Investors Facing Risk:

Splitting Money Between Risky and Risk-Free Assets

Erick W. Rengifo∗ Emanuela Trifan†

October 6, 2006

Abstract

This paper studies the impact of loss aversion on decisions regarding the alloca-

tion of wealth between risky and risk-free assets. We use a Value-at-Risk portfolio

model with endogenous desired risk levels that are individually determined in an

extended prospect theory framework. This framework allows for the distinction be-

tween gains and losses with respect to a subjective reference point as in the original

prospect theory, but also for the influence of past performance on the current per-

ception of the risky portfolio value. We show how the portfolio revision frequency

impacts investor decisions and analyze the role of past gains and losses in the cur-

rent wealth allocation. The perceived portfolio value exhibits distinct evolutions in

two frequency segments delimitated by what we consider to be the optimal revision

horizon of one year. Our empirical results suggest that previous research relying on

fixed confidence portfolio risk levels underestimates the loss aversion of individual

investors.

Keywords : prospect theory, Value-at-Risk, loss aversion, portfolio revision

JEL Classification: C32, C35, G10

∗Fordham University, New York, Department of Economics. 441 East Fordham Road, Dealy Hall,

Office E513, Bronx, NY 10458, USA, phone: +1(718) 817 4061, fax: +1(718) 817 3518, e-mail: rengi-

fomina@fordham.edu†Darmstadt University of Technology, Institute of Economics, Department of Applied Economics and

Econometrics, 15 Marktplatz, 64283 Darmstadt, Germany, phone: +49(0) 6151 162636, fax: +49(0) 6151

165652, e-mail: trifan@vwl.tu-darmstadt.de

1

1 Introduction

Optimal portfolio allocation models represent important tools in helping investors to de-

cide upon how to split their wealth among assets. The goal of such models is to find what

is called the optimal allocation, i.e. the one that maximizes expected portfolio returns at

a given risk level. One of the most well known and broadly used portfolio optimization

settings is the mean-variance model, which suggests the variance of portfolio returns as

risk measure. Recent research employs other ways of quantifying risk, such as the so called

Value-at-Risk (VaR), defined as the highest expected loss from financial investments over

a specified time horizon and subject to a certain confidence level.

Acting on the VaR-concept, Campbell, Huisman, and Koedijk (2001) develop a model

for maximizing expected returns subject to both a budget and a VaR constraint, where

the latter requires the maximum expected loss to meet an exogenously specified VaR-limit

(the so called desired VaR, shortly VaR*). One important feature of this model is that,

as in the classical mean-variance framework, the two fund separation theorem applies

namely, neither the investors’ initial wealth nor the desired VaR*, affect the maximiza-

tion procedure. Thus, investors interested in allocating wealth among risky assets can

first determine the risky portfolio composition, and then decide upon the extra amount of

money to be borrowed or lent (i.e. invested in risk-free assets), according to the individual

degree of risk aversion measured by the selected VaR*. In practice, the former decision

is often made by professional portfolio managers in charge of the construction of an op-

timal risky portfolio for their clients. These clients, usually non-professional investors,

concentrate sooner on the second decision step, by choosing the amount of money to be

invested in the risky portfolio as a whole, and implicitly by fixing the level of the risk-free

investment.1

In this context, the focus of our paper is on the decisions of non-professional investors.

The first and most important of these decisions refers to the formulation of VaR*. Its

value decides then on the optimal portfolio composition, hence on the sum of money to

be invested in risk-free assets. As mentioned above, finding the optimal risky portfolio

represents the task of professional portfolio managers and was extensively studied in

previous research on portfolio optimization. However, the resulting values of the total

risky vs. the risk-free investment as percentages of total wealth directly concerns non-

professional investors, becoming thus an object of study in the present work.

Our model builds on the portfolio optimization setting with exogenous desired VaR*

1In other words, non-professional investors consider the risky portfolio as exogenously given (fixed by

the manager). They are exclusively concerned with determining the final position in risky vs. risk-less

assets (i.e. how much money to put in the risky portfolio as a whole, while the rest is allocated to risk-free

assets), according to their own level of risk aversion.

2

in Campbell, Huisman, and Koedijk (2001) that we extend by explicitly accounting for the

formation of the individual VaR*-levels. They rely on the subjective perception of non-

professional investors over the risky portfolio performance and over utility in general. In

other words, we analyze how non-professional investors set their subjective VaR* and how

this (now endogenous) VaR* impacts on the wealth allocation between risk-free assets and

the risky portfolio. Our paper also incorporates the idea that individual risk perception

is affected by the previous evolution of financial wealth, so that facing past gains (losses)

induces a more (less) aggressive behavior, hence an increase (decrease) in risky portfolio

holdings. Thus, we present evidence for how different investment decisions of individual

investors can be interpreted as a consequence of different financial performance histories,

how these decisions change subject to the individual degree of loss aversion, and how our

results conform with previously documented findings.

Our paper comes in line with previous research aiming at studying the influence of

behavioral aspects on financial decisions. Benartzi and Thaler (1995) develop a plausible

explanation for the equity premium puzzle that relies on the interaction between loss

aversion and frequent portfolio evaluations, denoted as “myopic loss aversion”. Their

findings support the idea that, when investors review the performance of their portfolios

yearly, the resulting empirical equity premium is consistent with the loss aversion values

estimated in the standard prospect theory framework. Furthermore, Barberis, Huang, and

Santos (2001) apply the main concepts of the prospect theory to asset pricing, showing

that investors derive utility not only from consumption but also from variations in the

perceived value of financial investments. Moreover, investors not only distinguish between

gains and losses with respect to a subjective reference and warily avoid losses, as stated

in the original Kahneman and Tversky’s prospect theory, but also their perception over

losses appears to be affected by previous portfolio performance, i.e. by gains and losses

accumulated from past trades and referred as “cushions”.

We assume that, at the beginning of the trade, non-professional investors already

hold well diversified portfolios such as a market index (i.e. the empirical part of our

paper considers the SP500 index as proxy for the risky portfolio). Thus, the problem

they actually face reduces to the allocation of wealth between this risky portfolio (as a

whole) and the risk-free investment alternative. This separation of the risky and risk-

free investments complies with the concept of “mental accounting”, as first introduced

in Thaler (1980). According to Thaler (1992), people manifest the tendency to frame

(i.e. code and evaluate) outcomes in several non-fungible mental categories or accounts

(such as accounts for current income, current wealth or future income) with different

consumption propensities. This mental categorization decides upon the perceived utility

of those outcomes.

3

In our model, investors find the optimal solution to their decision problem by maxi-

mizing subjectively perceived utility. This utility is assumed to be derived merely from

changes in financial wealth.2 In line with the prospect theory (introduced in Tversky and

Kahneman (1979) and extended in Kahneman and Tversky (1992)), the perceived value

of risky investments is denoted as the prospective value. It relies on the subjective value

generated by one unit of risky project that is captured by the so-called value function.

Yet, in line with Barberis, Huang, and Santos (2001), we reconsider the original prospect

theory definition of the value function in order to account for the possible influence of past

performance. The idea that past gains and losses may change the current risk aversion,

hence financial decisions, is supported by an empirically observed phenomenon denoted

as the “house money effect” and documented in Thaler and Johnson (1990). Accordingly,

subjects who made money in past gambles appear to behave less risk aversely in subse-

quent bets. In other words, past gains make future losses less painful, while prior losses

may increase the risk aversion. A neurobiological explanation of this human reaction is

provided by the “somatic marker hypothesis” in Damasio (1994). Accordingly, preexist-

ing somatic (i.e. bodily) states can influence new ones by inducing modifications in the

level of activation (threshold) of the new state. As suggested in Bechara and Damasio

(2005), prior somatic states (in our case generated by past series of gains or losses) can

reinforce (impede) the perception of new ones (here, currently expected gains and losses)

by congruous (incongruous) valence (i.e. positivity/negativity). Also, even when prior

performance induces only weak somatic states (known as background states), it appears

to exert an impact on risk aversion. For instance, negative background states diminish

the risk aversion in face of sure losses (for that the fear of experience one more loss after a

series of past losses is higher and makes investors more risk loving in the hope of recovering

those losses), while positive background states enhance risk aversion in face of sure gains

(i.e. once several gains are experienced, investor predisposition to gambling diminishes).

In our framework, the desired Value-a-Risk (VaR*) is endogenously defined as the max-

imum expected loss perceived by individual investors and depends on past performance,

loss aversion, current value of the risky investment and the expected return premium.

We first compute the VaR* and then derive the desired level of investment in the risky

portfolio relative to the risk-free allocation. This allows us to draw a conclusion about the

investor risk aversion and to provide a comparison with the exogenous VaR* framework

in Campbell, Huisman, and Koedijk (2001).

We design two ways of assessing the prospective utility (which reduces in our setting to

2In other words, investors are interested only in the (perceived) value of their financial investments

(and not in other determinants of utility, such as consumption). This can rely on the fact that investors

narrow frame, i.e. put excessive emphasis on, the importance of financial investments and the utility they

generate. According to Barberis, Huang, and Thaler (2003), this is a common situation in practice.

4

the expected value of the risky investment). One definition relies on the original prospect

theory, and another one answers what we call a “worst case scenario”, where investors

are assumed to be concerned with the maximum sustainable (and not with the expected)

loss in the risky investment. Moreover, we study how investor decisions change according

to different market conditions, as captured by the prospect theory-part of the utility

function, and how different ways of representing loss aversion can influence utility. For

instance, we expect that risk averse investors reduce their risky holdings, shifting their

positions to more secure investment alternatives. Also, we derive equivalent significance

levels for VaR* at each trading time and compare them to the corresponding significance

levels used in the original model of Campbell, Huisman, and Koedijk (2001). In addition,

we introduce the notions of actual loss aversion and of global first order risk aversion that

we consider better suited than the simple coefficient of loss aversion to measuring the real

investor attitude to losses.

Finally, we study how investment decisions change under different portfolio revision

horizons (one day, one month, one quarter, four months, one year, two years, and up to

eight years), in other words how the revision frequency exerts influence on the risk percep-

tion and wealth allocation. In this context, we estimate the evolution of the prospective

value, as well as of the actual loss aversion and of the global first order risk aversion, as

functions of the revision frequency and address the problem of optimal revision horizons.

The theoretical findings in the first part of our paper are implemented and amended

in the subsequent empirical part. In so doing, we use real market data, such as the SP500

index as proxy for the risky portfolio and the US 10-year bond accounting for the risk-free

investment alternative.

Our empirical findings lead to several interesting conclusions. First, the risky holdings

of non-professional investors performing annual portfolio revisions represent on average

42% of the investors’ wealth and decrease to a value very close to zero for the revision

horizon of one day. Second, even when the coefficient of loss aversion remains constant

over trading dates, financial wealth fluctuations determined by the success of previous

decisions play a key role in the current portfolio allocation, Thus, when investors are

unable to accumulate positive significant cushions (calculated as the difference between

the risky return and the reference point of the value function represented here by the

risk-free return), most of their wealth is directed to the risk-free investment. Third, the

creation of positive and significant cushions is inversely related to the revision frequency.

As this frequency increases, the ability to accumulate profits decreases and a lower wealth

portion will be invested in risky assets. Fourth, as in Benartzi and Thaler (1995), our

results support the idea that one year appears to be the most plausible revision frequency

to be used by non-professional loss averse investors in practice. Further estimations show

5

that the evolution of the perceived portfolio utility (i.e. of the prospective value) for

different revision frequencies can be decomposed into two distinct intervals, namely one

for high frequency revisions (less than one year), and a second for low frequency revisions

(more than one year). The prospective value on the first interval can be analytically

represented as a quadratic convex function of the revision horizon, while on the second

segment, its evolution is rather upward sloping and of higher degree (yet, a simple line

represents a satisfactory approximation). A similar segmentation can be observed also

for the actual loss aversion and the global first order risk aversion. Fifth, the VaR*-levels

assessed from real data point out that, in practice, the loss aversion of real non-professional

investors may be higher than the values obtained for common confidence levels such as

90%, 95% and 99% in previous theoretical papers. Finally, the equivalent average loss

aversion coefficient computed for fixed confidence levels of 99% and 90% lie far below

the widely documented and empirically supported value of 2.25. Again, this implies that

previous research considering these confidence levels underestimates the aversion to losses

manifest in practice. We also note the inverse relationship between the revision frequency

and the associated loss aversion coefficient. For the maximal considered frequency of one

day, the coefficient of loss aversion becomes one, which implies that investors treat gains

and losses in the same way.

The remainder of the paper is organized as follows. Section 2 presents the main

theoretical considerations. We start by a brief review of the optimal portfolio selection

model with exogenous VaR* as in Campbell, Huisman, and Koedijk (2001), on which we

build our own theoretical structure. Section 2.2 takes on the reformulation of the value

functions in Barberis, Huang, and Santos (2001), out of which we derive what we define

as actual loss aversion. In Section 2.3 we introduce the notions of VaR* and of global first

order risk aversion and propose different ways of quantifying the endogenous VaR*. The

subsequent Section 2.4 frames distinct ways of assessing the value of the risky portfolio

as perceived by individual investors, while Section 2.5 treats the influence of a variable

revision frequency on the prospective value, the actual loss aversion and the global risk

aversion. Section 3 illustrates the empirical implementation of our theoretical model.

Section 3.1 discusses the impact of variable revision frequencies and of the cushion on the

evolution of wealth percentages invested in the risky portfolio. In Section 3.2, we analyze

the evolution of the prospective utility in time, as well as in the revision frequency domain.

Finally, Section 3.3 restates our model in terms of previous research with exogenous

VaR*, in that equivalent significance levels of portfolio risk and of loss aversion coefficients

that result from the average VaR* computed from our data and according to our model

equations are inferred. Section 4 concludes and provides an outlook for future research.

Graphics and further results are included in the Appendix.

6

2 Theoretical model

This section contains the main theoretical considerations of our work. We take up pre-

senting the model of portfolio selection with VaR as the risk measure and an exogenous

desired risk aversion (VaR*) of Campbell, Huisman, and Koedijk (2001). This model has

motivated us in extending the analysis for the case with endogenous VaR*. Subsequently,

we formulate our own model referring the individual perception of risky projects, detailing

the construction of the endogenous measure of risk aversion VaR* and its implications for

individual investor decisions. More precisely, we show how the investor desired VaR* can

be formulated and how it flows into the prospective value of the risky investment that

investors aim at maximizing. Also, we detail the definition of the real investor attitude to

losses, by introducing the notions of actual loss aversion and of global first order risk aver-

sion. Moreover, we analyze how the prospective value and these two further risk attitude

measures vary subject to different portfolio revision frequencies.

2.1 Optimal portfolio selection with exogenous VaR*

Let us first refresh the portfolio selection model with exogenous VaR* introduced in

Campbell, Huisman, and Koedijk (2001). Accordingly, financial assets are allocated by

maximizing the expected return subject to the common budget constraint, as well as to an

additional risk constraint, where risk is measured by the so-called Value-at-Risk (VaR).

The optimal portfolio is derived such that the maximum expected loss does not exceed

the VaR* indicated by non-professional investors, for a chosen investment horizon and at

a given confidence level. Additionally, investors can borrow or lend money at the fixed

market interest rate.

We denote by Wt the investor wealth at time t, by Bt the amount of money to borrow

(Bt > 0) or to lend (Bt < 0) at the fixed risk-free gross return rate Rf , and by VaR*

the individually desired VaR (specified later in this section). Let the risky portfolio

consist in i = 1 . . . n financial assets with single time t prices pi,t and define Ωt ≡ [wt ∈Rn :

∑ni=1 wi,t = 1] the set of portfolio weights at time t, such that xi,t = wi,t(Wt +

Bt)/pi,t represents the number of shares of the asset i contained in the portfolio at time

t. Obviously, the portfolio gross return at next trade (Rt+1) depends on the portfolio

composition at the current date wt. With the budget constraint:

Wt + Bt =n∑

i=1

xi,tpi,t = x′tpt, (2.1)

the value of the portfolio at t + 1 results in:

Wt+1(wt) = (Wt + Bt)Rt+1(wt)−BtRf . (2.2)

7

The VaR is defined as the maximum expected loss over a given investment horizon

and for a given confidence level 1-α, thus:

Pt[Wt+1(wt) ≤ Wt − VaR∗] ≤ 1− α, (2.3)

where Pt is the conditional probability on the available information at time t and VaR∗

is the the investor desired VaR level. Equation (2.3) represents the risk constraint that

(professional) investors have to take into account in addition to the budget constraint

(2.1), when searching for optimal portfolio weights.

The portfolio optimization problem can be now expressed in terms of the maximization

of expected portfolio returns Et[Wt+1(wt)], subject to both the budget restriction and the

VaR-constraint:

w∗t ≡ arg max

wt

(Wt + Bt)Et[Rt+1(wt)]−BtRf, s.t. (2.1) and (2.3). (2.4)

Here, Et[Rt+1(wt)] represents the expected return of the portfolio given the information

at time t.

The optimization problem may be rewritten in an unconstrained way, by replacing

(2.1) in (2.2) and taking expectations:

Et[Wt+1(wt)] = x′tpt(Et[Rt+1(wt)]−Rf ) + WtRf . (2.5)

Equation (2.5) points out that risk-averse investors are going to put a fraction of their

wealth in risky assets if the expected risky portfolio return is higher than the risk-free

rate Et[Rt+1(wt)] ≥ Rf .

Substituting (2.5) (before taking expectation) in (2.3) gives:

P [x′tpt(Rt+1(wt)−Rf ) + WtRf ≤ Wt − VaR∗] ≤ 1− α,

so that

P

[Rt+1(wt) ≤ Rf − VaR∗ + Wt(Rf − 1)

x′tpt

]≤ 1− α, (2.6)

defines the quantile qt(wt, α) of the distribution of portfolio returns for a given confidence

level 1-α or probability of occurrence (α).

Thus, the budget constraint can be restated as:

x′tpt =VaR∗ + Wt(Rf − 1)

Rf − qt(wt, α). (2.7)

Finally, substituting (2.7) in (2.5) and dividing by the initial wealth Wt, we obtain a

new expression to be maximized:

Et[Wt+1(wt)]

Wt

=VaR∗ + Wt(Rf − 1)

WtRf −Wtqt(wt, α)(Et[Rt+1(wt)]−Rf ) + Rf . (2.8)

8

Given that at moment t of maximization, Wt is known and Rf is fixed, the optimal

portfolio composition can be derived as:

w∗t ≡ arg max

wt

Et[Rt+1(wt)]−Rf

WtRf −Wtqt(wt, α). (2.9)

Equation (2.9) shows that, similarly to the traditional mean-variance framework, the

two fund separation theorem applies, i.e. neither the (non-professional) investor’s initial

wealth nor the desired VaR* affect the maximization procedure. In other words, investors

can first allocate wealth inside of the risky portfolio (i.e. among different risky assets)

and second fix the extra amount money to be borrowed or lent (i.e. invested in risk-free

assets). The latter reflects by how much the portfolio VaR varies according to the investor

degree of risk aversion, which is measured by the selected (desired) VaR* level. Replacing

(2.1) in (2.7), we further derive:

Bt =VaR∗ + VaR

Rf − qt(w∗t , α)

(2.10a)

VaR = Wt[qt(w∗t , α)− 1]. (2.10b)

2.2 The value function

Coming from the main ideas of the Campbell, Huisman, and Koedijk (2001)’s setting,

our model goes a step further by asking how individual investors set their desired level

of risk aversion VaR*. We elaborate on the construction of an endogenous VaR* and its

implications for the wealth allocation between risky and the risk-free assets.

Investors’ desires depend on their perception upon the value of financial investments.

Prospect theory suggests how individual perceptions of financial performance can be for-

malized by means of the so-called value function. According to Tversky and Kahneman

(1979) and Kahneman and Tversky (1992), to human minds, the actual carriers of value

are not absolute outcomes of a project, but deviations from an individual reference point,

where the deviations above (below) this reference are labelled as gains (losses). Thus,

the value function is kinked at the reference point and exhibits distinct evolution in the

domains of gains and losses, i.e. it is steeper for losses (a property known as loss aversion).

Also, it shows diminishing sensitivity in both domains (namely, it is concave for gains but

convex for losses).

As noted in Barberis, Huang, and Santos (2001), the view of the original prospect

theory over individual perceptions of risky investments can be enriched by accounting for

the potential impact of past performance (i.e. in addition to the mental distinction be-

tween gains and losses). Accordingly, the value function additionally reflects the influence

of a so-called cushion, that is defined as the difference between the current value of the

risky investment St and a benchmark level from the past Zt (e.g. the purchasing price

9

of the stock). When this difference is positive, investors made money from past risky

investment, otherwise they accumulated losses.

Our approach relies on the formulation of the value function proposed in Barberis,

Huang, and Santos (2001). In their Equations (15) and (16), the reference point changes

with the past performance (from ztRft for zt ≤ 1 to Rft for zt > 1, where zt = Zt/St). We

restate these definitions, in order to obtain identical reference points and similar courses in

the loss domain for both considered cases with positive and negative cushions, like in the

original prospect theory formulation, where gains are defined as the difference between the

value function argument (here Rt+1) and the reference point. Thus, we fix the reference

value in both cases (with prior gains zt ≤ 1 and prior losses zt > 1) to Rft and rearrange

the terms in Equations (15) and (16) in Barberis, Huang, and Santos (2001), obtaining:

v =

St(Rt+1 −Rft) , for Rt+1 ≥ Rft

λSt(Rt+1 −Rft) + (λ− 1)(St − Zt)Rft , for Rt+1 < Rft

, for zt ≤ 1(⇔ Zt ≤ St),

(2.11)

and

v =

St(Rt+1 −Rft) , for Rt+1 ≥ Rft

λSt(Rt+1 −Rft) + k(Zt − St)(Rt+1 −Rft) , for Rt+1 < Rft

, for zt > 1(⇔ Zt > St).

(2.12)

Here, λ is the coefficient of loss aversion, while the parameter k > 0 captures the

influence of previous losses on the perception of current ones (i.e. the larger the previous

loss is, the more painful the next losses become). We observe that, while the gain branches

of both value functions are invariable to past performance zt, the loss branches contain

a first term that resembles the original prospect theory, i.e. λSt(Rt+1 − Rft), but also a

second one revealing the impact of the cushion St−Zt. Moreover, the time t + 1-value of

the risky investment is derived as:

St+1 = (Wt + Bt)Rt+1. (2.13)

Of note is also that the simultaneous impact of the loss aversion coefficient λ and of

past portfolio performance k changes the actual investor aversion to losses. This can be

easily deduced by merging Equations (2.11) and (2.12) to:

v =

Stxt+1 , for xt+1 ≥ 0

[λSt − (1− πt)k(St − Zt)]xt+1 + πt(λ− 1)Rft(St − Zt) , for xt+1 < 0,

where πt = Pt(zt ≤ 1) is the probability of experiencing past gains and xt+1 = Rt+1−Rft

the equity return premium. According to the original prospect theory, the loss aversion

turns out to be risk aversion of first order in the loss domain, hence identical to the

10

derivative of the loss branch of the value function. Let us define the risk aversion of first

order as:

Λ(xt+1) =∂v

∂xt+1

=

St , for xt+1 ≥ 0

λSt − (1− πt)k(St − Zt) , for xt+1 < 0.(2.14)

Clearly, the actual loss aversion is given by the loss-branch of the above Function

(2.14), i.e. λSt− (1−πt)k(St−Zt), which is more than the simple loss aversion coefficient

λ from the original prospect theory. In other words, the aggregation of positive past

results (i.e. a positive cushion St − Zt ≥ 0) lowers the actual investor aversion to losses,

for that they are more confident in being able to cover prospective losses by past gains.

Henceforth, in line with the original prospect theory formulation, we mostly refer as “loss

aversion” the loss aversion coefficient λ (and not to the actual loss aversion derived above).

2.3 The endogenous VaR*

Our first goal is to formulate the maximum loss expected by individual investors, i.e. the

individual desired VaR*. This value will subsequently enter the optimization problem

when deciding between borrowing or lending.

To this end, we start from the literal definition of VaR*, concentrating on the notions of

“maximum”, “loss”, and “individual”. First, VaR* quantifies losses. However, according

to the prospect theory, what actually counts for individual investors is not the absolute

magnitude of a loss, but rather the subjectively perceived one, as captured by the value

function. Hence, VaR* should rely on the subjective value (or utility) of losses expressed in

the loss branches of the value functions (2.11) and (2.12). It depends on individual investor

characteristics (originated in the subjective view over gains and losses) and can vary over

time. Second, VaR* should represent a (subjective) expectation, because the next period

returns Rt+1 on which the evaluation of risky investments relies, are still unknown at the

decision time t. Third, we are looking for a maximal value, such that in calculating VaR*,

investors must ascribe a maximal occurrence probability Pt(Et[Rt+1] < Rft) = 1 to the

losses in the value function.

Therefore, we propose that VaR* accounts for the maximum expectation of sustainable

losses as resulting from individual valuations of the risky investment. However, we consider

investors to be sophisticated enough in order to consider that not only the mean, but also

the variation of prospective losses should be considered in order to accurately ascertain

the maximum acceptable loss level. Thus, in a second approximation, we enlarge the

VaR*-definition by adjusting for loss variance.

Henceforth, we consider that value functions are weighted by the pure probabilities

of occurrence (and not by non-linear probability functions as stated in the cumulative

11

prospect theory). According to Equations (2.11) and (2.12), we then derive:

Et[loss-utilityt+1] = πt[λSt(Et[Rt+1]−Rft) + (λ− 1)(St − Zt)Rft]

+ (1− πt)[λSt(Et[Rt+1]−Rft) + k(Zt − St)(Et[Rt+1]−Rft)]

= λSt(Et[Rt+1]−Rft) + [πt(λ− 1)Rft − (1− πt)k(Et[Rt+1]−Rft)](St − Zt)

(2.15a)

V art[loss-utilityt+1] = Et[loss-utility2t+1]− E2

t [loss-utilityt+1]

= πt[λSt(Et[Rt+1]−Rft) + (λ− 1)(St − Zt)Rft]2

+ (1− πt)[λSt(Et[Rt+1]−Rft) + k(Zt − St)(Et[Rt+1]−Rft)]2

− E2t [loss-utilityt+1]

= [λSt(Et[Rt+1]−Rft)]2

+ [πt(λ− 1)2R2ft − (1− πt)k

2(Et[Rt+1]−Rft)2](St − Zt)

2

+ 2[πt(λ− 1)Rft − (1− πt)k(Et[Rt+1]−Rft)]λSt(Et[Rt+1]−Rft)(St − Zt)

− [λSt(Et[Rt+1]−Rft)]2

− [πt(λ− 1)Rft − (1− πt)k(Et[Rt+1]−Rft)]2(St − Zt)

2

− 2[πt(λ− 1)Rft − (1− πt)k(Et[Rt+1]−Rft)]λSt(Et[Rt+1]−Rft)(St − Zt)

= πt(1− πt)[(λ− 1)Rft + k(Et[Rt+1]−Rft)]2(St − Zt)

2. (2.15b)

Note that, while the first term of the expected losses (2.15a) is similar to the loss

formulation in the original prospect theory, the remaining terms point out the influence

of the cushion accumulated over past trades. In contrast, the variance of losses (2.15b)

is exclusively dictated by the cushion-part, as individually perceived by investors, and

depends on the probability of having made gains or losses in the past, on the variance of

expected returns with respect to the reference risk-free rate, and on the squared cushion.

As mentioned above, in a first approximation, we stick to the literal definition of VaR*

as an expectation and design VaR* as the maximum expected loss:

VaR∗1t+1 = Et[loss-utilityt+1]

= λSt(Et[Rt+1]−Rft) + [πt(λ− 1)Rft − (1− πt)k(Et[Rt+1]−Rft)](St − Zt).

(2.16)

However, investors may consider loss-variance as an equally important parameter for

determining the maximal sustainable loss. Then, assuming that VaR* follows a cer-

tain distribution (i.e. normal or Student-t)3 with the value ϕ, we introduce the second

3Although VaR is a very popular measure of risk, it has been criticized because it does not satisfy

one of the four properties for coherent risk measure, namely subadditivity (see Artzner, Delbaen, Eber,

and Heath (1999), Rockafellar and Uryasev (2000) and Szego (2002)). However, according to Embrechts,

McNeil, and Straumann (1999), VaR becomes subadditive and can be considered as a coherent risk

12

(variance-adjusted) VaR* definition:

VaR∗t+1 = Et[loss-utilityt+1]− ϕ

√V art[loss-utilityt+1], (2.17)

which, according to Equations (2.15a) and (2.15b), results in:

VaR∗t+1 = λSt(Et[Rt+1]−Rft)

+ [(πt − ϕ√

πt(1− πt))(λ− 1)Rft − (1− πt + ϕ√

πt(1− πt))k(Et[Rt+1]−Rft)](St − Zt).

(2.18)

Again, expression (2.18) encompasses the twofold loss effect stemming from the loss

aversion coefficient of the original prospect theory and from the cushion introduced in

Barberis, Huang, and Santos (2001).

It is worth noting that, for sure gains (i.e. when πt = Pt(zt ≤ 1) = 1), both VaR*

expressions (2.16) and (2.18) reach a common upper bound:

VaR∗1,upt+1 = VaR∗up

t = λSt(Et[Rt+1]−Rft) + (λ− 1)Rft(St − Zt), (2.19)

while for sure losses (i.e. when πt = Pt(zt ≤ 1) = 0), the lowest value of:

VaR∗1,lot+1 = VaR∗lo

t = λSt(Et[Rt+1]−Rft)− k(Et[Rt+1]−Rft)(St − Zt) (2.20)

is attained.

The definition of VaR* serves to determining the optimal level of borrowing or lending

(Bt) from Equation (2.10a). When VaR* lies “to the right” of the portfolio VaR (i.e. it

is higher in absolute value than VaR), Bt is negative, hence investors become more risk

averse and save money. On the contrary, for a VaR* lower than VaR in absolute value,

investors augment their risky investment by borrowing extra money. Thus, an empirical

analysis of the evolution of Bt (as conducted in Section 3) can shed some light on the

investor risk behavior.

Also, one interesting topic to invetigate consists in estimating the equivalent loss aver-

sion parameter λ∗t that can be obtained for a VaR∗t+1 = VaR∗ that is fixed for commonly

used significance levels such as 1, 5 or 10%. The derivation can be readily done from the

definition (2.18) of VaR*:

λ∗t+1 =VaR∗ + [(πt − v

√πt(1− πt))Rft + (1− πt + v

√πt(1− πt))k(Et[Rt+1]−Rft)](St − Zt)

St(Et[Rt+1]−Rft) + (πt − v√

πt(1− πt))Rft(St − Zt).

(2.21)

Moreover, since λ∗t+1 depends on the fixed (thus exogenous) VaR∗, there should exist

no further causal relationship between past and future losses, such that we can set k = 0.

Accordingly, Equation (2.21) becomes:

λ∗t+1 =VaR∗ + [(πt − v

√πt(1− πt))Rft](St − Zt)

St(Et[Rt+1]−Rft) + (πt − v√

πt(1− πt))Rft(St − Zt). (2.22)

measure, if used in the case of elliptic joint distributions, such as normal and Student-t with finite

variance.

13

2.4 The prospective value of the risky investment

The estimation of the maximum acceptable individual loss level represents only the first

step of our analysis. As shown in Section 2.1, one of its consequences with direct impact

on non-professional investors resides in the determination of the optimal borrowing level.

This results as a byproduct of the optimization inside the risky portfolio that is undertaken

by the professional manager. For the non-professional client, it amounts to the optimal

choice in terms of wealth percentages invested, between risky and risk-less assets.

When investors decide about the optimal sum of money to be put in the risky portfolio

(equivalently in risk-free assets), they might not exclusively think in terms of VaR*,

but sooner aim at maximizing the utility generated by their financial investments. This

utility is encompassed in the prospect theory by the so called prospective value of the

risky investment Vt+1.4 Denoting the expected equity return premium by Et[xt+1] =

Et[Rt+1] − Rft and the probability of a positive premium by ωt = Pt(Et[Rt+1] ≥ Rft) =

Pt(Et[xt+1] ≥ 0), the prospective value of the risky investment can be formulated as:

Vt+1 = [ωt+(1−ωt)λ]StEt[xt+1]+(1−ωt)πt(λ−1)Rft−(1−πt)kEt[xt+1](St−Zt). (2.23)

Furthermore, we resolve to analyze the evolution of the prospective value for different

portfolio evaluation frequencies, on the grounds that revising portfolio performance at

different time intervals, implies drawing back on distinct return values, hence on different

return premia. This implicitly changes the values of several model parameters such as St,

Zt, πt, or ωt affecting the prospective value (2.23), a topic that detailed in Section 2.5.

Yet, in practice, risk averse investors may rely on a slightly different method for eval-

uating expected values of risky prospects. For instance, they may continue to consider

gains as unsure events and account for them as “wishes” (i.e. expectations). However,

losses would be assessed at their maximal impact, so to speak in a “worst case scenario”.

This being the case, gains flow into the definition of the prospective value as expected

gains, exactly as in Equation (2.23), while losses take the form of VaR*. In other words,

investors are sufficiently wary in order to count on the possibility of experiencing a max-

imal loss, hence to put an upper bound (in absolute value) on expected losses. It is this

upper bound that now generates utility (value) to the individual investor, and not the

expected loss. These considerations entail an alternative definition V ∗t+1 of the prospective

4Remember that our investors are not concerned with consumption and derive utility merely from the

financial wealth fluctuations.

14

value:

V ∗t+1 = ωtStEt[xt+1] + VaR∗

t+1

= (ωt + λSt)Et[xt+1]

+ [(πt − ϕ√

πt(1− πt))(λ− 1)Rft − (1− πt + ϕ√

πt(1− πt))kEt[xt+1]](St − Zt),

(2.24)

where the latter expression was derived according to Equation (2.18). In Section 3.2.2,

we investigate the evolution and implications of both prospective value definitions stated

here.

Before closing this section, we introduce a new definition that provides a basis to the

empirical analysis in Section 3.2.1. Similarly to the actual loss aversion defined in the

above Equation (2.14), a corresponding notion of global first order risk aversion can be

assessed from the prospective value as:

Λ(xt+1) =∂V

∂xt+1

= [ωt + (1− ωt)λ]St − (1− ωt)(1− πt)k(St − Zt). (2.25)

2.5 The impact of the portfolio revision frequency

As shown in the previous sections, the expected portfolio returns Et[Rt+1] (hence the

expected return premium Et[xt+1]) play a major role in the formulation of the value

function and consequently of almost all other variables of interest in our model (such as

VaR*, the prospective value, the optimal borrowing level, and also future cushions, gain

probabilities, etc.). Therefore, it is essential to notice that the value of returns directly

depends on the time horizon τ over which they are computed, i.e. or on the portfolio

revision frequency 1/τ . We hypothesize that different revision frequencies impact on

investor risk behavior leading to different investment decisions. The main reason for this

resides in the dependence of the computed performance of the risky portfolio on expected

returns, which further gives rise to the dependency of the investor attitude towards the

risky deposit and of the money invested in it on the portfolio revision frequency. The

higher the frequency of evaluation of risky investments is, the less likely is that risky

returns lie above risk-less ones, thus the more pronounced the investor disappointment

concerning the risky portfolio performance. Thus, given that, according to the prospect

theory, registered losses are perceived as more painful than gains of similar size, risky

investments become even less attractive.

The idea that the joint effect of narrow framing (myopia) over financial decisions and

reluctance to losses can dramatically impact risk perception and hence the subjective de-

sirability of risky investments, comes in line with the so-called myopic loss aversion. This

notion was was firstly introduced in Benartzi and Thaler (1995), and subsequently tested

in an experimental context in Thaler, Tversky, Kahneman, and Schwartz (1997), Gneezy

15

and Potters (1997), and Gneezy, Kapteyn, and Potters (2003). According to Barberis and

Huang (2004a), p. 4, myopia refers strictly to annual evaluations of gains and losses, hence

the term of narrow framing is better suited in describing the underlying phenomenon. In

a financial context, narrow framing illustrates the isolate evaluation of stock market risk

(i.e. unrelated to overall wealth risk). As underlined in Barberis and Huang (2004b), this

isolate evaluation determines an underestimation of the stock desirability, even though,

viewed in a wide utility-risk frame, they represent a good diversification modality.

According to Barberis and Huang (2004b), narrow framing can be interpreted as a

consequence of regret of not having taken another decision (non-consumption utility ex-

planation). Another explanation relies on the (higher) accessibility of (financial) infor-

mation that justifies its over-important role in final decisions. As referred in Kahneman

(2003), the easily accessible information is very appealing for the intuitive (for that spon-

taneous, effortless) way in which people use to make decisions. Our work draws upon

the latter motivation, namely accessibility. We consider it as better suited to financial

decisions, for that nowadays, investors are exposed to a tremendously high quantity of

financial information and need to make decisions in a fast changing financial environment.

Consequently, they tend to perform more frequent checks of their investments.

The empirical part of our paper (Section 3) analyzes closely the impact of various

revision horizons (ranging from one day to eight years) on the risk-free investment and

on the prospective value, where the focus lies on high revision frequencies (hence the ones

that are more plausible in practice), such as one day, one week, one month or one quarter.

In Section 3.2.1, we also plot and empirically assess the analytical form of the actual loss

aversion from Equation (2.14) and of the global risk aversion of first order from Equation

(2.25). However, in order to better understand how the revision frequency impacts the

prospective value and the investor attitude to risk, further explanations are necessary and

the rest of this section is dedicated to detailing this problem.

We start by noting that the first variable affected by the revision horizon τ is the

gross return value Rt(τ) = log(Pt/Pt−τ ) that accounts for the price variation over the

time interval τ . Therefore, the expected return premium Et[xt+1(τ)] = Et[Rt+1(τ)]−Rft

depends on the revision frequency.5 For instance, if prices are highly volatile in the short

run but do not change very much in mean in the long-run, a higher τ should generate

higher returns. However, even though there are more parameters (such as St, Zt, πt, etc.)

that are computed from Rt(τ), being thus affected by τ , in our (empirical) analysis, we

assume all λ, k, and Rft as fixed (i.e. independent of τ). Therefore, the changes of the

prospective value Vt+1 documented in Section 3.2.1 are a consequence of a chain impact

5For simplicity reasons, we henceforth drop most of the time-indices at places where we discuss the

dependence of the variables calculated at (the fixed) time t on τ .

16

whose very first seed is the revision horizon, but that does not imply the loss aversion

coefficient λ.6 Obviously, this chain reaction (hence its source, τ) also affects the actual

loss aversion and the first order risk aversion of first, but does not change the simple

coefficient of loss aversion:

Λloss(τ) = λSτ − (1− πτ )k(Sτ − Zτ ) (2.26a)

Λ(τ) = [ωτ + (1− ωτ )λ]Sτ − (1− ωτ )(1− πτ )k(Sτ − Zτ ). (2.26b)

In addition, we address a further theoretical issue which is closely related to the impact

of the portfolio revision frequency discussed above. Given that this frequency appears to

affect the investor risk perception, thus the level of risky investments, could the reverse

causality hold as well? In other words, for a certain loss aversion value (at date t), is there

a revision frequency that is optimal in terms of maximization of the prospective value?

In order to answer this question, we analyze the direct impact of Rt(τ) on the utility

maximization problem of individual investors. To this end, the c.p. dependence of the

prospective value V (x) from Equation (2.23)7 on x(τ) at time t is taken into account.

In other words, we study the direct dependence of utility on returns, but discard the

indirect effects generated by other model parameters influenced by returns.8 Under this

assumptions, the prospective value V (τ) at time t is linear in the return premium x,

with the proportionality coefficient δ = [ω + (1− ω)λ]S − (1− ω)(1− π)k(S − Z).9 The

prospective value is therefore maximized at each t for one of the extreme values of x(τ)

(the highest or the lowest, depending on the sign of δ) and, assuming there exists an

invertible computable form for the function x(τ), then τopt = x−1(τ). Naturally, if we

can find an invertible function V (τ), then the optimal revision horizon τ can be directly

computed at each trade t from the maximization problem of the prospective value. In fact,

this last simplified method is adopted in the empirical Section 3.2.1, where we suggest an

optimal value for the portfolio revision horizon τ based on practical considerations and

our data set.

6This chain reaction takes place in successive steps: (1) τ → Et[xt+1] =: xτ , (2) xτ → St =: Sτ , (3)

S1, S2, ...St → St − Zt =: Sτ − Zτ .7Or the corresponding V ∗(x) from Equation (2.24).8In essence, this can be considered a plausible assumption. The choice of an optimal current τ takes

place at the fixed time t where the model parameters indirectly affected by τ (i.e. St, Zt, πt), depend

on past values of x. The only exception is ωt(τ) that depends on τ through Et[xt+1(τ)], but assuming

that investors also assess ωt on the basis of past experience (e.g. as the frequency of past positive return

premia), we can confine ourselves to analyze the isolate role of Et[xt+1(τ)] in the prospective value

function.9The corresponding proportionality coefficient for V ∗(τ) yields δ∗ = (ω + λ)S − [1 − π +

ϕ√

π(1− π)]k(S − Z).

17

3 Empirical results

This chapter presents empirical findings complying with the theoretical results derived in

Section 2.

We consider that non-professional investors perceive risky investments according to

the value functions in Equations (2.11) and (2.12), and calculate the maximum loss level

according to Equation (2.18). The empirical analysis is based on daily data for the SP500

and the 10-year bond nominal returns (considered as the risky and the risk-free investment

alternative, respectively), ranging from 01/02/1962 to 03/09/2006 (11,005 observations).10

From this data set, we construct weekly, monthly, three-monthly, quarterly, yearly and

further lower frequency returns (ranging from two to eight years with an one-year in-

crement). We divide our sample in two equal parts and use the first one to estimate the

empirical mean and the standard deviation of the portfolio returns. The second half of the

data allows us to run the model on the basis of Sections (2.1) and (2.2) and to derive the

desired VaR*, as well as the wealth proportion invested in the risky portfolio (considered

here to be the index SP500). The remaining money is assumed to be automatically put

in the risk-free 10-year bond. Investors are assumed to start trading with an even initial

wealth allocation between the risky portfolio and the bond.

3.1 The evolution of the risky investment

3.1.1 The impact of the revision frequency

According to Benartzi and Thaler (1995), loss averse investors who evaluate the perfor-

mance of their portfolios once a year and employ a linear value function with standard

prospect theory parameter values, give rise to a market evolution that can explain the

equity premium observed in practice. We are interested in how varying the frequency

of portfolio evaluation can change investor decisions, hence the market evolution in our

setting.

We start by computing the portfolio VaR in Equation (2.10b) for either normally or

Student-t distributed portfolio gross returns and for a significance level of 5%. Then,

taking λ = 2.25 and k = 3 as in Barberis, Huang, and Santos (2001), as well as πt

identical to the empirical frequency of the cases where zt ≤ 1, we derive VaR∗t+1 according

to Equation (2.18). In this context, we consider different alternatives for the estimation of

expected portfolio returns, namely as the unconditional mean returns until the last date

before decision time, a zero mean process, or an AR(1) process. The value Zt of past

portfolio performance that impacts the valuation of current losses is taken to be identical

10Descriptive statistics can be found in Tables 8 and 9 of the Appendix.

18

to the last period risky asset holding Zt = St−1. The derived VaR∗t+1 is then plugged into

Equation (2.10a) in order to determine the optimal level Bt of borrowing or lending.

Table 1 presents the average percentages of wealth invested in the risky portfolio

St/Wt, where St is derived according to Equation (2.13), for different portfolio evaluation

horizons τ , normally and Student-t distributed portfolio returns Rt, and expected returns

Et[Rt+1] computed as the unconditional mean of past returns.11

Table 1: Percentage investments in SP500 for myopic cushions

Revision frequencyExpected returns

Normal Student-t

1 year 34.46 42.91

4 months 15.79 22.32

3 months 13.79 16.10

1 month 7.25 8.53

1 week 3.66 3.72

1 day 1.79 1.91

Accordingly, when investors are loss averse and use the VaR∗t+1 from Equation (2.18) as

measure of the maximal acceptable risk, higher portfolio revision frequencies entail lower

investments in the risky portfolio. This result is consistent with previous findings, such

as Benartzi and Thaler (1995) and Barberis, Huang, and Santos (2001). For normally

distributed returns, the risk aversion of investors appears to be higher compared to the

case with Student-t distribution.

3.1.2 The impact of the cushion

In a next step, we are interested in the interdependence among risky portfolio returns,

cushions and wealth percentages invested in the risky portfolio. In order to analyze this

issue, we fix the revision frequency at one year and plot annual returns on the SP500,

the evolution of the cushion St − Zt generated by series of past gains or losses, and the

resulting yearly percentage of wealth invested in the risky portfolio. As mentioned above,

the past performance benchmark is derived as the risky investment value in the previous

year Zt = St−1. The sample covers the 22 years of analysis (from 1983 to 2005).

At a first inspection, Figure 1 points to a positive correlation of the three variables

(SP500 yearly returns, yearly cushions, and yearly percentage investments in the risky

11Similar results are obtained when expected returns are derived as the zero mean or the AR(1) process.

See Tables 10 and 11 in the Appendix.

19

portfolio). The proportion of wealth invested the risky portfolio reaches its maximum

of 64.4% at the time when the yearly cushion attains its highest value, which is mainly

generated by the previous bull market observable in the SP500 returns. The importance of

the cushion for investor decisions can be traced back on Equation (2.18) that emphasizes

the twofold structure of the individual VaR*. The first term on the left-hand side of

this equation accounts for the expectation of future portfolio returns weighted by the loss

aversion parameter λ, while the second term is responsible for the influence of previous

performance (as encompassed by the cushion St − Zt). We denote them as the PT-term

and the cushion-term, respectively. Accordingly, positive expectations with respect to the

future evolution of the risky portfolio coupled with a positive cushion (i.e. past gains)

should render investors less loss averse. Consequently, given that VaR* directly enters Bt

and hence St, the wealth proportion invested in the risky portfolio St/Wt increases, as

illustrated in Figure 1. This effect is reversed when both return expectations and cushions

become negative.

Moreover, it is interesting to observe that small changes in the cushion at the beginning

of the effective trade period12 entails high variations in the portfolio allocation. This first

investor reaction turns strongly against investing money in risky assets, but the increase

in cushion makes it smooth over time, so that it ends by following fairly close the cushion

evolution. This result is again in line with the concept of loss aversion, i.e. the lower the

cushion of wealth accumulated in past trades is, the more loss averse investors become,

because they dispose of less back-up for later contingent losses. This lowers the wealth

fraction invested in risky assets.13

Further, we ask which is the impact of different ways of assessing the cushion on

investor decisions. In line with Barberis, Huang, and Santos (2001), we assume that

investors account for cumulative cushions (instead of the yearly cushions considered up to

this point), where the cumulation of past performance is carried out since the beginning

of the estimation sample. Table 2 presents the resulting wealth percentages invested in

the risky portfolio for various revision frequencies and different return distributions.

Interestingly, the yearly results with cumulative cushions under the normal distribution

exactly match the so called TIAA-CREF typical allocation mentioned in Benartzi and

Thaler (1995). This time, investors appear to be less risk averse for normally than for

Student-t distributed returns. Moreover, when investors make very frequent evaluations

(e.g. daily), they become extremely loss averse and end up by putting all money in the

12Remember that the effective trade (i.e. the observations that effectively underly the estimation

procedure) begins in the second half of our sample.13Gneezy and Potters (1997) test for the influence of experienced gains and losses on risk behavior, but

find no significant effect. However, as noted on p. 641, their experimental framework deviates from real

market settings, as considered in our model.

20

0 5 10 15 20 25−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(a) Yearly SP500 returns.

0 5 10 15 20 25−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

(b) Yearly cushions.

0 5 10 15 20 250.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(c) Yearly percentage investments in SP500.

Figure 1: Evolution of yearly SP500 returns, yearly cushions, and yearly percentages

invested in the risky portfolio.

21

Table 2: Percentage investments in SP500 for cumulative cushions

Revision frequencyExpected returns

Normal Student-t

1 year 49.93 41.53

4 months 14.55 14.52

3 months 5.68 5.89

1 month 1.87 1.32

1 week 0.50 0.10

1 day 0.00 0.00

risk-free asset.

Comparing Tables 1 and 2, we find out that, for a revision frequency of one year,

investors who rather cumulate than myopically treat past gains and losses to cushions, are

less risk averse. However, this situation reverses as the frequency of evaluation increases

(in fact, this phenomenon already occurs at a four month revision horizon). Also, the

speed at which the risk aversion increases is much higher for cumulated cushions. Thus,

investors relying only on short-term cushions continue to allocate modest but positive sums

to the risky portfolio, even when they check performance very often (e.g. every day). In

contrast, the net risky investment quickly converges to zero when cushions are based on all

previous trades. In general, these results support the idea that loss averse investors who

narrow frame financial projects and perform high frequency revisions become extremely

risk averse.

At this point, a further interesting empirical question arises: how long does it take

for an investor performing frequent evaluations to quit the risky market? In order to

answer this question, let us further assume that investors start with an initial investment

in risky assets of 50% of the total wealth. Figure 2 points out the dramatic effect of a high

revision frequency, i.e. when portfolio performance is checked every single day, investors

get out of the risky market in only few days. This behavior can be also explained in the

context of Equation (2.18), according to which, highly volatile SP500-returns and very low

cumulative cushions (as generated by the daily change in position) result in an enhanced

acceptable risk level VaR*. This captures the picture of an extremely risk averse investor.

3.1.3 An analysis with unadjusted VaR*

Finally, we analyze the investor behavior for a VaR* that exclusively accounts for maxi-

mum expected losses as in Equation (2.16). The results are quite similar to the previous

ones obtained for the variance adjusted VaR* from Equation (2.18). The explanation

22

0 1000 2000 3000 4000 5000 6000−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

(a) Daily SP500 returns.

0 1000 2000 3000 4000 5000 6000−60

−40

−20

0

20

40

60

(b) Daily cushions.

0 1000 2000 3000 4000 5000 60000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(c) Daily percentage investments in SP500.

Figure 2: Evolution of daily SP500 returns, daily cushions, and daily percentages invested

in the risky portfolio.

23

becomes apparent in Figure 3, that illustrates the evolution of the probability of accu-

mulating prior gains πt = Pt(zt ≤ 1) for yearly and daily revision horizons, respectively.

Clearly, πt lies very close to one (especially for daily revisions) which renders the term√πt(1− πt) close to zero, hence makes the difference between the formulas (2.16) and

(2.18) disappear. Thus, both ways of estimating the VaR* introduced in Section 2.3

lead to similar results, more precisely close to the upper bound from Equation (2.19). For

daily revisions, the probabilities πt are almost one during the whole trading interval, which

yields zt = Zt/St ≤ 1 but also St − Zt ≈ 0. In other words, for this revision frequency,

the influence of the cushion St − Zt in Equations (2.16) or (2.18) becomes insignificant,

implying that the value of the VaR* depends solely on the first term in the definition

(i.e. what we denoted as the PT-term) λStEt[xt+1]. This underlines again the important

impact of cushions on investment decisions. When cushions are almost constant in time,

the PT-term (that is determined by the loss aversion coefficient λ and the risk premium

Et[xt+1] = Et[Rt+1] − Rft) dominates the maximum acceptable risk level VaR*. Thus,

loss aversion coupled with high market volatilities makes investors to renounce their risky

portfolios.

24

0 5 10 15 20 250.925

0.93

0.935

0.94

0.945

0.95

0.955

0.96

0.965

(a) Yearly revisions.

0 1000 2000 3000 4000 5000 60000.9997

0.9997

0.9998

0.9998

0.9999

0.9999

(b) Daily revisions.

Figure 3: Evolution of the probability of prior gains πt = Pt(zt ≤ 1).

25

3.2 The evolution of the prospective value

3.2.1 The impact of the revision frequency

According to the results in Section 3.1, the measured performance of the risky portfolio

varies with the revision horizon τ .

In order to closer analyze the impact and to determine an optimal value of τ at

each decision time t, we first recall the observation made in Section 2.5 that τ exerts

direct influence on the expected returns, thus on the expected return premium Et[xt+1] =

Et[Rt+1] − Rft. Therefore, the revision time affects the prospective value of the risky

investment from Equation (2.23).14 Here we distinguish between two terms with relevant

contribution to the formation of V (Et[xt+1]), namely the first term on the right hand side

of the Equation (2.23) that stands for the prospective value as considered in the original

prospect theory (that we denote as the PT-effect), and the second one (called the cushion-

effect) generated by the cushions of past gains or losses suggested in Barberis, Huang, and

Santos (2001).

Figure 4 illustrates the evolution and the contribution of these two effects to the final

prospective value in Equation (2.23), for revision frequencies of one day and one year,

respectively. The cushion is again myopically assessed, i.e. Zt = St−1.

At a first inspection of the upper panel in Figure 4, we find that the prospective value

V (Et[xt+1]) relies mainly on the PT-effect, that appears to be negative for the first years

of the total trading period (for the first 500 days approximately). This figure provides

also an additional explanation of why daily revisions result in lower risky investments.

The daily volatility does not allow to investors to accumulate significant cushions from

trading risky assets, hence the cushion-effect cannot overcome the PT-effect. Indeed, the

upper panel points out that, for portfolio checks performed more often than once a year,

the cushion is most of the time positive, but very low. However, the opposite occurs for

yearly revisions (lower panel), where the cushion-effect has a much stronger impact on

the prospective utility.

In the subsequent Figure 5, we plot again the prospective value and its two components

(the PT- and the cushion-effect) now as a function of the revision horizon τ , which ranges

from one day, one week, one month, three months, four months, one year, up to eight

years.15

Let us have a closer look at the curve in Figure 5. First, we note a visible kink of V (τ)

at τ = 1 year (corresponding to the fifth point on the horizontal axis16) that delimitates

14The effects of the alternative prospective value definition (2.24) are discussed in Section 3.2.2.15A revision frequency of eight years implies that investors can only make three portfolio checks during

our estimating sample. Therefore, a further increase of the revision time becomes senseless.16Each point on the abscissa stays for one of the considered revusion frequencies from one day to eight

26

0 1000 2000 3000 4000 5000 6000−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

V(t)

Cushion−Effect

PT−Effect

(a) Daily revisions.

0 5 10 15 20 25−1500

−1000

−500

0

500

1000

1500

2000

2500

V(t)

Cushion−Effect

PT−Effect

(b) Yearly revisions.

Figure 4: Prospective value evolution for daily and yearly revisions.

27

0 2 4 6 8 10 12 14−2000

0

2000

4000

6000

8000

10000

12000

Cushion−Effect

V(t)

PT−Effect

Figure 5: Prospective value evolution for different revision frequencies.

two segments of different evolution. This reinforces the idea that, in practice, one year

represents indeed a “critical” evaluation frequency. As documented in Benartzi and Thaler

(1995), a decade ago, investors used to perform yearly portfolios checks. Nowadays, due to

the high amount of information available at almost no cost and to the enhanced dynamic of

the market events, we claim that investors manifest the tendency to reconsider the problem

of splitting their money between risky and risk-free assets much more often. Yet, one year

remains an important anchor in the investor minds, given that, on one hand, various

events (such as release of annual activity reports) take place with this frequency and,

on the other hand, non-professional investors may not be sufficiently impatient (perhaps

because they do not dispose of enough time and financial resources) to perform more often

portfolio checks.

The evident segmentation of the prospective value for revision frequencies lower and

higher than one year motivates us to have a closer look at the two separate frequency

segments, as encompassed in Figure 6. Our goal is to find an analytical form that underlies

this evolution and that would allow us to determine the optimal revision frequency.

Optically, for τ < 1 year, the prospective value function V (τ) appears to exhibit a

parabolic (convex) form, while for τ ≥ 1 year, it rather evolves in periodical waves of small

amplitude following an ascending trend. In effect, fitting an analytical model to the data

yields a parabola for τ < 1 year and a polynomial of fifth degree for τ ≥ 1 year, whose

curvature coefficient estimates are given in Table 3. Of note is that already a linear model

of the type a1x + a2, where a1 = 1, 292 (with 95%-CI (1, 135; 1, 450)), and a2 = −1, 089

(with 95%-CI (−1, 884;−293.3)), fits the data in the right segment reasonably well (R-

square: 0.9853, Adjusted R-square: 0.9829, RMSE: 417.2) with respect to the best fit

obtained for the left revision horizon segment.

years.

28

1 1.5 2 2.5 3 3.5 4 4.5 5−10

0

10

20

30

40

50

60

70

80

90

V(t)

Cushion−Effect

PT−Effect

(a) Daily, weekly, quarterly and four-monthly revisions.

1 2 3 4 5 6 7 8−2000

0

2000

4000

6000

8000

10000

12000

Cushion−Effect

V(t)

PT−Effect

(b) Revisions from one to eight years.

Figure 6: Prospective value evolution on the two relevant revision frequency segments.

29

Table 3: Estimated prospective value evolution as a function of the portfolio revision

frequency for λ = 2.25, k = 3.

Fitted model Coefficient 95%-CI Goodness of fit

τ < 1 year year

a1x2 + a2x + a3

a1 = 7.152 (1.709; 12.6) R-square: 0.9909

a2 = −22.46 (−55.75; 10.83) Adjusted R-square: 0.9819

a3 = 15.47 (−28.21; 59.15) RMSE: 4.734

τ ≥ 1 year

b1 = 4.076 (−6.174; 14.33)

b2 = −84.45 (−315.7; 146.8)

b1x5 + b2x

4 + b3x3 b3 = 630.9 (−1, 305; 2, 566) R-square: 0.9993

+b4x2 + b5x + b6 b4 = −1, 994 (−9, 365; 5, 377) Adjusted R-square: 0.9975

b5 = 3, 518 (−8, 885; 15, 920) RMSE: 159.1

b6 = −1, 343 (−8, 397; 5, 711)

30

In sum, table 3 suggests the following analytical form of the prospective value as a

function of the portfolio revision horizon:

V (τ) =

a1τ2 + a2τ + a3, for τ ≤ 1 year

b1τ5 + b2τ

4 + b3τ3 + b4τ

2 + b5τ + b6, for τ > 1 year,(3.1)

which is plotted in the subsequent Figure 7. We note that there is a high jump (kink)

in the prospective value at what we consider to be the reference frequency of τ = 1 year,

namely V (1 year−) = 0.16 ¿ 731.53 = V (1 year+)). For the linear model fitting the right

segment of the curve, we obtain V (1 year+) = 203 À V (1 year−).

1 1.5 2 2.5 3 3.5 4 4.5 5

0

10

20

30

40

50

60

70

80

(a) Daily, weekly, quarterly and four-monthly revisions.

1 2 3 4 5 6 7 8

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

(b) Revisions from one to eight years.

Figure 7: Curve fitting for the prospective value on the two relevant revision frequency

segments.

Interestingly, the two-segment evolution illustrated in Figure 5 and formally assessed

in Equation (3.1) offers an original perspective over the prospective value in the revision

frequency domain. It relies on the visual similarity to the value function in terms of gains

and losses, as formulated in the original prospect theory. More precisely, the prospective

31

value of the risky portfolio in time units appears to mirror the value function in money

units. For better understanding this claim, recall first that, in the revision horizon do-

main, the individually perceived value of prospective risky investments exhibits a twofold

evolution, namely it unfolds convexly for revision horizons lower than one year, while its

course is (almost) linear for revision times higher than one year. In other words, the

prospective value drops much faster for low revision horizons (in the left time segment)

than for high ones (in the right time segment). Thus, in the left revision horizon segment,

individual investors appear to upgrade much faster risky prospective values for revision

horizons approaching the kink-value of one year, while in the right segment there is almost

no significant difference in the valuation change for different frequencies. However, the

slope of the prospective value in revision time domain appears to be lower to the left than

to the right of the reference horizon of one year (i.e. in Equation (3.1), a2 for τ < 1 year

is lower in absolute value than b5 for τ > 1 year), as opposed to the value function where

loss aversion becomes manifest in the left domain (λ > 1 for losses). However, low revision

horizons are equivalent to high revision frequencies, so we conclude on a steeper slope for

low frequencies in revision time domain, similar to the value function evolution for neg-

ative monetary units. We denote this phenomenon as loss aversion in revision frequency

domain. Also, the two frequency segments are separated by the critical revision frequency

of one year. According to the practical considerations (concerning real market conditions,

information overflow and impatience of real investors, as presented below), this critical

frequency can be considered as an anchor (reference point) that provides mental support

to individual investors who resolve to invest in risky portfolios.

Section 5.3 in the Appendix presents the results of various robustness checks performed

for further values of the loss aversion coefficient λ and of the cushion sensitivity parameter

k. Figures 11, 12, and 13 confirm the important contribution of the cushion-effect to the

prospective value, as well as the segmentation for revision horizons lower and higher than

one year. Interestingly, loss loving investors (i.e. λ < 1) appear to perceive the risky

prospect as having a negative and concavely decreasing value in the revision horizon

domain. Formally, this draws back on Equation (2.23), where λ < 1 and k > 0 entail a

negative cushion-effect. Given that this effect dictates the course of the entire prospective

value and that it increases in absolute value for higher revision horizons, the prospective

value decreases in τ . For loss neutral investors (i.e. λ = 1), the cushion effect becomes

nil, hence the PT-effect controls the evolution of the prospective value. However, note

that magnitude of the PT-effect is similar to the case with λ = 2.25. Again, Equation

(2.23) and the probability of experiencing past gains (πt) which lies in our data set very

close to one (see Figure 3) provide explanations of this fact. Finally, the prospective value

obtained for λ = 3 exhibits similar evolution to the original case with λ = 2.25, with

32

a higher peak at the revision horizon of eight years and a more pronounced sinusoidal

component. In sum, the attitude to losses captured by the loss aversion coefficient λ

appears to decide upon which of the cushion- or PT-effects has the major contribution to

the final prospective value, thus affects the current perception of the risky investment at

different revision frequencies.

In addition, an analogous curve fitting procedure performed for these all three further

cases reinforces these conclusions. For each λ ∈ 0.5; 1; 3, a parabola continues to best

describe the prospective value evolution in the high revision frequency segment. In con-

trast, for the low frequency segment, a parabola attains the best fit for loss loving investors

λ = 0.5, but more complex dependencies emerge for higher loss aversion, namely a sixth

order polynomial for loss neutral investors λ = 1 and a fourth order one for extremely

loss averse investors λ = 3. Overall, the kink at the reference revision horizon of one year

is again evident and the slope of the fitted curves remains smaller in the low-frequency

domain underlining the same phenomenon that we called the loss aversion in revision

frequency domain.

Tables 14 and 15 show that the parameter k exerts no significant influence on the

evolution of the prospective value. Confirming the qualitative results in Figures 11, 12,

and 13, the prospective value is negative and decreases in the revision horizon for λ ≤ 1,

but becomes positive and growing for loss averse investors λ > 1.

Furthermore, as discussed in Section 2.5, the portfolio revision frequency reflects on

the actual loss aversion defined in Equation (2.14) as well as on the global first order risk

aversion from Equation (2.25). Subsequently, we conduct an empirical analysis similar

to the above investigations with respect to the evolution of the prospective value. First,

we observe that both actual loss aversion and global first order risk aversion exhibit

resembling patterns, being derived from two interrelated functions, namely the value

function and the prospective value (that results from the value function), respectively.

Also, the obtained values for both measures of the real investor risk attitude are of the same

order of magnitude as the prospective value in the revision frequency domain. As depicted

in Figure 8, there is an apparent similar segmentation around the revision frequency of

one year (corresponding to the fifth point on the horizontal axis).

33

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Actual loss aversion

Global loss aversion

(a) All frequency revisions.

1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

9000

Actual loss aversion

Global loss aversion

(b) Daily, weekly, quarterly and four-monthly revisions.

1 2 3 4 5 6 7 81.5

2

2.5

3

3.5

4

4.5x 10

4

Actual loss aversion

Global loss aversion

(c) Revisions from one to eight years.

Figure 8: Evolution of the actual loss aversion and global first order risk aversion on the

two relevant revision frequency segments.

34

In mathematical terms, the evolution on the left (right) revision horizon segment is one

order of complexity higher than the prospective value, namely it corresponds to a third

(sixth) order polynomial. The curvature parameters estimates are presented in Table

4 (and Table 16 in the Appendix), while and the corresponding theoretical courses are

illustrated in Figure 9.

Table 4: Estimated evolution of the actual loss aversion as a function of the portfolio

revision frequency for λ = 2.25, k = 3.

Fitted model Coefficient 95%-CI Goodness of fit

τ < 1 year

a1 = −122.3 (−694; 449.5)

a1x3 + a2x

2 a2 = 1, 337 (−3, 841; 6, 515) R-square: 0.9992

+a3x + a4 a3 = −2, 415 (−16, 370; 11, 540) Adjusted R-square: 0.9968

a4 = 1, 959 (−8, 714; 12, 630) RMSE: 170.8

τ ≥ 1 year

b1 = 49.17 (−141.1; 239.5)

b2 = −1, 343 (−6, 487; 3, 801)

b1x6 + b2x

5 + b3x4 b3 = 14, 150 (−40, 470; 68, 760) R-square: 0.9956

+b4x3 + b5x

2 + b6x + b7 b4 = −71, 930 (−360, 000; 216, 200) Adjusted R-square: 0.969

b5 = 180, 000 (−603, 600; 963, 700) RMSE: 1,327

b6 = −196, 600 (−1, 215, 000; 822, 100)

b6 = 96, 460 (−381, 500; 574, 500)

35

1 1.5 2 2.5 3 3.5 4 4.5 5

1000

2000

3000

4000

5000

6000

7000

8000

actual loss aversion

global risk aversion

(a) Daily, weekly, quarterly, four-monthly, and yearly revisions.

1 2 3 4 5 6 7 8

2

2.5

3

3.5

4

4.5

x 104

actual loss aversion

global risk aversion

(b) Revisions from two to eight years.

Figure 9: Curve fitting for the actual loss aversion and the global first order risk aversion

on the two relevant revision frequency segments.

36

Finally, we return to the question concerning the optimal revision frequency. It appears

natural to assume that investors concerned exclusively with financial investments (and not

with other sources of utility such as consumption) attempt to maximize the prospective

value of their risky portfolios. In fact, smart investors could look for an optimal revision

frequency, i.e. that maximizes the prospective value at a given decision time t. According

to the functional forms fitted to our data set for the prospective value (3.1), as well as for

the actual loss aversion and global first order risk aversion (from Tables 4 and 16), these

variables describing the investor risk behavior are increasing in τ in each of the two relevant

segments. Thus, the optimal τ lies at the upper end of the corresponding segment (i.e.

is identical to the maximal τ of the segment).17 Consequently, in each revision frequency

segment, (loss averse) investors should check the performance of their risky investment

as seldom as possible in order to maximize the corresponding prospective value of this

investment.18 Again, the highest revision frequency of one day entails a minimal expected

value of the risky portfolio, pushing investors to step out of the risky market and to

allocate all their money to risk-free assets. Yet, we know that practical reasons (such as

the huge amount of financial data available almost at no costs to every individual investor

and the high interest raised by financial events in general) entail an increase of the revision

frequency below the limit of one year. Thus, investor perceptions lie sooner in the left

revision horizon domain and the crossover to the second segment is improbable. Therefore,

accounting for the today’s financial market evolution and importance, we consider τ = 1

year as the most reasonable revision time that would increase the perceived returns of

risky investments.

3.2.2 An analysis under the “worst case scenario”

For the “worst case scenario” described in Section 2.4, investors may use a slightly different

definition of the prospective value, namely according to Equation (2.24). Figure 10 plots

the evolution of both Vt+1 in line with the original prospect theory and the new V ∗t+1,

that appear to be almost identical. Thus, the discussion on the evolution in the revision

frequency domain in Section 3.2.1 is also valid the “worst case scenario”. We conclude

that the hypothesis that prudent investors perceive risks in this “worst case” appears to

be acceptably realistic in the context of variable revision frequencies, because it generates

17The situation is inverted for λ ≤ 1, as presented in Section 5.3 of the Appendix. However, the same

arguments hold if we consider that loss loving and loss neutral investors attempt to maximize the absolute

prospective value of the risky investment.18In fact, the optimality of the revision frequency should be sooner understood from the viewpoint of

portfolio managers, whose interest is to attract more clients willing to invest money in risky assets. Rec-

ommending to these clients to undertake performance checks in the “optimal” frequency should maximize

the budget at manager’s disposal.

37

results that are similar those stemming from considerations of the original prospect theory.

0 5 10 15 20 25−1000

−500

0

500

1000

1500

2000

2500

3000

V(t)*

V(t)

Figure 10: Prospective values in the original prospect theory-formulation vs. the “worst

case scenario” for yearly revisions.

3.3 A comparison to the portfolio optimization framework

3.3.1 VaR*-equivalent significance levels

One further question of interest arises from the use of the VaR∗ as a measure of risk in

the portfolio optimization model in Section 2.1. Statistically, VaR∗ represents the lower

quantile of portfolio returns at a given (i.e. fixed) significance level α (or confidence

level 1 − α), where usually α ∈ [0.01, 0.1]. The individually optimal VaR∗t+1 (that is in

fact previously derived by investors on the basis of subjective considerations according

to Equation (2.18)), is compared to the portfolio VaR in Equation (2.10b), in order to

determine how investor wealth is going to be split between the risky portfolio and the

risk-free bond (where the sum to be invested in risk-free assets is formalized in Equation

(2.10a)). We denote by α∗t the significance level that corresponds to the VaR∗t+1 com-

puted in our model. Thus, if the portfolio VaR at time t corresponds to an α < α∗t (or

equivalently, to a confidence level 1 − α > 1 − α∗t ), then the sign of Equation (2.10a) is

negative. In words, too much risk would arise by putting the entire wealth in the risky

portfolio, so that, in order to accommodate the desired (lower) risk level, a percentage of

the investor wealth should be lent, i.e. invested in the risk-free asset (Bt < 0). On the

contrary, if α > α∗t , then the portfolio risk meets the individual risk requirements (being

lower than the subjective risk threshold) and investors borrow extra money (Bt > 0) in

order to increase their SP500-holdings.

In this section, we determine the significance level corresponding to the value of VaR∗t+1

as derived from Equation (2.18), using either normal or the Student-t return distributions

38

and considering again the initial setting with short-term cushions Zt = St−1. Table 5

presents equivalent significance levels averaged over time and denoted as α∗.

Table 5: Equivalent significance levels of the estimated VaR∗t+1.

Revision frequencyα∗

Normal Student-t

1 year 34.94 43.95

4 months 40.49 46.50

3 months 43.18 46.34

1 month 45.23 46.67

1 week 47.69 48.01

1 day 48.90 49.06

Table 5 emphasizes an interesting empirical fact. As stated above, classical portfolio

selection models based on VaR assume that the significance level chosen by investors is

low (i.e. α ∈ [0.01, 0.1]). By contrast, our results (i.e. the much higher values α∗ > 0.3

obtained for any revision frequency higher than one year) show that this assumption does

not match real market data and that investors might be in practice substantially more loss

averse than considered in theory. The lowest α∗ in Table 5 is obtained for the standard

revision frequency of one year but lies still far above commonly assumed significance levels

(between 1−10%). Also, when investors revise their portfolios more frequently (e.g. every

day), α∗ enhances tremendously (up to approximatively 50%) pointing out an excessive

risk aversion.

3.3.2 VaR*-equivalent loss aversion levels

In the same context of equivalency, we now address the impact of an exogenous VaR* as

originally employed in Campbell, Huisman, and Koedijk (2001), on the values of the loss

aversion coefficient λ∗t+1, computed according to Equation (2.22) in our model. To this

end, we go back on the commonly used significance levels of 10% and 1% and estimate

an equivalent exogenous VaR* as derived from Equation (2.10b). Then, we compute the

corresponding λ∗t+1 from Equation (2.22).

Tables 6 and 7 present the wealth percentages invested in the risky portfolio subject

to the assumed portfolio returns distribution (i.e. either normal or Student-t) as resulting

from our model, as well as the generated average equivalent loss aversion coefficient λ∗ for

the two significance levels mentioned above (10% and 1%, respectively). Remember that

the portfolio VaR in Equation (2.10b) is estimated using a 5% significance level that is

going to be considered as the benchmark for the values in these tables.

39

Table 6: Percentage investments in SP500 and the average λ∗ for yearly cushions, expected

returns = unconditional mean and α = 0.10.

Revision frequency

Expected returns

Normal Student-t

Wealth % λ∗ Wealth % λ∗

1 year 77.21 1.43 72.80 1.29

4 months 77.66 1.20 73.07 1.21

3 months 77.73 1.07 73.13 1.08

1 month 77.90 1.08 73.25 1.09

1 week 77.94 0.86 73.28 0.86

1 day 77.94 0.93 73.27 0.93

Table 7: Percentage investments in SP500 and the average λ∗ for yearly cushions, expected

returns = unconditional mean and α = 0.01.

Revision frequency

Expected returns

Normal Student-t

Wealth % λ∗ Wealth % λ∗

1 year 136.50 1.32 160.49 1.40

4 months 139.76 1.18 164.80 1.26

3 months 140.14 1.15 165.29 1.23

1 month 140.91 1.09 166.30 1.14

1 week 141.25 1.05 166.75 1.07

1 day 141.35 1.02 166.89 1.03

40

Tables 6 and 7 show that the equivalent recommendations of our model for 10% (1%)

significance lie well below (above) the benchmark VaR at 5%. This points out a higher

(lower) risk aversion as resulting in our endogenous VaR*-framework (after restating it

in terms of the exogenous VaR model) relative to the portfolio risk measured by VaR.

However, these percentages are still much higher than those in Table 1, where VaR* is

treated as endogenous. This difference is also observed in the lower values of the equivalent

loss aversion coefficient λ∗ implied by significance levels commonly assumed in exogenous

settings.

Moreover, according to the value function from Equations (2.11) and (2.12), for λ = 1

(and recalling that k = 0), gains and losses exhibit identical values in the investor percep-

tion. An equivalent extreme situation (with λ ≈ 1) can be observed in Tables 6 and 7 for

high revision frequencies (over one month). In the rest of the cases, even for low frequency

revisions (e.g. one year), λ∗ is higher than one (indicating slight loss aversion) but lower

than 2.25 (the empirical level estimated in the original prospect theory and largely used in

previous empirical research19). This reinforces our earlier claim that assuming low signif-

icance levels (as it is the common case in previous portfolio optimization research) entails

an underestimation of the real investor loss aversion captured by the specific coefficient

λ.

4 Conclusions

In this paper we investigate the risk behavior of non-professional investors faced with the

problems of fixing a maximal acceptable level of financial losses and of splitting money

between risk-free assets and a risky portfolio. We assume that these investors are loss

averse, narrow frame financial investments and perceive future portfolio returns subject

to past performance.

We extend the portfolio allocation model developed in Campbell, Huisman, and Koedijk

(2001) in order to incorporate the effect of a desired VaR*, that is now subjectively assessed

by individual loss averse investors. Thus, the first task of non-professional investors con-

sists in fixing the VaR*-level that is subsequently communicated to professional portfolio

managers in charge of finding the optimal portfolio composition. The portfolio optimiza-

tion procedure delivers also the optimal sum of money to be invested in risk-free assets,

which represents another important decision variable for the non-professional investor.

In modelling the investor perception over the risky investment that yields the subjec-

tive VaR*, we rely on the notion of myopic loss aversion introduced in Benartzi and Thaler

(1995) and employ the extended subjective valuation of prospective risky investments pro-

19Such as Barberis, Huang, and Santos (2001), Benartzi and Thaler (1995).

41

posed in Barberis, Huang, and Santos (2001). We integrate these behavioral explanations

in the portfolio decision framework mentioned above, enriching the two models with orig-

inal findings that stem both from theoretical consideration and empirical results obtained

on the basis of real market data (such as SP500 and US 10-year bond price series).

Considering that investors are concerned merely with financial investments, we theo-

retically model their perception over the utility of risky assets and define the maximum

individually sustainable level of financial losses VaR*. This level serves to deciding upon

the optimal amount of money to be invested in the risky portfolio. Also, we assess the

utility of risky prospect captured by the prospective value and suggest two ways of quan-

tifying actual risk attitudes. Moreover, we investigate the influence of different revision

frequencies on the prospective value and the actual risk attitude and derive an optimal

performance check horizon under consideration of practical constraints.

The theoretical results are supported and extended by our empirical findings which, in

sum, show that non-professional investors allocate the main part of their wealth to risk-free

assets. A smaller sum is put into the risky portfolio for increasing frequencies of revising

its performance. Also, financial wealth fluctuations determined by the success of previous

decisions exert a significant impact on the current portfolio allocation, making investors

without substantial gain cushions to firmly refuse holding risky assets. One year appears

to be a critical revision frequency, optimal from the viewpoint of maximizing risky holdings

and commonly used in practice. This revision frequency splits individual perceptions over

risky investments (captured by the prospective value) and over market risk in general

(captured by the actual loss aversion and the global first order risk aversion) in two

qualitatively different segments with distinct evolutions. Estimates of the analytical form

of these functions point out a more pronounced slope for revision frequencies higher than

one year, which we denote as loss aversion in the revision frequency domain. Moreover, the

computation of equivalent loss aversion values for confidence levels commonly assumed

in previous research suggests an underestimation of the loss aversion coefficient of real

non-professional investors.

42

5 Appendix

5.1 Descriptive statistics

Table 8: Log-difference of the SP500 index for quarterly and yearly portfolio revisions.

SP500Revision frequency

Quarterly Yearly

Mean 0.017 0.073

Median 0.017 0.070

Std.Dev. 0.006 0.026

Kurtosis 0.623 0.974

Skewness 0.951 1.042

Max. 0.036 0.142

Min. -0.009 0.037

Obs. 175 43

Table 9: Log-difference of the 10-year bond for quarterly and yearly portfolio revisions.

10-yearRevision frequency

Quarterly Yearly

Mean 0.017 0.066

Median 0.018 0.071

Std.Dev. 0.079 0.137

Kurtosis 2.661 -0.659

Skewness -0.671 -0.205

Max. 0.290 0.345

Min. -0.302 -0.207

Obs. 175 43

43

5.2 Percentage invested in SP500 using different expected re-

turns

Table 10: Percentage investment in SP500 for myopic cushions and expected returns =

zero mean.

Revision frequencyExpected returns

Normal Student-t

1 year 29.63 24.42

4 months 15.55 12.85

3 months 12.43 10.28

1 month 7.45 6.14

1 week 3.70 3.04

1 day 1.79 1.46

Table 11: Percentage investment in SP500 for myopic cushions and expected returns =

AR(1)0.

Revision frequencyExpected returns

Normal Student-t

1 year 31.55 25.31

4 months 16.22 13.29

3 months 14.01 11.36

1 month 7.27 6.02

1 week 0.00 0.00

1 day 0.00 0.00

5.3 The prospective value evolution as a function of the portfolio

revision frequency for different parameter values

44

0 2 4 6 8 10 12 14−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500

PT−Effect

Cushion−Effect

V(t)

(a) All revision frequencies.

1 1.5 2 2.5 3 3.5 4 4.5 5−25

−20

−15

−10

−5

0

PT−Effect

Cushion Effect

V(t)

(b) Daily, weekly, quarterly, and four-monthly revisions.

1 2 3 4 5 6 7 8−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500

PT−Effect

Cushion−Effect

V(t)

(c) Revisions from one to eight years.

Figure 11: Prospective value evolution for different revision frequencies, λ = 0.5, and

k = 3.

45

0 2 4 6 8 10 12 14−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

200

Cushion−Effect

PT−Effect = V(t)

(a) All revision frequencies.

1 1.5 2 2.5 3 3.5 4 4.5 5−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Cushion−Effect

PT−Effect = V(t)

(b) Daily, weekly, quarterly, and four-monthly revisions.

1 2 3 4 5 6 7 8−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

200

Cushion−Effect

PT−Effect = V(t)

(c) Revisions from one to eight years.

Figure 12: Prospective value evolution for different revision frequencies, λ = 1, and k = 3.

46

0 2 4 6 8 10 12 14−5000

0

5000

10000

15000

20000

Cushion−Effect

V(t)

PT−Effect

(a) All revision frequencies.

1 1.5 2 2.5 3 3.5 4 4.5 5−20

0

20

40

60

80

100

120

140

160

V(t)

Cushion−Effect

PT−Effect

(b) Daily, weekly, quarterly, and four-monthly revisions.

1 2 3 4 5 6 7 8−5000

0

5000

10000

15000

20000

Cushion−Effect

V(t)

PT−Effect

(c) Revisions from one to eight years.

Figure 13: Prospective value evolution for different revision frequencies, λ = 3, and k = 3.

47

Table 12: Estimated prospective value evolution as a function of the portfolio revision

frequency for τ < 1 year.

Fitted model Coefficient 95%-CI Goodness of fit

λ = 0.5, k = 3

a1x2 + a2x + a3

a1 = −1.957 (−3.972; 0.05797) R-square: 0.9861

a2 = 5.561 (−6.763; 17.88) Adjusted R-square: 0.9722

a3 = −3.471 (−19.64; 12.7) RMSE: 1.752

λ = 1, k = 3

a1x2 + a2x + a3

a1 = −0.06707 (−0.1443; 0.01011) R-square: 0.9847

a2 = 0.1749 (−0.2971; 0.6469) Adjusted R-square: 0.9695

a3 = −0.1023 (−0.7217; 0.517) RMSE: 0.06712

λ = 3, k = 3

a1x2 + a2x + a3

a1 = 14.69 (5.439; 23.94) R-square: 0.9932

a2 = −48.4 (−105; 8.154) Adjusted R-square: 0.9863

a3 = 34.82 (−39.4; 109) RMSE: 8.042

48

Table 13: Estimated prospective value evolution as a function of the portfolio revision

frequency for τ ≥ 1 year.

Fitted model Coefficient 95%-CI Goodness of fit

λ = 0.5, k = 3

b1x2 + b2x + b3

b1 = −54.5 (−86.1;−22.89) R-square: 0.9888

b2 = −15.41 (−306.8; 276) Adjusted R-square: 0.9844

b3 = −171.2 (−742.8; 400.3) RMSE: 159.4

λ = 1, k = 3

b1 = 2.875 (−7.904; 13.66)

b2 = −81.35 (−372.8; 210.1)

b1x6 + b2x

5 + b3x4 b3 = 893.2 (−2201; 3987) R-square: 0.9972

+b4x3 + b5x

2 + b6x b4 = −4808 (−21, 130; 11, 510) Adjusted R-square: 0.9801

+b7 b5 = 13, 150 (−31, 250; 57, 540) RMSE: 75.19

b6 = −16, 940 (−74, 650; 40, 770)

b7 = 7781 (−19, 300; 34, 860)

λ = 3, k = 3

b1 = 60.27 (−72.43, 193)

b1x4 + b2x

3 + b3x2 b2 = −1071 (−3470; 1329) R-square: 0.8996

+b4x + b5 b3 = 6536 (−8232; 21, 300) Adjusted R-square: 0.7658

b4 = −14, 570 (−49, 980; 20, 840) RMSE: 1774

b5 = 11, 000 (−15, 440; 37, 510)

49

1 1.5 2 2.5 3 3.5 4 4.5 5

−25

−20

−15

−10

−5

0

(a) Daily, weekly, quarterly, four-monthly, and yearly revisions.

1 2 3 4 5 6 7 8−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

(b) Revisions from two to eight years.

Figure 14: Curve fitting for the prospective value on the two relevant revision frequency

segments for λ = 0.5, k = 3.

50

1 1.5 2 2.5 3 3.5 4 4.5 5

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

(a) Daily, weekly, quarterly, four-monthly, and yearly revisions.

1 2 3 4 5 6 7 8

−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

200

(b) Revisions from two to eight years.

Figure 15: Curve fitting for the prospective value on the two relevant revision frequency

segments for λ = 1, k = 3.

51

1 1.5 2 2.5 3 3.5 4 4.5 5

0

20

40

60

80

100

120

140

160

(a) Daily, weekly, quarterly, four-monthly, and yearly revisions.

1 2 3 4 5 6 7 8

0

2000

4000

6000

8000

10000

12000

(b) Revisions from two to eight years.

Figure 16: Curve fitting for the prospective value on the two relevant revision frequency

segments for λ = 3, k = 3.

52

Table 14: Prospective value evolution for different revision frequencies (lower than one

year) and different parameter values.

Revision frequency

1 day 1 week 1 month 3 months 4 months

λ = 0.5

k = 3 -0.03 -0.35 -2.90 -14.38 -23.93

k = 10 -0.03 -0.35 -2.90 -14.38 -23.93

k = 20 -0.03 -0.35 -2.90 -14.38 -23.93

λ = 1

k = 3 0.00 -0.02 -0.13 -0.55 -0.88

k = 10 0.00 -0.02 -0.13 -0.55 -0.88

k = 20 0.00 -0.02 -0.13 -0.55 -0.88

λ = 2.25

k = 3 0.08 0.85 7.91 44.45 80.55

k = 10 0.08 0.85 7.91 44.45 80.55

k = 20 0.08 0.85 7.91 44.45 80.55

λ = 3

k = 3 0.13 1.42 13.69 82.36 158.27

k = 10 0.13 1.42 13.69 82.36 158.27

k = 20 0.13 1.42 13.69 82.36 158.27

53

Table 15: Prospective value evolution for different revision frequencies (higher than one

year) and different parameter values.

Revision frequency

1 year 2 years 3 years 4 years 5 years 6 years 7 years 8 years

λ = 0.5

k = 3 -151.83 -534.59 -741.86 -1054.10 -1740.59 -1957.99 -3065.79 -3794.99

k = 10 -151.83 -534.59 -741.86 -1054.10 -1740.59 -1957.99 -3065.79 -3794.99

k = 20 -151.83 -534.59 -741.86 -1054.10 -1740.59 -1957.99 -3065.79 -3794.99

λ = 1

k = 3 -7.65 -99.63 104.55 -185.69 -373.27 26.59 -245.66 -1553.26

k = 10 -7.65 -99.63 104.55 -185.69 -373.27 26.59 -245.66 -1553.26

k = 20 -7.65 -99.63 104.55 -185.69 -373.27 26.59 -245.66 -1553.26

λ = 2.25

k = 3 736.51 1510.73 2540.90 3618.29 5345.62 6437.69 7744.96 9872.30

k = 10 736.51 1510.73 2540.90 3618.29 5345.62 6437.69 7744.96 9872.30

k = 20 736.51 1510.73 2540.90 3618.29 5345.62 6437.69 7744.96 9872.30

λ = 3

k = 3 2063.66 -65.86 3512.46 2208.50 7055.31 5088.58 6904.74 11504.94

k = 10 2063.66 -44.52 3786.25 2271.02 7055.31 5297.73 7043.24 11504.94

k = 20 2063.66 89.26 3590.36 2360.35 7055.31 5596.52 7241.11 11504.94

54

Table 16: Estimated evolution of the first order global risk aversion as a function of the

portfolio revision frequency for λ = 2.25, k = 3.

Fitted model Coefficient 95%-CI Goodness of fit

τ < 1 year

a1 = −110.8 (−613; 391.4)

a1x3 + a2x

2 a2 = 1, 207 (−3, 342; 5, 755) R-square: 0.9992

+a3x + a4 a3 = −2, 175 (−14, 430; 10, 080) Adjusted R-square: 0.997

a4 = 1, 766 (−7, 610; 11, 140) RMSE: 150

τ ≥ 1 year

b1 = 27.87 (−107.2; 163)

b2 = −759.2 (−4, 412; 2, 893)

b1x6 + b2x

5 + b3x4 b3 = 7, 876 (−30, 910; 46, 660) R-square: 0.9974

+b4x3 + b5x

2 + b6x + b7 b4 = −38, 690 (−243, 300; 165, 900) Adjusted R-square: 0.9815

b5 = 90, 249 (−466, 200; 646, 700) RMSE: 942.5

b6 = −83, 320 (−806, 700; 640, 100)

b6 = 43, 670 (−295, 700; 383, 100)

55

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