All the remaining assumptions made by Mongrut and Ramirez (2006) are also made in this derivation namely, it is
an individual optimization meaning that each individual must define the parameters of the resulting discount rate expression
according to his risk aversion and the specifics of the project. However, one can also define the lower bound expression
assuming the less risk-averse entrepreneur, which implies an absolute risk aversion coefficient equal to 1.
One starts by optimizing the following two-period objective function (one also assumes that the utility function is
time separable):
(
)
−
+
−
+
−
+
−
=
−
+
−
+
1
1
1
1
1
1
,
1
1
1
1
γ
γ
γ
α
γ
γ
δ
γ
α
γ
γ
t
t
t
t
t
C
E
C
C
C
U
Subject to the following constraint:
(
)
t
t
t
t
C
W
C
R
E
−
=
+
+
1
1
Where:
(
)
:
1
+
t
R
E
Represents the project expected return in period “t+1”
t
W
:
The entrepreneur initial wealth level.
t
C
:
The entrepreneur’s consumption from today
1
+
t
C
: The entrepreneur’s consumption from tomorrow
The stochastic discount factor (SDF) of the previous problem is equal to:
γ
γ
γ
α
γ
α
δ
−
+
+
+
+
=
1
1
1
1
t
t
t
C
C
m
(7)
In order to estimate the parameters of the SDF for this individual, one may state the following system of equations
(Mongrut and Ramirez, 2006):
14
Convention