Prospect Theory in the Automated Advisory Process

JONATAN WERNER JONAS SJÖBERG

Master of Science Thesis Stockholm, Sweden 2016

Prospektteori i en automatiserad rådgivningsprocess

JONATAN WERNER JONAS SJÖBERG

Examensarbete Stockholm, Sverige 2016

Prospect Theory in the Automated Advisory Process

Jonatan Werner Jonas Sjöberg

Master of Science Thesis INDEK 2016:58 KTH Industrial Engineering and Management

Industrial Management SE-100 44 STOCKHOLM

Prospektteori i en automatiserad rådgivningsprocess

av

Jonatan Werner Jonas Sjöberg

Examensarbete INDEK 2016:58 KTH Industriell teknik och management

Industriell ekonomi och organisation SE-100 44 STOCKHOLM

Master of Science Thesis INDEK 2016:58

Prospect Theory in the Automated Advisory Process

Jonatan Werner

Jonas Sjöberg

Approved

2016-06-01 Examiner

Gustav Martinsson Supervisor

Tomas Sörensson Commissioner

Erik Penser Bank Contact person

Daniel Rosfors

Abstract With robo-advisors and regulation eventually changing the market conditions of the financial advisory industry, traditional advisors will have to adapt to a new world of asset management. Thus, it will be of interest to traditional advisors to further explore the topic of how to automatically evaluate soft aspects such as client preferences and behavior, and transform it into portfolio allocations while retaining stringency and high quality in the process. In this thesis, we show how client preferences and behavioral aspects can be translated into quantitative parameters, suitable for an asset allocation model based on prospect theory. A risk profiler, a type of questionnaire, is found to be an appropriate tool to use in this process. Further, we show that the impact of the parameters on the resulting portfolio allocations is consistent with prospect theory and the preferences of the investor. Finally, we conclude that the optimized portfolio allocation generated by the model suit the investor's preferences. Keywords: prospect theory, portfolio allocation, robo-advising, risk profiling, investor preferences

Examensarbete INDEK 2016:58

Prospektteori i en automatiserad rådgivningsprocess

Jonatan Werner

Jonas Sjöberg

Godkänt

2016-06-01

Examinator

Gustav Martinsson

Handledare

Tomas Sörensson Uppdragsgivare

Erik Penser Bank Kontaktperson

Daniel Rosfors

Sammanfattning Allteftersom robotrådgivning och regleringar förändrar marknadsvillkoren för finansiell rådgivning kommer traditionella aktörer behöva anpassa sig till helt nya förutsättningar. Därmed är det av intresse för traditionella rådgivare att ytterligare undersöka hur man automatiskt kan utvärdera mjuka faktorer, såsom kunders preferenser och beteende, och omvandla dem till portföljallokeringar samtidigt som man bibehåller stringens och hög kvalitet i processen. I denna avhandling visar vi hur kundpreferenser och beteendemässiga aspekter kan översättas till kvantitativa parametrar för en allokeringsmodell baserad på prospektteori. En riskprofilerare, en typ av frågeformulär, visar sig vara ett bra verktyg att använda i processen. Vidare visas att parametrarnas effekt på de resulterande portföljerna är förenliga med prospektteori och investerarens preferenser. Slutligen drar vi slutsatsen att den optimerade allokeringen passar investerarens preferenser. Nyckelord: prospektteori, portföljallokering, robotrådgivning, riskprofilering, investerar-preferenser

Acknowledgements

First and foremost, we would like to thank our supervisor Tomas Sorensson for his valu-able insight and guidance, constructive suggestions and interesting discussions.

Further, we would like to extend our thanks to our friends and fellow students at TheRoyal Institute of Technology whose helpful input, criticism and ideas have been muchappreciated.

Stockholm, June 2016

Jonas Sjoberg & Jonatan Werner

Contents

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 The Current Market for Robo-Advisors . . . . . . . . . . . . . . . 41.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Literature Review 82.1 Decision Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Mean-Variance Analysis . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Expected Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Prospect Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Behavioral Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Assessing Investor Preferences and Behavioral Aspects . . . . . . . . . . . 16

2.3.1 The Risk Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 General Considerations and Common Pitfalls . . . . . . . . . . . . 20

2.4 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Methodology 213.1 Designing the Risk Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Risk Ability: Determining Investment Constraints . . . . . . . . . 223.1.2 Risk Preference: Determining Parameter Values . . . . . . . . . . 233.1.3 Risk Perception: Ensuring Behavioral Coherence . . . . . . . . . . 243.1.4 The Final Risk Profiler . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 The Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Results & Analysis 304.1 The Impact of Risk Profiler Answers on Parameters . . . . . . . . . . . . 304.2 The Impact of Parameter Values on Portfolio Characteristics . . . . . . . 33

4.2.1 Optimal Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.2 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Case study: Out-of-sample Testing on Real Investors . . . . . . . . . . . . 384.4 Reliability & Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Conclusion 415.1 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Appendices 47A Empirical Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47B Empirical Conditional Value at Risk . . . . . . . . . . . . . . . . . . . . . 47C Maximum Drawdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47D Case: Investors’ Risk Profiler Answers . . . . . . . . . . . . . . . . . . . . 48

1

List of Figures

1 List of the Largest Robo-Advisors . . . . . . . . . . . . . . . . . . . . . . 42 Estimated Yearly AUM of Robo-Advisors . . . . . . . . . . . . . . . . . . 53 The E�cient Frontier, Market Portfolio and Capital Market Line . . . . . 94 The Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 The Probability Weighting Function . . . . . . . . . . . . . . . . . . . . . 146 Visualization of Pompian’s Principles I and II . . . . . . . . . . . . . . . . 177 Return and Risk for Considered Asset Classes . . . . . . . . . . . . . . . . 298 The Impact of Profiler Answers on ↵+ and ↵� . . . . . . . . . . . . . . . 319 The Impact of Profiler Answers on � . . . . . . . . . . . . . . . . . . . . . 3210 The Impact of Parameter Values on the Value Function . . . . . . . . . . 3211 The Impact of � on Portfolio Allocation . . . . . . . . . . . . . . . . . . . 3412 The Impact of ↵+ on Portfolio Allocation . . . . . . . . . . . . . . . . . . 3413 The Impact of ↵� on Portfolio Allocation . . . . . . . . . . . . . . . . . . 3514 The Impact of RP on Portfolio Allocation . . . . . . . . . . . . . . . . . . 3615 The Impact of Parameter Values on Portfolio Return and Risk . . . . . . 3716 Case: Allocations and Performance . . . . . . . . . . . . . . . . . . . . . . 39

List of Tables

1 The Axioms of Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 The Final Risk Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Initial Allocation Portfolio Characteristics . . . . . . . . . . . . . . . . . . 274 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Performance Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Case: Investors’ Parameters and Constraints . . . . . . . . . . . . . . . . 387 Case: Investor A Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 Case: Investor B Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Case: Investor C Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010 Case: Investor D Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2

1 Introduction

This chapter starts with a brief background to asset allocation and continues with adescription of the problems financial advisors face when allocating assets for their clients.Further, it describes a market situation which is increasingly favorable for automatedadvisory models. The chapter finishes by stating the research questions and providingthe purpose of this study.

1.1 Background

Investing in financial markets and allocating assets is for many investors a tiresome anddi�cult process. Financial advisory and wealth management has for long been a valuableservice for individuals who do not have the interest or the time to manage their invest-ments. However, recent development in the financial industry towards strict regulationcauses di�culty in running an advisory business. Financial advisors are worried thiswill increase costs of running the service, thus making it available only to very wealthyindividuals. Clients with less wealth risk becoming unadvised on their personal financialissues.

During the global financial crisis 2007, investors lost approximately $8 billion from mak-ing impulse financial decisions, according to Winchester et. al. (2011). Moreover, theauthors claim that investors purchasing advisory services are fifty percent more likelyto adhere with their long time financial plan than investors who do not. Also, investorswith a written financial plan are twice more likely to make optimal long-term investmentdecisions. Hence it is important to note that the initial allocation of a portfolio is onlyone part of the investment process, maintaining the long-term strategy is just as impor-tant but also very di�cult. This fact emphasises the importance of having an initialallocation avoiding the investor being uncomfortable with her investments, leading toirrational decisions.

In recent years, technology has paved the way for a new type of investment advisoryservice. The so-called robo-advisors have entered the financial field and software-based,automated and standardized portfolio management is available to investors in an increas-ing number of countries. Low cost and accessibility are two characteristics that makerobo-advising very interesting (Fein, 2015). Consequently, financial advisors in Sweden,where robo-services are not yet available on a large scale, are interested in how such ser-vices can be incorporated in their businesses. The regulatory aspect further intensifiesthe interest in the topic, since robo-advising is not as strictly regulated and not regardedan advisory service in the same way that traditional advisory is (Financial Conduct Au-thority, 2016).

MiFID II, which regulates the financial advisory business, is coming. In order to main-tain advisory service toward less wealthy clients, robo-advising or automated advisorymight be a viable option. For traditional advisories, a service that accounts for bothinvestor biases and the qualitative views of the advisor is to strive for, so the clients stillfeel they receive quality advice even though the service has been (at least partly) auto-mated. It is our belief that such a service can be achieved through combining behavioral

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finance concepts with mathematical tools for portfolio optimization in a rigorous andmore extensive way than what is currently available in the market.

1.1.1 The Current Market for Robo-Advisors

This section intends to provide the interested reader with a broader understanding ofthe current market situation for robo-advisors and its impact on the future of wealthmanagement. The robo-advisor industry is growing rapidly and some compare it to the1990s travel industry, when the travel agent model lost ground to online services. Thistime, technology has opened up for a business model relying on the combination of thebasic components of a wealth management o↵ering together with simple user interfaceswith relevant content, automated technology, greater transparency and lower pricing.The robo-advisors are here to stay, so traditional players need to determine if, or ratherhow, they want to approach the rise of the robos (Ernst & Young, 2015).

Robo-advisors manage around $19 billion (Vincent, Laknidhi, Klein, & Gera, 2015)compared to the $225 trillion in global investable assets (Novick et al., 2014). In otherwords, the current estimated market share of robo-advisors amounts to less than 0.01%.Regardless of their small percentage of the total AUM (assets under management), thereare indications of significant influence on the wealth management landscape. Severalrobo-advisors already have more than $1 billion under management. A non-exhaustivelist of some of the largest robo-advisors, based on their AUM, are presented in Figure1. The current market leaders are Betterment, Wealthfront and Personal Capital withcollective AUM of $7.1 billions (Karz, 2015). However, the market is still young andmost entrants struggle to be profitable and are still reliant on venture capital funding(Chappuis Halder & Co, 2015).

3 0032 613

1 518686

600245

1578770513121196532

0 500 1000 1500 2000 2500 3000

BettermentWealthfront

PersonalCapitalAssetBuilder

FutureAdvisorRebalance IRA

BlooomAcornsSigFig

Smart401kCovestor

WiseBanyanHedgeable

MarketridersTradeKing Advisors

InvessenceUpside Advisor

Millions

Figure 1: List of some of the largest robo-advisors, based on their AUM (Karz,2015).

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The future market situation for robo-advisors is all but certain. However, based on theinfluence they have already had on the wealth management industry, with traditionalfirms embracing the technology in one way or another, many agree that robo-advising willgrow. Growth will origin partly from the shift of already invested assets to robo-advisoryservices, and partly from non-invested assets, i.e. from new investors. Estimates suggestthat the total AUM of robo-advisors will grow to US$2.2 trillion by 2020 (Epperson,Hedges, Singh, & Gabel, 2015). Estimated yearly AUM of robo-advisors is presented inFigure 2.

0,30,5

0,9

1,5

2,2

0

0,5

1

1,5

2

2,5

2016E 2017E 2018E 2019E 2020E

Trillions

Figure 2: Estimated yearly AUM of robo-advisors (Epperson, Hedges, Singh, &Gabel, 2015).

Robo-advisors provide a way to serve smaller accounts and increase advisor productivity(Ludden, Thompson, & Mohsin, 2015). Scale is crucial since every new client brings ad-ditional revenue at little extra cost (The Economist, 2015). Traditional wealth managerswill eventually have to face the challenges of a new market situation and have mainlythree ways to address this issue – partner with an existing robo-advisor, acquire oneor develop an in-house solution (Vincent, Laknidhi, Klein, & Gera, 2015). Some wealthmanagers are already participating in these activities. Fidelity is partnering with Better-ment (Chappuis Halder & Co, 2015), Blackrock is acquiring FutureAdvisor (Blackrock,2015) and Vanguard launched its own robo-advisor in May 2015 (Chappuis Halder &Co, 2015). However, as more and more players enter the robo-advisory market, someservices, such as asset allocation, may end up a commodity service. In the longer run,technology might drive down the price – eventually to free – which puts the profitabilityof robo-advisory services into question (Chishti & Barberis, 2016). Still, all wealth man-agers may be forced to provide robo-advisory services to remain competitive. Certainly,there are other options but dismissing the challanges of a changing market situationmight prove disastrous.

1.2 Problem Statement

For traditional financial advisors and wealth managers, the main problem with currentrobo-advising is that it is too simplistic and would not meet the high fiduciary standardsthat governs such firms (Fein, 2015). Many existing robo-advisory services provide the

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investor with a basic allocation between asset classes, depending on financial situationand risk preferences. Typically, these factors are measured through surveys where theinvestor answers questions about her income, wealth, investment horizon and risk prefer-ence. These variables are then translated into a variance constraint for a mean-varianceoptimization. Entering robo-advising, traditional advisory firms may consequently stillseek to attain better understanding of investor preferences and insight into investors’decision-making processes. This requires more advanced tools to assess client behaviorand preferences than the existing robo-advisory services provide.

Further, experience from wealth management and financial markets demonstrate theprevalence of irrational decision making in finance. Thus, Pompian (2011) recognizesthat the growing field of behavioral finance is ideally positioned to assist in the everydayprocesses of professionals in the field and we believe this applies just as well to automatedfinancial advice. Investor behavior and psychological traits will play an important rolein the outcome of an investment, avoiding irrational sell-o↵s or other deviations fromthe investment strategy. Thus, it might be of high interest to incorporate the conceptsof behavioral finance into an automated asset allocation model.

There are many client assessment methods readily available, but incorporating one intoan automated asset allocation model puts extra demand on the method. Further, itmust still comply with current legislation and take risk preferences and behavior intoaccount. Also, the output of the assessment needs to be incorporated in an asset allo-cation model, which means it needs to be quantifiable. It also needs to be accurate, sonot to generate allocations diverging from the client’s preferences. Linciano & Soccorso(2012) conclude that in practice, ”most questionnaires are not aligned with the economicand psychological literature”. However, a few di↵erent attempts to address this issuehave been made. For example, Grable and Lytton (1999) develop a multidimensionalrisk-assessment method using 20 questions, which covers a variety of types of risk prefer-ences and combines them to an index score. The problem with scoring risk preferences isthat when optimizing mathematically, the broad spectrum of di↵erent risk preferences isone-dimensionalized in the model, to a single score number. This means that the modelwill generalize the di↵erent types of risk preferences, e.g. loss aversion and risk aversion,which leads to a less customizable model for the individual investor.

Markowitz’s (1952) modern portfolio theory has been widely used by practitioners tryingto optimize asset portfolios, but in recent years the basic assumptions of his theory havebeen widely challenged. The mean-variance model assumes normally distributed returns,an assumption that does not apply to many modern financial assets and instruments.Also, the mean-variance model has a limited ability to incorporate client preferencesapart from the actual mean-variance trade-o↵. Finding an optimization method that ismore flexible to client behavior might be valuable. In Kahneman and Tversky’s prospecttheory (1979), they present a value function that provides asymmetric utilities and pos-sibly di↵erent curvature in the positive and negative domain. This is to better accountfor investors’ preferences in a more dynamic way. However, the authors do not proposeany specific way to customize the value function to a specific investor. We can thus notethat there are several ways of measuring risk preferences, quantifying them and usingthem to find an asset allocation. However, we need to find a way to better integratethe di↵erent aspects in a way that suits both investors’ complex preferences and the

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automated advisory environment.

If these issues are resolved, the advisors still need to contribute with something in theprocess that cannot be provided by competing robo-advisors. This will be vital to staycompetitive during the automation of the financial advisory industry. However, this par-ticular issue lies beyond the scope of this study. We will focus on the issue of establishingan automated asset allocation process that incorporates client behavior and preferencesin order to produce suitable asset allocations.

1.3 Research Questions

The objective of this study is to answer the following questions:

• How can investor preferences and behavioral aspects be translated into quantitativeparameters, suitable for an asset allocation model?

• How do the parameter values a↵ect the allocation and the portfolio characteristics?

• To what extent does the final allocation suit the investor and her preferences?

1.4 Purpose

The purpose of this thesis is to investigate how to design an asset allocation processthat is entirely automated. Further, the purpose is to develop an investor assessmentmethod that accurately translates investors’ behavior and preferences into quantitativemeasures for an allocation model. The main idea is thus to integrate existing modelsand theories within e.g. portfolio theory and behavioral finance in a new way, to bettersuit the complex needs of a firm striving to o↵er automated yet high quality financialadvice to its clients. The model should accurately provide allocations that the clientsare comfortable with regardless of changing market conditions.

An accurate investor assessment method is valuable for several reasons. From a clientperspective, a good method will enable a better understanding of the client’s needs,behavior and preferences, and potentially produce a matching investment portfolio. Abetter understanding of the client will also benefit the advisor as it enables the advisorto build a deeper and more thorough relationship with the clients. This is an importantbusiness aspect for financial advisory firms that want build a sustainable competitiveadvantage.

An automated asset allocation process that incorporates client preferences and behaviorwill be a valuable tool for financial advisors for several reasons. First, it has the advantagethat it improves the consistency of advisory services across the firm and over time. Thiswill reduce the risk that the client perceives advice as arbitrary. Second, new regulationsin the financial industry might not apply to automated services in the same way it willapply to traditional advisory.

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2 Literature Review

In this section, we will present relevant literature for designing an automated assetallocation process. In Section 2.1, we will present some of the most essential developmentsin Decision Theory, leading up to Prospect Theory. In Section 2.2, we will outline anddiscuss several biases and how financial advisors may moderate or adapt to such biasesin the advisory process. Finally, in Section 2.3, we will present the beneficial aspects ofrisk profiling and what has to be considered in the process of designing a reliable riskprofiler.

2.1 Decision Theory

In its broadest definition, decision theory is concerned with the choices of individuals.Here, we will focus on the area within decision theory that considers decisions involvingrisk and uncertainty. Such decisions can be seen as selecting and combining lotteries(Hens & Rieger, 2010). Financial assets can be viewed as di↵erent lotteries with uniquerisk-reward characteristics. This is a simple way of understanding the dynamics of thelogical and psychological factors a↵ecting investors’ portfolio allocations. In this section,we will present three renowned models of decision theory and how they can be appliedto portfolio allocation, in order to gain further insight in their usefulness for automatedasset allocation. In Section 2.1.1, we will introduce Markowitz’s Modern Portfolio Theory(MPT), which was groundbreaking when it was published in 1952 and has been widelyused (and criticized!) by practitioners. In Section 2.1.2, we present Expected UtilityTheory, and in Section 2.1.3 Prospect Theory, which is developed from Expected UtilityTheory but puts more emphasis on psychology and behavioral aspects of the investor.

2.1.1 Mean-Variance Analysis

Modern Portfolio Theory (MPT), or mean-variance analysis, is a foundation for mathe-matical analysis of asset allocation and was invented by Harry Markowitz (1952). Amongthe basic assumptions in this theory are that investors are rational, that returns fromsecurities are normally distributed and that investors avoid (unnecessary) risk if possi-ble. While these assumptions later have been challenged by findings in the behavioralfinance field (see e.g. Kahneman & Tversky, 1979), they still provide a useful foundationfor mathematical analysis and the derivation of quite elegant results that are still usedtoday.

The equation formulated by Markovitz is known as mean-variance optimization and looksas follows:

minimizew

w

T⌃w

subject to w

T

R = µ

X

wi

= 1

Where ⌃ is the covariance matrix of the assets, w is a vector of asset weights and R

is the expected returns of the assets and µ is the target portfolio return. Solving theoptimization problem suggested by Markowitz, for several values of µ, results in a phe-nomenon known as the e�cient frontier. The e�cient frontier is a curve (see Figure 3) of

8

optimal portfolio allocations given a certain level of risk (volatility, which is the standarddeviation of the portfolio return). Consequently, according to MPT, there is a uniqueoptimal way of combining a given set of risky assets at a specific level of volatility (risk).

�

µ

The Efficient Frontier

The Capital Market Line

The Market Portfolio

Assets

Figure 3: Visualization of several assets and their e�cient frontier. The e�cientfrontier is given by the return of the optimal portfolio allocation for any given levelof risk. The e�cient frontier is a concept in Modern Portfolio Theory introducedby Harry Markowitz in 1952. The linear combination of the market portfolio, withexpected return µ and volatility �, and the risk-free asset is called the CapitalMarket Line (CML).

The value of diversification is an essential result of the mean-variance analysis, and wasextended by Tobin (1958). Introducing a risk-free asset, one assumes that investorscan borrow and lend at a risk-free interest rate, r

f

. From this, Tobin derives the two-fund-separation theorem, which says that it is always optimal for an investor to invest acertain ratio in the risk-free asset and the rest in a fully diversified portfolio, the marketportfolio. The linear combination of the market portfolio, with expected return µ andvolatility �, and the risk-free asset is called the Capital Market Line (CML). Its slope,called the Sharpe ratio, is given by

µ� r

f

�

. (1)

Due to widespread criticism of the theory, it is clear that the results from mean-varianceanalysis are not strictly utilized by practitioners today (see eg. Feldstein, 1978, Taleb,2007). This may be partly due to impracticality but mainly due to the weaknesses inthe underlying assumptions of the model. However, Markowitz’s results form a soundtheoretical foundation to build upon and to benchmark more recent theories against, asresearch on the subject is very thorough.

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2.1.2 Expected Utility Theory

Another widely used tool for analyzing preferences under risk is expected utility theory.One appealing aspect of the theory is that it can be derived from axioms that most peo-ple can agree upon, regarding rationality of choice (Von Neumann & Morgenstern, 2007).Note the distinction between decision being made under risk, that is the outcome beingrandom but with known probability, or the probabilities are formed as beliefs by thedecision maker. In the latter case, a second degree of uncertainty occurs, the uncertaintyof having formed correct beliefs. In this case one speaks of ambiguous situations. Gener-ally, expected utility theory deals with the case where the probabilities are known ex ante.

A set of mathematical axioms of rationality are necessary to distinguish between a gooddecision and a lucky decision (since under randomness, any decision can have a favorableoutcome, rational or not) and in order to describe these axioms we need to introduce theconcept of preferences (Von Neumann & Morgenstern, 2007). A preference x ⌫ y meansthat alternative x is at least as good as alternative y. x � y means that alternative x

is strictly better than alternative y. Further, x ⇠ y means the alternatives are equallygood. Now, the axioms of rationality can be represented as follows:

Axiom Assumption

Completeness Given two possible alternatives the individual has awell-defined preference, either x � y, x � y or x ⇠ y.

Transitivity x � y, y � z =) x � z.

Continuity The preferences cannot exhibit erratic behavior suchas sudden ”jumps” caused by minor changes in thedata.

Independence For 0 < ↵ < 1,x � y =) ↵x+ (1� ↵)z ⌫ ↵y + (1� ↵)z.

Table 1: The axioms of rationality by Von Neumann and Morgenstern (2007)are completeness, transitivity, continuity, and independence. Preferences of anindividual satisfy these axioms can be represented by an expected utility function.

If the axioms in Table 1 – completeness, transitivity, continuity and independence – aresatisfied, they can be represented by an expected utility function, where the followingholds: x � y =) E[u(x)] > E[u(y)].

Another important concept within expected utility theory is the certainty equivalent.The certainty equivalent of a lottery is the amount to receive with certainty that wouldgive the same utility as participating in a given lottery. If for instance a lottery has anexpected payo↵ of $100, but the certainty equivalent is $90, the investor is risk averseand the risk premium is ten dollars. To mirror this preference mathematically, one needsa concave utility function. Similarly, a risk loving utility function is convex while arisk neutral utility function is linear. Indeed, a concave utility function is in line with

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the classic economic theory of decreasing marginal benefit from consumption - here weconsume risk to receive expected returns.

2.1.3 Prospect Theory

In 1979, Kahneman and Tversky present a critique of expected utility theory as a de-scriptive framework for the way people make choices in the face of risk and uncertainty.They develop an alternative model, which they call prospect theory. What distinguishesprospect theory from expected utility theory is that value is assigned to gains and lossesrather than to final wealth (Shefrin & Statman, 2000). Thus, the utility function ofthe expected utility theory is replaced by the value function that has three importantproperties:

Property I: Investors assess gains and losses relative to a reference point.Individuals think in terms of gains and losses rather than in total wealth, whether acertain outcome is a gain or a loss depends on the individual’s reference point, whichcan be absolute or relative. A typical reference point is the individual’s status quo,i.e. her initial wealth (Hastie & Dawes, 2010). This assumption is compatible withbasic principles of perception and judgement. When we respond to stimuli, we evaluatedi↵erences in relation to a reference point based on past and present context, ratherthan absolute magnitudes. Consider for example brightness – your eyes may be verysensitive to light as you step out of a dark room but that same amount of brightnessseem comfortable as soon as your eyes have adapted. Kahneman and Tversky (1979)suggest the same principle applies to non-sensory attributes, such as wealth. At any givenpoint, we are more accustomed to changes in attributes such as brightness, loudness, andtemperature than we are to their absolute magnitudes (Barberis, 2013).

Property II: Attitude towards risk di↵er in the domains of gains and losses.According to Kahneman and Tversky (1979) attitudes toward risk di↵er in the contextof gains and losses. They suggest decision-makers are risk averse in the face of gains andrisk seeking in the face of losses. This element of prospect theory is known as diminishingsensitivity because it implies that, while replacing a $100 gain (or loss) with a $200 gain(or loss) has a significant utility impact, replacing a $1,000 gain (or loss) with a $1,100gain (or loss) has a smaller impact (Barberis, 2013). This also implies that the valuefunction curve is normally concave for gains and convex for losses (Kahneman & Tversky,1979).

Property III: Investors are loss averse.The idea behind loss aversion is that people are much more sensitive to losses thanto gains of the same magnitude, i.e. “losses loom larger than gains” (Kahneman &Tversky, 1979). A typical empirical estimate indicates that losses are approximatelytwice as painful than gains are pleasurable. This results in a value function that issteeper for losses than for gains (Barberis, 2013).

There are several possibilities to modelling the value function, which is a combinationof utility functions. One example of a viable class of functions is the piecewise powerfunction of Tversky and Kahneman (1992). Let �x represent the gain (loss) relativeto the reference point, ↵ represent the individual’s risk aversion and � represent lossaversion.

11

Reference pointx

v(�x)

Figure 4: A visualization of the value function, based on Prospect Theory byKahneman and Tversky (1979). The value function is defined on gains and lossesrelative to a reference point. It is normally concave for gains and convex forlosses, which represent risk aversion in the positive domain and risk seeking inthe negative domain. Further, the value function is generally steeper for lossesthan for gains.

V (x) =

(�x

↵

, for �x � 0

��(��x)↵, for �x < 0(2)

Another viable function is the piecewise quadratic value function (Hens & Bachmann,2011). Let �x represent the gain (loss) relative to the reference point. Let ↵+ and ↵�

represent the investor’s risk aversion for gains and losses, respectively, and let � representher degree of loss aversion. Then, the value function can be expressed as:

V (x) =

8>>>>>>>><

>>>>>>>>:

1

2↵+, for �x � 1

↵

+

�x� ↵

+

2(�x)2, for 1

↵

+

> �x � 0

�

✓�x� ↵

�

2(�x)2

◆, for 1

↵

� < �x < 0

1

2↵� , for �x 1↵

�

(3)

The constraints �x � 1↵

+

and �x 1↵

� prevent the utility from falling after a gainreaches 1

↵

+

and from increasing after a loss reaches 1↵

� . Thereafter, we assume a con-stant utility of V (x) = 1

2↵+

and V (x) = 12↵� , respectively. Note that for ↵+

> 0 and↵

�< 0 the function is s-shaped, implying that the function is concave for �x > 0 and

12

convex for �x < 0. This is consistent with the properties described above and visualizedin Figure 4.

One advantage of the piecewise power function is its monotonic properties. We expect↵ to be less than one here in order for the function to be concave and consistent withprospect theory. The disadvantage is that the model does not o↵er di↵erent curvatureparameters in the positive and negative domain, which the piecewise quadratic functiono↵ers. Further, the piecewise power function does not provide robust solutions undera mean-variance optimization framework, something that is o↵ered by the piecewisequadratic function. Also, in the special case where � = 1 and ↵+ = ↵

� and the refer-ence point is set to the expected return of the portfolio, the piecewise quadratic valuefunction will generate the same portfolio as a standard mean-variance investor (Hens &Bachmann, 2011), a property that might prove useful. According to the authors, themedian investor with the piecewise quadratic value function has preferences describedby the parameter values ↵+ = 2.15114,↵� = �1.84688,� = 2.25.

Another departure from the expected utility theory framework is the replacement ofprobabilities by decision weights. Kahneman and Tversky (1979) found empirically thatpeople tend to overweight small probabilities and underweight moderate and high prob-abilities. However, a biased perception of probabilities may lead to irrational decision-making since it motivates decisions which are in contradiction with the expected utilitytheory and in particular with the independence axiom. This may be avoided by sep-arating beliefs from risk attitudes. While probability weighting should not be used inthe asset allocation process, it may be quite useful in explaining actions and behavior ofactors in financial markets (Hens & Bachmann, 2011). Tversky and Kahneman (1992)suggest that the psychological probability weight should be analytically calculated usingthe following probability weighting function

w(p) =p

�

(p� + (1� p)�)1/�(4)

where p is the actual probability and � describes a distortion in the perception of prob-abilities. The lower the parameter, the stronger is the distortion.

2.2 Behavioral Biases

In this section, we will outline and discuss several biases and how financial advisors maymoderate or adapt to such biases in the advisory process. Even if the advisory processbecomes fully automated, the importance of dealing with investors’ biases will remainand be a vital part of the investment process.

One of the central assumptions supporting the Capital Asset Pricing Model (CAPM)(Sharpe, 1964), is that each individual player on the financial market acts rationally.Rationality, in this case, means that the investors have a set of consistent preferences,which satisfy the axioms of rational choice (Von Neumann & Morgenstern, 2007). Inorder to make a rational decision, one must correctly evaluate and process the flow ofinformation. Clearly, this is easier said than done and cognitive as well as emotionalbiases a↵ect the interpretation of information as well as the choices of both retail andprofessional investors. For a financial advisor, it is essential to be aware of these biases

13

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

p

w(p)

� = 1� = 0.5

Figure 5: Visualization of the probability weighting function by Kahneman andTversky (1992). In prospect theory, probabilities are replaced by decision weights.Generally, decision weights tend to be lower than the true probabilities, exceptfor low probabilities.

and how to address them (Pompian, 2011). The importance of bias awareness andhandling should reasonably extend to the automated advisory and allocation process.The main categories of biases are listed below, along with some examples of such biases.The section ends with a discussion concerning which biases are the most important, mostprevalent and how they can be addressed in an automated allocation model.

Information selection biases

• Availability bias: judging the relevance of information through how easily re-membered or easily accessed it is. Selecting stocks that are in the news, or havehigh daily returns which catches attention is a typical example. There is no evi-dence such stocks outperform the market. Further, overreaction to company newsamong investors is a consequence of availability bias. Indeed, stocks punished bynews tend to outperform stocks that gained a lot from news on a three year horizon(Barber & Odean, 2008).

Information processing biases

• Framing: the answer to a rational question with the same outcomes depends onthe setting. E.g. bond/equity ratio of a pension fund might depend on how manyalternatives are provided in the two categories (Tversky & Kahneman, 1981).

• Overconfidence: overestimating accuracy of predictions or own ability. Over-all, overconfident decision-makers overestimate the precision of information signals.

14

This makes them feel more secure than they are. As a consequence, individuals areready to take more risks than they actually can a↵ord (Barber & Odean, 2001).

Decision biases

• Mental accounting: thinking di↵erently about di↵erent types of investments e.g.eating home choosing the cheap alternatives and eating out choosing the mostexpensive (Thaler, 1985).

• Disposition e↵ect: Not realizing losses but realizing gains too early. This mentalaccounting between winners and losers hinders the client in facing his economicreality, planning his investment in hindsight rather than looking ahead (Hens &Vlcek, 2011).

• House money e↵ect: The willingness to gamble with money recently earned dueto the utility being associated with the recent gain and therefore does not feel likea loss if it turns out badly (Thaler & Johnson, 1990).

Decision evaluation biases

• Regret aversion: one regrets more what one has done (error of commission) thanwhat one has not done (error of omission) (Seiler, Seiler, Traub, & Harrison, 2009).

Hens and Bachmann (2011) provide some very concrete ways to deal with the specificbiases listed above. They first point out that e.g. choices driven by risk or loss aversionare not necessarily irrational. However, the biases listed above, called cognitive biases,are irrational and can be dealt with. It is thus an important task for the advisor todistinguish rational from irrational behavior. Then the advisor needs to help the clientmake rational decisions. The so called risk profiler, which will be discussed further inthe following section, is at the heart of such a process.

According to Hens and Bachmann (2011), the most important actions to take is to showlong term asset behavior and point out recent glamour stories that failed, to make surethe client correctly interprets and does not neglect statistical data and to update theclient’s expectations according to recent fundamentals, etc. This is to make sure theclient has realistic expectations and avoid for example availability bias and anchoring.We believe that one way of implementing this into an automated asset allocation modelcould be to provide the investor with information regarding the characteristics of hersuggested portfolio. When confronted with the risk figures and expected return of ahypothetical portfolio, the investor might be pushed toward more realistic expectations.

Pompian (2011) proposes another approach to mitigating client biases. He recognizesthat the growing field of behavioral finance is ideally positioned to assist financial ad-visors. However, he also states that only a few of the biases identified in behavioralfinance research today are commonly considered by advisors in the allocation processand wonders why behavioral finance is still only rarely used in wealth management.He defines what he calls the best practical allocation of a portfolio. It builds on thenotion that client’s interests and preferences stem from their behavioral characteristicsand hence, the client might not be best served by the output of a mean-variance modeloptimization. Such an allocation might result in a scenario where the client, in responseto market turmoil, demands her allocation to be changed. The best practical allocation

15

may generate a slightly underperforming portfolio, but a portfolio that the client can becomfortable with. However, in some cases the best practical allocation may contradicta client’s natural psychological tendencies. Then, the client may be better served byaccepting some uncomfortable risks in order to maximize expected returns.

To construct the best practical allocation, Pompian (2011) puts forth two guiding prin-ciples.

Principle I: Advisors should moderate biases in less-wealthy clients and adapt to biasesin wealthier clients.The intuition behind the first principle is to protect clients from devastating impacts ontheir personal finance. A portfolio that performs poorly because the allocation adaptstoo much to the biases of the client may constitute a serious threat to the personal financeof a less-wealthy individual and should therefore be moderated. For a wealthy client,whose day-to-day security is not threatened by an underperforming portfolio, adaptingto the client’s biases may be the appropriate course of action.

Principle II: Advisors should moderate cognitive biases and adapt to emotional biases.Behavioral biases fall into two broad categories. Cognitive biases originate from clients’inadequate reasoning while emotional biases stem from instinct or feeling. Since cog-nitive biases originate from inadequate reasoning, they can often be corrected throughinformation and advice. Emotional biases, on the other hand, build on emotions and areconsequently much more di�cult to rectify.

Comparing the approaches proposed by Hens and Bachmann (2011) and Pompian (2011),the former o↵er more of a practical approach as to how clients can be made aware oftheir biases and, from an advisor’s point of view, how to adapt the portfolio allocationwith respect to the biases. Pompian, on the other hand, provides a scheme for advisorsas to which biases are important to moderate and which the advisor can adapt to. Thesimilarities are that the biases referred to by him as cognitive biases are to be moderated,which is in line with Hens and Bachmann (2011) who point out that such biases leadto irrational decisions. Pompian (2011) also provides a client satisfaction aspect, whereemotional biases among wealthier clients can be adapted to as they are important to theclient even though they might not maximize gains. For the purpose of making a clientquestionnaire for automated advisory, there are important takeaways from both theseaspects. However, Hens and Bachmann (2011) provide more practical insights that canbe directly implemented into a questionnaire.

2.3 Assessing Investor Preferences and Behavioral Aspects

Establishing the risk an investor is willing and able to take is a key part of the advisoryprocess (Financial Services Authority, 2011). In this section, we will present the conceptof risk profiling and how such a tool can be used to determine an investor’s behavioralcharacteristics and attitude towards risk. Linciano and Soccorso (2012) conclude thatin practice, ”most questionnaires are not aligned with the economic and psychologicalliterature”. However, a few di↵erent attempts to address this issue have been made. Forexample, Grable and Lytton (1999) develop a multidimensional risk-assessment methodusing 20 questions, which covers a variety of types of risk preferences and combinesthem to an index score. What is problematic with scoring is that when optimizing

16

Moderate & Adapt Adapt

Moderate Moderate & Adapt

High level of wealth

Cognitive biases

Low level of wealth

Emotional biases

Figure 6: Visualization of Pompian’s (2011) Principles I and II. The principlessuggest that advisors should moderate biases in less-wealthy clients and adapt tobiases in wealthier clients. Also, advisors should moderate cognitive biases andadapt to emotional biases.

mathematically, the broad spectrum of di↵erent risk preferences is one-dimensionalizedin the model, to a single score number. In Kahneman and Tversky’s (1979) prospecttheory, they present a value function that provides asymmetric utilities and possiblydi↵erent curvature in the positive and negative domain. This is to account for investors’preferences in a more dynamic way. However, the authors do not propose any specificway to customize the value function to a specific investor, i.e. some sort of risk profiler.Hens and Bachmann (2011) show some general ways in which a risk profiler can beconnected to a mathematical model. These authors as well as Grable and Lytton (1999),divide risk preferences into di↵erent categories. While the latter provide a very thoroughassessment leading to a risk ”score”, the former show a variety of di↵erent questionnairesthat are viable. It is evident that there are numerous ways of doing risk profiling or justmeasuring risk tolerance. Due to its appealing properties, we will further explore theconcept of the risk profiler in the next section.

2.3.1 The Risk Profiler

The main features of the client’s risk profile are her risk ability, risk preferences, andrisk perception. A risk profiler is a questionnaire which assesses these characteristics in

17

a systematic way (Roszkowski, Davey, & Grable, 2005). In other words, the risk profilerprovide structure and consistency to the client assessment process (Financial ServicesAuthority, 2011). There are several reasons to why a detailed and accurate risk profileris valuable.

From a client perspective, a good risk profiler will enable a better understanding of theclient’s needs, behavior and preferences, and potentially produce a matching investmentportfolio. Further, it may reduce the impression that the advice a client gets is arbitrary(Hens & Bachmann, 2011). A better understanding of the client will also benefit theadvisor as it enables the advisor to build a deeper and more thorough relationship withclients. This is an important business aspect for financial advisory firms that want tobuild a sustainable competitive advantage. A further benefit is that a risk profiler can as-sess clients quickly and accurately (Grable & Lytton, 1999). Despite some arguments tothe contrary, a client’s risk preferences can be measured accurately by a questionnaire,provided that the questionnaire has been developed in accordance with psychometricprinciples (Roszkowski, Davey, & Grable, 2005).

Another advantage of the risk profiler is it’s ability to support and improve the con-formability of financial advisory services with the requirements of the European Marketsin Financial Instruments Directive (MiFID) (Roszkowski, Davey, & Grable, 2005). Thedirective was designed to improve the competitiveness of EU financial markets by cre-ating a single market for investment services and activities, and ensuring a high degreeof harmonised protection for investors in financial instruments (European Union, 2004).For advisors, the directive seeks to improve service quality, especially by aligning finan-cial advice and client needs. Advisors will be required to pay greater attention to theassessment of the client’s needs and preferences in order to ensure that the suggestedasset allocation is suitable to the client and her investment objectives.

Further, research has shown that the continued use, evaluation, and adaptation of arisk profiler has a positive impact on the daily practices of financial advisors, and mostimportantly, on the lives of wealth management clients (Grable & Lytton, 1999).

Risk abilityThe risk ability is a purely financial parameter that describes the client’s capacity for lossor, in other words, her ability to absorb falls in the value of her investment. It may beuseful to categorize the client’s capital which is necessary to keep up the client’s lifestyleor wealth which may enhance her lifestyle (Hens & Bachmann, 2011). Specifically, aloss of capital that would have materially detrimental e↵ect on her standard of living isvery important to assess accurately (Financial Services Authority, 2011). Consequently,the risk ability does not describe the amount of risk a client should take, but rather theamount of risk she may take without jeopardizing her standard of living. Hence, it maybe seen as a financial constraint.

There are generally two ways to view the risk ability (Hens & Bachmann, 2011): either,a certain amount shall not be risked in any case and then we say that the client needssafety first (Roy, 1952). In such a situation one would simply allocate that amount to arisk-free asset, i.e. a bank account. The client might receive no (or negative) return forthe moment, but one can not argue that bond investments are risk-free, and the Swedish

18

deposit insurance actually guarantees up to EUR 100,000 for the investor if the bankwould default. Thus this is the closest we can get to a risk-free asset.

Else, a certain small probability of risking the amount is acceptable to the client. Insuch a situation it might be suitable to implement a probability constraint into theallocation model, such as Value at Risk (VaR) or Conditional Value at Risk (CVaR),the latter is also known as Expected Shortfall (for a detailed explanation of these riskmeasures, see appendices A and B). Both measure an amount which will not be lostwith a set probability (e.g. 99%). VaR simply measures the 99-quantile value of the lossdistribution, while CVaR measures the average loss conditional on the outcome beingin the 99th quantile. This means that for heavy tailed distributions of asset returns,the VaR measure will not adequatly take the catastrophic outcomes into account (Hult,Lindskog, Hammarlid, & Rehn, 2012). CVaR is a more conservative risk measure andthus preferable when implementing a risk ability constraint. An even more conservativerisk measure is the maximum drawdown (see appendix C), which is the maximum lossfrom a peak to a trough of an investment, i.e. the worst possible outcome over a specifictime period.

Risk preferenceUnlike risk ability, which is a financial parameter, risk preference is a psychologicalparameter. A client’s risk preference describes how an individual feels about takingrisk and may be viewed as a measure of her risk and loss aversion (Guillemette, Finke,& Gilliam, 2012). Research suggests that risk preferences tend to be relatively stableover time. Sahm (2007) concludes that even though risk preference is subject to somesystematic changes (e.g. declining with age and tendency to co-move positively withmacroeconomic conditions) the persistent di↵erences between individuals’ risk prefer-ences account for more than 80% of the variation in measured risk preference. Further,according to Roszkowski (2010) preferences was not drastically a↵ected by the economiccircumstances of 2008. There was, however, clearly a change in people’s risk perception.This will be discussed in the next paragraph.

Risk perceptionIndividuals perceive risk di↵erently. Risk perception is the subjective judgement thatpeople make about the characteristics and severity of a risk, and are often inconsistentwith true statistical probabilities (Roszkowski & Davey, 2010). Especially new investors,are usually not well-informed about investment risk, particularly the relationship be-tween risk and return, and the range and likelihood of possible outcomes, including thepossibility of extreme events (Davey & Resnik, 2012). Generally, there is a tendencyamong investors to weigh losses more heavily than gains. Also, people tend to over-weight low probability events and to underweight high probability events (Roszkowski& Davey, 2010). Moreover, investors tend to underestimate risk in a rising market andoverestimate it in a falling market (Davey & Resnik, 2012).

One’s personality, past experiences, culture, and world view play a significant role inthe interpretation of the mathematical information (Roszkowski & Davey, 2010). Foradvisors, it is helpful to know whether a client is not acting (or acting) because of mis-perception of the risk or a reluctance (or eagerness) to make a risky decision (Roszkowski& Davey, 2010). It is the advisor’s duty to respond to clients’ preferences and correctbiases if necessary. Prospect theory provides some useful insights on how to perform thistask successfully (Kahneman & Tversky, 1979).

19

2.3.2 General Considerations and Common Pitfalls

Below follows a list of some general considerations and common pitfalls one may en-counter in the process of developing a client profiler.

Recency e↵ect Clients often recall and emphasize recent events and observations morethan those that occurred in the past. Before applying a risk profiler, the client shouldbe introduced to the risks and rewards of various asset classes over a long horizon. Thisis important in order to remove a potential bias where her risk preference will dependtoo much on her recent investment experiences (Pompian, 2011).

Inappropriate focus Some client profilers focus solely on the risk a client is willingto take and fail to su�ciently consider her other needs, objectives and circumstances(Financial Services Authority, 2011).

Validation questions Introducing validation question improves the quality of the riskprofiler but will increase length. It is considered good practice to include such questions(Financial Services Authority, 2011).

Quantitative or qualitative Precise answers to quantitative questions provide thebest condition for good advisory. However, some clients may be uncomfortable or notused to thinking quantitatively. A designer might consider simplifying quantitative ques-tions or sorting di↵erent clients into separate profilers (Hens & Bachmann, 2011).

Question evaluation Questions in a profiler should be evaluated for their understand-ability and answerability, and their ability to di↵erentiate between di↵erent client types(Roszkowski, Davey, & Grable, 2005).

Reliability A profiler that measures consistently, with known accuracy is consideredreliable (Roszkowski, Davey, & Grable, 2005). When designing a profiler, it is importantto judge how consistently findings emerge from one measurement to another (Grable &Lytton, 1999).

Validity A valid profiler measures what it claims to measure. This is generally moredi�cult to achieve than reliability since complex behavior is determined by more thanone factor (Roszkowski, Davey, & Grable, 2005). Particularly, questions which lead toan answer depending on both risk ability and risk preference are not valid (Linciano &Soccorso, 2012). Thus, risk ability, risk preference and risk perception should be assessedseparately.

The above list is not exhaustive and provides only some guidance in the design process.There are several other considerations and pitfalls that may be of particular interestdepending on the character and purpose of the profiler.

2.4 Summing Up

Above, we have reviewed the relevant literature for this study. In Section 2.1, we wentthrough the concepts of MPT, expected utility theory and prospect theory. We canconclude that whereas MPT was designed to capture how a perfectly rational investorshould invest, expected utility theory and prospect theory are more in line with our goalto incorporate behavioral finance in the asset allocation process. Especially, the concepts

20

of the prospect theory investor with the value function that has di↵erent curvature inthe positive domain and negative domain, are very appealing. Further, expected utilitytheory provides valuable lessons for designing questions for our risk profiler, with regardsto how the questions should calibrate and shape the value function to be consistent withan investors risk ability, risk preferences and risk perception.

In Section 2.2 we outlined and discussed several biases and how financial advisors maymoderate or adapt to such biases in the advisory process. Even if the advisory processbecomes fully automated, the importance of dealing with investors’ biases will remainand be a vital part of the investment process. We conclude that Hens and Bachmann(2011) provide some very concrete ways to deal with the specific biases listed above,specifically in their so called risk profiler, a structured way of assessing client preferencesand biases. We deem the risk profiler a useful tool for determining client preferences andbehavior, and it can be designed in detail for our needs. Pompian (2011) proposes an-other approach to mitigate client biases. He suggests advisors should moderate cognitivebiases and, where possible, adapt to emotional biases. Further, it is more important tomoderate biases for less wealthy clients. These insights will be valuable when designingthe risk profiler.

Finally, in section 2.3, we review the concept of the risk profiler. The main features ofthe client’s risk profile are her risk ability, risk preferences, and risk perception. A riskprofiler is a questionnaire which assesses these characteristics in a systematic way. Weconclude there are several benefits to the risk profiler, among those are reduction of sub-jectivity in the assessment process and that it can assess clients quickly and accurately.Another advantage of the risk profiler is it’s ability to support and improve the con-formability of financial advisory services with the requirements of the European Marketsin Financial Instruments Directive (MiFID). We conclude this section by providing a listof general considerations and pitfalls to be considered in the process of designing a riskprofiler, according to the literature and authorities.

In the following section we outline the methodology we use in determining the optimalasset allocation for an investor, given preferences and behavioral aspects.

3 Methodology

To determine a suitable asset allocation may be viewed as a three step process. First, weneed to assess the clients’ preferences and restrictions, which is essential for providingcustomized advice. This is done through the risk profiler. Second, we need to translateclient preferences into quantitative measures, by forming equations based on the client’sanswers. Third, we incorporate the client specific parameters in an allocation optimiza-tion model in order to determine a suitable allocation that the client is comfortable withand that fits her needs. We will describe these steps in detail in the following subsections.

In the first subsection, we describe the development of a risk profiler that assesses theclient’s risk ability, her risk preference, and her risk perception in a systematic way. Asdiscussed in section 2.3, the risk profiler is a good way to determine client preferencesand suitable for our choice of modelling as it is flexible and can be translated into several

21

quantitative measures a↵ecting the shape of the value function. Further, we also describehow that translation of the investor’s answers into parameters is done.

Last, we provide a walk-through of how the results from the risk profiler, via the pa-rameters estimated, is used in an optimization model with historical data, to generateportfolios that are in accordance with the client specific behavioral and economic aspects.

3.1 Designing the Risk Profiler

As previously concluded, it is essential in a financial advisory process to determine theclient’s risk ability, risk preference and risk perception. In an automated asset allocationprocess these steps will be of even greater importance, as the human interaction betweenclient and advisor will be minimal. In this section, we describe the approach that willbe used to gather the information necessary for the asset allocation model. We designa so called risk profiler, a questionnaire designed to capture the investor’s risk ability,risk preferences and risk perception. In context of the allocation model, risk ability putsan absolute limit to the downside risk, risk preference determines parameter values andthe reference point, and risk perception validates that the investor’s risk preferences arecoherent with her behavioral traits (it is di�cult for an investor to precisely know herrisk preference, as discussed in section 2.3.1).

Many existing robo-advising services o↵er a mean-variance optimization where the in-vestor decides a level of risk on a scale. We find this method problematic in several ways,e.g. that investor’s do not know their risk level themselves and that there are di↵erentkinds of risk, where preferences can di↵er. The advantage of using such a simple model isof course the minimal time and e↵ort required from the investor. We deem a risk profilermore suitable, as it can measure di↵erent kinds of preferences, is fully customizable, andmore resembles the traditional advisory process. Many di↵erent models for risk profil-ing have been made, most try to give the investor a score of risk tolerance, which putsher somewhere on the e�cient frontier (i.e. on a volatility constraint). We prefer anapproach that better captures the individual risk preferences and behavior in the actualmodel.

3.1.1 Risk Ability: Determining Investment Constraints

As discussed in section 2.3.1, risk ability boils down to determining how much of hercapital the investor is able to put at risk and further, several ways of implementing itinto the allocation model. While the CVaR measure is more appealing from a theoreti-cal point of view, it can be practical to simply hold a cash amount one is not willing tolose. In that case, however, the investor would have to give up some substantial return,depending on how big the risk ability is compared to the total amount of investments.A maximum drawdown constraint could be a good trade-o↵ due to its very conservativenature.

The risk profiler and allocation model will make sure the investor stays out of financialdistress by constraining the maximum drawdown and the empirical estimate of the CVaRso that any amount of the investment that might be needed for future expenses willremain even in a worst case scenario.

22

3.1.2 Risk Preference: Determining Parameter Values

The risk preferences of a an investor may be described by the piecewise quadratic valuefunction presented in section 3.2. To let the value function represent the preferences of aparticular investor is a matter of determining ↵+, ↵� and �, where ↵+ and ↵� representthe individual’s risk aversion for gains and losses, respectively, and � represents her de-gree of loss aversion. Further, it is important to determine what the investors considersa loss and what she considers a gain, i.e. what her reference point is.

The risk profiler contains questions that seek to quantitatively determine these parame-ters. Questions regarding the client’s attitude to uncertainty in gains and losses respec-tively may help determining her risk aversion. E.g. what certain gain the client wouldprefer over the opportunity to gain a certain amount while otherwise breaking even withequal probabilities, can provide valuable insight into the client’s risk aversion in gains.Similarly, how much the client is willing to pay to prevent a possible loss of a certainamount, while otherwise breaking even with equal probabilities, can tell something aboutthe client’s risk aversion to losses. Prospect theory suggests that attitudes toward riskdi↵er in the context of gains and losses (Kahneman & Tversky, 1979). This is why it isimportant to determine risk aversion for gains and losses separately.

The magnitude of the gains and losses asked for will be based on total amount invested,since a gamble decision on e.g. 10 000 SEK might be very di↵erent depending on wealth.Recall the piecewise quadratic value function in section 3.2. The value in the positivedomain is given by

�x� ↵

+

2(�x)2. (5)

Building on the approach of Hens and Bachmann (2011), in order to determine ↵+ wecan ask what certain gain, �x

c

, the client would prefer to the opportunity of gaining anamount �x

u

, while otherwise breaking even with equal probabilities. This situation canbe modeled with the equation

�x

c

� ↵

+

2(�x

c

)2 = 0.5(�x

u

� ↵

+

2(�x

u

)2) (6)

which gives

↵

+ =�x

c

� 12�x

u

(�xc)2

2 � (�xu)2

4

. (7)

The same approach can be used to determine ↵� for the negative domain where thevalue is given by

�

✓�x� ↵

�

2(�x)2

◆. (8)

We can ask what certain loss, �x

c

, the client would prefer to the uncertainty of losingan amount �x

u

, while otherwise breaking even with equal probabilities. This can bemodeled as

23

�

✓�x

c

� ↵

�

2(�x

c

)2◆

= 0.5�

✓�x

u

� ↵

�

2(�x

u

)2◆

(9)

which gives

↵

� =�x

c

� 12�x

u

(�xc)2

2 � (�xu)2

4

. (10)

Questions regarding the client’s attitude to losses may help determining her loss aversion.Kahneman and Tversky (1979) suggest that people are much more sensitive to lossesthan to gains of the same magnitude. Thus, it is relevant to ask what size of a loss�x

L

is acceptable in order to gain a certain amount �x

G

, with equal probabilities. Thissituation can be modeled as

�x

G

� ↵

+

2(�x

G

)2 � �

✓�x

L

� ↵

�

2(�x

L

)2◆

= 0 (11)

which gives

� =�x

G

� ↵

+

2 (�x

G

)2

�x

L

� ↵

�

2 (�x

L

)2. (12)

3.1.3 Risk Perception: Ensuring Behavioral Coherence

The risk profiler also measures the investor’s risk perception in order to identify potentialbehavioral and cognitive biases which might influence the answers about risk abilityand risk preference. This part of the profiler will consist of validation questions, andcoherence between the validation questions and previous answers will help identify anypossible biases. There are several approaches as to implementing and/or handling riskperception in an automated advisory service. Here we choose to take an educationalapproach – if the investor’s perception of risk is unrealistic the model will help herreconsider her answers to attain a more realistic risk-reward view.

3.1.4 The Final Risk Profiler

In this section, we present the design of the risk profiler, what questions are included torepresent the di↵erent aspects of investor preferences and why. The questions are shownin Table 3.1.4, and have been numbered 1-12 for reference. Here, we will discuss andmotivate the questions briefly.

The first two questions in the risk profiler are to determine the investor’s risk ability.Question (1) simply asks how large the amount intended for investment is. This is a nec-essary background question of course, as well as important for the upcoming questions,since every question involving numbers will be based on the answer to this question.Question (2) determines the amount needed for future expense. Together, these twoquestions provide a good basic understanding of the investor’s financial situation with-out going too much into detail. Later on, these amounts will be compared to the modeloutput and presented to the investor, in order to have her re-evaluate her risk level ifnecessary. There are a lot of other relevant questions with regard to the investor that

24

Question Answer

type

Attribute

(1) What is the amount you intend to invest? SEK Risk ability

(2) Of this amount, how much do you require for future expense(e.g. business needs, lifestyle, and unexpected family events)?

SEK Risk ability

(3) What is your main objective with this investment? Multiplechoice

Risk preference

(4) Over how many years do you intend to invest? # years Risk perception

(5) What yearly return are you aiming to achieve with this in-vestment?

% Risk perception

(6) In a worst case scenario, what is the greatest loss you expectthis investment could incur?

% Risk perception

(7) All of the asset classes (see. Table 4) will be consideredfor your portfolio. Please indicate if you have a strong feel-ing that you do not want to hold more than a certain mini-mum/maximum investment in a particular asset class.

% Risk preference

(8) Suppose you have a 50% chance of gaining x% (X SEK)on your investment in a year, while otherwise you gain nothing.What sure gain would you prefer over this opportunity?

SEK Risk preference

(9) Under a year of unfavorable market conditions, what sureloss would you be willing to take in order to avoid the uncer-tainty of losing x% (X SEK) or losing nothing with equal prob-ability?

SEK Risk preference

(10) Consider a strategy that has a 50% chance of gaining x%(X SEK) on your investment in a year. In the other 50% cases,what is the maximal loss would you find acceptable in order tofollow the strategy?

SEK Risk preference

(11) In 1 out of 20 years, unfavorable market conditions maycause your portfolio to perform extraordinarily bad. What is thegreatest loss you expect in such a scenario to still be comfortablewith your investment?

SEK Risk preference

(12) What do you feel is the maximum loss in any time periodthat you would be comfortable with?

SEK Risk preference

Table 2: The final risk profiler consists of 12 questions that aim to measurean investor’s risk ability, risk preference and risk perception. The answers pro-vided by the investor are used to calibrate the value function in Section 3.2 andconstitute constraints in the optimization problem described in Section 3.3.2.

could be, and sometimes are required to, be asked in an advisory situation (Oxenstierna,2015). However, we choose the most relevant in order to keep the risk profiler reason-ably short, as for instance Roszkowski and Bean (1990) have shown that the responserate is inversely related to questionnaire length, and that shorter questionnaires are al-most always better. Also, it is possible that automated advice might not necessarily beconsidered an advisory situation by law. Thus the questions asked have been deemedsu�cient, but could easily be complemented with more thorough client analysis if nec-essary, although that would not drastically change the model output.

The two initial questions are followed by questions determining the investor’s risk prefer-ences. As discussed in Sections 2.3 and 3.1.2, risk preferences are central in the investor’sdecisions. In short they measure how the investor values risk versus reward in di↵erentsituations. In our model, this is used to calibrate the value function, as explained inSection 3.1.2. Question (3) determines the investor’s reference point, as described inSection 2.1.3. The higher gains she is expecting, the higher the reference point. Thereference point is a central part of Kahneman and Tversky’s (1979) prospect theory and

25

thus important to include in the profiler. Question (4) determines time horizon, which isan important factor when deciding the level of risk suitable for the investment (Butler &Domian, 1991). As we have learned in Section 2.3, investor’s have di↵erent perceptionsof risk which are sometimes unrealistic. Thus, question (5) and (6) investigates riskperception by asking what return the investor is expecting and what loss she expects tomake in a worst case scenario, given the expected return. Further, the worst case loss isused later in the risk profiler as a benchmark level for gambles (which will be explainedlater in this section), to make those questions more customized for each individual.

Question (7) is of a more practical type. The investor is asked to indicate if she hasany preferences with regards to holding specific asset classes. This will be considered inthe model, as much as possible given the risk ability and expected return. In line withPompian (2011), one can then depending on the client’s financial situation either adaptto, or moderate these preferences. Reasonably, the model will try to adapt as muchas possible to the client’s preferences or home bias, as long as it does not compromisethe risk-return level. Questions (8)-(9) ask the investor what sure gain/loss she wouldprefer over a fifty-fifty gamble between a gain/loss where the size depends on question(6), and gaining/losing nothing. These questions are based on utility theory (which isalso incorporated in prospect theory) as described in Section 2.1.2, where we are askingfor the investor’s certainty equivalent of the gambles. Question (8) determines the cer-tainty equivalent in the positive domain whereas question (9) determines the certaintyequivalent in the negative domain. If the certainty equivalent is less than the expectedvalue of the gambles, the investor is risk averse and has a concave utility function, andvice versa. This is followed by question (10) which measures the degree of loss aversion,an integral part of the investor’s preferences, according to prospect theory (Kahneman& Tversky, 1979), also described in Section 2.1.3.

After the first ten questions have been answered, the model will provide a preliminaryportfolio allocation. For this allocation, the investor will be provided with the infor-mation in Table 3. Table 3 describes how the initially suggested portfolio would haveperformed historically, e.g. during the financial crisis. It also compares the investor’sexpectations to realistic expectations to show the investor how she perceives risk. Afterreading this information, she will be given a chance to modify her level of risk in question(11) and (12). This is intended to have an educational e↵ect on the investor, in orderto slightly help moderating a biased risk perception. Question (11) will ask how muchthe investor is willing to risk losing under unfavorable market conditions 1 out of 20years. This answer will be related to a CVaR constraint. Question (12) will ask for themaximum loss acceptable at any time, and related to a maximum drawdown constraint.Then another optimization is performed and the final allocation is provided to the in-vestor. This way of handling risk perception and biases, suggested by e.g. Pompian’s(2011) framework of cognitive versus emotional biases, where this is to be considered acognitive bias, we deem very suitable for an automated asset allocation model.

3.2 The Value Function

To evaluate any asset allocation for the investor we will use the piecewise quadratic valuefunction presented in Section 2.1.3. The piecewise quadratic value function appropriatelycaptures the three properties of the investor according to prospect theory, and is more

26

Portfolio characteristics

You indicated the main objective of your investment to be capital preservation.This corresponds to an expected yearly return of r1 - r2% and a recommendedinvestment horizon of up to T1 years. You indicated your target yearly return tobe x% (X SEK) and your investment horizon to be T2 years.

The expected yearly return of your suggested portfolio is r% (R SEK).

In 1 in 20 years, the portfolio risks losing l% (L SEK) or more. The expected lossin such a scenario is l% (L SEK).

In a worst case scenario, the greatest loss the portfolio is expected to incur ina single year is l% (L SEK). You indicated your expected greatest loss is l% (LSEK).

During the period of worst performance for the suggested portfolio, the incuredloss would have been l% (L SEK).

During the financial crisis 2007/2008, the suggested portfolio would have incureda loss of l% (L SEK).

During the european debt crisis in 2011, the suggested portfolio would have in-cured a loss of l% (L SEK).

Table 3: Portfolio characteristics for the initial optimized allocation. This in-formation is provided to the investor together with the initial allocation. Theinformation is intended to describe how the initially suggested portfolio wouldhave performed historically, e.g. during the financial crisis. It also compares theinvestor’s expectations to realistic expectations to show her how she perceivesrisk. After this, she will be given a chance to modify her level of risk.

flexible than e.g. the piecewise power function proposed by Tversky and Kahneman(1992). Further, optimizing over the value function is preferred over e.g. mean-varianceoptimization as the latter assumes a completely rational investor. The value function,on the other hand, makes sure investor preferences and behavior are taken into accountin the portfolio optimization, in line with our purpose. Recall the mathematical repre-sentation of the piecewise quadratic value function in Equation 3, where �x representsthe gain (loss) relative to the reference point, ↵+ and ↵

� represent the investor’s riskaversion for gains and losses, respectively, and � represents her degree of loss aversion.

3.3 Modeling Approach

In this section, we will provide a walk-through of the implementation of the risk profiler,the mathematical model and how they are combined to provide portfolio allocations. InSection 3.2 we presented an investor’s value function. From the questions in the riskprofiler presented in Section 3.1, we are able to determine the risk and loss parameters(↵+, ↵� and �) of the value function and establish risk ability constraints. To proceed,we need to set up the portfolio optimization problem and define the empirical data setto be considered in the optimization problem.

27

3.3.1 Data Set

Historical data for di↵erent asset classes will be represented by indices that are retrievedthrough Bloomberg’s database. All indices are gross dividends and are represented asmonthly returns for the period January 1999 to March 2016. The motivation for the timeperiod is that it is the longest possible period for which data for all asset classes exists.Notably, the asset allocations generated will be a↵ected of the performance during thetimeframe used. Thus, we believe an as large sample size as possible is good to preventthis e↵ect. Likely, the 17 years of data will contain some irregularities, and portfoliosoptimized on our data will perform very well in historical comparisons. For that reason,we will also perform some out of sample tests on allocations generated to ensure that theportfolio still performs reasonably. The full set of asset classes, corresponding indiciesand Bloomberg tickers under consideration are listed in Table 4. The average yearlyreturn and standard deviation for each of the asset classes are visualized in Figure 7.

Asset class Index Name Ticker

Swedish Equity OMXSB Index SBXCAP

Global Equity MSCI ACWI NDUEACWF

European Equity MSCI Europe Index MSDEE15N

American Equity MSCI USA Index NDDLUS

Japanese Equity MSCI Japan NDDLJN

Emerging Markets MSCI Emerging Markets Index NDUEEGF

Alternative investments HFRX Global Hedge Fund Index HFRXGL

Swedish T-Bills OMRX T-BILL RXVX

Swedish T-Bonds STO RXBO Index RXBO

European Credit Markit IBOXX Liquid Corp. IB8A

Commodities RJ/CRB Commodity TR Index CRYTR

Table 4: Asset classes under consideration, their index representations andBloomberg tickers.

The 207 monthly returns will be used to perform a historical simulation in order toproduce a series of simulated yearly returns. We will use the approach outlined by Hultet al. (2012). Consider our sample of monthly return vectors

�R�n+1, . . . , R0

. We

draw with replacement T = 12 vectors from the sample and form the componentwiseproduct of these vectors, denoted by R

T

i

. This procedure is repeated m = 1000 timesto obtain the sample

�R

T

1 , . . . , RT

m

of fictive return vectors over time periods of length

T . As long as the original return vectors are independent and identically distributed, anassumption we will make here, then the vectors R

T

1 , . . . , RT

m

are identically distributedbut not independent (but conditionally independent given R�n+1, . . . , R0). Many usestatistical distributions without understanding probability, like parrots memorizing sen-tences. Historical simulation does not require any distributional assumptions. It is areasonable approach if we believe the mechanism that produced returns in the past isthe same that will produce returns in the future (Hult, Lindskog, Hammarlid, & Rehn,2012). This is an assumption we prefer to make here over assigning distributions andparameters to the asset classes’ returns.

28

Swedish Equity

Global Equity

European Equity

American Equity

Emerging Markets

Japanese Equity

Alternative Investments

Swedish T-Bills

Swedish T-BondsEuropean Credits

Commodities

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

0% 5% 10% 15% 20% 25%

Ave

rage

Yea

rly

Ret

urn

Risk (Standard Deviation)

Figure 7: Average yearly return and standard deviation (volatility) for each ofthe considered asset classes. The graph shows that low risk assets, such as bonds,are generally associated with low return. To achieve higher expected return, it isnecessary to increase risk.

3.3.2 Portfolio Optimization

In order to capture the concepts of behavioral finance, we employ the behavioral valuefunction and take a slightly di↵erent approach to the optimization problem. Given theparameter values, estimated as in Section 3.1, we proceed to find the portfolio weightsmaximizing the expected utility based on the empirical distribution of portfolio outcomes.

Here the risk ability is accounted for by the maximum drawdown constraint. Let w bea vector of asset weights, R

i

is a vector from our sample of fictive yearly returns, thescalar be the investor’s maximum drawdown level over time T (risk ability), the scalar be the client’s maximum CVaR on the p confidence level and let V (x) be the piecewisequadratic value function. The optimization problem becomes as follows:

maximizew

1

m

mX

i=1

V (wT

R

T

i

)

subject to CVaRp

(wT

R)

MDD(T ) X

wi

= 1 , wi

� 0.

(13)

29

Shefrin and Statman (2000) describe a method which is not very di↵erent from MPT,when solving what they call the behavioral portfolio selection problem. Instead of opti-mizing in the (µ,�)-space as in classic MPT, they optimize in the (E[W ], P (W A)-space, where W is the individual’s wealth. Thus, looking at expected wealth comparedto the risk of substantial loss. However, in our approach, we do not make any distribu-tional assumptions and we look at the utility of every single yearly gain/loss, which webelieve better mimics an individuals actual utility from holding a portfolio. Optimizingthe utility rather than total wealth better accounts for investor preferences such as dif-ferent attitudes toward risk in positive and negative domain, the aspect of loss aversionetc., as discussed in Section 2.1.3. Hence we prefer this approach.

4 Results & Analysis

In this section we provide a walkthrough of how the model actually performs. We showhow the risk profiler answers a↵ect the parameter values of the value function. Then,we proceed to show how those parameters shape the value function, and in turn howthe characteristics of di↵erent variants of the function a↵ects portfolio allocation andcharacteristics. Indeed, one can see that more risk-averse investors do get conservativeallocations through the model, and vice versa. Finally, we conduct a case study andout-of-sample test on an actual investor to see if the suggested portfolio satisfies theinvestor’s needs.

4.1 The Impact of Risk Profiler Answers on Parameters

In this subsection, we show the parameter values obtained for di↵erent answers to thequestions in the risk profiler, in accordance with the method described in 3.1.2. In short,↵

+ is decided by the choice between a gamble with 50% chance of gaining e.g. 20%and a 50% chance gaining 0%, or a sure gain between 0% and 20%. The question isdesigned so that the respondent decides the sure gain she would prefer to the gamble.The ↵� parameter is decided in a similar manner, but with a 20% loss instead of a 20%gain in the lottery. ↵� is then decided by the the sure loss she would prefer to the gamble.

In calculating �, the respondent is asked for what level of loss she finds acceptable ina 50-50 situation where the gain is fixed. For instance, what level of loss would be ac-ceptable given 50% chance to gain 20% on an investment. In Figure 8, we show how theparameter values vary with di↵erent answers to the trade-o↵ questions.

The ↵+-parameter governs the quadratic curvature in the positive domain of the valuefunction. In prospect theory, investors are expected to be risk-averse on the positivedomain, thus we expect the curve to be concave and ↵+ to be positive (recall the valuefunction in section 3.2). This is true for all answers to the trade-o↵ between a suregain and a gamble that are lower than the expected gain of the gamble. Hence, ↵+

decreases as the investors preference for sure gain approaches the expected value of thegamble (then the positive domain value function becomes linear). Should the investorbe risk-seeking on the upside, ↵+ will be negative and the positive domain of the valuefunction will be convex, in discord with prospect theory. The relation between ↵+ andthe trade-o↵ preference is shown in Figure 8.

30

0% 10%0

10

↵

+

0% 10%�10

0

↵

�

Figure 8: Left: ↵

+ as a function of the risk profiler answer. A lower certaintyequivalent (the horizontal axis), gives a higher ↵

+, which in turn correspondsto a more concave value function in the positive domain, i.e. a more risk averseinvestor in the context of gains. Right: ↵� as a function of the risk profiler answer.A lower certainty equivalent (the horizontal axis), gives a lower ↵�, which in turncorresponds to a more convex value function in the negative domain, i.e. a morerisk seeking investor in the context of losses.

The ↵�-parameter governs the negative domain quadratic curvature of the value func-tion. In prospect theory, investors are expected to be risk-seeking on the downside, thuswe expect the curve to be convex and thus ↵� to be negative. This is true for all answersto the trade-o↵ between a sure loss and a gamble that are lower than the expected lossof the gamble. Hence, the magnitude of ↵� decreases as the investors preference forsure loss approaches the expected value of the gamble (then the negative domain valuefunction becomes linear). Should the investor be risk-averse on the downside and thusaccept a greater sure loss than the expected value of the gamble, ↵� will be positive andthe negative domain of the value function will be concave, in discord with prospect theory.

The rationale behind the �-parameter is rather straightforward. The lower loss that isacceptable to the respondent, given a fixed level of gain, the larger the loss aversion andthus the larger the �. This simply means that the value function will decrease faster inthe negative domain than it increases in the positive domain. In Figure 9 we see how thevalue of the �-parameter is decreasing with the size of loss acceptable to the investor, andquickly increasing as the acceptable loss approaches zero. An investor with an accept-able loss approaching zero is extremely risk averse. Note also that a completely rationalinvestor, who considers a loss equally painful as a gain of the same size is rewarding., hasa �-parameter value of 1. This is consistent with the general assumptions of prospecttheory.

31

0% 5% 10% 15% 20%0

5

10

15�

Figure 9: � as a function of the risk profiler answer. A smaller acceptableloss (the horizontal axis), gives a higher � which corresponds to a steeper valuefunction in the negative domain, i.e. a more loss averse investor.

x

v(�x)x

v(�x)

Figure 10: Left: The piecewise quadratic value function for parameter values↵

+ = 1, 2, 3, 4 and ↵

� = -1, -2, -3, -4. A higher value of ↵+ corresponds to a moreconcave value function in the positive domain, i.e. a more risk averse investor inthe context of gains. A higher value of ↵� corresponds to a more convex valuefunction in the negative domain, i.e. a more risk seeking investor in the contextof losses. Right: The piecewise quadratic value function for parameter values �

= 1, 2, 3, 4. A higher value of � corresponds to a steeper value function in thenegative domain, i.e. a more loss averse investor.

32

To illustrate the parameters’ impact on the value function, we provide two graphs be-low. In the first, we see how the curvature of the value function decreases when the↵-parameters approach zero. In other words, the investor is becoming more rational asthe ↵-parameters approaches zero, since she is becoming more risk averse in the negativedomain and less risk averse in the positive domain. In the second graph, we see how theslope in the negative domain increases with �. We can see that the connection betweenrisk profiler response to calibration of the value function is properly in accordance withour theoretical framework. In the next subsection, we will examine that the optimizationwith the di↵erent value functions provide portfolio allocations that are aligned with thepreferences.

4.2 The Impact of Parameter Values on Portfolio Characteristics

In this section, we optimize portfolios with the behavioral value function as objectivefunction, for di↵erent parameter values. We show that the parameters properly a↵ectthe portfolio characteristics, i.e. consistent with the prospect theory investor’s risk pref-erences. First, we investigate the impact of the parameter values on the allocations andthe logical coherence of changing allocations as parameter values change. Then, we con-duct a more quantitative investigation of the same allocations in order to determine ifthe parameter values a↵ects the optimal portfolio in a way that is consistent with ourtheoretical framework.

4.2.1 Optimal Allocations

We start by letting ↵+ = 2.15114 and ↵� = �1.84688 and the reference point is set tothe initial wealth (or a gain of 0%), all in line with the median investor according toHens and Bachmann (2011). The loss aversion parameter, �, is varied between 1 and 4.The results are shown in Figure 11 and as can be seen, a higher � results in a more con-servative portfolio allocation. This is very much to be expected as a loss averse investorwants a more conservative portfolio, and that is what a higher � is meant to accountfor. An investor with � = 4 will allocate over three quarter of her wealth to long-termbonds, dividing the rest between Swedish equity and emerging markets. It should benoted here that the risk characteristics of the portfolio can be varied quite well usingonly these three broad asset classes. In the particular data set that the calculations arebased on, the risk-reward for these asset classes are the highest.

We proceed by fixing � = 2.25 and ↵

� = �1.84688, and letting ↵+ range from 1 to4. The ↵+-parameter determines the curvature of the value function in the positive do-main. Higher values leads to a more concave shape. As one might expect, higher valuesof ↵+ leads to a more conservative allocation, which can be seen in Figure 12. Clearly itmakes sense that for an investor with a high ↵+, the magnitude of the gain becomes lessand less important (the marginal utility is decreasing) and hence it will be more fruitfulwith a lower, steadier income i.e. a conservative portfolio. Note that the opposite holdsfor ↵+ between zero and one, which corresponds to a risk-loving investor. We considersuch investors quite rare, thus we choose to show portfolio allocations in the range of ↵+

mentioned above.

33

Emerging Markets

47%Swedish Equity41%

Swedish T-Bonds

12%

β = 1

Emerging Markets

21%

Swedish Equity15%

Swedish T-Bonds

64%

β = 2

Emerging Markets

13% Swedish Equity10%

Swedish T-Bonds

77%

β = 3

Emerging Markets

9% Swedish Equity10%

Swedish T-Bonds

81%

β = 4

Figure 11: Optimized allocations for � = 1, 2, 3, 4. As � increases, the allocationto low risk assets (bonds) increases and the allocation to riskier assets (equity andcommodities) decreases.

Emerging Markets

35%

Swedish Equity26%

Swedish T-Bonds

39%

α+ = 1

Emerging Markets

19%

Swedish Equity14%Swedish

T-Bonds67%

α+ = 2

Emerging Markets

13%Swedish Equity12%

Swedish T-Bonds

75%

α+ = 3

Emerging Markets

12% Swedish Equity10%

Swedish T-Bonds

78%

α+ = 4

Figure 12: Optimized allocations for ↵

+ = 1, 2, 3, 4. As ↵

+ increases, theallocation to low risk assets (bonds) increases and the allocation to riskier assets(equity and commodities) decreases.

34

Next, we fix � = 2.25 and ↵

+ = 2.15114, and letting ↵� range from 1 to 4. The ↵�-parameter determines the curvature of the value function in the negative domain. Highernegative values leads to a more convex shape and hence a riskier portfolio allocation, seeFigure 13. As we can see, the negative domain di↵ers in that the higher the curvature,the riskier the portfolio allocation. This is because the more the curve flattens out inthe negative domain, the more willing the investor is to take a gamble with his losses,as losses are not heavily punished by the value function. Given the other parameters attheir median level, ↵� of the magnitudes tested here do not severely change the portfolioallocation.

Emerging Markets

17%Swedish Equity13%

Swedish T-Bonds

70%

α- = –1

Emerging Markets

18%

Swedish Equity14%Swedish

T-Bonds68%

α- = –2

Emerging Markets

19%

Swedish Equity15%Swedish

T-Bonds66%

α- = –3

Emerging Markets

20%

Swedish Equity16%

Swedish T-Bonds

64%

α- = –4

Figure 13: Optimized allocations for ↵

� = -1, -2, -3, -4. As ↵

� approaches 0,the allocation to low risk assets (bonds) increases and the allocation to riskierassets (equity and commodities) decreases.

Finally, we vary the reference point from 0% to 6% gain, with the other parameter fixedat the median investor values as above. Figure 14 shows that the allocation to low-risk assets is initially increasing with the reference point. For higher reference points,however, we see that an increasing reference point yields a riskier portfolio allocation,as the investor needs to take more risk in order to achieve what she perceives as a gain.Generally one would thus probably expect the portfolio’s riskiness to increase when thereference point increases. Here we have an interesting outcome, where the stable incomefrom allocating more to bonds at RP 2% than at RP 0% is suggested by the model. Webelieve that is because when the utility reference point is shifted, at first its better toplay it safe because losses hurt more. However, it will not be possible to achieve enoughexpected gains to reach the 4% and 6% RP . Thus for those portfolios, the riskiness willhave to increase in order to reach the RP and more is allocated to the equity classes ofinvestments.

35

Emerging Markets

18%

Swedish Equity14%Swedish

T-Bonds68%

RP = 0%

Emerging Markets

14%Swedish Equity13%

Swedish T-Bonds

73%

RP = 2%

Emerging Markets

16%

Swedish Equity15%

Swedish T-Bonds

70%

RP = 4%

Emerging Markets

22%

Swedish Equity22%

Swedish T-Bonds

57%

RP = 6%

Figure 14: Optimized allocations for RP = 0%, 2%, 4 %, 6%. As the referencepoint increases, the allocation to low risk assets (bonds) initially increases butthen decreases. The allocation to riskier assets (equity and commodities) initiallydecreases but then increases.

4.2.2 Performance Metrics

In this section, we provide several performance metrics for the allocations generated inthe previous section, to see how the performance corresponds to investor preferences.The investor preferences will be represented by the parameter vector (↵+

,↵

�,�, RP ),

where RP is the reference point for gains and the other parameters as in previous sec-tions. We begin the comparison with the median investor, where the parameter vectoris (2.15114,�1.84688, 2.25, 0), and then proceed to modify the parameters. Recall thatincreasing alphas corresponds to a more risk averse investor, greater � means a more riskaverse investor and an increase in reference point requires more risk in order to achievethe higher required gain.

In Table 5, we see the characteristics of portfolios generated by some example sets of pa-rameters. The first row describes the expected yearly return (µ), the standard deviation(�), the Sharpe ratio (SR), the Conditional Value at Risk (CVaR) and the maximumdrawdown (MDD) of the portfolio generated for the median investor. Figure 15 visu-alizes the relationship between (µ), risk (�) and Sharpe ratio, and ↵

+, ↵�, � and thereference point, respectively. For simplicity, we assume that the risk-free rate, r

f

, is 0when computing Sharpe ratios. We can see how increasing the magnitude of ↵+ and�, as well as decreasing the magnitude of ↵�, gives a less risky portfolio. Conversely,doing the opposite will give a riskier portfolio. Increasing the reference point from 0%to 2% gives a slightly less risky portfolio, which might seem counterintuitive, given thathigher required return usually implies greater risk taking. However, low risk assets such

36

Parameters

(↵+,↵

�,�, RP ) µ � SR CVaR MDD

(2.2,�1.85, 2.25, 0) 6.5% 5.9% 1.1 5.5% 12.7%

(4.0,�1.85, 2.25, 0) 5.9% 4.2% 1.4 2.5% 7.0%

(2.2,�4.0, 2.25, 0) 6.7% 6.6% 1.0 6.7% 14.8%

(1.0,�1.85, 2.25, 0) 10.8% 20.0% 0.5 27.0% 47.0%

(2.2,�1.0, 2.25, 0) 6.3% 5.5% 1.1 4.8% 11.4%

(2.2,�1.85, 1.0, 0) 10.0% 17.3% 0.6 23.3% 41.7%

(2.2,�1.85, 3.5, 0) 5.8% 4.0% 1.5 2.2% 6.5%

(2.2,�1.85, 2.25, 0.02) 6.1% 5.0% 1.2 3.9% 9.7%

(2.2,�1.85, 2.25, 0.06) 7.2% 8.1% 0.9 9.3% 19.2%

Table 5: Performance statistics measuring return (µ), risk (�, CVaR and MDD)and Sharpe ratio (SR) for optimized portfolios for example sets of parameters.As ↵+ increases, risk statistics decreases. As the magnitude of ↵� increases, riskincreases. As � increases, risk decreases. As RP increases, risk initially decreasesbut then increases.

Return (µ) Risk (σ) Sharpe ratio

0,0

0,5

1,0

1,5

2,0

2,5

0%

5%

10%

15%

20%

25%

0 5 10

Shar

pe ra

tio

Ret

urn

& R

isk

α+

0,0

0,5

1,0

1,5

0%

5%

10%

15%

20%

25%

-10 -5 0

Shar

pe ra

tio

Ret

urn

& R

isk

α-

0,0

0,5

1,0

1,5

2,0

0%

5%

10%

15%

20%

25%

0 5 10

Shar

pe ra

tio

Ret

urn

& R

isk

β

0,0

0,5

1,0

1,5

0%

5%

10%

15%

20%

0% 5% 10%

Shar

pe ra

tio

Ret

urn

& R

isk

RP

Figure 15: The above charts show how the portfolio characteristics, return (µ),risk (�) and Sharpe ratio, are a↵ected when ↵

+, ↵�, � and the reference pointare varied, respectively. As ↵

+ increases, risk decreases. As the magnitude of↵

� increases, risk increases. As � increases, risk decreases. As RP increases,risk initially decreases but then increases. As each parameter is varied, the otherparameters are those of the median investor described in Section 2.1.3.

37

as bonds provide a stable return above the reference point. This will be rewarded by thevalue function, as returns lower than 2% (e.g. a bad stock market year) are punishedwhen the reference point is shifted. Thus, as long as there exists low risk assets thatgenerate returns above the reference point, the optimization will allocate more capitalto these assets. On the other hand, when the reference point is shifted to 6%, we haveto move out on the risk scale to even be able to achieve positive utility. Thus, as thereference point increases further, the optimization algorithm will allocate more capitalto risky assets. We can also observe how the Sharpe ratio, i.e. risk-adjusted return, isinversely related to risk, consistent with previous research (see e.g. Blitz & Van Vliet,2007). As the portfolio risk decreases, the Sharpe ratio increases. Further, these re-sults are all consistent with the results in Section 4.2.1 and consequently consistent withprospect theory and the preferences of the investor. Also we can note that none of theportfolios evaluated have a strictly worse risk-reward trade-o↵ than the others, whichindicates (but does not prove) that the optimization algorithm is producing e�cientportfolios.

4.3 Case study: Out-of-sample Testing on Real Investors

In this section, we will investigate how the optimal portfolios of four real investors be-haves in unknown market development. We will gather information and answers from theinvestors and optimize their portfolios over data for the period January 1999 to March2011. Then, we will analyze the performance of their portfolios during the following 5years, i.e. from April 2011 to March 2016.

Consider the four investors Investor A, Investor B, Investor C and Investor D. Theiranswers to the risk profiler questions are given in Appendix D. The answers for eachinvestor corresponds to the model parameters in Table 6. The final portfolio allocationsand their estimated and actual performance metrics are summarized in Figure 16.

Parameter A B C D

↵

+ 7.2 0.7 6.7 3.9

↵

� -7.9 -2.6 -5.0 -5.2

� 2.1 2.9 1.5 1.5

RP 0% 7% 5% 5%

CVaR 10% 13%

MDD 15% 13%

Table 6: Parameters and constraints for Investor A, B, C and D correspondingto their respective answers of the risk profiler. See Appendix D for the actualanswers.

We can conclude that no portfolio meet expectations when it comes to yearly return,which is much lower than expected for each portfolio. Also, the standard deviation(volatility) is lower for each portfolio. This is normally a good thing, since investorsgenerally dislike volatility. The reason for the lower standard deviation can probablybe derived for the extreme volatility during a part of the sample period, which includes

38

Emerging Markets

13%

Commodities14%

Swedish T-Bonds

73%

Investor A

Emerging Markets

29%

Swedish Equity40%

Commodities31%

Investor B

Swedish Equity17%

Alternative Investments

37%

Swedish T-Bonds

46%

Investor C

Swedish Equity20%

Commodities20%

European Credit60%

Investor D

Investor A Investor B Investor C Investor D

Metric Est. Out. Est. Out. Est. Out. Est. Out.

Yearly return 6.9% 2.0% 12.1% 0.8% 6.3% 3.1% 7.1% 2.8%

Standard deviation 4.2% 3.1% 17.6% 10.5% 5.3% 3.6% 6.4% 4.8%

Sharpe ratio 1.6 0.6 0.7 0.1 1.2 0.8 1.1 0.6

Maximum drawdown 4.6% 7.4% 38.0% 20.9% 15.0% 5.7% 13.5% 8.1%

Figure 16: Allocation, estimated performance metrics and the actual outcomefor suggested portfolios for Investor A, B, C and D.

the financial crisis 2008/2009. The maximum drawdown is lower than the estimatedmaximum drawdown for Investors B, C and D (the case of Investor A will be addressedbelow). This is an important result that indicates that the model produces portfolioallocation that satisfy the investor’s preferences when it comes to downside risk.

The question remains whether this is positive or negative results. The problem withportfolio performance in out-of-sample methods is that the allocation is optimized fora known period of time. Then the portfolio is expected to perform well in unchartedterritory, i.e. during unknown market development. It is virtually impossible that theoptimal portfolio for the sample period is also optimal for the out-of-sample period.Thus, to evaluate to what extent the final allocation suit the investors and their prefer-ences, a humble approach may be to analyze and discuss the outcome from a qualitativeperspective.

As we already concluded, the yearly return did not meet expectations. However, fromthe beginning, the investors were aware that their investments were associated with risk.It is more relevant to determine if the investors are still comfortable with their portfoliosand their development. Ceteris paribus, the lower standard deviation is something theinvestors should be comfortable with, since investors generally dislike volatility. Also,the out-of-sample maximum drawdown is lower than estimated for Investor B’s, C’s, andD’s portfolios. The maximum drawdown for Investor A is higher than expected, but stilllower than Investor A’s expected maximum drawdown (10%, see Appendix D). Thus,considering the downside risk for the portfolios, the investors should all be comfortablewith their investments (even though they may not be satisfied). The conclusion is thateven though the portfolios did not yield the expected return, they are still suitable foreach investor.

39

4.4 Reliability & Validity

The quality of our method is determined by its reliability and validity. A reliable riskprofiler measures consistently, with known accuracy. As the required answers generallyare of a quantitative nature, the reliability of this risk profiler is deemed relatively high.The answers provided by investors are precise and will correspond to a particular pref-erence or behavior in a certain situation. Then, they are converted into parameters andincorporated in the piecewise quadratic value function in order to model an investor’srisk preferences. This process is a purely mathematical exercise and will consequentlygive accurate results. Also, the same answers to the risk profiler will always yield thesame parameter values and corresponding value function. Hence, we can conclude thatthe process of gathering answers, converting answers into parameters and incorporatingthem into the value function is a very reliable method that should generate consistentoutcomes for any given investor.

A valid risk profiler measures what it claims to measure. We make a bold move intrying to explain the full spectrum of an investor’s risk ability, risk preferences and riskperception with twelve simple questions. As a matter of fact, we determine an investor’sparticular preferences or behavior only in certain situations and generalize the resultsto apply in all situations. Hence, it is reasonable to argue that the validity of the riskprofiler may be low. As risk profiler designers, however, a trade-o↵ we have to make isto balance between the overall quality and the length of the profiler. By constructingquestions that are relevant and realistic, we believe our generalization of the investor’sbehavioral aspects is as accurate as possible. Further, our method leans on vast researchin e.g. the area of prospect theory, where investor’s risk preferences and behavior havebeen found to be well explained by the value function, which strengthens its validity. Wehave, based on utility theory, shown how we interpret answers from the risk profiler andwhy they should a↵ect the parameter values and hence shape the value function as theydo. In sum, when trying to describe human behavior one always has to make assumptionsand simplifications. We have outlined the logic behind our choices to increase the validityof our results.

4.5 Practical Considerations

For the practitioner who is interested in incorporating the results of this thesis in theiradvisory process, there are several practical issues to consider. The most prominent iswhether to use a fully automated service integrated in an online platform or incorporateit as a complement in the current advisory process. A fully automated service will havethe ability to reach a broader audience but keeping the human touch in the process willmake it easier to retain high service quality. Other considerations will be how to handlereallocation of the client’s assets over time, and whether a new optimization should beconducted at every reallocation or only at the time of investment.

Another very important issue that will arise is about the optimized allocation producedby the model. Many of the resulting allocations in this thesis could be di�cult tomotivate for clients or di�cult for the advisor to provide for some reason. For example,we believe many investors will question the rationale behind a portfolio consisting of73% Swedish T-Bonds, 14% commodities and 13% emerging markets equity, especiallyin today’s economic environment. To approach this, one option could be to provide a

40

fixed set of allocations and evaluate these with the client’s individual value function.Or if the advisor have a series of standardized products, it could be possible to letthe model produce a linear combination of these products, i.e. optimize the weightingbetween products instead of individual assets. Thus, the advisor will be able to provideportfolios that are comfortable for both the advisor and the client, and easy to motivatefor the client – while still accounting for her preferences and behavioral aspects. Theseapproaches also reduces the complexity of the model.

5 Conclusion

With robo-advisors and regulation eventually changing the market conditions of the fi-nancial advisory industry, traditional advisors will have to adapt to a new world of assetmanagement. Thus, it is of interest to traditional advisors to further explore the topic ofhow to automatically evaluate soft aspects such as client preferences and behavior, andtransform it into portfolio allocations while retaining stringency and high quality in theprocess. Consequently, the purpose of this thesis has been to create an automated assetallocation model, combining the concepts of portfolio theory and prospect theory. Apart of that purpose was to design a client assessment method that accurately translatesclient behavior and preferences into quantitative measures for the allocation model.

In order to answer our first research question, we have shown how client preferences andbehavioral aspects can be translated into quantitative parameters, suitable for an assetallocation model. A risk profiler, a type of questionnaire, was found to be an appropriatetool to use in this process. An investor can answer questions given in the profiler, andher answers can be converted into quantitative parameters. We chose to keep the riskprofiler short and concise, though su�cient to gain information from which the param-eter values could be extracted. The parameters were used to calibrate a value functionthat describes investor preferences. The value function is used to evaluate portfolio allo-cations in an optimization problem that seeks to determine an allocation that matchesthe investor’s preferences and behavioral aspects.

In order to answer our second research question, we have shown that the impact of theparameters on the resulting allocations is consistent with prospect theory and the pref-erences of the investor, by investigating how the allocation and portfolio characteristicschange when parameters are varied. A more risk averse answer in the profiler, whetherit describes risk aversion in the positive or negative domain, or loss aversion, resultsin a portfolio that allocates more capital to low-risk assets, such as bonds (comparedto equity), and vice versa. Performance metrics (standard deviation, CVaR, MDD) forportfolios show the same relationship – risk averse answers in the profiler results in port-folios with lower values of the performance metrics, i.e. a less risky portfolio, and viceversa.

In order to answer our third research question, we have shown that portfolio risks werewithin expected levels for all portfolios as we performed out-of-sample case analyses forreal investors. While some asset classes performed poorly and some well, no risk limitwas breached for the portfolio as a whole, which indicates that the model produces al-locations suitable to the investors. However, it is hard to draw any conclusions from

41

this, as models based on historical data are always biased by performance during theestimation period. The returns of assets in the estimation period is not necessarily rep-resentative of the future.

We can conclude that the final allocation generated by the model should suit the investor,under the assumption that the investor answers the questions in the risk profiler accordingto her actual preferences. The questions have been designed with the intention to besimple to understand and require no or little investment experience. Questions relatedto risk perception has been included to validate investors’ answers and moderate riskperception biases if necessary. Hence, we find it reasonable to assume the investor’sanswers are consistent with her actual preferences. Of course, whether the assumptionholds partly depends on the validity and reliability of the risk profiler and partly on thequality of the sample answers.

5.1 Further Research

In this study, we have focused on combining concepts from traditional financial advisory,portfolio theory and behavioral finance to develop an automated asset allocation model.How this model, more specifically, will generate value in a traditional financial advisoryfirm i.e. how it will be implemented in the business, priced, who the target customeris, etc., is an interesting topic of further research. Moreover, in the process of providingadditional value, a firm might be interested in incorporating their own market views inthe automated model by adding additional complexity to the model, e.g. as suggestedby Black and Litterman (1992).

Concerning the statistical method, we employed empirical historical simulations to gen-erate data that the optimization is based on. Another possibility would be to estimatea parametric distribution of returns and draw outcomes from that distribution. Thelatter approach makes distributional assumptions rather than drawing from historicaloutcomes, which might generate di↵erent results. Also, parametric distributions couldoften reduce required computational power.

The questionnaire used in our model serves its purpose well and is useful for shapingthe value function in order to provide an allocation model that takes risk preferencesand behavioral aspects into account. However, since we have combined knowledge fromdi↵erent academic disciplines in order to create our model, the questionnaire itself isstill open for further research. A more thorough investigation of how preferences andbehavioral aspects correspond to answers to questions is an interesting topic of researchfor anyone interested in the more psychology related subjects within behavioral finance.

42

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Appendices

A Empirical Value at Risk

The Value at Risk (VaR) at level p 2 (0, 1) of a portfolio with value X at time 1 is

VaRp

(X) = min{m : P(mR0 +X < 0) p}.That is, VaR

p

(X) is the value such that the probability that the loss on the portfolioover the given time horizon exceeds this value is p. VaR may also be described by thefollowing expression

VaRp

(X) = F

�1L

(1� p).

where the loss at time 1 is L = �X/R0, R0 is the risk-free rate and F

�1L

is the inverseof the distribution function F

L

.

To establish an empirical estimate of the VaR, consider the sample L1, ..., Ln

of inde-pendent copies of L. The empirical estimate of VaR

p

(X) is

dVaRp

(X) = L[np]+1,n

where L1,n � ... � L

n,n

is the ordered sample. I.e. the VaR is nothing more than theempirical (1� p)-quantile of L

k

(Hult, Lindskog, Hammarlid, & Rehn, 2012).

B Empirical Conditional Value at Risk

The Conditional Value at Risk (VaR) at level p 2 (0, 1) of a portfolio with value X attime 1 with risk-free rate R0 is

CVaRp

(X) =1

p

Zp

0

VaRu

(X)du, L = �X/R0.

To establish an empirical estimate of the CVaR, we replace VaRp

(X) by its empirical

estimator dVaRp

(X) = L[np]+1,n (see Appendix A), where L1,n � ... � L

n,n

is the orderedloss sample. Then the empirical estimator of the CVaR is

dCVaRp

(X) =1

p

Zp

0

L[nu]+1,ndu =1

p

[np]X

k=1

L

k,n

n

+

✓p� [np]

n

◆L[np]+1,n

!

(Hult, Lindskog, Hammarlid, & Rehn, 2012).

C Maximum Drawdown

Consider the random process X = (X(t), t � 0) with X(0) = 0. The (current) drawdownat time T , denoted D(T ), is defined as

D(T ) = max

⇢0, max

t2(0,T)X(t)�X(T )

�.

The maximum drawdown, denoted MDD(T ), is the maximum of the drawdown D(T )during the entire history of the variable. Formally,

47

MDD(T ) = max⌧2(0,T)

max

t2(0,⌧)X(t)�X(⌧)

�.

D Case: Investors’ Risk Profiler Answers

Questions: Investor A Answer

(1) What is the amount you intend to invest? 1 000 000

(2) Of this amount, how much do you require for future expense (e.g.business needs, lifestyle, and unexpected family events)?

300 000

(3) What is your main objective with this investment? Capital preser-vation

(4) Over how many years do you intend to invest? 7 years

(5) What yearly return are you aiming to achieve with this investment? 3%

(6) In a worst case scenario, what is the greatest loss you expect thisinvestment could incur?

10%

(7) All of the asset classes (see. Table 4) will be considered for yourportfolio. Please indicate if you have a strong feeling that you do notwant to hold more than a certain minimum/maximum investment in aparticular asset class.

Nopreferences

(8) Suppose you have a 50% chance of gaining 20% (100 000 SEK) on yourinvestment in a year, while otherwise you gain nothing. What sure gainwould you prefer over this opportunity?

37 000

(9) Under a year of unfavorable market conditions, what sure loss wouldyou be willing to take in order to avoid the uncertainty of losing 20% (100000 SEK) or losing nothing with equal probability?

35 000

(10) Consider a strategy that has a 50% chance of gaining 20% (100 000SEK) on your investment in a year. In the other 50% cases, what is themaximal loss would you find acceptable in order to follow the strategy?

35 000

(11) In 1 out of 20 years, unfavorable market conditions may cause yourportfolio to perform extraordinarily bad. What is the greatest loss youexpect in such a scenario to still be comfortable with your investment?

(12) What do you feel is the maximum loss in any time period that youwould be comfortable with?

Table 7: The risk profiler questions 1-12 and corresponding answers for InvestorA. The answers provided by the investor are used to calibrate the value functionin Section 3.2 and constitute constraints in the optimization problem describedin Section 3.3.2.

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Questions: Investor B Answer

(1) What is the amount you intend to invest? 6 800 000

(2) Of this amount, how much do you require for future expense (e.g.business needs, lifestyle, and unexpected family events)?

0

(3) What is your main objective with this investment? Above averagegrowth

(4) Over how many years do you intend to invest? 10 years

(5) What yearly return are you aiming to achieve with this investment? 10%

(6) In a worst case scenario, what is the greatest loss you expect thisinvestment could incur?

40%

(7) All of the asset classes (see. Table 4) will be considered for yourportfolio. Please indicate if you have a strong feeling that you do notwant to hold more than a certain minimum/maximum investment in aparticular asset class.

Min40% SwedishEquity

(8) Suppose you have a 50% chance of gaining 40% (2 720 000 SEK) onyour investment in a year, while otherwise you gain nothing. What suregain would you prefer over this opportunity?

1 250 000

(9) Under a year of unfavorable market conditions, what sure loss wouldyou be willing to take in order to avoid the uncertainty of losing 40% (2720 000 SEK) or losing nothing with equal probability?

750 000

(10) Consider a strategy that has a 50% chance of gaining 50% (2 720 000SEK) on your investment in a year. In the other 50% cases, what is themaximal loss would you find acceptable in order to follow the strategy?

1 000 000

(11) In 1 out of 20 years, unfavorable market conditions may cause yourportfolio to perform extraordinarily bad. What is the greatest loss youexpect in such a scenario to still be comfortable with your investment?

(12) What do you feel is the maximum loss in any time period that youwould be comfortable with?

Table 8: The risk profiler questions 1-12 and corresponding answers for InvestorB. The answers provided by the investor are used to calibrate the value functionin Section 3.2 and constitute constraints in the optimization problem describedin Section 3.3.2. The last two questions are not answered since Investor B iscomfortable with the initial allocation.

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Questions: Investor C Answer

(1) What is the amount you intend to invest? 500 000

(2) Of this amount, how much do you require for future expense (e.g.business needs, lifestyle, and unexpected family events)?

200 000

(3) What is your main objective with this investment? Capital growth

(4) Over how many years do you intend to invest? 5 years

(5) What yearly return are you aiming to achieve with this investment? 4%

(6) In a worst case scenario, what is the greatest loss you expect thisinvestment could incur?

15%

(7) All of the asset classes (see. Table 4) will be considered for yourportfolio. Please indicate if you have a strong feeling that you do notwant to hold more than a certain minimum/maximum investment in aparticular asset class.

No EmergingMarkets, NoCommodities

(8) Suppose you have a 50% chance of gaining 15% (75 000 SEK) on yourinvestment in a year, while otherwise you gain nothing. What sure gainwould you prefer over this opportunity?

22 000

(9) Under a year of unfavorable market conditions, what sure loss wouldyou be willing to take in order to avoid the uncertainty of losing 15% (75000 SEK) or losing nothing with equal probability?

27 000

(10) Consider a strategy that has a 50% chance of gaining 15% (75 000SEK) on your investment in a year. In the other 50% cases, what is themaximal loss would you find acceptable in order to follow the strategy?

30 000

(11) In 1 out of 20 years, unfavorable market conditions may cause yourportfolio to perform extraordinarily bad. What is the greatest loss youexpect in such a scenario to still be comfortable with your investment?

50 000

(12) What do you feel is the maximum loss in any time period that youwould be comfortable with?

75 000

Table 9: The risk profiler questions 1-12 and corresponding answers for InvestorC. The answers provided by the investor are used to calibrate the value functionin Section 3.2 and constitute constraints in the optimization problem describedin Section 3.3.2.

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Questions: Investor D Answer

(1) What is the amount you intend to invest? 750 000

(2) Of this amount, how much do you require for future expense (e.g.business needs, lifestyle, and unexpected family events)?

350 000

(3) What is your main objective with this investment? Capital growth

(4) Over how many years do you intend to invest? 15 years

(5) What yearly return are you aiming to achieve with this investment? 6%

(6) In a worst case scenario, what is the greatest loss you expect thisinvestment could incur?

15%

(7) All of the asset classes (see. Table 4) will be considered for yourportfolio. Please indicate if you have a strong feeling that you do notwant to hold more than a certain minimum/maximum investment in aparticular asset class.

No EmergingMarkets, NoSwedish T-Billsor T-Bonds,max 20% Com-modities

(8) Suppose you have a 50% chance of gaining 15% (112 500 SEK) on yourinvestment in a year, while otherwise you gain nothing. What sure gainwould you prefer over this opportunity?

45 000

(9) Under a year of unfavorable market conditions, what sure loss wouldyou be willing to take in order to avoid the uncertainty of losing 15% (112500 SEK) or losing nothing with equal probability?

40 000

(10) Consider a strategy that has a 50% chance of gaining 15% (112 500SEK) on your investment in a year. In the other 50% cases, what is themaximal loss would you find acceptable in order to follow the strategy?

70 000

(11) In 1 out of 20 years, unfavorable market conditions may cause yourportfolio to perform extraordinarily bad. What is the greatest loss youexpect in such a scenario to still be comfortable with your investment?

100 000

(12) What do you feel is the maximum loss in any time period that youwould be comfortable with?

100 000

Table 10: The risk profiler questions 1-12 and corresponding answers for InvestorD. The answers provided by the investor are used to calibrate the value functionin Section 3.2 and constitute constraints in the optimization problem describedin Section 3.3.2.

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