Prospect Theory in the Automated Advisory Process

Prospect Theory in the

Automated Advisory Process

Prospect Theory in the

Automated Advisory Process

Jonatan Werner

Jonas Sjöberg

Master of Science Thesis INDEK 2016:58

Prospect Theory in the

Automated Advisory Process

Jonatan Werner

Jonas Sjöberg

2016-06-01

Gustav Martinsson

Tomas Sörensson

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

Examensarbete INDEK 2016:58

Prospektteori i en automatiserad

rådgivningsprocess

Jonatan Werner

Jonas Sjöberg

2016-06-01

Gustav Martinsson

Tomas Sörensson

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,

Acknowledgements

First and foremost, we would like to thank our supervisor Tomas S¨orensson 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 The Royal Institute of Technology whose helpful input, criticism and ideas have been much appreciated.

Stockholm, June 2016

Contents

1 Introduction 3

1.1 Background . . . 3

1.1.1 The Current Market for Robo-Advisors . . . 4

1.2 Problem Statement . . . 5 1.3 Research Questions . . . 7 1.4 Purpose . . . 7 2 Literature Review 8 2.1 Decision Theory . . . 8 2.1.1 Mean-Variance Analysis . . . 8

2.1.2 Expected Utility Theory . . . 10

2.1.3 Prospect Theory . . . 11

2.2 Behavioral Biases . . . 13

2.3 Assessing Investor Preferences and Behavioral Aspects . . . 16

2.3.1 The Risk Profiler . . . 17

2.3.2 General Considerations and Common Pitfalls . . . 20

2.4 Summing Up . . . 20

3 Methodology 21 3.1 Designing the Risk Profiler . . . 22

3.1.1 Risk Ability: Determining Investment Constraints . . . 22

3.1.2 Risk Preference: Determining Parameter Values . . . 23

3.1.3 Risk Perception: Ensuring Behavioral Coherence . . . 24

3.1.4 The Final Risk Profiler . . . 24

3.2 The Value Function . . . 26

3.3 Modeling Approach . . . 27

3.3.1 Data Set . . . 28

3.3.2 Portfolio Optimization . . . 29

4 Results & Analysis 30 4.1 The Impact of Risk Profiler Answers on Parameters . . . 30

4.2 The Impact of Parameter Values on Portfolio Characteristics . . . 33

4.2.1 Optimal Allocations . . . 33

4.2.2 Performance Metrics . . . 36

4.3 Case study: Out-of-sample Testing on Real Investors . . . 38

4.4 Reliability & Validity . . . 40

4.5 Practical Considerations . . . 40

5 Conclusion 41 5.1 Further Research . . . 42

Appendices 47 A Empirical Value at Risk . . . 47

B Empirical Conditional Value at Risk . . . 47

C Maximum Drawdown . . . 47

List of Figures

1 List of the Largest Robo-Advisors . . . 4

2 Estimated Yearly AUM of Robo-Advisors . . . 5

3 The Efficient Frontier, Market Portfolio and Capital Market Line . . . 9

4 The Value Function . . . 12

5 The Probability Weighting Function . . . 14

6 Visualization of Pompian’s Principles I and II . . . 17

7 Return and Risk for Considered Asset Classes . . . 29

8 The Impact of Profiler Answers on ↵+ and ↵ . . . 31

9 The Impact of Profiler Answers on . . . 32

10 The Impact of Parameter Values on the Value Function . . . 32

11 The Impact of on Portfolio Allocation . . . 34

12 The Impact of ↵+ _{on Portfolio Allocation . . . 34}

13 The Impact of ↵ on Portfolio Allocation . . . 35

14 The Impact of RP on Portfolio Allocation . . . 36

15 The Impact of Parameter Values on Portfolio Return and Risk . . . 37

16 Case: Allocations and Performance . . . 39

List of Tables

2 The Final Risk Profiler . . . 25

3 Initial Allocation Portfolio Characteristics . . . 27

4 Data Set . . . 28

5 Performance Statistics . . . 37

6 Case: Investors’ Parameters and Constraints . . . 38

7 Case: Investor A Answers . . . 48

8 Case: Investor B Answers . . . 49

9 Case: Investor C Answers . . . 50

1

Introduction

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

1.1

Background

Investing in financial markets and allocating assets is for many investors a tiresome and difficult process. Financial advisory and wealth management has for long been a valuable service 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 regulation causes difficulty in running an advisory business. Financial advisors are worried this will increase costs of running the service, thus making it available only to very wealthy individuals. Clients with less wealth risk becoming unadvised on their personal financial issues.

During the global financial crisis 2007, investors lost approximately$8 billion from

mak-ing impulse financial decisions, accordmak-ing to Winchester et. al. (2011). Moreover, the authors claim that investors purchasing advisory services are fifty percent more likely to adhere with their long time financial plan than investors who do not. Also, investors with a written financial plan are twice more likely to make optimal long-term investment decisions. Hence it is important to note that the initial allocation of a portfolio is only one part of the investment process, maintaining the long-term strategy is just as impor-tant but also very difficult. This fact emphasises the importance of having an initial allocation avoiding the investor being uncomfortable with her investments, leading to irrational decisions.

In recent years, technology has paved the way for a new type of investment advisory service. 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 make robo-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 intensifies the interest in the topic, since robo-advising is not as strictly regulated and not regarded an advisory service in the same way that traditional advisory is (Financial Conduct Au-thority, 2016).

finance concepts with mathematical tools for portfolio optimization in a rigorous and more 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 of the current market situation for robo-advisors and its impact on the future of wealth management. The robo-advisor industry is growing rapidly and some compare it to the 1990s travel industry, when the travel agent model lost ground to online services. This time, technology has opened up for a business model relying on the combination of the basic components of a wealth management o↵ering together with simple user interfaces with relevant content, automated technology, greater transparency and lower pricing. The robo-advisors are here to stay, so traditional players need to determine if, or rather how, 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 other

words, 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), there are indications of significant influence on the wealth management landscape. Several

robo-advisors already have more than$1 billion under management. A non-exhaustive

list of some of the largest robo-advisors, based on their AUM, are presented in Figure 1. The current market leaders are Betterment, Wealthfront and Personal Capital with

collective AUM of $7.1 billions (Karz, 2015). However, the market is still young and

most entrants struggle to be profitable and are still reliant on venture capital funding (Chappuis Halder & Co, 2015).

3 003 2 613 1 518 686 600 245 157 87 70 51 31 21 19 6 5 3 2 0 500 1000 1500 2000 2500 3000 Betterment Wealthfront PersonalCapital AssetBuilder FutureAdvisor Rebalance IRA Blooom Acorns SigFig Smart401k Covestor WiseBanyan Hedgeable Marketriders TradeKing Advisors Invessence Upside Advisor Millions

The future market situation for robo-advisors is all but certain. However, based on the influence they have already had on the wealth management industry, with traditional firms embracing the technology in one way or another, many agree that robo-advising will grow. Growth will origin partly from the shift of already invested assets to robo-advisory services, and partly from non-invested assets, i.e. from new investors. Estimates suggest that 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 in Figure 2. 0,3 0,5 0,9 1,5 2,2 0 0,5 1 1,5 2 2,5

2016E 2017E 2018E 2019E 2020E

Tr

ill

ion

s

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). Traad-ditional wealth managers will eventually have to face the challenges of a new market situation and have mainly three ways to address this issue – partner with an existing robo-advisor, acquire one or develop an in-house solution (Vincent, Laknidhi, Klein, & Gera, 2015). Some wealth managers 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, some services, 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 profitability of 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 situation might prove disastrous.

1.2

Problem Statement

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

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

There are many client assessment methods readily available, but incorporating one into an automated asset allocation model puts extra demand on the method. Further, it must still comply with current legislation and take risk preferences and behavior into account. 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, so not to generate allocations diverging from the client’s preferences. Linciano & Soccorso (2012) conclude that in practice, ”most questionnaires are not aligned with the economic and psychological literature”. However, a few di↵erent attempts to address this issue have been made. For example, Grable and Lytton (1999) develop a multidimensional risk-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 preferprefer-ences is that when optimizing mathematically, the broad spectrum of di↵erent risk preferences is one-dimensionalized in the model, to a single score number. This means that the model will 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.

automated advisory environment.

If these issues are resolved, the advisors still need to contribute with something in the process that cannot be provided by competing robo-advisors. This will be vital to stay competitive 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 establishing an automated asset allocation process that incorporates client behavior and preferences in 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 quantitative parameters, 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 process that is entirely automated. Further, the purpose is to develop an investor assessment method that accurately translates investors’ behavior and preferences into quantitative measures for an allocation model. The main idea is thus to integrate existing models and theories within e.g. portfolio theory and behavioral finance in a new way, to better suit the complex needs of a firm striving to o↵er automated yet high quality financial advice to its clients. The model should accurately provide allocations that the clients are comfortable with regardless of changing market conditions.

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

2

Literature Review

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

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 involving risk 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 unique risk-reward characteristics. This is a simple way of understanding the dynamics of the logical 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 applied to portfolio allocation, in order to gain further insight in their usefulness for automated asset 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 widely used (and criticized!) by practitioners. In Section 2.1.2, we present Expected Utility Theory, and in Section 2.1.3 Prospect Theory, which is developed from Expected Utility Theory 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). Among the basic assumptions in this theory are that investors are rational, that returns from securities are normally distributed and that investors avoid (unnecessary) risk if possi-ble. While these assumptions later have been challenged by findings in the behavioral finance field (see e.g. Kahneman & Tversky, 1979), they still provide a useful foundation for mathematical analysis and the derivation of quite elegant results that are still used today.

The equation formulated by Markovitz is known as mean-variance optimization and looks as follows: minimize w w T_⌃w subject to wTR = µ X wi = 1

optimal portfolio allocations given a certain level of risk (volatility, which is the standard deviation of the portfolio return). Consequently, according to MPT, there is a unique optimal 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 efficient frontier. The efficient frontier is given by the return of the optimal portfolio allocation for any given level of risk. The efficient frontier is a concept in Modern Portfolio Theory introduced by Harry Markowitz in 1952. The linear combination of the market portfolio, with expected return µ and volatility , and the risk-free asset is called the Capital Market Line (CML).

The value of diversification is an essential result of the mean-variance analysis, and was extended by Tobin (1958). Introducing a risk-free asset, one assumes that investors

can borrow and lend at a risk-free interest rate, rf. From this, Tobin derives the

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

µ rf

. (1)

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 being random but with known probability, or the probabilities are formed as beliefs by the decision maker. In the latter case, a second degree of uncertainty occurs, the uncertainty of 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 good decision and a lucky decision (since under randomness, any decision can have a favorable outcome, rational or not) and in order to describe these axioms we need to introduce the

concept of preferences (Von Neumann & Morgenstern, 2007). A preference x_{⌫ y means}

that 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 equally}

good. Now, the axioms of rationality can be represented as follows:

Axiom Assumption

Completeness Given two possible alternatives the individual has a

well-defined preference, either x y, x y or x⇠ y.

Transitivity x y, y z =) x z.

Continuity The preferences cannot exhibit erratic behavior such

as sudden ”jumps” caused by minor changes in the data.

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 an individual satisfy these axioms can be represented by an expected utility function.

If the axioms in Table 1 – completeness, transitivity, continuity and independence – are satisfied, they can be represented by an expected utility function, where the following

holds: 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 would give the same utility as participating in a given lottery. If for instance a lottery has an

expected payo↵ of $100, but the certainty equivalent is $90, the investor is risk averse

the classic economic theory of decreasing marginal benefit from consumption - here we consume 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 distinguishes prospect theory from expected utility theory is that value is assigned to gains and losses rather than to final wealth (Shefrin & Statman, 2000). Thus, the utility function of the expected utility theory is replaced by the value function that has three important properties:

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 a certain outcome is a gain or a loss depends on the individual’s reference point, which can 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 with basic principles of perception and judgement. When we respond to stimuli, we evaluate di↵erences in relation to a reference point based on past and present context, rather than absolute magnitudes. Consider for example brightness – your eyes may be very sensitive to light as you step out of a dark room but that same amount of brightness seem 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 given point, we are more accustomed to changes in attributes such as brightness, loudness, and temperature 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 context of gains and losses. They suggest decision-makers are risk averse in the face of gains and risk seeking in the face of losses. This element of prospect theory is known as diminishing

sensitivity 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 $1,100

gain (or loss) has a smaller impact (Barberis, 2013). This also implies that the value function 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 than to gains of the same magnitude, i.e. “losses loom larger than gains” (Kahneman & Tversky, 1979). A typical empirical estimate indicates that losses are approximately twice as painful than gains are pleasurable. This results in a value function that is steeper for losses than for gains (Barberis, 2013).

There are several possibilities to modelling the value function, which is a combination of utility functions. One example of a viable class of functions is the piecewise power

function of Tversky and Kahneman (1992). Let x represent the gain (loss) relative

to the reference point, ↵ represent the individual’s risk aversion and represent loss

Reference point x v( x)

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

her 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 gain

reaches 1

↵+ and from increasing after a loss reaches _↵1 . Thereafter, we assume a

con-stant utility of V (x) = 1

2↵+ and V (x) = _2↵1 , respectively. Note that for ↵+ > 0 and

convex for x < 0. This is consistent with the properties described above and visualized in 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 with prospect theory. The disadvantage is that the model does not o↵er di↵erent curvature parameters in the positive and negative domain, which the piecewise quadratic function o↵ers. Further, the piecewise power function does not provide robust solutions under a mean-variance optimization framework, something that is o↵ered by the piecewise

quadratic 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 value function will generate the same portfolio as a standard mean-variance investor (Hens & Bachmann, 2011), a property that might prove useful. According to the authors, the median investor with the piecewise quadratic value function has preferences described

by the parameter values ↵+_{= 2.15114, ↵ = 1.84688, = 2.25.}

Another departure from the expected utility theory framework is the replacement of probabilities by decision weights. Kahneman and Tversky (1979) found empirically that people 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 utility theory 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 in the asset allocation process, it may be quite useful in explaining actions and behavior of actors in financial markets (Hens & Bachmann, 2011). Tversky and Kahneman (1992) suggest that the psychological probability weight should be analytically calculated using the 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 may moderate or adapt to such biases in the advisory process. Even if the advisory process becomes fully automated, the importance of dealing with investors’ biases will remain and be a vital part of the investment process.

0 0.2 0.4 0.6 0.8 1 0 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 and Tversky (1992). In prospect theory, probabilities are replaced by decision weights. Generally, decision weights tend to be lower than the true probabilities, except for low probabilities.

and how to address them (Pompian, 2011). The importance of bias awareness and handling 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, most prevalent 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 have high daily returns which catches attention is a typical example. There is no evi-dence such stocks outperform the market. Further, overreaction to company news among investors is a consequence of availability bias. Indeed, stocks punished by news 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 on the setting. E.g. bond/equity ratio of a pension fund might depend on how many alternatives are provided in the two categories (Tversky & Kahneman, 1981). • Overconfidence: overestimating accuracy of predictions or own ability.

This makes them feel more secure than they are. As a consequence, individuals are ready 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 most expensive (Thaler, 1985).

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

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

Decision evaluation biases

• Regret aversion: one regrets more what one has done (error of commission) than what 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 specific biases listed above. They first point out that e.g. choices driven by risk or loss aversion are 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 to distinguish rational from irrational behavior. Then the advisor needs to help the client make rational decisions. The so called risk profiler, which will be discussed further in the following section, is at the heart of such a process.

may generate a slightly underperforming portfolio, but a portfolio that the client can be comfortable with. However, in some cases the best practical allocation may contradict a client’s natural psychological tendencies. Then, the client may be better served by accepting 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 biases in wealthier clients.

The intuition behind the first principle is to protect clients from devastating impacts on their personal finance. A portfolio that performs poorly because the allocation adapts too much to the biases of the client may constitute a serious threat to the personal finance of 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, adapting to 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 through information and advice. Emotional biases, on the other hand, build on emotions and are consequently much more difficult 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 of their biases and, from an advisor’s point of view, how to adapt the portfolio allocation with respect to the biases. Pompian, on the other hand, provides a scheme for advisors as to which biases are important to moderate and which the advisor can adapt to. The similarities 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 lead to irrational decisions. Pompian (2011) also provides a client satisfaction aspect, where emotional biases among wealthier clients can be adapted to as they are important to the client even though they might not maximize gains. For the purpose of making a client questionnaire for automated advisory, there are important takeaways from both these aspects. However, Hens and Bachmann (2011) provide more practical insights that can be directly implemented into a questionnaire.

2.3

Assessing Investor Preferences and Behavioral Aspects

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 principles suggest that advisors should moderate biases in less-wealthy clients and adapt to biases in wealthier clients. Also, advisors should moderate cognitive biases and adapt to emotional biases.

mathematically, the broad spectrum of di↵erent risk preferences is one-dimensionalized in the model, to a single score number. In Kahneman and Tversky’s (1979) prospect theory, they present a value function that provides asymmetric utilities and possibly di↵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 specific way 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 be connected 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 thorough assessment leading to a risk ”score”, the former show a variety of di↵erent questionnaires that are viable. It is evident that there are numerous ways of doing risk profiling or just measuring risk tolerance. Due to its appealing properties, we will further explore the concept of the risk profiler in the next section.

2.3.1 The Risk Profiler

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

From a client perspective, a good risk profiler will enable a better understanding of the client’s needs, behavior and preferences, and potentially produce a matching investment portfolio. 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 the advisor as it enables the advisor to build a deeper and more thorough relationship with clients. This is an important business aspect for financial advisory firms that want to build 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 to the contrary, a client’s risk preferences can be measured accurately by a questionnaire, provided that the questionnaire has been developed in accordance with psychometric principles (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 Markets in Financial Instruments Directive (MiFID) (Roszkowski, Davey, & Grable, 2005). The directive 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 degree of 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 the assessment of the client’s needs and preferences in order to ensure that the suggested asset allocation is suitable to the client and her investment objectives.

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

The risk ability is a purely financial parameter that describes the client’s capacity for loss or, in other words, her ability to absorb falls in the value of her investment. It may be useful to categorize the client’s capital which is necessary to keep up the client’s lifestyle or wealth which may enhance her lifestyle (Hens & Bachmann, 2011). Specifically, a loss of capital that would have materially detrimental e↵ect on her standard of living is very 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 the amount of risk she may take without jeopardizing her standard of living. Hence, it may be seen as a financial constraint.

deposit insurance actually guarantees up to EUR 100,000 for the investor if the bank would 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. In such a situation it might be suitable to implement a probability constraint into the allocation 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 risk measures, see appendices A and B). Both measure an amount which will not be lost with a set probability (e.g. 99%). VaR simply measures the 99-quantile value of the loss distribution, while CVaR measures the average loss conditional on the outcome being in 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 and thus preferable when implementing a risk ability constraint. An even more conservative risk measure is the maximum drawdown (see appendix C), which is the maximum loss from a peak to a trough of an investment, i.e. the worst possible outcome over a specific time period.

Risk preference

Unlike risk ability, which is a financial parameter, risk preference is a psychological parameter. A client’s risk preference describes how an individual feels about taking risk 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 stable over time. Sahm (2007) concludes that even though risk preference is subject to some systematic changes (e.g. declining with age and tendency to co-move positively with macroeconomic 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 economic circumstances of 2008. There was, however, clearly a change in people’s risk perception. This will be discussed in the next paragraph.

Risk perception

Individuals perceive risk di↵erently. Risk perception is the subjective judgement that people make about the characteristics and severity of a risk, and are often inconsistent with 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 the possibility of extreme events (Davey & Resnik, 2012). Generally, there is a tendency among investors to weigh losses more heavily than gains. Also, people tend to over-weight low probability events and to underover-weight high probability events (Roszkowski & Davey, 2010). Moreover, investors tend to underestimate risk in a rising market and overestimate it in a falling market (Davey & Resnik, 2012).

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 more than those that occurred in the past. Before applying a risk profiler, the client should be introduced to the risks and rewards of various asset classes over a long horizon. This is important in order to remove a potential bias where her risk preference will depend too much on her recent investment experiences (Pompian, 2011).

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

Validation questions Introducing validation question improves the quality of the risk profiler 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 the best condition for good advisory. However, some clients may be uncomfortable or not used 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 answerunderstand-ability, and their understand-ability to di↵erentiate between di↵erent client types (Roszkowski, Davey, & Grable, 2005).

Reliability A profiler that measures consistently, with known accuracy is considered reliable (Roszkowski, Davey, & Grable, 2005). When designing a profiler, it is important to 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 more difficult to achieve than reliability since complex behavior is determined by more than one factor (Roszkowski, Davey, & Grable, 2005). Particularly, questions which lead to an 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 assessed separately.

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 interest depending on the character and purpose of the profiler.

2.4

Summing Up

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

In Section 2.2 we outlined and discussed several biases and how financial advisors may moderate or adapt to such biases in the advisory process. Even if the advisory process becomes fully automated, the importance of dealing with investors’ biases will remain and 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 preferences and biases. We deem the risk profiler a useful tool for determining client preferences and behavior, 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 cognitive biases and, where possible, adapt to emotional biases. Further, it is more important to moderate biases for less wealthy clients. These insights will be valuable when designing the risk profiler.

Finally, in section 2.3, we review the concept of the risk profiler. The main features of the client’s risk profile are her risk ability, risk preferences, and risk perception. A risk profiler is a questionnaire which assesses these characteristics in a systematic way. We conclude 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 Markets in Financial Instruments Directive (MiFID). We conclude this section by providing a list of general considerations and pitfalls to be considered in the process of designing a risk profiler, according to the literature and authorities.

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

3

Methodology

quantitative measures a↵ecting the shape of the value function. Further, we also describe how 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 generate portfolios 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 the client’s risk ability, risk preference and risk perception. In an automated asset allocation process these steps will be of even greater importance, as the human interaction between client and advisor will be minimal. In this section, we describe the approach that will be used to gather the information necessary for the asset allocation model. We design a 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 puts an absolute limit to the downside risk, risk preference determines parameter values and the reference point, and risk perception validates that the investor’s risk preferences are coherent with her behavioral traits (it is difficult for an investor to precisely know her risk 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↵erent kinds of risk, where preferences can di↵er. The advantage of using such a simple model is of course the minimal time and e↵ort required from the investor. We deem a risk profiler more suitable, as it can measure di↵erent kinds of preferences, is fully customizable, and more 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 puts her somewhere on the efficient frontier (i.e. on a volatility constraint). We prefer an approach that better captures the individual risk preferences and behavior in the actual model.

3.1.1 Risk Ability: Determining Investment Constraints

As discussed in section 2.3.1, risk ability boils down to determining how much of her capital the investor is able to put at risk and further, several ways of implementing it into the allocation model. While the CVaR measure is more appealing from a theoreti-cal point of view, it can be practitheoreti-cal to simply hold a cash amount one is not willing to lose. 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 conservative nature.

3.1.2 Risk Preference: Determining Parameter Values

The risk preferences of a an investor may be described by the piecewise quadratic value function presented in section 3.2. To let the value function represent the preferences of a

particular investor is a matter of determining ↵+_{, ↵ and , where ↵}+_{and ↵ represent}

the 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 considers a 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 would prefer over the opportunity to gain a certain amount while otherwise breaking even with equal 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 certain amount, while otherwise breaking even with equal probabilities, can tell something about the client’s risk aversion to losses. Prospect theory suggests that attitudes toward risk di↵er in the context of gains and losses (Kahneman & Tversky, 1979). This is why it is important 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 positive domain is given by

x ↵

+

2 ( x)

2_{. (5)}

Building on the approach of Hens and Bachmann (2011), in order to determine ↵+ _we

can ask what certain gain, xc, the client would prefer to the opportunity of gaining an

amount xu, while otherwise breaking even with equal probabilities. This situation can

be modeled with the equation

xc ↵ + 2 ( xc) 2_{= 0.5( x} u ↵ + 2 ( xu) 2_{) (6)} which gives ↵+= xc 1 2 xu ( xc)2 2 ( xu)2 4 . (7)

The same approach can be used to determine ↵ for the negative domain where the

value is given by

✓

x ↵

2 ( x)

2◆_{. (8)}

We can ask what certain loss, xc, the client would prefer to the uncertainty of losing

an amount xu, while otherwise breaking even with equal probabilities. This can be

✓ xc ↵ 2 ( xc) 2 ◆ = 0.5 ✓ xu ↵ 2 ( xu) 2 ◆ (9) which gives ↵ = xc 1 2 xu ( 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 losses than to gains of the same magnitude. Thus, it is relevant to ask what size of a loss

xL is acceptable in order to gain a certain amount xG, with equal probabilities. This

situation can be modeled as xG ↵+ 2 ( xG) 2 ✓ xL ↵ 2 ( xL) 2 ◆ = 0 (11) which gives = xG ↵+ 2 ( xG) 2 xL ↵₂ ( xL)2 . (12)

3.1.3 Risk Perception: Ensuring Behavioral Coherence

The risk profiler also measures the investor’s risk perception in order to identify potential behavioral and cognitive biases which might influence the answers about risk ability and risk preference. This part of the profiler will consist of validation questions, and coherence between the validation questions and previous answers will help identify any possible biases. There are several approaches as to implementing and/or handling risk perception in an automated advisory service. Here we choose to take an educational approach – if the investor’s perception of risk is unrealistic the model will help her reconsider 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 to represent the di↵erent aspects of investor preferences and why. The questions are shown in Table 3.1.4, and have been numbered 1-12 for reference. Here, we will discuss and motivate the questions briefly.

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? Multiple

choice

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 expect

this investment could incur?

% Risk perception (7) All of the asset classes (see. Table 4) will be considered

for 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 sure

loss 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 to follow the strategy?

SEK Risk preference

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

SEK Risk preference

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

SEK Risk preference

Table 2: The final risk profiler consists of 12 questions that aim to measure an 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 and constitute 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 response rate 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 be considered an advisory situation by law. Thus the questions asked have been deemed sufficient, but could easily be complemented with more thorough client analysis if nec-essary, although that would not drastically change the model output.

thus important to include in the profiler. Question (4) determines time horizon, which is an 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 perceptions of risk which are sometimes unrealistic. Thus, question (5) and (6) investigates risk perception by asking what return the investor is expecting and what loss she expects to make in a worst case scenario, given the expected return. Further, the worst case loss is used later in the risk profiler as a benchmark level for gambles (which will be explained later 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 has any preferences with regards to holding specific asset classes. This will be considered in the model, as much as possible given the risk ability and expected return. In line with Pompian (2011), one can then depending on the client’s financial situation either adapt to, or moderate these preferences. Reasonably, the model will try to adapt as much as possible to the client’s preferences or home bias, as long as it does not compromise the risk-return level. Questions (8)-(9) ask the investor what sure gain/loss she would prefer 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 is also incorporated in prospect theory) as described in Section 2.1.2, where we are asking for the investor’s certainty equivalent of the gambles. Question (8) determines the cer-tainty equivalent in the positive domain whereas question (9) determines the cercer-tainty equivalent in the negative domain. If the certainty equivalent is less than the expected value of the gambles, the investor is risk averse and has a concave utility function, and vice 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 preliminary portfolio 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 have performed historically, e.g. during the financial crisis. It also compares the investor’s expectations to realistic expectations to show the investor how she perceives risk. After reading 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 order to slightly help moderating a biased risk perception. Question (11) will ask how much the investor is willing to risk losing under unfavorable market conditions 1 out of 20 years. This answer will be related to a CVaR constraint. Question (12) will ask for the maximum 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 a cognitive bias, we deem very suitable for an automated asset allocation model.

3.2

The Value Function

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 recommended

investment horizon of up to T1 years. You indicated your target yearly return to

be 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 loss in such a scenario is l% (L SEK).

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

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

During the financial crisis 2007/2008, the suggested portfolio would have incured a 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. The information is intended to describe how the initially suggested portfolio would have performed historically, e.g. during the financial crisis. It also compares the investor’s expectations to realistic expectations to show her how she perceives risk. 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-variance optimization as the latter assumes a completely rational investor. The value function, on the other hand, makes sure investor preferences and behavior are taken into account in the portfolio optimization, in line with our purpose. Recall the mathematical

repre-sentation of the piecewise quadratic value function in Equation 3, where x represents

the gain (loss) relative to the reference point, ↵+ _{and ↵ represent the investor’s risk}

aversion 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. In Section 3.2 we presented an investor’s value function. From the questions in the risk profiler 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,}

3.3.1 Data Set

Historical data for di↵erent asset classes will be represented by indices that are retrieved through Bloomberg’s database. All indices are gross dividends and are represented as monthly returns for the period January 1999 to March 2016. The motivation for the time period 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 the timeframe used. Thus, we believe an as large sample size as possible is good to prevent this e↵ect. Likely, the 17 years of data will contain some irregularities, and portfolios optimized 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 the portfolio still performs reasonably. The full set of asset classes, corresponding indicies and Bloomberg tickers under consideration are listed in Table 4. The average yearly return 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 and Bloomberg tickers.

The 207 monthly returns will be used to perform a historical simulation in order to produce a series of simulated yearly returns. We will use the approach outlined by Hult

et 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 componentwise

product of these vectors, denoted by RT

i . This procedure is repeated m = 1000 times

to obtain the sample RT

1, . . . , RTm of fictive return vectors over time periods of length

T . As long as the original return vectors are independent and identically distributed, an

assumption we will make here, then the vectors RT

1, . . . , RTm are identically distributed

but not independent (but conditionally independent given R n+1, . . . , R0). Many use

Swedish Equity Global Equity European Equity American Equity Emerging Markets Japanese Equity Alternative Investments Swedish T-Bills Swedish T-Bonds European Credits Commodities 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 0% 5% 10% 15% 20% 25% A ve ra ge Y ea rl y R et ur n

Risk (Standard Deviation)

Figure 7: Average yearly return and standard deviation (volatility) for each of the 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 is necessary to increase risk.

3.3.2 Portfolio Optimization

In order to capture the concepts of behavioral finance, we employ the behavioral value function and take a slightly di↵erent approach to the optimization problem. Given the parameter values, estimated as in Section 3.1, we proceed to find the portfolio weights maximizing 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 be

a vector of asset weights, Ri is a vector from our sample of fictive yearly returns, the

scalar 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 piecewise quadratic value function. The optimization problem becomes as follows: