1. finance
2. personal finance
3. financial planning
4. asset and portfolio management

1. (20 marks) Consider an economy with two periods and ONE state of the world at time 1 (i.e. there is
no uncertainty in this economy) and one perishable good. There are two agents in this economy with the
same utility function
 (0  1 ) = ln 0 +  ln 1 
where 0    1100 is the agents’ time discount factor, and endowments
¡
¡
¢
¢
1 = 10  11 and 2 = 20  21 
¡
¢
with  = 1 + 2 = 10 + 20  11 + 21  0 Suppose there is only one riskless asset with payoff 1 at time
1
1 and price 1+
at time 0, where  is the interest rate.

(a) (12 marks) Find the equilibrium interest rate  for the general endowments
¡
¡
¢
¢
1 = 10  11 and 2 = 20  21 
(b) (8 marks) Using the equilibrium interest rate  derived in part (a), find the equilibrium interest
rates  for the following two particular endowments
1∗
1∗∗
= (100 1) and 2∗ = (100 1) and
= (100 1) and 2∗∗ = (1 100) 
Are the two interest rates equal?
2. (20 marks) Given the price-payoff couple ( ) as follows:
⎞ ⎛
⎛
25
1
⎜ 2 ⎟ ⎜ 15
⎟ ⎜
=⎜
⎝ 3 ⎠ = ⎝ 05
025
4
and
=
¡
1
2
3
4
(a) (2 marks) Are the markets complete?
⎛
1
¢ ⎜ 2
=⎜
⎝ 3
4
⎞
⎟
⎟
⎠
0
1
2
3
0
0
1
2
⎞
0
0 ⎟
⎟
0 ⎠
1
(b) (3 marks) Can the Arrow-Debreu security 2 be replicated? If yes, find a portfolio  such that
⎛
⎞
0
⎜ 1 ⎟
⎟
 = 2 = ⎜
⎝ 0 ⎠
0
(c) (5 marks) Do there exist arbitrage opportunities in the markets? Why or why not? State your
reasons clearly.
(d) (10 marks) If there are arbitrage opportunities in the markets, is it possible to change asset 4’s price
only such that there are no more arbitrage opportunities? Why or why not?
3. (20 marks) Consider a two periods, 0 and 1, and Ω states of the world economy in which there is a risky
asset  available at time 0 and its payoffs at time 1 are also denoted by  Suppose one agent with utility
function  (0  1 ) has consumption good endowment (0  1 ) ∈ R1+Ω
++ and  units of the asset endowment,
respectively.
(a) (6 marks) Let  = 0 Suppose that the agent is going to sell one unit of the asset short at time 0.
What is the minimum price CE() he is going to ask?
(b) (6 marks) Let  = 1 Suppose that the agent still wants to buy one more unit of the asset at time 0.
What is the maximum price  () he is going to bid?
1
(c) (8 marks) Let  = 1 At time 0, suppose that the agent is going to buy insurance for protecting the
loss at some situations. What is the maximum risk premium  () he is going to pay?
(Hint: For each of the questions, just specify the equation that the price should satisfy.)
4. (20 marks) Consider an economy consisting of (i) two periods and two states of the nature at time 1 with
the same probability 12 and (ii)  (≥ 2) strictly monotone and strictly risk averse agents with the total
endowment

P
 = (0  1 ) = (0  11  12 ) = (10 100 20) 
=
=1
There are one riskfree asset with the riskfree interest rate   0 and two risky assets in the
economy as follows:
25 if state 1
%
•
1
&
5
if state 2
and
2
•
%
&
5
if state 1
25 if state 2
¡ ¢
¡ ¢
¡
¢
¡
¢
(a) (10 marks) Calculate E 1  E 2  cov 1  1  and cov 1  2 , respectively.
(b) (10 marks) Suppose there is a representative agent in the economy. Is it possible that the equilibrium
prices of the two risky assets are the same? Why or why not? State your reasons clearly. (Hint: Try
to use CCAPM (Consumption Based Capital Asset Pricing Model) to explain this.)
5. (20 marks) Consider an economy consisting of two time periods, 0 and 1, Ω states of nature at time 1, one
perishable consumption good, and  ( Ω) assets.
(a) (4 marks) To express one agent’s feasible budget set  ( )  how many inequalities are needed?
Please write down the agent’s feasible budget set  ( ) clearly, where  = (1  2       )0 is price
vector of the  assets and  ∈ R1+Ω
+ \ {0} is the agent endowment.
(b) (16 marks) Suppose there are two equilibria with allocations
¡ ∗1 ∗2
¢
¡
¢
        ∗ and ∗∗1  ∗∗2      ∗∗   ≥ 2
in the economy.
Is that possible that one is non Pareto efficient and the other one is Pareto efficient? Why or
why not? State your reasons clearly.
2
Solution to the final exam
1. (a) To derive the equilibrium interest rate   we should find the two agents’ demands first.
Given   an agent’s total income is
1
0 (1 +  ) + 1
=
1 + 
1 + 
 = 0 +
and his feasible budget set is
½
1+1
: 0 +
 (  ) = (0  1 ) ∈ +
¾
1
≤ 
1 + 
To find his optimal demands, the agent solves
max
(0 1 )∈( )
 (0  1 ) = ln 0 +  ln 1 
Obviously, the Inada condition is satisfied, there is no corner solution. Thus, we just need to solve
the Lagrangean problem as follows:
µ
¶
1
L =  (0  1 ) −  0 +
−
1 + 
µ
¶
1
− 
= ln 0 +  ln 1 −  0 +
1 + 
The first-order conditions are:
L
0
L
1
1
−=0
0

1
−
=0
1
1 + 
=
=
and
1
= 
1 + 
(1)
1
(1 +  ) 
 1 =



(2)
0 +
Accordingly,
0 =
Substituting (2) into (1) yields
1
(1 +  ) 
+
  (1 +  )
=
 =
1+
=

1

1+
=⇒ =



1+
Thus,
0 (1+ )+1
1+
0 (1 +  ) + 1
(1 +  ) (1 + )
0
=
1

=
=

1+
1
=
 (1 +  ) 0 1+
(1 +  ) 
=

1+
1+
=
 (1+ )+1
=
(0 (1 +  ) + 1 ) 

1+
Thus,
10 =
10 (1 +  ) + 11
2 (1 +  ) + 21
and 20 = 0
(1 +  ) (1 + )
(1 +  ) (1 + )
In equilibrium,
10 + 20 = 10 + 20 
3
Accordingly,
10 (1+ )+11
20 (1+ )+21
1
2
(1+ )(1+) + (1+ )(1+) = 0 + 0 ⇐⇒
¡
¢
1
1
2
2
0 (1 + ¡ ) + 1 +
0 ¡(1 +  )¢+ ¡1 = 10 +
20 (1 +  ) (1 + ) ⇐⇒
¢
¢
1
2
1
2
1
2
¡(1 1+  2) ¢ 0 + 0 + 11+ 12 = 0 + 0 (1 +  ) (1 + ) ⇐⇒
0 + 0 (1 +  )  = 1 + 1 ⇐⇒
1 +2
1 +  =  11 +12 ⇐⇒
( 0 0)
(1 +2 )−(1 +2 )
1 +2
 =  11 +12 − 1 = 1 1 1 +2 0 0
(
0
0
)
(
0
0
)
That is, the equilibrium interest rate is
¢
¡
¢
11 + 21 −  10 + 20
 =
 (10 + 20 )
¡
(b) For endowment
1∗ = (100 1) and 2∗ = (100 1) 
2∗
1∗
2∗
1∗
0 + 0 = 200 and 1 + 1 = 2
thus, the equilibrium interest rate is
¢
¡
¢
¡ 1∗
2∗
−  1∗
1 + 2∗
2 − 200
1 − 100
1
0 + 0
∗
 =
=
=
1∗
2∗
 (0 + 0 )
200
100
For endowment
1∗∗ = (100 1) and 2∗∗ = (1 100) 
+ 2∗∗
= 101 and 1∗∗
+ 2∗∗
= 101
1∗∗
0
0
1
1
thus, the equilibrium interest rate is
¢
¡
¢
¡ 1∗∗
+ 2∗∗
−  1∗∗
1 + 2∗∗
101 −  × 101
1−
1
0
0
∗∗
 =
=
=
2∗∗ )
 (1∗∗
+

101

0
0
Obviously, the two interest rates are different.
2. (a) Yes, the markets are complete since
¯
¯
¯
¯
¯
¯
¯
¯
1
2
3
4
0
1
2
3
0
0
1
2
0
0
0
1
¯
¯
¯
¯
¯ = 1
¯
¯
¯
and the rank of  is 4 Thus, the maximum number of linearly independent column vectors is equal
to the number of the number of the states of nature. Accordingly, the markets are complete.
(b) Yes since the markets are complete. To find  we solve
 = 2 
That is,
⎛
1
⎜ 2
⎜
⎝ 3
4
1
21 + 2
31 + 22 + 3
41 + 32 + 23 + 4
0
1
2
3
=
=
=
=
0
0
1
2
⎞⎛
0
1
⎜ 2
0 ⎟
⎟⎜
0 ⎠ ⎝ 3
1
4
0 =⇒ 1
1 =⇒ 2
0 =⇒ 3
0 =⇒ 4
4
⎞
⎛
⎞
0
⎟ ⎜ 1 ⎟
⎟=⎜
⎟
⎠ ⎝ 0 ⎠
0
=0
= 1 − 21 = 1
= −31 − 22 = −2
= −41 − 32 − 23 = −3 + 4 = 1
(c) By the definition, we have
 = 1 1 + 2 2 + 3 3 + 4 4   = 1 2 3 4
Thus,
1
2
3
4
11 1 + 12 2 + 13 3 + 14 4
21 1 + 22 2 + 23 3 + 24 4
31 1 + 32 2 + 33 3 + 34 4
41 1 + 42 2 + 43 3 + 44 4
=
=
=
=
= 1 + 22 + 33 + 44
= 0 + 12 + 23 + 34
= 0 + 0 + 13 + 24
= 0 + 0 + 0 + 14 
We solve them from the last one equation,
4
3 + 24
2 + 23 + 34
1 + 22 + 33 + 44
=
=
=
=
4
3
2
1
= 025 =⇒ 4 = 4 = 025
= 05 =⇒ 3 = 3 − 24 = 05 − 24 = 0
= 15 =⇒ 2 = 2 − 23 − 34 = 15 − 0 − 075 = 075
= 25 =⇒ 1 = 1 − 22 − 33 − 44 = 25 − 15 − 0 − 1 = 0
Though
 ≥ 0
however,
 6À 0
there are arbitrage opportunities in the security markets.
(d) It is impossible. From
   = 
we have
That is
⎛
1
⎜ 2
⎜
⎝ 3
4
⎛
1
⎜ 0
⎜
⎝ 0
0
0
1
2
3
⎞ ⎛
0
⎜
0 ⎟
⎟ ⎜
0 ⎠ ⎝
1
0
0
1
2
2
1
0
0
3
2
1
0
⎞⎛
4
⎜
3 ⎟
⎟⎜
2 ⎠⎝
1
1
2
3
4
1
2
3
4
⎞
⎞
25
⎟ ⎜ 15 ⎟
⎟=⎜
⎟
⎠ ⎝ 050 ⎠ 
4
⎞
⎛
⎛
⎞
25
⎟ ⎜ 15 ⎟
⎟=⎜
⎟
⎠ ⎝ 050 ⎠ 
4
Since the first matrix of the right hand side is an upper triangle matrix, we solve them from the
bottom
4
3 + 24
2 + 23 + 34
1 + 22 + 33 + 44
=
=
=
=
Equation 6− equation 5 give
1 + 2 + 3 + 4 = 1
and Equation 5− equation 4 give
2 + 3 + 4 = 1
Accordingly,
1 = 0
That, there is arbitrage opportunity in the economy!
 À 0
there are no arbitrage opportunities in the security markets.
5
4
05
15
25
(3)
(4)
(5)
(6)
3. (a)  = 0
The agent has two options at time 0: Doing nothing about the asset and selling one unit of the asset
short. The price CE() should make the agent to be indifferent with the two options. That is, the
price CE() is a solution to
 (0  1 ) =  (0 + CE ()  1 − ) 
(b)  = 1
The agent has two options at time 0: Buying one more unit of the asset and not buying it. The
price  () should make the agent to be indifferent with the two options. That is, the price  () is
a solution to
 (0  1 + ) =  (0 −  ()  1 + 2) 
(c)  = 1
The agent has two options: Buying the insurance and not buying it at time 0. The risk premium  ()
should make the agent to be indifferent with the two options. That is, the price  () is a solution to
 (0  1 + ) =  (0 −  ()  1 + E ()) 
4. (a)
E1 = E2 =
25 + 5
= 15
2
Since
¡
¢
¢
¡
cov 1   = E (1 − E1 )  − E
¢
¡
= E 1  − E1 E   = 1 2
¡
¢
we calculate E 1  and E1 
100 + 20
= 60
2
¢
¡
100 × 25 + 20 × 5
E 1 1 =
= 1300
2
¢
¡
100 × 5 + 20 × 25
E 1 2 =
= 500
2
E1 E1 = E1 E2 = 60 × 15 = 900
¢
¡
cov 1  1 =
=
¡
¢
cov 1  2 =
=
E1
=
¡
¢
¡
¢
E (1 − E1 ) 1 − E1 = E 1 1 − E1 E1
1300 − 900 = 400  0
¡
¢
¡
¢
E (1 − E1 ) 2 − E2 = E 1 2 − E1 E2
500 − 900 = −400  0
(b) No.  2 should be bigger than  1  When tomorrow (state 1) is booming, you will have more income
from your other resource, and will have lower marginal utility. That is, in booming period, you don’t
“really” need more money. However, asset 1 still gives you more money when you don’t “really”
need it. When tomorrow is in recession, you will have less income from your other resource, and
will have higher marginal utility. That is, in recession period, you do “really” need more money.
However, asset 1 does not give you more money when you do “really” need it.
That is, when you do not “really” need more money, asset 1 gives you more money; However, when
you “really” need more money, asset 1 does not give you more money. Somehow, owning asset 1
is equivalent to buying insurance against the booming. However, we don’t need such insurance. In
other words, asset 1 is a “bad” asset. Therefore, you would not like to pay more for it.
We can we use CCAPM to explain this as following: Asset 1 and the underlying economy have
positive covariance
¶
µ 0
¡
¢
1 (1 ) 1
cov 1  1  0 =⇒ cov
0


00 (0 )
6
(Can you imagine why? Since 01 is strictly decreasing from 001  0) As a result,
´
³ 0
 ( )
cov 10 (10 )  1  0
0
³ 0
´ E 1
E[1 ]
[ ]
 ( )
1 = 1+ + cov 10 (10 )  1  1+ 
0
That is, for such a “bad” asset 1 , the risk premium is negative.
How about asset 2 ?
When tomorrow is booming, you will have more income from your other resource, and will have lower
marginal utility. That is, in booming period, you don’t “really” need more money. And asset 2
also does not give you more money when you don’t “really” need it. When tomorrow is in recession,
you will have less income from your other resource, and will have higher marginal utility. That is, in
recession period, you do “really” need more money. Fortunately, asset 2 will give you more money
when you do “really” need it.
That is, when you do not need more money, asset 2 also does not give you more money; Fortunately,
when you do need more money, asset 2 also gives you more money. Somehow, owning asset 2
is equivalent to buying insurance against the recession. In other words, asset 2 is a “good” asset.
Therefore, you would like to pay more for it.
Similarly, we can use CCAPM to explain this. Asset 2 and the underlying economy have negative
covariance
¶
µ 0
¡
¢
1 (1 ) 2
2
cov 1    0 =⇒ cov
  0
00 (0 )
As a result,
³
´
01 (1 )
2
0
00 (0 )  
³ 0
´
E[2 ]
 ( )
2 = 1+ + cov 10 (10 )  2
0
cov

E[2 ]
1+ 
That is, for such a “good” asset 2 , the risk premium is positive.
Thus,
£ ¤
£ ¤
E 2
E 1
=
 1 
2 
1 + 
1 + 
5. (a) Since the number  of assets is less than the number Ω of states, the markets are incomplete for
sure. Thus, only the consumption bundle (0  1 ) = (0  11  12  · · ·  1Ω ) satisfying
⎧
0 = 0 −  > 
≥0
⎪
⎪
⎨
11 = 11 + 1·  ≥ 0
···
⎪
⎪
⎩
1Ω = 1Ω + Ω·  ≥ 0
is feasible. That is, the feasible budget set is the following
⎫
⎧
0 = 0 −  >  ⎬
⎨
: 1 = 1 + 
 ( ( )) = (0  1 ) ∈ R1+Ω
+
⎭
⎩
 ∈ R
consisting of Ω + 1 equalities, rather than a unique equality.
(b) Since the markets are incomplete and ¡the existence of ¢market equilibrium, therefore, there must be
no arbitrage in equilibrium allocation ∗1  ∗2      ∗ . As a result, there are at least two different
2 Ω
state price vectors {1 }Ω
buddles. Thus, agent 1 may use
=1 and { }=1 at equilibrium¡consumption
¢
1 Ω
1∗
{ }=1 to decide his equilibrium consumption buddle 1∗


from
solving the following equations
0
1

1∗
10
1
1 (1 )
1∗ =    = 1 2 · · ·  Ω;
10
0 (0 )
¡ 2∗ 2∗ ¢
from solving
while agent 2 may use {2 }Ω
=1 to decide his equilibrium consumption buddle 0  1
the following equations
20 (2∗ )
= 2   = 1 2 · · ·  Ω
  110 1
0 (2∗
)
0
7
Thus,
1∗
20 (2∗ )
10
1 (1 )
6= 120 1
when 1 6= 2 
10
1∗
0 (0 )
0 (2∗
)
0
In other words, the two agents may have different ratios of marginal utilities and the equilibrium
allocation {∗ }=1 in an incomplete market may be not Pareto efficient.
2 Ω
Though there are at least two different state price vectors {1 }Ω
=1 and { }=1 at equilibrium
consumption buddles in the incomplete market, however, it is still possible that all the agents may
the ¢same state price vector, for example, {1 }Ω
=1  to decide their equilibrium consumption buddle
¡use
∗
∗
from solving the following equations
0  1

1∗
0
1 (1 )
= 1   = 1 2 · · ·  Ω
1∗
0
0 (0 )
Thus,
1∗
0
0 (2∗ )
1 (1 )
= 10 1
for all  and 
0
1∗
0 (0 )
0 (2∗
0 )
In other words, all the agents may have the same ratios of marginal utilities and the equilibrium
allocation {∗ }=1 in an incomplete market may be Pareto efficient.
8