Lecture 9,time and assets
market
Contents
? Inter-temporal preferences
– Two periods
– Several periods
? Asset market
– CAPM
– APT
– Complete market
– Pure arbitrage
Inter-temporal preferences
? Utility function of inter-temporal
? Every period consumption ct depend on
how much he consumed and invested in
period t-1,
1
1
1
( ) ( )
T
t
Tt
t
U c c u c? ?
?
? ?L
Inter-temporal preferences
? Two periods model,
? In the case with out any uncertainty
? First order condition,
? If means
0 1 0 1
01
m a x (,) ( ) ( )
., ( ) ( 1 )
U c c u c u c
s t w c r c
???
? ? ?
0
1
() (1 )
()
uc r
uc ?
?
?
? ??
?
1
1 r? ? ?
01cc???
Inter-temporal preferences
? Two periods model with uncertainty
investment,
– Endowment wealth w,
– Period1,consume c1,invest the rest wealth in
two assets,(1-x) percentage has a certain
return of R0 and x pays a random return of
– Period2,
– Utility function,
1R%
2 2 1 1 0 1( ) [ ( 1 ) ] ( )c w w c R x R x w c R? ? ? ? ? ? ?%%%%
1 2 1 2(,) ( ) ( )U c c u c E u c???%%
Inter-temporal preferences
? Two periods model,
– Indirect utility function of period 1 with w,
– First order condition,1
11,( ) m a x ( ) ( )cxV w u c E u w c R?? ? ? %
12
2 1 0
( ) ( )
( ) ( ) 0
u c E u c R
E u c R R
?? ?
? ??
%%
%%
Inter-temporal preferences
? several periods model
– Period t,consume ct,invest the rest wealth in
two assets,(1-xt) percentage has a certain
return of R0 and xt pays a random return of
– Periodt+1,
– Utility function,
1R%
11 ()t t t tc w w c R??? ? ? %%%
1
0
(,) ( )
T
t
Tt
t
U c c E u c?
?
? ?% % %L
Inter-temporal preferences
? Several periods model,
– Indirect utility function of period T-1,
– First order condition,
111 1 1 1 1,
( ) m a x ( ) ( )
TTT T T T Tcx
V w u c E u w c R?
??? ? ? ? ?
? ? ? %
1
10
( ) ( )
( ) ( ) 0
TT
T
u c Eu c R
Eu c R R
??? ?
? ??
%%
%%
Inter-temporal preferences
? Several periods model,
– For period T-2,when we got then
– So
– The first order condition,
1 2 2()T T Tw w c R? ? ??? %%
22(,)TTcx??
222 2 2 1 2 2,
( ) m a x ( ) ( )
TTT T T T T Tcx
V w u c E V w c R?
??? ? ? ? ? ?
? ? ? %
21
1 1 0
( ) ( ) 0
( )( ) 0
TT
T
u c E V w R
E V w R R
???
?
????
?? ??
%%
%
Asset market
? CAPM,Capital Asset Pricing Model
– Consumption of the next period depend on
how to invest the wealth in different assets,
– is the return of asset a and is the
percentage of it,Asset 0 is the no risky,
00
01
( ) ( ) [ ]
AA
a a a a
aa
c w c x R w c x R x R
??
? ? ? ? ???%%%
aR ax
Asset market
? CAPM,
– Mean-variance efficient,minimize the
Variance when the Means are same,0
00
0
0
m in
..
1
A
AA
a b a b
xx
ab
A
aa
a
A
a
a
xx
s t x R R
x
?
??
?
?
?
?
??
?
?
L
Asset market
? CAPM,
– The first order condition,
– If a portfolio is mean-variance
efficient,means invest 100% in asset e and 0
in others is M-V efficient too,Then we got,
0
20
A
b a b a
b
xR? ? ?
?
? ? ??
1()eeAe x x? L
20a e aR? ? ?? ? ?
Asset market
? CAPM,
– For a=0 and a=e we got,
– Then we have,
– That means if we have a M-V efficient
portfolio asset an a risky-free asset,we can
achieve efficient portfolio set by taking convex
combinations of them,(see the fig.)
0
00
22;e e e e
ee
R
R R R R
??????
??
00()
ae
ae
ee
R R R R??? ? ?
Asset market
? CAPM,
– Let e=m,where is the
market portfolio of risky assets,
– And
11
II
m a i a
a a i a i
iii
pqx p q w
w ???? ??
0 0 0 0( ) ( )
am
a m a m
mm
R R R R R R R? ??? ? ? ? ? ?
Asset market
? APT,Arbitrage pricing theory
– One factor,
? Construct a portfolio of two assets a and b with x
and (1-x)
? The return will be
0 1 1a a a n a n aR b b f b f ?? ? ? ? ?%%% %L
0 1 1a a aR b b f?? %%
0;ijE f f i j? ? ?%%
0 0 1 1 1( 1 ) ( 1 ) ( ( 1 ) )a b a b a bx R x R x b x b x b x b f? ? ? ? ? ? ? ? %%%
Asset market
? APT,Arbitrage pricing theory
– One factor,
? If this portfolio is risky-free,means
? And,
? We have or
? So
? Finally we got,
11(1 ) 0abx b x b? ? ?
1
11
b
ba
bx
bb
? ?
?
0 0 0(1 )abx b x b R??? ? ?
0 0 0 0
1 1 1
b b a
b b a
b R b b
b b b
???
?
0 0 0 0
1 1 1
a a b
a a b
b R b b
b b b
???
?
00
1
1
a
a
bR
b ?
? ?
0 0 1 1a a aR b R b ?? ? ?
Asset market
? APT,Arbitrage pricing theory
– Two factor,
– Construct a risky-free portfolio
– The matrix must be singular
– So
0 1 1 2 2a a a aR b b f b f? ? ?%%%
(,,)a b cx x x
0 0 0 0 0 0
1 1 1
2 2 2
0
0
0
a b c a
a b c b
a b c c
b R b R b R x
b b b x
b b b x
? ? ?? ? ? ? ? ?
? ? ? ? ? ??
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
33()ijb ?
0 1 1 2 2a a aR R b b??? ? ?
Asset market
? Expect utility,
– We have got,
– Rubinstein(1976)
– For individual i,
0
1 c o v (,( ) )
()aaR R R u cE u c ??? ?
% %
%
c o v (,( ) ) ( ) c o v (,)aaR u c E u c R c? ???%%% % %
( ) / ( )i i i i ir E u c E u c?? ??? %%
0
1( ) c o v (,)
a a i
i
R R R cr?? % %
1
00
11( ) c o v (,) [ ] c o v (,)
a a a a
iiii
R R R C R R R Crr ?? ? ? ? ??? %%%%
Asset market
? Expect utility,
– When a=c,
– We got
11 0
0
11[ ] v a r ( ) [ ] c
c
iii i c c
RRR R C
rr ?
?? ?? ? ? ??? %
0 0 0 0( ) ( )
cca
a c a c
cc
R R R R R R R? ??? ? ? ? ? ?
Asset market
? Complete market,
– An asset a valued Vas in state s,how much is
this asset worth in period 0?
– Construct a portfolio,holding Vas unit Arrow-
Debreu Security s,means a A-D security
valued $1 while the portfolio valued Vas in
state s,so,
1Sa s a ssp p V?? ?
Asset market
? Complete market,
– Let be the probability of s
– it’s the value of a portfolio that pays off $1 for
certain,and R0 be the risk-free return of it,
Then,
1
( ) c o v (,)
S
s
a a s s a a a
s s
p p p pp V E V V E E V?
? ? ? ??? ? ? ??
% % %% % %
% % %
11
SS
s
ss
sss
ppEp?
?? ??????
%
%
s?
0
1pE
R? ?
%
%
Asset market
? Complete market,
0
c o v (,)aaaV ppVR ??? % %%
Asset market
? Pure arbitrage,
– No opportunity for pure arbitrage,
– No arbitrage principle,if then
– There exist price such that the value
of any asset a is given by,
– Let we have,
0?VX 0?pX
1(,)S??L
1
S
a s a s
s
pV?
?
? ?
/ssZ ???%
00
00
c ov (,) c ov (,)
c ov (,)
aa
a a a
aa
VV
p Z V Z V R R Z
pp
R R R Z R
? ? ? ? ?
? ? ?
%
% % %
%%
Assignment
? Textbook:ex.19.1;ex.20.1
Two period with out any uncertainty
Consumption
of period 0
Consumption
of period 1 (1 )wr?
w
01(,)U c c
01( ) (1 )w c r c? ? ?
1c?
0c?
1
0
() 1
( ) 1
ucs l o p e
u c r
? ???
? ?
efficient portfolio set
aR
a?
0R
eR
e?