1. Static Decision Making
Objectives:
Understanding of
•
Individual decision making under uncertainty
•
Individual evaluation of goods with uncertain future values
•
Behavior towards risk
•
The concept of the risk premium
•
Preferences on assets with normal distributed returns
•
Alternatives to the Expected Utility Hypothesis
Concepts
•
Expected Utility Hypothesis
•
Risk Averseness, Risk Neutrality, Risk Loving
•
Markowitz Risk Premium and Certainty Equivalent
•
Arrow-Pratt Approximation of Risk Premium
•
Mean-Variance Preferences
•
Prospect Theory
Contents:
1.1
Contingent Goods
1.2
Structure of the Model
1.3
Expected Utility Hypothesis
1.4
Behavior towards Risk
1.5
Risk Premium
1.6
Mean Variance Criterion
1.7
Static Alternatives to the Expected Utility Hypothesis
Financial Modeling I
Static Decision Making
2
1.1 Contingent Goods
The concept of contingent goods developed by Kenneth Arrow is very useful to characterize
the future value of goods in an uncertain world. In the following we apply it only to financial
assets.
•
Under certainty goods are characterized by their physical quality, by the time and the
location of their availability.
•
Under uncertainty we characterize goods additionally by the incidence of future
(events that influence the future) value of goods.
•
It is convenient to declare the incidence of a certain event as a “state of the world” and
the future value of the good is thus contingent upon this “state of the world”.
•
In this case the future value of goods is a random variable. The random variable may
have finite or infinite, but numerable realizations.
Event Tree
In an inter-temporal context the concept of states of the world can be presented with an event
tree. The event tree represents the complete sequence of the realizations of a discrete random
variable from the present to a future date.
Figure 1.1: 3-period event tree
In this graph we observe a simple example of an economy that lasts three periods and can
take three randomly determined states. The random generator may influence the level of
business activities at each node (yellow point) of the tree. The level of activity may be high in
the left (blue) branch, medium in the middle (red) branch and low in the right (green) brunch.
In this example we have 33 = 27 endpoints of the tree.
-
In general, if an economy lasts t= 1,…T periods and the random influence generates
ω= 1,…Ω possible outcomes in each point of realization the tree has Ω T different
endpoints.
-
Each endpoint is associated with a complete history of random realizations from the
present to a certain future date.
-
Each endpoint of the tree is representing a state of the world at date T. There are
s= 1,… S of them with S = Ω T .
-
The complete history of these realizations is of economic relevance. Think of a sequence
of good or bad economic developments.
Financial Modeling I
Static Decision Making
3
One Period Model
We assume that in the 1-period future the economy assumes s= 1,… S states of the world.
Figure 1.2: 1-period event tree
Information
We do not discuss any information problems. We can imagine different types of information
problems, such as insider information, or lower costs of public or private available
information. These are very interesting topics. However, it is beyond Financial Modeling I.
Value of Assets
When we discuss assets we think of financial assets in most cases. However, most of the
insights apply also for real assets. Because of depreciation and storage costs some minor
modifications and extensions are necessary.
Moreover, we restrict our analysis to a one-period analysis. In this case we can characterize
assets by their current and future values – including (intra-period) payoffs – or even more
conveniently by their returns. Therefore we use the following conventions:
-
The value of an asset X at the beginning of the period is X 0 .
The random future value of an asset - at the end of the current (or the beginning of the
 1 with the probability distribution g X
 1 . (The
future) period – is denoted as X
( )
circumflex is attached to random variables.)
-
In discrete presentations it is convenient to express the random future values of assets
and their probability of occurrence as functions of state of the world s. In this case
X 1 ( s ) denotes the future value of the asset in the state s that occurs with probability
gX 1 (s) .
-
The random generator governing the future values of assets (associated with certain
states of the world) may follow any probability distribution. However, many models
assume that future values of assets are lognormal distributed. Lognormal, since the
minimal value of a many assets – stocks, plain vanilla bonds etc. – is zero.
Return on Assets
According to convenience the returns of assets are defined as arithmetic or geometric rate of
 X 1 of the asset X is defined as
returns. The future (gross) return R
Financial Modeling I
Static Decision Making
4
X 1 (s)
or
RX ( s ) =
X0
X
In some models we use the (net) rates of return, which are defined as
(1.1)
 X 1 − 1 ≈ ln R
 X1 ,
rX 1 ≡ R
(1.2)
1
 X1 = X ,
R
0
1
or
rX 1 ( s ) ≡ RX 1 ( s ) − 1 ≈ ln RX 1 ( s )
As the future value of the assets is random the return is, of course, also a random variable. We
express the probability distribution as f (rX 1 ) , or f  rX 1 ( s )  , respectively. The probability
functions, f (rX 1 ) and f  rx1 ( s )  , are approximately normal distributed if the probability
 1 and g  X 1 ( s )  are lognormal distributed. (For the definition of normal or
functions g X


( )
lognormal probability distributions see Appendix A.2)
Simplification of the Notation
Utility functions can be defined on total wealth, on contingent goods, or contingent values of
assets or portfolios of assets. In the second and third section we show that these insights apply
to all objects of individuals’ preferences. Thus we omit the indication of specific goods or
assets in these sections.
Since we restrict our analysis to the one-period analysis during these lectures we simplify
notation by omitting the time indices. To simplify notations we omit the time indices. We
 , or X ( s ) ,
denote the present value of the asset x as X , and its random future values as X
 X and RX ( s ) , or rX and rX ( s ) , respectively.
and returns as R
1.3 Expected Utility Hypothesis
John von Neumann and Oskar Morgenstern proved that under some specific assumptions (see
Appendix A.1.3) individual preferences concerning lotteries (games) can be presented as the
expected utility of the payoffs of the lottery.
1.3.1 Definitions and Concepts
A twice continuously differentiable (concave) utility function u (· ) is defined over the
(future) value of the asset.
-
Future value of the asset can be grasped by a lottery (game) that is defined by the
 and the probability function g X
 , or in a state depending
random future payoffs X
( )
presentation by =
X ( s )  X ( 1) ,… , X =
( S ) and g X  g X (1) ,g X ( 2 )… ,g X ( S ) . This
presentation can be applied to total wealth and to future value of assets or portfolios of
assets as well.
-
( )
 ;g X
  or G  X ( s ) ; g X ( s )  . For simplicity we
We write a lottery (game) as G  X



omit the specification of the probability distribution.
Financial Modeling I
Static Decision Making
5
-
The utility of a lottery (game) presents the individual preferences over the lottery
 ; g  , or U G ( X ( s ) ; g )  .
(game), i.e. U G X



-
The expected utility of the lottery (game) is defined as the expected value of the utility
of the payoffs of a lottery (game), i.e.
(
(
)
)
{ ( ) }
 ;g  = E U X
 ; g , or
U G X

U G
=
( X ( s ) ; g ) g (1) u  X (1) +…+ g ( S ) u  X ( S ) .
-
(1.3)
The expected utility of a lottery (game) has to be distinguished from the utility of the
expected value (of the payoffs) of the lottery (game). The latter is defined as
{
(
) } { ( )} , or
 ;g  = u E X

U E G X

{
}
U E G (=
X ( s ) ; g )  u  g ( 1) X ( 1) +…+ g ( S ) X ( S ) 
(1.4)
In order to present the expected utility hypothesis in a graphical exposition and a small
numerical example it is recommended to introduce a binomial world with two states of the
world, s = 1,2 , with the payoffs X ( 1) and X ( 2 ) which occur with probabilities g ( 1) , and
g ( 2 )= 1 − g ( 1) .
 as µ X , the variance as
In order to simplify notation we express the expected value of X
σ XX , and the standard deviation as σ X .
The expected value and the variance of these payoffs are
=
µ X g ( 1) X ( 1) + g ( 2 ) X ( 2 ) ,
(1.5)
and
=
σ XX g ( 1)  X ( 1) − µ X  + g ( 2 )  X ( 2 ) − µ X  .
2
2
(1.6)
The expected utility of this game is
{
}
E U=
 X ( s ) ; p  g ( 1) u  X ( 1)  + g ( 2 ) u  X ( 2 )  ,
(1.7)
and the utility of the expected value is
{
}
U E  X ( s ) ; g  = u  g ( 1) X ( 1) + g ( 2 ) X ( 2 )  = u ( µ X ) .
(1.8)
In the subsequent graphic the expected utility is presented as a convex combination of the
payoffs somewhere along the red line depending on the probabilities of the payoffs. The
utility of the single payoffs and moreover the utility of the expected value of the game
(lottery) are located along the utility function. If the utility function is concave the utility of
the expected value must exceed the expected utility of the game (lottery).
Financial Modeling I
Static Decision Making
6
Figure 1.3: Expected Utility
The expected utility of the game presented in figure 1.3 can be illustrated by a simple
numerical example. We assume that the utility function is a square root u  X ( s )  = X ( s ) .
Moreover we specify the payoffs of the lottery as x ( 1) = 100 , and x ( 2 ) = 10000 , and the
binomial probabilities as g ( 1) = 3 4 , and therefore g ( 2 ) =
1 − g ( 1) =
1 4 . The expected
value of the game is thus µ X =
3 4* 100 + 1 4* 10000 =
2,575 .
The expected utility of the game is thus defined as
{
}
E U  X ( s ) ; g  = 3 4* 100 + 1 4* 10000 = 7.5 + 25 = 32.5 .
The utility of the expected value of the game is
{
}
U E  X ( s ) ; g  = 3 4* 100 + 1 4* 10000 = 75 + 2500 = 2575 ≈ 50.74 .
Obviously the utility from the game with uncertain outcomes (the expected utility) is
significantly smaller than the utility from the deterministic expected value of the game.
1.3.2 An Alternative Presentation of the Expected Utility Hypothesis
 into expected value of
Sometimes it is convenient to decompose the random variable X
 and a lottery ε .
E X
( )
( )
 E X
 + ε ,
=
X
(1.9)
Obviously, ε is a fair lottery with an expected value of zero and a constant variance equal to
 :
the variance of X
Financial Modeling I
Static Decision Making
7
(
)
 − µX = 0
E ( ε ) = E X
{
(1.10)
(
)
} {
( )
}
Var ( ε =
) E  X − µ X − E X − µ X  = E  X − E X  = σ XX .
2
2
(1.11)
In the binomial case with two states, s = 1,2 , and the probabilities g ( s ) the values of the
assets deviate from its mean by ε ( s ) . The asset value can be presented by the sum of the
expected value of the initial game plus the game of the stochastic deviations. The expected
value and the variance of the initial game can be expressed as
E {µ X + ε } = µ X + g ( 1) ε ( 1) + g ( 2 ) ε ( 2 ) = µ X ,
(1.12)
and
Var ( µ X =
+ ε ) g ( 1)  µ X + ε ( 1) − µ X − µε  + g ( 2 )  µ X + ε ( 2 ) − µ X − µε 
2
2
= g ( 1) ε ( 1) + g ( 2 ) ε ( 2 ) .
2
2
(1.13)
Thus the expected utility of the game can be expressed as
{
}
E u  µ X + G ε ( 1) ,ε ( 2 ) ; g =
  g ( 1) u  µ X + ε ( 1)  + g ( 2 ) u  µ X + ε ( 2 )  .
(1.14)
Also this version of the expected utility can be illustrated graphically:
Figure 1.4: Expected Utility Fair Game
In the numerical example of the previous section the stochastic deviations of the payoffs from
the mean are defined as
ε ( 1) =
X ( 1) − µ X =
100 − 2575 =
−2475 ,
Financial Modeling I
Static Decision Making
8
and
ε ( 2 )= X ( 2 ) − µ X= 10000 − 2575= 7425 .
Thus the mean of the deviations equals
µε = g ( 1) ε ( 1) + g ( 2 ) ε ( 2 ) = 3 4* ( −2475 ) + 1 4* 7425 = 0 .
1.3.3 The Existence of a von Neumann Morgenstern Utility Function
If preferences are defined over the entire range of lotteries and if these preferences satisfy
some specific assumptions (see Appendix A.1.3) a utility function with the following
property can be defined over the range of lotteries:
{
}
S
U G ( X ( s ) ; g )  E U G ( X ( s ) ; g )  ≡ ∑ g ( s ) u ( X ( s ) )
=
(1.15)
s =1
Under the von Neumann Morgenstern axiomatic, the preferences over lotteries can be
represented by the expected utility of the outcomes of the lottery.
1.4 Behavior towards Risk
Without explanation we used a concave utility function that implies risk averse investors. In
this section we will discuss this topic explicitly. We will classify the individuals’ preferences,
their attitudes towards risk, and the shape of the utility function.
1.4.1 Characterization of the Behavior towards Risk
An individual’s behavior towards risk can be classified into three categories:
-
Risk-averse behavior
Risk-loving behavior
Risk-neutral behavior
Risk Averse Behavior
The individual is risk-averse if it prefers the expected value of the payoffs of a lottery (game)
rather than the lottery itself. In terms of utilities this means the utility of the expected value of
the payoffs of a lottery is greater than the utility of the lottery (game). This condition is
obviously fulfilled in case of a concave utility function.
 S
 S
u [µX =
] u ∑ g ( s ) X ( s ) > ∑ g ( s ) u  µ X + ε ( s )= E {u [ µ X + ε ]} .
=
 s 1=
 s1
(1.16)
Depending on the probabilities the expected utilities are along the red lines in the graphs
below whereas the utility of the expected value is somewhere along the utility function.
Financial Modeling I
Static Decision Making
9
Figure 1.5: Utility functions of risk averse, risk loving, and risk neutral individuals
Risk Loving Behavior
If the individual prefers the lottery (game) rather than the expected value of the payoffs of the
lottery (game) it is a risk lover. In terms of utilities this means the utility of the expected
value of the payoffs of a lottery is smaller than the utility of the lottery (game). This condition
is satisfied if the utility function is convex.
 S
 S
u [µX =
] u ∑ g ( s ) X ( s ) < ∑ g ( s ) u  µ X + ε ( s )= E {u [ µ X + ε ]}
=
 s 1=
 s1
(1.17)
Risk Neutral Behavior
The individual is risk neutral if it is indifferent between the expected value of the payoffs of a
lottery and the lottery itself, or, if the utility of the expected value of the payoffs of a lottery
equals the utility of the lottery (game). This is the case if the utility function is linear.
 S
 S
u [µX =
] u ∑ g ( s ) X ( s )= ∑ g ( s ) u  µ X + ε ( s )= E {u [ µ X + ε ]}
=
 s 1=
 s1
(1.18)
1.4.2 Local measures of risk behavior
In the economic analysis we use two measures of behavior towards risk:
-
Absolute Risk-Aversion (ARA)
-
Relative Risk-Aversion (RRA)
As both ARA and RRA are local measures we should apply them only for small changes of
wealth. (In this section we omit the indices of time and state of the world because risk
aversion is a time independent property of the utility function rather than the random
realization of future wealth.)
The absolute risk-aversion is defined as:
ARA ≡ −
u ′′ ( X )
u′ ( X )
(1.19)
ARA is a measure of the curvature of the utility function. It decreases with its slope and it
increases with the change of the slope.
Financial Modeling I
Static Decision Making
10
For many economic problems it is of great importance how ARA changes with increasing
wealth. The answer can be seen from the derivative of ARA with respect to wealth:
u ′′ ( X )
2
u ′′′ ( X ) u ′ ( X ) − u ′′ ( X ) 
u′ ( X )
dARA
≡−
=−
2
dX
dX
u ′ ( X ) 
d
(1.20)
As the denominator is positive the sign of the change of ARA is determined by the following
condition:
dARA
≤ (≥) 0
dX
⇔
u ′′′ ( X ) u ′ ( X ) ≥ ( ≤ ) u ′′ ( X ) 
2
(1.21)
Since u ′′ ( X ) < 0 we can rewrite the condition as follows:
dARA
≤ (≥) 0
dX
⇔
u ′′′ ( X )
u ′′ ( X )
≤ (≥)
u ′′ ( X )
u′ ( X )
(1.21’)
Therefore absolute risk-aversion decreases with increasing wealth if the second derivative is
less elastic than the first derivative of the utility function.
The relative risk aversion is defined as the elasticity of the marginal utility, i.e.
RRA ≡ −
u ′′ ( X )
=
X ARA ⋅ X
u′ ( X )
(1.22)
In order to investigate how the relative risk aversion changes with increasing wealth we
calculate its respective derivation:
 u′′ ( X ) 
d −
X
u′ ( X )  d ( ARA ⋅ X )
dRRA
dARA

=
=
= ARA + X
dX
dX
dX
dX
(1.23)
u ′′ ( X ) u ′ ( X ) + u ′′′ ( X ) u ′ ( X ) X − u ′′ ( X )  X
= −
2
u ′ ( X ) 
2
u ′′ ( X )
u ′′′ ( X )
dRRA
≤ (≥) 0 ⇔ 1 −
X ≤ (≥) −
X
dX
u′ ( X )
u ′′ ( X )
(1.24)
1.4.3 Two Useful Examples
Constant Absolute Risk Aversion (CARA) utility function
u ( x ) = −e − bx
′ (· ) be − bx > 0 , u ′′ (· ) =
−b 2 e − bx < 0 ,
u=
(1.25)
−b 2 e − bx
ARA ( x ) =
b>0
−
=
be − bx
dARA ( x )
=0
dx
(1.26)
Obviously, this utility function exhibits CARA (Constant ARA).
Financial Modeling I
Static Decision Making
11
dRRA ( x )
−b 2 e − bx
−
RRA ( x ) =
x=
bx > 0
= b>0
− bx
be
dx
(1.27)
Obviously, RRA increases with increasing outcome (goods).
Constant Relative Risk Aversion (CRRA) utility function
In many models economists use an utility function with constant relative risk aversion.
u ( x) =
x 1− γ
1−γ
−γ x −γ −1 < 0 .
x ) x −γ > 0 , u ′′ ( x ) =
with u ′ (=
(1.28)
The absolute risk aversion of this utility function is therefore decreasing in x:
−γ x −γ −1
1
ARA =
−
=
γ >0
−γ
x
x
dARA
1
=
−γ 2 < 0
dx
x
(1.29)
The relative risk aversion is obviously a constant
RRA=
γ x − γ −1
x −γ
x= γ > 0
(1.30)
1.5 Risk premium
The concept of the risk premium has been developed by Harry M. Markowitz. He defines the
risk premium as the difference between the expected wealth of a lottery (game) and the
certainty equivalence of the lottery (game), i.e. the certain amount of wealth (money) for
which an individual is indifferent against the lottery. Therefore we can view the risk premium
as the costs of a lottery (game).
1.5.1 Definition of the Markowitz Risk Premium
Following this idea the risk premium π [ µ X ,ε ] can be defined as
u  µ X − π [ µ X ,ε ] =E {u [ µ X + ε ]} ,
(1.31a)
or
 S
 S
u  ∑ g ( s ) X ( s ) − π  µ X ,ε ( s ) ,g ( s )   =
∑ g ( s ) u  µ X + ε ( s )
 s 1=
 s1
(1.31b)
The risk premium is defined as the amount of money that can be deducted from the expected
value in order to equalize the utility of the residual with the expected utility of the game. The
risk premium has to equalize the utility of the certainty equivalent, i.e. the difference of the
expected payoffs and the risk premium, and the expected utility of the game (lottery).
For the numerical example and the graphic presentation we use again the binomial
1) µ X + ε ( 1) , and
simplification in the fair game version with the payoffs X (=
X ( 2=
) µ X + ε ( 2 ) . In this model the certainty equivalent is defined as
CE =
g ( 1)  µ X + ε ( 1)  + g ( 2 )  µ X + ε ( 2 )  − π  µ X ,ε ( s ) ,g ( s )  =
µ X − π  µ X ,ε ( s ) ,g ( s )  ,(1.32)
Financial Modeling I
Static Decision Making
12
and the risk premium is implicitly determined by the equation
u [CE
=
] g ( 1) u ( µ X + ε ( 1) ) + g ( 2 ) u ( µ X + ε ( 2 ) ) .
(1.33)
The risk premium can thus be expressed as
π  µ X ,ε ( s ) ,g ( s )  =
µ X − u −1  g ( 1) u ( µ X + ε ( 1) ) + g ( 2 ) u ( µ X + ε ( 2 ) )  .
(1.34)
From this presentation it is obvious that the size of the risk premium depends on the mean of
the game, the stochastic deviations of the payoffs from the mean, and the probabilities of
these stochastic deviations.
In the subsequent graph the risk premium is displayed by the green line.
Figure 1.6: Markowitz risk premium
The graphic representation shows that the height of the risk premium depends on four factors:
-
The concavity of the utility function
-
The location of the expected value
-
The stochastic deviations of the payoffs from the mean
-
The probabilities of the stochastic payoffs
Of course, we can determine the risk premium in the numerical example. It is determined by
the equation
3 4* 100 + 1 4* 10000 − =
π (· ) 3 4* 100 + 1 4* 10000 ,
or
Financial Modeling I
Static Decision Making
13
2575 − π (· ) =
32.5 .
If we square the equation, and solve for the risk premium we receive
π (· ) =
2575 − 1056.25 =
1518.75 .
The size of the risk premium is due to the strong concavity of the root-function, the big
stochastic deviations from the mean, and the high probability of the low payoff in state 1.
1.5.2 A local approximation of the Markowitz risk premium
Kenneth Arrow and John W. Pratt developed a useful and simple measure of this risk
premium. The measure can be derived by a Taylor approximation of π ( ⋅) from the equation
u  µ X − π [ µ X ,ε ] =E {u [ µ X + ε ]} .
(1.31a)
We expand the left side in order to receive
u  µ X − π ( µ X ,ε )  ≅ u ( µ X ) − π ( µ X ,ε ) u ′ ( µ X ) .
(1.35)
Expanding the right hand side around E ( ε ) = 0 gives
1


E u ( µ X + ε )  ≅ E u ( µ X ) + ε u ′ ( µ X ) + ε 2u ′′ ( µ X ) 
2


1
=
u ( µ X ) + E ( ε ) u ′ ( µ X ) + Var ( ε ) u ′′ ( µ X )
2
(1.36)
1
= u ( µ X ) + Var ( ε ) u ′′ ( µ X )
2
The second order term is necessary since the middle term vanishes because of E ( ε ) = 0 .
If we insert (1.35) and (1.36) into (1.31a), and solve for π ( µ X ,ε ) we receive the so-called
Arrow-Pratt approximation of the risk premium:
u ′′ ( µ X ) 1
1
−
Var ( ε ) =
ARA Var ( ε ) .
π ( µ X ,ε ) =
u′ ( µ X ) 2
2
(1.37)
Of course it is only a local measure of the risk premium around µ X . From this Arrow-Pratt
approximation we acknowledge the central determinants of a risk premium:
•
The level of wealth
•
The (absolute) risk aversion of the individual
•
The volatility of the lottery (game)
The Markowitz risk premium of the CRRA utility function
The risk premium of the CRRA utility function is:
Financial Modeling I
Static Decision Making
14
π ( µ X ,ε ) =
γ 1
Var ( ε )
µX 2
(1.38)
1.6 Mean Variance Criterion
Many financial models are based on the assumption that the future values of assets are
lognormal and/or their returns are normal distributed. In these cases the von Neumann
Morgenstern utility function can be redefined on the mean and the variance of a lottery.
1.6.1 Return and Risk of Assets
 X − 1 the (expected) return µr equals
rX X
As the rate of return of an asset is defined as=
( )


X
 E X
µX
− 1 =
− 1=
−1,
X
X
X

µr ≡ E 
(1.39)
and the variance denoted as σ rr is
{
}
2
  X


2
X
   1
1
 −E X
=
 ≡ 1 σ XX ,

E  X
Var X
σ rr E   − 1 − E  − 1 =
=
 
2
2

X
X2
X
   X
  X
( )
( )
(1.40)
where σ XX is the variance of the value of the asset.
1.6.2 The EUH in terms of the Rate of Return
For some applications it is useful to define the utility over the rate of return rather than the
wealth. If we denominate the utility of the rate or return as u (r ) and the probability function
of the normal distributed rate of return as f (r ; µr ,σ rr ) with the return µr and the variance
σ rr (or the standard deviation σ r ) the expected utility of the rate of return can be expressed as
E u (r )  =
∞
∫ u (r ) f (r ; µ ,σ ) dr .
r
(1.41)
rr
−∞
 as
It is useful to use the standard normal distributed random variable η
η =
r − µr
.
σr
(1.42)
 is zero and the variance is equal to one:
The expected value of η
 r − µr  E (r ) − µr
µη E=
=

 = 0
σr
 σr 
σ ηη
(1.43)
2

 
2


 r − µr     E (r − µr )
 r − µr
=
−E
E 

  =
σ r   
σ rr

 σ r


 
 
=0
Financial Modeling I
{
}  =
1

Static Decision Making
(1.44)
15
r µr + σ rη we can write dr = σ r dη , and taking into account
Having in mind the relation =
f (r ; µr ,σ rr ) =
 ;0,1)
f (η
σr
,
(1.45)
we can rewrite the expected utility of the rate of return in terms of the standard normal
 :
variable η
∞
E u (r )  =
∫ u ( µr + σ rη )
 ;0,1)
f (η
σr
−∞
∞
σ r dη =
∫ u ( µr + σ rη ) f (η ;0,1) dη
(1.46)
−∞
1.6.3 Indifference curves of a Risk-Averse Investor
In order to calculate the µ − σ - indifference curves it is recommended to understand the
expected rate of return as an implicit function of the variance of the rate of return. Therefore,
we derive the expected utility with respect to the standard deviation to receive
dE u (r ) 
dσ r
∞
 dµ

0.
=
∫−∞ u′ ( µr + σ rη )  dσ rr + η  f (η ;0,1) dη =
(1.47)
It is convenient to decompose the right hand side of the double equation as
d µr
dσ r
∫
∞
−∞
 ;0,1) dη + ∫
u ′ ( ⋅) f (η
∞
−∞
.
 f (η ;0,1) dη =
u ′ ( ⋅ )η
0
(1.48)
Figure 1.7: Mean-Variance Preferences
 as η + and negative values as η − the concavity of the utility
If we denote positive values of η
function implies for η + = η −
u ′ ( ⋅)η1+ f (η1 ;0,1) < u ′ ( ⋅) η1− f (η1 ;0,1) .
(1.49)
From (1.48) we derive the slope of the indifference curve as
Financial Modeling I
Static Decision Making
16
∞
d µr
dσ r
∫ u′ ( µ
 )η f (η ;0,1) dη
+ σ rη
r
=
− −∞∞
d  E (U )  =0
∫ u′ ( µ
r
 ) f (η ;0,1) dη
+ σ rη
=
−
A
>0.
B
(1.50)
−∞
(1.49) implies A < 0 . The denominator B is positive since each element of the integrand is
positive. Thus, the slope of the µ − σ indifference curve is positive.
The slope of a µ − σ - indifference curve is increasing if (if and only if) the individual is risk
averse. This can be seen if we derive (1.50) with respect to σ R . Applying the fraction rule we
receive
dµ

dµ

.
  η f (η ;0,1) dη − A ⋅ ∫ u′′ ( ⋅)  r + η  f (η ;0,1) dη
B ⋅ ∫ u ′′ ( ⋅)  r + η
d
d
σ
σ
d µr
 r

 r

=
−
>0
dσ r2
B2
2
(1.51)
Equation (1.51) can be rewritten as
dµ
d 2 µr
A d µr A 
1
 + η 2 −
  f (η ;0,1) dη
=
−   ∫ u ′′ ( ⋅)  r η
− η
2
dσ r
d
σ
B
d
σ
B
B
 r

r
.
(1.52)
Using equation (1.50) we can express (1.52) as
.
d 2 µr
 1  ′′  A   2 A A A   
=−   ∫ u ( ⋅)  − η + η +
− η f (η ;0,1) dη
dσ r2
B B B 
B
 B
(1.53)
Applying the binomial formula we can simplify (1.53) to
.
(1.54)
d µr
1
 A 
 ;0,1) dη > 0
=
−   ∫ u ′′ ( ⋅) η
−  f (η
2
dσ r B
B



 

2
2
<0
<0
Thus, the slope of an indifference curves is positive and increasing.
Example of a corresponding utility function
A simple example of a utility function defined on the return und the risk of an asset can be
given as
u (r , µr ,σ rr=
) µr − ασ rr 2 ,
(1.55)
with ∂µr ∂σ =
2ασ rr > 0 , and ∂ 2 µr ∂σ rr =2α > 0 .
rr
2
Financial Modeling I
Static Decision Making
17
1.7 Static Alternatives to the Expected Utility Hypothesis
During the last three decades different alternatives of the expected utility have been
developed. This new developments are based on empirical anomalies discovered by Maurice
Allais and proceeded by Daniel Kahneman and Amos Tversky and other psychologists.
Kahneman and Tversky developed the Prospect Theory (PT). They proposed to substitute the
overall concave value function associated with the EUH by a S-shaped value function which
is concave in the positive range and convex at the negative range of the payoffs centered
around the mean (median) of the payoffs. Moreover, probabilities are replaced by decision
weights.
=
U G ( X ( s ) ; g )  w  g ( 1)  u  X ( 1)  +…+ w  g ( S )  u  X ( S )  .
(1.56)
The centered value functions of PT and EUH are presented in the subsequent graphs:
Figure 1.8a: PT Value Function
Figure 1.8b: EUH Value Function
Obviously, the PT Value Function implies strong loss aversion, much stronger than the EUH.
The decision weights are defined on the probabilities associated with the positive and
negative payoffs centered around the mean (median):
Figure 1.9: PT Decision Weights
Financial Modeling I
Static Decision Making
18
The decision weights are lower than the probabilities except for payoffs (states of the world)
with low probabilities. You can imagine these decision weights as a kind of a cognitive bias.
PT is appealing since it solves some empirical anomalies. The shortcoming of the approach is
its ad hoc character. The theory misses an axiomatic of the preferences. Moreover, it creates
problems in macroeconomic models.
In addition there are different models of habit formation. Habit formation means that the
evaluation of present consumption depends on experienced consumption streams in the past.
Finally, there are different specifications of recursive preferences. Despite big differences in
the specification of the utility functional these approaches have in common that the evaluation
of consumption streams in one period depends somehow on the expected consumption
streams of subsequent periods. From a purely formal view these approaches lead to
comprehensive dynamic evaluation of consumption streams. The dynamics are backward or
forward oriented. Since we do not apply these approaches in the asset pricing models we
examine in this class we don’t discuss these preferences in detail in this class.
Financial Modeling I
Static Decision Making
19