework in Markowitz (1952)by blending investors’ private expectations, known as Black-Litterman (BL) model. A Bayesianstatistical interpretation of BL model is proposed in He and Litterman (2002) and an inverseoptimization perspective is derived in Bertsimas et al. (2012). Most of these mean-variance basedapproaches assume an investor’s risk preference is known. For example, in Bertsimas et al. (2012),the risk-award trade-off δ is denoted by the ratio between expected profit and variance. As theymention: Even though there are various proposals in the literature, there is no consensus on howto fit δ . (Bertsimas et al. 2012) Later, in their experiments, δ is exogenously set to 1.25 based onthe suggestion of He and Litterman (2002). Suggested empirical risk aversion values also vary fromexpert to expert. Ang (2014) suggests a range from 1 to 10 for retail investors and believes it israre to have risk aversion greater than 10 (Ang 2014). Fabozzi et al. (2007), however, states thatrisk aversion value should be somewhere between 2 and 4.In expected utility theory, risk aversion relies on the choice of (usually nonlinear) a von Neu-mann–Morgenstern utility function (von Neumann and Morgenstern 1947) u ( c ), where c representsvalue change in wealth. Absolute risk aversion − u ′′ ( c ) /u ′ ( c ) and relative risk aversion − u ′′ ( c ) c/u ′ ( c )(Arrow 1971, Pratt 1964) are used to measure how much utility an investor gains (or loses) as theincrease (or decrease) of wealth, and how risk aversions are compared across different individuals.Unfortunately, selection of such utility functions to fit representative investors is very challengingbecause the exact form and parameters of utility functions are generally unknown, and their selec-tions depend on the objectives and preferences of the investor (Warren 2018). This is essentially asubjective process, and the literature has reached no consensus over which utility function providesthe best description of individual behavior (Starmer 2000).In CAPM (Captial Asset Pricing Model) (Treynor 1961, Sharpe 1964, Lintner 1969, Mossin1966), the relationship between risk and expected return is modeled through beta , an indicator hi, Yuxin, and Chaosheng: Learning Time Varying Risk Preferences from Investment Using Inverse Optimization measures relative volatility of the target security/portfolio comparing to market. Market benchmarkportfolio has a beta of 1.0, and larger beta value usually means the security/portfolio can potentiallyoutperform the average market return in a larger margin, and thus allow investors to gauge whetherthe cost (price) is consistent with such a likely return. References
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