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1 Derivation of eq. (10.79) in Ljungqvist & Sar-
gent - p. 262
First, L&S make the assumption that the returns and consumption growth
follow the following process:
c
t+1
c
t
=¯c
?
exp
©
?
c,t+1
??
2
c
/2
ª
(1)
1+r
i
t+1
=
¡
1+¯r
i
¢
exp
©
?
i,t+1
??
2
i
/2
ª
; i = s,b (2)
It is assumed that the means of the innovations (?
t+1
) are 0. And also that
they are normally distributed.
The basic asset pricing equation is:
1=?E
"Ã
1+r
i
t+1
µ
c
t+1
c
t
¶
??
!#
(3)
Using the assumed processes for the returns and consumption growth (eqs.
(1) and (2) yields:
1=?
¡
1+¯r
i
¢
¯c
??
?
E
©
exp
£
?
i,t+1
??
2
i
/2??
¡
?
c,t+1
??
2
c
/2
¢¤ª
(4)
Note that eq.(1) has simply been raised to the power of ??.Thetermin
braces can be written as:
?
i,t+1
???
c,t+1
??
2
i
/2+??
2
c
/2 (5)
That last two terms are constants. Define the variable x as:
x
t+1
= ?
i,t+1
???
c,t+1
(6)
Since ?
i
and ?
c
are jointly normally distributed...

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