# Dynamic Estimates Of The Arrow-Pratt Absolute And Relative Risk Aversion
Coefficients

George Samartzis
Department of Banking and Financial Management
University of Piraeus
Piraeus, 18534
<EMAIL_ADDRESS>

Nikitas Pittis
Department of Banking and Financial Management
University of Piraeus
Piraeus, 18534
<EMAIL_ADDRESS>

###### Abstract

We derive a closed-form expression capturing the degree of Relative Risk
Aversion (RRA) of investors for non-”fair” lotteries. We argue that our
formula is superior to earlier methods that have been proposed, as it is a
function of only three variables. Namely, the Treasury yields, the returns and
the market capitalization of a specific market index. Our formula, is tested
on CAC 40, EURO, S&P 500 and STOXX 600, with respect to the market
capitalization of each index, for different time periods. We deduce that the
investors in these markets exhibit Decreasing Absolute Risk Aversion (DARA)
through all the different time periods that we consider, while the degree of
RRA has altered between being constant, decreasing or increasing. Furthermore,
we propose a simple and intuitive way to measure the degree to which a wrong
assumption with respect to the utility function of an investor will affect the
structure of his portfolio. Our method is built on a two asset portfolio
framework. Namely, a portfolio consisting of one risky and one risk-free
asset. Applying our method, the empirical findings indicate that the weight
invested in the risky asset varies substantially even among utility functions
with similar characteristics.

Index terms  ARA, RRA

##  1 Introduction

The risk attitude of investors is considered to be a crucial factor in
Portfolio and Decision Theory. A rational investor in the Von Neumann-
Morgenstern sense [19], will always choose the lottery that maximizes his/her
expected utility function. Based on that decision criterion, Arrow and Pratt
[2], [16] developed the concepts of absolute and relative risk aversion
(hereafter; ARA and RRA) as ways to indicate the level of risk-aversion of the
investor. Determining the trends of ARA and RRA is crucial for pointing at a
specific class of utility functions.

The existing literature has done a pile of work on the recovery of degree of
ARA and RRA of investors. In terms of ARA, the literature universally finds
evidence of decreasing ARA (DARA) as wealth increases. This is in agreement
with Arrow [2] who claimed that the DARA ”seems supported by everyday
observation”. On the other hand, the empirical works on RRA derive mixed
conclusions. Some of the earliest studies that found evidence of DRRA with
respect to wealth are [7], [13], [12], [10], [14] and [15]. While evidence of
increasing RRA (IRRA) was found in [18], [4], [3] and [8]. Most of these
works, based their findings on cross-section data, which implicitly assumes
that all investors have the same utility function, a rather unrealistic
assumption. Works like that of [12], [4], [3] and [8] used both experimental
and econometric methods to extract RRA. A totally different approach was used
by [11] and [15] who utilized information from Options markets. The most
recent part of the literature, argues that investors reveal constant RRA
(CRRA). Some of the works supporting this evidence are [17], [5] and [6]. The
advantage of these studies is that they all use panel data. So, the implicit
assumption of a common utility function among investors is no longer made.

Our work, revolves around the definitions of ARA and RRA and the concept of a
lottery that is not ”fair”. Considering the fact that markets are not ”fair”
lotteries the value of such an approach becomes evident. We proceed with
deriving the formulae of the absolute and relative risk premium which
subsequently lead to ARA and RRA when the underlying lottery is not ”fair”.
These formulae can be used to extract the risk attitude of investors across
different markets, specifically, in terms of ARA and RRA.

Using monthly returns and market cap from CAC 40, EURO, S&P 500 and STOXX 600
as well as 10-year Treasury yields, we find evidence of DARA and IRRA for 2012
to 2022, for all markets. Moreover, our results, capture the impact of the QE
program announced by Fed in March 2020. We also test our findings using a
rolling-window approach with 5 and 15-year window sizes to assess their
robustness. The new results do not deviate from our prior conclusions. Then,
we use data for different time periods, namely, 1993-1998 and 1999-2005 to
determine whether or not the investors’ risk attitude is consistent through
years. Our findings, point at DARA and DRRA for 1993-1998 and CRRA/IRRA and
DARA for 1999-2005. As a result, we conclude that through the years the
markets became more risk-averse with respect to their level of wealth.

The evidence of DARA and IRRA for 2012-2022 is satisfied by a specific class
of utility functions. This leads us to our next question which has to do with
measuring the impact of a wrong assumption with respect to the utility
function of an investor. For that, we consider an asset manager who mistakenly
assumes that his customer is characterized by a quadratic utility function
while in reality the customer’s true utility function is logarithmic. Assuming
that the asset manager diversifies the investor’s wealth between a risky and a
risk-free asset we propose a way to measure the differences in the asset
manager’s decisions on behalf of the investor for different utility functions.
The results predicate that the asset manager will mislead his customer
considerably, in case he assumes quadratic instead of logarithmic utility
functions.

Our main findings split into four parts. (i) We derive a closed-form
expression for the degree of RRA for a lottery with nonzero mean. (ii) We
apply our formula on four different markets, namely CAC 40, EURO, S&P 500 and
STOXX 600, to deduce that all of them exhibit DARA and IRRA. (iii) We also
find evidence that the slope of the RRA has changed through different time
periods, while the slope of ARA has always been decreasing. (iv) Finally, we
introduce a way to measure the differences in portfolio diversification among
different utility functions.

##  2 Theoretical Framework

###  2.1 Degree Of Risk Aversion For A ”Fair” Lottery

In this section, we delve into the different notions underlying Decision
Theory. Consider the following case. A decision maker (or investor) with level
of wealth $w_{0}$ is asked whether or not he wants to participate in a game
(or lottery) represented by $Z$. Assuming that his binary relation ($\succeq$)
satisfies the axioms posed by VN-M for any lottery, we can safely say that his
decision criterion arises from the Representation Theorem of VN-M. Namely, he
should always choose the lottery that maximizes his expected utility function.
Now, in order to participate in the game the investor demands a risk premium
represented by $\rho(w_{0},Z)$. The risk premium is based on the notion of
Certainty Equivalent (hereafter; CE). The CE is defined below.

######  Definition 2.1.

The Certainty Equivalent of lottery $Z$ is the amount $z_{0}$ which makes the
decision maker indifferent between lottery $Z$ and this certain amount
$z_{0}$. Alternatively, if the decision maker owns lottery $Z$, then $z_{0}$
represents the minimum amount he is willing to sell the lottery.
Mathematically,

$\zeta_{z_{0}}\sim
Z\overset{VN-M}{\Leftrightarrow}U(z_{0})=E[U(Z)]\Leftrightarrow
z_{0}=U^{-1}(E[U(Z)]),\quad\forall Z$

From the definition, we see that the CE results from the decision maker’s
subjective characteristics, but is also dependent on the objective
(probabilistic) characteristics of lottery $Z$. Thus, the level of ”fear” the
decision maker has with respect to risk will determine the magnitude of CE for
bearing the risk of $Z$. This leads to the definition of risk premium.

######  Definition 2.2.

The risk premium is the difference between the expected return of lottery $Z$
and the Certainty Equivalent the decision maker asks for it.

$\rho(Z)=E[Z]-z_{0}=\mu_{Z}-z_{0}.$

The notion of risk premium translates in two ways. (i) If the decision maker
owns lottery $Z$, $\rho(Z)$ represents the expected units of return he is
willing to ”sacrifice” in order to avoid its risk. (ii) Alternatively, if the
decision maker considers owning lottery $Z$, $\rho(Z)$ represents the expected
units of return he ”demands” in order to bear the risk of lottery $Z$. In our
frame, we will focus exclusively in the case of risk-averse investors with an
increasing utility function $U^{\prime}>0$. That implies that the following
should hold:

  * •

The utility function $U$ is concave, i.e. $U^{\prime\prime}<0$

  * •

The risk premium is positive $\rho(Z)>0,\ \forall Z$

Now, assume that on a later stage of his life the investor has a new higher
level of wealth $w_{0}^{+}$ ($w_{0}^{+}>w_{0}$). What is the new risk premium
$\rho(w_{0}^{+},Z)$ the investor demands assuming that the objective
characteristics of lottery $Z$ remain unchanged? Although, it can be the same
we shall consider the case where

$\rho(w_{0}^{+},Z)\neq\rho(w_{0},Z).$

Since the mean and variance of lottery $Z$ remain the same what is the
causative factor for the change in $\rho$? There should be a function of
$w_{0}$ which we will express by $r(w_{0})$ that affects the risk premium.
Based on Arrow and Pratt this function is called local degree of absolute risk
aversion (hereafter; ARA) of the investor with respect to his level of wealth.
This measure is defined below.

######  Definition 2.3.

The local degree of Absolute Risk Aversion, $r(w)$, at a level of wealth $W$
is defined as

$r(W)=-\frac{U^{\prime\prime}(W)}{U^{\prime}(W)}$

Interpretation of ARA:

  * •

For risk-averse investors we have $U^{\prime\prime}<0$. The larger
$|U^{\prime\prime}|$ it is, the more concave the utility function of the
investor will be. In other words, the investor will be more risk-averse.

  * •

According to VN-M, any positive linear transformation of $U$ should still
satisfy the Representation theorem and by extension lead to the same $r(W)$.
That justifies the division by $U^{\prime}$.

  * •

ARA overcomes an important limitation of the risk premium. Namely, in order to
compare the degree of risk-aversion of two investors, we would need to compare
their risk premiums for any lottery $Z$ and any level of wealth $W$. On the
contrary, $r(W)$ needs to be tested just for any level of wealth $W$.

  * •

This degree of risk aversion is measured in absolute terms (dollars).

Now that we specified the factor that generates the differences in the
aforementioned example, we will derive $\rho(w_{0},Z)$ in terms of $r(w_{0})$.
In fact, we assume that $Z$ is a ”fair” lottery in the sense that $\mu_{Z}=0$
and $\sigma_{Z}$ is relatively small. Let also, the final level of wealth of
the decision maker expressed by

$W_{1}=w_{0}+Z.$

Using the definition of CE together with 2nd degree Taylor series we derive

$\rho(W_{1})=\rho(w_{0},Z)=\rho(w_{0}+Z)=\frac{1}{2}r(w_{0})\sigma_{Z}^{2}.$
(1)

Interpretation of $\rho(w_{0},Z)$:

  * •

The subjective factor, $r(w_{0})$, remains unchanged regardless of any changes
in the objective characteristics of lottery $Z$. However, the risk premium
varies for different lotteries. This is due to the objective factor of the
lottery, $\sigma_{Z}^{2}$.

  * •

If $r(w_{0})$ is decreasing (DARA), increasing (IARA) or constant (CARA) for
any level of wealth $w_{0}$, then $\rho(w_{0},Z)$ will also be decreasing,
increasing, or constant, respectively.

Within the current context, lottery $Z$ is independent of any change in
$w_{0}$. Consider the following case. A decision maker with $w_{0}=1000\$$ is
asked to participate in a coin toss game with the following set of outcomes.
If the coin lands ”Heads” the decision maker will earn $100\$$ while if it
lands ”Tails” he will have to pay $100\$$. The decision maker decides that
this lottery is too risky for his risk appetite. Assume that the decision
maker’s degree of risk aversion is DARA. So, at a later stage of his life his
wealth is $100,000\$$ and thus he feels comfortable to participate in exactly
the same game. Now, assume that the same decision maker is asked to
participate in a different coin toss game $Z^{\prime}$ in which if the coin
lands ”Heads” he will earn $10\%$ of his wealth $w_{0}$ while if it lands
”Tails” he will have to pay $10\%$ of $w_{0}$. That means that for
$w_{0}=1000\$$ he will earn/lose $100\$$ while for $w_{0}=100,000\$$ earn/lose
$10,000\$$. Obviously, this concept may alter the decision made by the
decision maker. In particular, the fact that his wealth increased does not
necessarily mean that he will participate in $Z^{\prime}$ since this new game
changes with his level of wealth, i.e. $Z^{\prime}=w_{0}R$, where $R$ is a
lottery expressed in percentages.

To deal with such cases, Arrow and Pratt defined a new measure called the
local degree of Relative Risk Aversion (hereafter; RRA).

######  Definition 2.4.

The local degree of Relative Risk Aversion, $\lambda(w)$, at a level of wealth
$W$ is defined as

$\lambda(W)=-W\frac{U^{\prime\prime}(W)}{U^{\prime}(W)}=Wr(W)$

Interpretation of RRA:

  * •

RRA depends on ARA, since $\lambda(W)=Wr(W)$. If for example $r(W)$ is
decreasing at a faster rate than the rate at which $W$ is increasing, $r(W)$
will also be a decreasing function of $W$ (DRRA). On the other side, if $r(W)$
is decreasing at a slower rate than the rate at which $W$ is increasing,
$r(W)$ will be an increasing function of $W$ (IRRA).

  * •

This degree of risk aversion is measured in relative terms (percentages).

Under this new measure, consider a ”fair” lottery $Z=Rw_{0}$ with
$\mu_{Z}=\mu_{R}=0$ and $\sigma_{Z},\sigma_{R}$ is relatively small. Then, the
”relative” risk premium for $W_{1}$ will be

$\displaystyle\begin{split}&\rho(W_{1})=\rho(w_{0},Rw_{0})=\frac{1}{2}r(w_{0})\sigma_{Rw_{0}}^{2}\\\
&\frac{\rho(w_{0},Rw_{0})}{w_{0}}=\frac{1}{2}\frac{w_{0}^{2}r(w_{0})\sigma_{R}^{2}}{w_{0}}\\\
&\tilde{\rho}_{w_{0}}(R)=\frac{1}{2}\lambda(w_{0})\sigma_{R}^{2}.\end{split}$
(2)

Based on the above analysis, the literature has done extensive research on the
Decision Making of investors under Risk. More specifically, as we will see in
the following subsections the literature focused on identifying the slope of
the ARA and the RRA of investors with respect to their level of wealth. We
will see that previous research works have followed multiple approaches in
trying to arrive at more reliable conclusions. Following our overview, we will
introduce a totally unique approach on this subject.

###  2.2 Decreasing Absolute Risk Aversion Literature

In 1965, Arrow [2] posed the hypothesis that investors reveal DARA and IRRA.
In fact, he claimed that the DARA ”seems supported by everyday observation”.
However, in terms of IRRA Arrow said that: ”the hypothesis of increasing
relative risk aversion is not easily confrontable with intuitive evidence”.
From that point on, an abundance of research works have attempted to recover
the relative risk preferences of the investors, while DARA is universally
accepted. Extracting RRA, has been proven to be particularly challenging. We
will review the research work on RRA in the subsequent sections.

Haim Levy (1994) [12], applying time-series analysis found that ”only six
subjects out of 62 significantly contradict the DARA property”. As a matter of
fact, Levy concluded that the evidence is significantly strong and supports
Arrow’s assertion. We will analyze Levy’s approach in the next section where
we review his findings on RRA.

Some other studies that find supportive evidence of DARA are [3], [11] and
[8]. Going on, the literature considered DARA to be the norm and focused more
on RRA. This universally accepted property of individual risk preferences
plays a crucial role in many applications of the expected utility theory.

###  2.3 Decreasing Relative Risk Aversion Literature

Let us focus on the bibliography that shows evidence of DRRA. Cohn et al. [7]
focused on the information from 588 different portfolio allocations provided
by customers of a retail brokerage firm. In particular, each customer provided
information with respect to the share he had in common stocks, corporate
bonds, etc., in percentages. The authors established two alternative
classifications for long-term fixed-income securities. In the first one, they
treated Savings Account, Checking Account, Personal Residence, Personal
Property and Other Assets as the ”risk-free” assets. While, in the second
Classification they also included Preferred Stock, Corporate Bonds and
Government Bonds in this category. However, there is no compelling reason to
treat Corporate Bonds as riskless assets. The level of wealth was also given
two different definitions. With the first one being the Total Assets and the
second one being Total Assets less Personal Residence. Although, they
acknowledged the fact that a preferred proxy of wealth would be the Net Worth
of each customer, which was not possible as the customers did not provide any
data on their Liabilities. Such an approach could conceivably alter their
conclusions. The authors found strong evidence of DRRA even when they
controlled for demographic factors like the age of each customer as well as
his/her marital status. In their study, they split the customers into
different wealth groups, in order to determine whether or not their findings
are consistent between different groups. The fact that their target group
pertains to active investors only, means that they do not cover a complete
cross-section of investors at each wealth level. Finally, the aforementioned
results depend on cross-sectional data which means that the authors did not
study the portfolio allocation of each individual in time. This implies that
they draw conclusions based on information from individuals with different
utility functions.

Morin and Suarez [13], used portfolio allocation data of 9,962 different
private households taken from the Survey of Consumer Finances (SCF) in Canada.
The authors argue that the SCF database used is more representative of
investors and broad-based in terms of range of wealth covered. One major
improvement in their methodology is that they use Net worth as a proxy of
wealth. This is due to the private households providing information with
respect to their debt obligations. In their framework, the ”risk-free” assets
include Cash, Deposit Account Balances, Canada Savings Bonds, and Personal
Property. As in Cohn et al., the households were split into different wealth
groups. What is interesting is that for poorer individuals they found evidence
of IRRA while for the middle-wealth individuals DRRA was supported and lastly
for the wealthiest group the findings pointed to CRRA. However, these findings
are based on interpersonal comparisons of utility. A similar approach was done
by Guiso et al. [10]. In fact, Guiso et al. revisited the RRA subject by doing
a cross-sectional analysis on a random sample of 8,274 Italian households who
provided information with respect to their portfolio allocation proportions.
Their findings showcased that the households revealed DRRA.

In 1994, Haim Levy [12] introduced a different approach in determining the
slope of RRA of individuals. Levy conducted an experiment in which 62 of his
MBA students participated. Each student was given an initial investment
allotment of $30,000\$$ ”paper” money and was offered stocks of 20 pure equity
firms as well as a ”risk-free” rate of $2\%$. The major difference between
Levy’s approach and the two aforementioned works is that he ran time-series
regressions for each individual and did not test cross-sectionally. What this
means, is that through his approach the results are not based on interpersonal
utility functions. Now, in terms of RRA, Levy argued that the results indicate
a decreasing trend with respect to wealth. On top of that, Levy also did
robustness checks studying the subjects’ portfolio allocation when the only
available assets are one risky and one risk-free asset. In this frame, the
conclusions did not deviate from the previous findings. Even though this new
approach overcomes some previous limitations an important caveat of this work
is the small sample size.

A whole different approach was proposed from Perignon and Villa [15] in 2002.
More specifically, they attempted to extract information from Put and Call
Options on CAC 40 with regards to the slope of RRA. The two authors, based
their methodology on Ait-Sahalia and Lo (2000) [1] who defined a pure-exchange
economy. Namely, in this economy and in equilibrium, the investor optimally
invests all his wealth in the single risky stock at every instant prior to the
terminal date and then consumes the terminal value of the stock at time $T$.
This implicitly says that the level of wealth of the individual is exactly
equal to the stock price at each time. In other words, the representative
agent consumes only at the final date and maximizes the expected utility of
the terminal wealth by choosing the amount invested in the stock at each
intermediary date. In this framework, Ait-Sahalia and Lo derive the ”implied”
RRA formula. Empirically, Perignon and Villa, obtain an ”implied” RRA for the
CAC 40 stock index and they conclude that as the index price (i.e. the wealth)
increases the ”implied” RRA decreases.

###  2.4 Increasing Relative Risk Aversion Literature

Contrary to the previous works, Siegel and Hoban [18] find that by restricting
the sample to higher wealth target groups will lead to DRRA or CRRA. But they
argue that ”the use of a broader based sample and a more comprehensive
measurement of wealth alters the conclusions and a pattern indicative of
increasing relative risk aversion emerges”. Siegel and Hoban used data from
the US National Longitudinal Surveys (NLS). More specifically, they acquired a
sample of 2,881 different sets of asset holdings of individual households. The
proxy for level of wealth was Net Worth. As for the ”riskless” assets they
used Cash, Deposit Account Balances, and U.S. Savings Bonds. Splitting into
different groups of level of wealth they concluded that for any group of
individuals there is evidence of IRRA rather than DRRA. We should keep in mind
that this paper constitutes a cross-sectional analysis as in [7] and [13].

In 2003, Eisenhauer and Ventura [8] found evidence of both DARA and IRRA. The
authors asked different households in Italy the following question: ”You are
offered the opportunity of acquiring a security permitting you, with the same
probabilities, either to gain 10 million lire or to lose all the capital
invested. What is the most you are prepared to pay for this security?”. The
answer to this question is denoted by $z$ and we can interpret it as the CE of
the respective household. The lower $z$ is the more risk-averse the household.
The proxy of level wealth, denoted by $w$, was the average income of each
household from 1993 to 1995. Now, the authors observed that 1,624 of the
households interpreted the 10 million lire as a gross gain meaning that in the
favourable case they get $w-z+10$ while in the other case their wealth becomes
$w-z$. A number of 1,730 households interpreted the 10 million lire as a net
gain meaning that in the favourable case they get $w+10$ while in the other
case their wealth remains to be equal to $w$. In any case, they found that
investors exhibit DARA and IRRA. Again, the most obvious limitations are that
the results are based on a hypothetical survey question as well as on a cross-
sectional analysis.

Two additional works finding evidence of IRRA are those of [4] and [3]. The
first one was based on a set of gamble questions that had to do with the
retirement decisions. The second one extracts the risk attitude of Israeli
farmers.

###  2.5 Constant Relative Risk Aversion Literature

The research works finding evidence of CRRA are more recent and better
substantiated. The first work finding evidence of CRRA was that of Friend and
Blume (1975) [9]. Using cross-section regressions based on household-level
data on asset holdings the authors conclude that CRRA characterizes household
behaviour. The main drawbacks of this study are that it is based on cross-
sectional data and the data is only focused on high-wealth households.

We will now discuss about the work of Sahm (2012) [17] which is the latest
work of our review but it had started since 2006. Sahm initialized the use of
panel data instead of cross-section data with regards to determining the slope
of RRA in terms of wealth. In particular, he used gamble responses across the
1992 to 2002 waves of the Health and Retirement Study (HRS). The gamble under
study is the following: ”Suppose that you are the only income earner in the
family. Your doctor recommends that you move because of allergies, and you
have to choose between two possible jobs. The first would guarantee your
current total family income for life. The second is possibly better paying,
but the income is also less certain. There is a 50-50 chance the second job
would double your total lifetime income and a 50-50 chance that it would cut
it by a third. Which job would you take - the first job or the second job?
Individuals who accept the first risky job then consider a job with a larger
downside risk of one-half, while those who reject the first risky job are
asked about a job with a smaller downside risk of one-fifth. If they reject
the first two risky jobs, individuals consider a third risky job that could
cut their lifetime income by one-tenth. Likewise, if they accept both risky
jobs, individuals consider a third risky job that could cut their lifetime
income by three-quarters.” Sahm finds no effect of wealth changes on relative
risk aversion and thus concludes CRRA.

The next work we are going to discuss is that of Brunnermeier and Nagel (2008)
[5]. The authors use household-level panel data from the Panel Study of Income
Dynamics (PSID), covering a period of 20 years between 1984 and 2003. The data
includes Asset holdings, Income and households characteristics. To identify
how wealth changes are related to market participation they use probit
regressions. In particular, the authors first split the data in two
subperiods, 1984 to 1999 and 1999 to 2003. The reason is that for 1984–1999
the time-span between successive waves of the PSID with wealth information is
$k=5$ years, and for the 1999–2003 sample the time-span between successive
waves of the PSID with wealth information is $k=2$ years. In practice, the
probit regressions derive the probability of the households who did not
participate in the stock market at time $t-k$ to enter until time $t$. They
also estimate the probability that a household that is participating at $t-k$
to exit the stock market until $t$. The authors find that for both subperiods,
namely 1984-1999 and 1999-2003, there is a $1\%$ probability to enter the
market when there is a $10\%$ increase in the wealth. Accordingly, the
probability of exiting the stock market when there is an increase in the
wealth is extremely low. Thus, they conclude that the households’ relative
risk aversion remains constant with respect to a change in wealth (CRRA). A
major improvement of this work is that it uses panel data. This way you avoid
the implicit assumptions made in the cross-sectional analysis. Namely,
assuming that the distributions of wealth and preferences are independent.

Another important work is that of Chiappori and Paiella (2011) [6]. Similar to
[5] and [17], they use panel data to showcase evidence of CRRA. In fact, they
prove that previous studies that supported their findings on cross-sectional
analysis are led to erroneous conclusions. Citing the authors: ”without a
priori restrictions on the joint distribution of wealth and preferences, the
form of individual preferences simply cannot be recovered from cross-sectional
data. In fact, any form of individual preferences is compatible with any
observed, joint distribution of wealth and risky asset shares provided that
one can freely choose the joint distribution of wealth and preferences.”. In
other words, estimating the joint distribution of wealth and risky asset
shares using cross-section data derives the joint distribution of wealth and
preferences only and only if you pre-assume that preferences are the same for
each investor. This of course is an over-simplification of reality, which
substantiates the use of panel data instead of cross-section data. The data
used is from the Survey of Household Income and Wealth (SHIW), which is a
large-scale household survey run every two years by the Bank of Italy from
which they get asset allocations for 1,332 households. Also, they exclude all
the households with a change in wealth less than $25\%$. As in [5] and [17],
they find evidence of CRRA.

##  3 Degree Of Risk Aversion For A Lottery With Non-Zero Mean

In the previous section, we derived the ”absolute” and ”relative” risk premium
for a ”fair” lottery. In reality however, stock prices are not a fair lottery
meaning that $E[Z]=\mu_{Z}\neq 0$. Equivalently, stock returns are not a
”fair” lottery and so $E[R]=\mu_{R}\neq 0$. Deriving ARA under $\mu_{Z}\neq 0$
leads to RRA through $\lambda(w_{0})=w_{0}r(w_{0})$. So, for a non-fair
lottery with $\mu_{Z}\neq 0$ we have

$\rho(W_{1})=\rho(w_{0}+Z)=E[w_{0}+Z]-z_{0}=w_{0}+\mu_{Z}-z_{0},$

where $z_{0}$ represents the CE of lottery $Z$ with $w_{0}$ level of wealth
(or equivalently the CE of lottery $w_{0}+Z$). Thus, the CE will be

$z_{0}=w_{0}+\mu_{Z}-\rho(w_{0}+Z).$

According to Definition (3.1), the following equation must be satisfied

$E[U(Z)]=U(z_{0}).$

In our case,

$E[U(w_{0}+Z)]=U(w_{0}+\mu_{Z}-\rho(w_{0}+Z)).$

At this point, we use Taylor expansions of 2nd-order around $w_{0}$ for both
sides of the equation. For the left side of the equation we derive

$\displaystyle\begin{split}E[U(w_{0}+Z)]&\approx
U(w_{0})+U^{\prime}(w_{0})E[w_{0}+Z-w_{0}]+\frac{U^{\prime\prime}(w_{0})}{2}E[(w_{0}+Z-w_{0})^{2}]\\\
&\approx
U(w_{0})+U^{\prime}(w_{0})\mu_{Z}+\frac{U^{\prime\prime}(w_{0})}{2}(\sigma_{Z}^{2}+\mu_{Z}^{2}).\end{split}$
(3)

Accordingly, for the right side of the equation we get

$\displaystyle\begin{split}U(w_{0}+\mu_{Z}-\rho(w_{0}+Z))&\approx
U(w_{0})+U^{\prime}(w_{0})(w_{0}+\mu_{Z}-\rho(w_{0}+Z)-w_{0})+\frac{U^{\prime\prime}(w_{0})}{2}(w_{0}+\mu_{Z}-\rho(w_{0}+Z)-w_{0})^{2}\\\
&\approx
U(w_{0})+U^{\prime}(w_{0})(\mu_{Z}-\rho(w_{0}+Z))+\frac{U^{\prime\prime}(w_{0})}{2}(\mu_{Z}-\rho(w_{0}+Z))^{2}\\\
&\overset{\text{drop 3rd
term}}{\approx}U(w_{0})+U^{\prime}(w_{0})(\mu_{Z}-\rho(w_{0}+Z)).\end{split}$
(4)

Now, combining (3) and (4) we get

$\displaystyle
U(w_{0})+U^{\prime}(w_{0})\mu_{Z}+\frac{U^{\prime\prime}(w_{0})}{2}(\sigma_{Z}^{2}+\mu_{Z}^{2})\approx
U(w_{0})+U^{\prime}(w_{0})(\mu_{Z}-\rho(w_{0}+Z))$
$\displaystyle\frac{U^{\prime\prime}(w_{0})}{2}(\sigma_{Z}^{2}+\mu_{Z}^{2})\approx-U^{\prime}(w_{0})\rho(w_{0}+Z)$
$\displaystyle\rho(w_{0}+Z)\approx-\frac{U^{\prime\prime}(w_{0})}{2U^{\prime}(w_{0})}(\sigma_{Z}^{2}+\mu_{Z}^{2})$
$\displaystyle\rho(w_{0}+Z)\approx\frac{1}{2}r(w_{0})(\sigma_{Z}^{2}+\mu_{Z}^{2}).$

So, compared to (1) the risk premium will satisfy the following equation

$\boxed{\rho(w_{0}+Z)\approx\frac{1}{2}r(w_{0})(\mu_{Z}^{2}+\sigma_{Z}^{2})}$
(5)

The main change is that the risk-premium is now positively related also to
$\mu_{Z}^{2}$. In other words, a higher mean in absolute terms, would increase
the required $\rho$. Now solving for the ARA term $r(w_{0})$ we get

$w_{0}+\mu_{Z}-z_{0}\approx\frac{1}{2}r(w_{0})\mu_{Z}^{2}+\frac{1}{2}r(w_{0})\sigma_{Z}^{2}$
$\boxed{r(w_{0})\approx\frac{2(w_{0}+\mu_{Z}-z_{0})}{\mu_{Z}^{2}+\sigma_{Z}^{2}}}$
(6)

The above formula represents a closed-form solution for ARA when $Z$ is not a
”fair” lottery. In relative terms, $Z$ can be expressed in the following way

$R=\frac{Z}{w_{0}}\Leftrightarrow Z=Rw_{0}.$

Therefore, we have

$\displaystyle\rho(w_{0}+Z)\approx\frac{1}{2}r(w_{0})w_{0}^{2}(\sigma_{R}^{2}+\mu_{R}^{2})$
$\displaystyle\rho(w_{0}+Z)\approx\frac{1}{2}r(w_{0})(\sigma_{Z}^{2}+\mu_{Z}^{2}).$

And so, the ”relative” risk premium for $\mu_{R}\neq 0$ takes the following
form

$\displaystyle\frac{\rho(w_{0},Rw_{0})}{w_{0}}=\tilde{\rho}(w_{0},R)$
$\displaystyle\approx\frac{1}{2}r(w_{0})w_{0}(\sigma_{R}^{2}+\mu_{R}^{2})$
$\displaystyle\approx\frac{1}{2}\lambda(w_{0})(\sigma_{Z}^{2}+\mu_{Z}^{2})$
$\boxed{\tilde{\rho}(w_{0},R)=\frac{1}{2}r(w_{0})w_{0}(\mu_{R}^{2}+\sigma_{R}^{2})}$
(7)

Now, we can derive the RRA term $\lambda(w_{0})$.

$\boxed{\boldsymbol{\lambda(w_{0})=\frac{2(1+\mu_{R}-\tilde{z}_{Rw_{0}})}{\mu_{R}^{2}+\sigma_{R}^{2}}}}$
(8)

We are going to use this last formula to extract the level of RRA of the
market. Observe that $\mu_{R}$ and $\sigma_{R}$ could represent the mean and
standard deviation of market returns, respectively. Likewise,
$\tilde{z}_{Rw_{0}}$ represents the risk-free rate. Finally, $w_{0}$ is the
market capitalization of the market. But before going any further, a careful
overview of the existing literature needs to be made.

##  4 Empirical Results

The previous sections showcased that there is no definitive answer as to if
and how much RRA is affected by a change in wealth. In our view, the largest
part of the literature has focused on determining the slope of RRA using an
approach with multiple perils. Namely, most studies have to deal with
demographic and socio-economic characteristics or over-simplifications. Even
controlling for those characteristics may lead to sub-optimal conclusions. The
most recent part of the literature finds evidence of CRRA using panel data. We
believe that these studies are more meaningful and better substantiated.

In our framework, we try to avoid the issues noted in cross-sectional data and
at the same time simplify even further the extraction of RRA from real data.
More specifically, we believe that one should focus on determining the
market’s ARA and RRA through the formulae we derived. These formulae are free
of any need of including and subsequently controlling for any subjective
characteristic of the investors such as educational status or age or even
level of wealth. The only subjective characteristic lies in the CE (or
equivalently, risk-free rate) which we will measure using 10-year Treasury
yields, since this is considered a logical proxy of the risk-free rate in the
literature.

We use monthly returns and market cap from four different markets, namely CAC
40, EURO, S&P 500 and STOXX 600, spanning from 1991 to 2021 excluding 2008.
The reason why we exclude 2008 is because this period is known for extreme
levels of volatility which may affect our concluding remarks. We believe that
the advantage of using monthly returns instead of daily returns is that they
do not exhibit volatility clustering. This means that $\mu_{R}$ and
$\sigma_{R}$ can be estimated recursively. Estimations start from 2012 using
all the past data and continue up until 2022. As a proxy of risk-free rate we
use the respective 10-year Treasury yields of each market. The reason we want
to extract RRA for all these indexes is to determine whether or not the
results are consistent across different markets.

The first thing we want to answer is how do ARA and RRA of the aforementioned
market indices change with respect to an increase in their market cap. From
Figure 5, we deduce that for all market indices ARA is decreasing (DARA). In
fact, the correlation is almost perfectly negative for all markets. What this
says, is that in all these indices the investors become more risk averse in
absolute terms. So, as the level of wealth (market cap) increases investors
are willing to take on more risk. This evidence is strongly supported by the
findings of the literature. Figure 6, presents the RRA of each market with
respect to each level of wealth. We observe that for the European indices RRA
displays an increasing trend with high positive correlation. When it comes to
S&P 500, we see that the correlation is $-36\%$, but we should not focus
entirely on it. Taking a closer look, we can see that up until the level of a
market cap of approximately $27$ trillion dollars the RRA exhibits an
increasing trend. More specifically, the correlation between the market cap
and the degree of RRA up until this point is approximately $90\%$. In fact, we
argue that the sudden drop at a market cap of approximately $27$ trillion
dollars is due to an important economic event that took place during the first
quarter of 2020, which we will discuss, in detail, in the following paragraph.
Thus, we maintain that we should not be distracted by a one-off event.

A closer look at Figure 1 justifies the sharp drop observed in the level of
relative risk aversion of the US market. In this figure, we plot RRA and
Market Cap in time. In February 2020, the Fed Chair Mr. Jerome H. Powell, in
an attempt to soothe the market which was extremely nervous due to the
COVID-19 pandemic, hinted a forthcoming rate cut. In March 2020, Fed announced
the new QE program through which it would purchase $600 billion in bank debt,
U.S. Treasury notes, and mortgage-backed securities (MBS) from member banks.
At the same time, it cut down the federal funds rate by a total of 1.5
percentage points. To comprehend the impact of such an action by the central
bank of United States we should refer to equation (7). The level of RRA is
negatively related to the CE (or equivalently risk-free rate) which is set by
the investors. So, assuming that the objective (probabilistic) characteristics
of the lottery remain unchanged, as CE gets smaller RRA should increase.
Equivalently, the investors would become more risk-averse and so they would
require a smaller CE in order to avoid the risky lottery. But this is not the
case in the graph. In fact, RRA plummets. To determine the reason behind this
drop we should break down the graph in two parts. Up until February 2020, it
was the market that had been setting the CE. But in March 2020, Fed set a new
lower level of risk-free rate. This new rate was not the result of a change in
the investors’ level of risk-aversion. It actually resulted from the
artificial cut in interest rates by the Federal Reserve Bank. This led to a
group of less risk-averse bondholders shifting to the stock market. So, the
drop in RRA gives the impression that all bondholders became less risk-averse,
which is not true. Assuming that Fed had not intervened, in which case would
we have an analogous shift from bonds to stocks by the bondholders? In case
the bondholders became more risk-averse. Thus, we conclude that the QE program
forced investors to move into risky assets and acted ”as-if” the bondholders
became more risk-averse. Furthermore, this action seems that affected the
other markets as well, which proves that markets are interconnected when it
comes to important economic events.

The discussion made in the previous paragraphs leads to the following
conclusion. The investors reveal DARA and IRRA for 2012-2022. It needs to be
highlighted that this conclusion applies for this time period and should not
be treated as a generalization on the investors’ risk preferences. In other
words, we imply that market participants may have had different risk
preferences during different time periods. In fact, we will see later on that
for different time periods the slope of the RRA of the different market
indices varies. These findings indicate that, currently, we could assume that
the investors’ class of utility functions should satisfy both DARA and IRRA.
The following utility functions are a part of this class.

$\displaystyle U(W)=(W+a)^{c},\ a>0,\ $ $\displaystyle 0<c<1,\quad
U(W)=-(W+a)^{-c},\ a>0,\ c>0,$ $\displaystyle U(W)=log(W+a),\ a>0$

Any utility function from this group serves as an appropriate proxy for the
true utility function of the markets. We proceed as follows. First, we will
compare our evidence of DARA and IRRA to a rolling-window approach. This will
serve as a robustness test of our evidence based on recursive estimations.
Then, we will extract the investors’ risk preferences through S&P 500 for
different time periods, namely, 1993-1998 and 1999-2005.

Figure 1: RRA: S&P 500

###  4.1 Rolling-Window Approach

As previously stated, our estimations with regards to $\mu_{R}$ and
$\sigma_{R}$ were done recursively. One could argue that such an approach is
problematic, reasoning that the estimations should better be updated using
more recent data. For that reason, we also estimate $\mu_{R}$ and $\sigma_{R}$
by employing a rolling-window approach. We test three different window sizes,
$M=60$ (5-years), $M=120$ (10-years) and $M=180$ months (15-years). These
three window sizes are regularly applied by the literature for monthly data.

The graphs in Figures 8 and 9 showcase that for a window size of $M=60$
months, or equivalently, a $5$-year window, there is strong evidence of DARA
and IRRA. In fact, the correlation between the ARA and the market cap is
highly negative and smaller than $-30\%$ for all markets. Accordingly, the
correlation between the RRA and the market cap is above $40\%$ for all markets
except for S&P 500, for which the magnitude of correlation is mainly affected
by the QE program announced by Fed. When we increase the window size to
$M=120$ motnhs, or equivalently, a $10$-year window, Figures 10 and 11, point
to the same direction with the exception of S&P 500, in which case it seems
that the RRA has a decreasing trend. Lastly, in Figures 12 and 13, it is
evident that we have similar conclusions to those made in the 5-years case.
More specifically, for a window size of $M=180$ months, or equivalently, a
$15$-year window, the evidence of DARA and IRRA is even stronger, with
correlations being higher in absolute terms. Overall, our results for
2012-2022 reveal strong evidence of DARA and IRRA.

###  4.2 RRA And ARA For S&P 500 In Different Time Periods

Now, we are going to study ARA and RRA of S&P 500 in different time periods.
Doing that will help us determining the risk attitude of US investors across
the years. On top of that, we would like to check whether or not the evidence
of DRRA and CRRA in earlier studies is supported by our model. For this, we
will recover ARA and RRA for 1993-1998 and 1999-2005.

Figure 2 displays the level of ARA and RRA for 1993-1998. What is worth
noticing is that investors’ risk attitude was rather different than currently
is, revealing DRRA instead of IRRA. This result supports the evidence of DRRA
found in [13], [12], [10] and [15] for earlier time periods. Although some of
these works refer to different time periods or different markets they are
indicative of the investors’ risk attitude overall. Now, the RRA graph on
Figure 3 is harder to interpret. The graph is indecisive between CRRA and
IRRA. Studies like those of [17], [5] and [6] found evidence of CRRA for this
time period. However, our graph does not offer a clear conclusion. In terms of
the ARA, both Figures reveal DARA which has been validated by most empirical
studies, for any time period.

Overall, our empirical evidence demonstrates the importance of studying the
risk attitude of investors across different periods. It seems that the
market’s utility function varies through years. More specifically, our
findings point to a utility function satisfying DRRA and DARA for years 1993
through 1998 meaning that investors were willing to take on more risk as they
became wealthier. For 1999 to 2005, investors started being more cautious as
they reached higher levels of wealth, revealing CRRA/IRRA and DARA. Currently,
the different markets seem to become continuously more risk-averse as their
market caps grow even further, revealing evidence of IRRA and DARA.

(a) ARA

(b) RRA

Figure 2: ARA and RRA: 1993-1998

(a) ARA

(b) RRA

Figure 3: ARA and RRA: 1999-2005

Following our empirical evidence in the previous sections, a subsequent
question would be: How much does a wrong assumption with respect to the
utility function of an investor, made by an asset manager, will affect the
structure of his portfolio? Could we find an easy and intuitive way to measure
the difference between the level of risk that the investor should incur, based
on his ”true” utility function, and the level of risk that the asset manager
recommends, based on his perception of the investor’s utility function? We
approach these questions in the following section.

###  4.3 Measuring The Differences In Portfolio Diversification Among
Different Utility Functions

Consider the following case. A financial advisor designs a survey to extract
the risk preferences of an investor. The asset manager misinterprets the
investor’s answers and concludes that the investor reveals IARA and IRRA. In
reality however, the investor reveals DARA and IRRA. How much would the
decisions made by the asset manager on behalf of the investor be affected?
Assuming IARA and IRRA leads to a very well-known utility function, the
quadratic $U(W)=W-bW^{2}$ with $b>0$. This utility function is considered by
the literature to be a sufficient condition for Markowitz’s Mean-Variance
Optimization method. So, by assuming a quadratic utility the asset manager
should use Markowitz’s method to diversify the investor’s portfolio. In the
following paragraph we are going to illustrate an intuitive way to measure the
effect of assuming a quadratic rather than a DARA and IRRA utility function
like one of those shown below.

$\displaystyle U(W)=(W+a)^{c},\ a>0,\ 0<c<1,$ $\displaystyle
U(W)=-(W+a)^{-c},\ a>0,\ c>0,$ $\displaystyle U(W)=log(W+a),\ a>0\ $
$\displaystyle U(W)=-e^{-c(W+a)},\ a>0,\ c>0$

Assume that at time $t=0$ the investor’s wealth is $w_{0}$. The asset manager
decides to invest $w_{s}$ in a risky asset with return $R$ and $w_{0}-w_{s}$
in a riskless asset with rate of return $R_{f}$. Both $w_{0}$ and $w_{s}$ are
measured in percentages. So, $w_{0}=100\%$ while $w_{s}$ can take any value.
The idea is that if $w_{s}>w_{0}$, the asset manager can borrow money to over-
invest in the risky asset, while if $w_{s}<0$ the asset manager will short-
sell the risky asset. So, at point $t=1$ the investor’s wealth will be

$\displaystyle W_{1}$ $\displaystyle=w_{s}(1+R)+(w_{0}-w_{s})(1+R_{f})$
$\displaystyle=w_{0}(1+R_{f})+w_{s}(R-R_{f}).$

Based on VN-M Representation Theorem the asset manager will simply need to
maximize the expected utility function of the investor for $W_{1}$.

$\underset{W_{1}}{\max}E[U(W_{1})]\Leftrightarrow E[U^{\prime}(W_{1})]=0$

Solving the above equation derives $w_{s}$. This percentage will differ among
different utility functions which means that we can measure the differences
between them. In our case, we will find $w_{s}^{quad}$ from the quadratic
utility function $U(W)=W-bW^{2}$ with $b=0.2,0.3,0.4$ and compare it with
$w_{s}^{log}$ from $U(W)=\log{W}$. Below, we derive the two quantities.
Namely, for $U(W)=W-bW^{2}$ we have

$w^{quad}_{s}=\frac{\mu_{R}-R_{f}-2b(1+R_{f})(\mu_{R}-R_{f})}{2b(\mu_{R}^{2}+\sigma_{R}^{2}-2\mu_{R}R_{f}+R_{f}^{2})}$

In terms of $U(W)=\log{(W)}$, we are unable to apply the approach we propose
directly on it, since we will end up with a fraction equal to $0$ that cannot
be solved. We can omit this hurdle by taking the Taylor expansion of 2nd-order
around $0$ as follows.

$\displaystyle U(w_{s})$ $\displaystyle\simeq
U(0)+w_{s}U^{\prime}(0)+\frac{w_{s}^{2}}{2}U^{\prime\prime}(0)$
$\displaystyle\simeq\log{(1+R_{f})}+w_{s}\frac{R-R_{f}}{1+R_{f}}-w_{s}^{2}\frac{(R-R_{f})^{2}}{2(1+R_{f})^{2}}$

Thus, we obtain

$w^{log}_{s}=\frac{(1+R_{f})(\mu_{R}-R_{f})}{\mu_{R}^{2}+\sigma_{R}^{2}-2\mu_{R}R_{f}+R_{f}^{2}}$

To measure the difference between the two utility functions we need to use
real data. For that reason, we are going to use monthly returns from S&P 500
and 10-year Treasury yields spanning from 2012 to 2022. Then, we will plot
only the weight invested in the risky asset, namely $w_{s}$, for both the
quadratic and the logarithmic utility functions, with respect to time.

Figure 4: Quadratic vs Logarithmic utility’s weight in the risky asset

The graph indicates that in case the asset manager had assumed one of the
depicted parametrizations of the quadratic utility function for the investor,
he would advise him to put significantly different portions of his wealth in
S&P 500 each month. Specifically, if $b=0.2$ the asset manager would
underestimate the ”true” level of risk-aversion of the investor, by advising
him to put $50\%$ more on S&P 500 than what his ”true” level of risk tolerance
suggests. While if $b=0.3,0.4$, the asset manager would overestimate the
”true” level of risk-aversion of the investor, by almost $50\%$ to $100\%$.
So, the asset manager’s misjudgment would lead the investor to incur
substantially more or less risk than he could actually tolerate. We should
highlight here that even if we consider the case in which the ”true” utility
function of the investor is quadratic with $b=0.3$, the weight invested in the
risky asset, namely $w_{s}$, is substantially higher through all years,
compared to the $b=0.4$ case. So, even if the asset manager correctly detects
that the investor’s utility is quadratic, there is still plenty of room to
underestimate (or overestimate) his true level of risk-aversion.

In the previous analysis, we compared the logarithmic utility function which
is known to correspond to investors exhibiting DARA and IRRA, with the
quadratic utility function which exhibits IARA and IRRA. So, although the
logarithmic and the quadratic utility functions are both increasing and
concave, they differ in the slope of the level of ARA. It would be interesting
to compare utility functions that exhibit DARA and IRRA. So, instead of the
quadratic utility function, we will now compare $\log{W}$ to $\sqrt{W}$ and
$-e^{-2W}$.

As in the case of $\log{W}$, we will need to use the Taylor expansion of 2nd-
order around $0$ for $\sqrt{W}$, as follows.

$\displaystyle U(w_{s})$ $\displaystyle\simeq
U(0)+w_{s}U^{\prime}(0)+\frac{w_{s}^{2}}{2}U^{\prime\prime}(0)$
$\displaystyle\simeq\sqrt{1+R_{f}}+w_{s}\frac{R-R_{f}}{2\sqrt{1+R_{f}}}-w_{s}^{2}\frac{(R-R_{f})^{2}}{8(1+R_{f})^{3/2}}$

Accordingly for $-e^{-2W}$, the Taylor expansion of 2nd-order around $0$ will
be

$\displaystyle U(w_{s})$ $\displaystyle\simeq
U(0)+w_{s}U^{\prime}(0)+\frac{w_{s}^{2}}{2}U^{\prime\prime}(0)$
$\displaystyle\simeq-e^{-2(1+R_{f})}+2w_{s}(R-R_{f})e^{-2(1+R_{f})}-2w_{s}^{2}(R-R_{f})^{2}e^{-2(1+R_{f})}$

Thus, we derive

$\displaystyle
w^{sqrt}_{s}=\frac{2(1+R_{f})(\mu_{R}-R_{f})}{\mu_{R}^{2}+\sigma_{R}^{2}-2\mu_{R}R_{f}+R_{f}^{2}}=2w^{log}_{s}$
$\displaystyle
w^{exp}_{s}=\frac{\mu_{R}-R_{f}}{2(\mu_{R}^{2}+\sigma_{R}^{2}-2\mu_{R}R_{f}+R_{f}^{2})}=\frac{w^{log}_{s}}{2(1+R_{f})}$

The above formulae measure exactly the differences between the weight invested
in the risky asset for each utility function. Namely, the square root utility
function requires double the weight invested in the risky asset, compared to
that of the logarithmic. Accordingly, the exponential utility function which
represents a more risk-averse investor requires approximately half the weight
invested in the risky asset, compared to that of the logarithmic. Below we
plot the weight invested in the risky asset, namely $w_{s}$, for the different
utility functions, with respect to time.

Figure 5: Logarithmic vs Square root vs Exponential utility’s weight in the
risky asset

To conclude, we designed a simple and intuitive method to measure the
differences between alternative utility functions, with relative accuracy. A
natural extension of this idea would be to study the above approach for DARA
and IRRA utility functions, by using higher orders of Taylor series.

##  5 Conclusions

In this work, we extensively overviewed the various ways the literature has
extracted the ARA and the RRA of investors through the years. We saw that, the
literature universally accepts DARA but is not conclusive with respect to the
RRA of investors. In fact, one part of previous research works reveals
evidence of DRRA, another part finds evidence of IRRA and the most recent
works point to CRRA. We asserted that using cross-section data as was
regularly done in earlier research works, conceals risks. More specifically,
cross-sectional analysis implicitly says that all investors have the same
utility function, which is a rather unrealistic assumption. Studies basing
their evidence on either panel data or Options markets data are better
supported.

We proposed a different approach in extracting both ARA and RRA. This
approach, is based on carefully analyzing the Theory of Arrow-Pratt together
with the different notions that underlie the Decision Theory under Risk. In
particular, we derived closed-form expressions for the ARA and the RRA of
investors, for non-”fair” lotteries. We proceeded with collecting data from
different markets which we then applied on our formulae. Our findings, pointed
out that through 2012-2022, European as well as US markets revealed evidence
of DARA and IRRA. We further showcased that our formula captures important
economic events as in the case of the QE program announced by Fed in March
2020, which caused a sharp drop in the investors’ level of relative risk-
aversion. Our results were further tested using a rolling-window approach for
different window sizes $M=60$, $M=120$ and $M=180$ months. The rolling-window
results supported our evidence of DARA and IRRA for 2012-2022. Then, we found
out that for different time periods, namely, 1993-1998 and 1999-2005 RRA may
vary. More specifically, for 1993-2005 we found strong evidence of DARA and
DRRA while for 1999-2005 we found strong evidence of DARA but in terms of RRA
our results were indecisive between CRRA and IRRA. In the last part of our
work, we proposed a simple way to measure the effect of a wrong assumption
with respect to the utility function of an investor. As it became clear, an
investor with a logarithmic utility function is led to a very different
portfolio structure compared to an investor with a quadratic utility function.

## Appendix

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 6: ARA with sorted wealth

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 7: RRA with sorted wealth

(a) CAC 40

(b) EURO

(c) STOXX 600

Figure 8: RRA in time

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 9: ARA with $M=60$

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 10: RRA with $M=60$

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 11: ARA with $M=120$

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 12: RRA with $M=120$

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 13: ARA with $M=180$

(a) CAC 40

(b) EURO

(c) STOXX 600

(d) S&P 500

Figure 14: RRA with $M=180$

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