function is not possible to obtain because
0
≠
α
and 1. One must remember that the constant absolute risk aversion
coefficient of this utility function is equal to
α
.
The case of a quadratic utility function appears when the following conditions are met:
1
−
→
γ
, k = -1 and
0
>
α
. By replacing these conditions in equation 12 yields:
(
)
( )(
)
(
)
−
−
−
−
≈
−
∑
−
−
=
−
+
−
+
+
+
1
1
1
1
1
1
1
1
1
1
,
1
k
t
t
t
t
t
f
t
f
t
t
C
W
R
R
R
R
Binomial
E
R
R
E
α
(13)
Simplifying the previous expression yields:
(
)
(
)
−
−
≈
−
+
+
+
α
t
t
t
f
t
t
f
t
t
C
W
R
R
R
E
R
R
E
1
1
1
(14)
Now, one must consider the following two conditions. The first one is related to the less risk-averse investor and
the second to the fact that early-stage entrepreneurs start with a small initial level of wealth.
0
1
1
1
1
→
−
=
⇒
=
−
=
t
t
t
W
C
C
A
α
α
α
Given these two conditions, equation 14 provides the lower bound for the discount rate for entrepreneurs with
quadratic preferences:
(
)
(
)
t
f
t
t
CV
RTV
R
R
E
−
+
≥
+
1
1
1
α
(15)
Where:
(
)
(
)
1
1
+
+
−
=
t
f
t
t
R
R
R
E
RTV
σ
is the reward-to-variability ratio and
(
)
(
)
1
1
1
+
+
+
=
t
t
t
t
R
E
R
CV
σ
is the coefficient of variation
16
Convention