978-0123865496 Chapter 4 Solution Manual

Type Homework Help

Pages 7

Words 346

Textbook Intermediate Financial Theory 3rd Edition

Authors Jean-Pierre Danthine, John B. Donaldson

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CHAPTER 5.

5.1. For full investment in the risky asset the first order condition has to satisfy the following:

   

 

0rr

1Y'UE f0 

   

 

   

 

   

 

   

 

   

 

       

1 2 3

1 1 1

1 2 3

U Y 1 exp bY

W a 1 exp b 2 a 1 exp b 2 1 exp b 2 a

W' a bexp b 2 a bexp b 2 a 0

ln b b 2 a ln b b 2 a

ln ln

a2b

U Y Y

1 1 1

W a 2 a 2 2 a

1 1 1



  

   

     

             

     

        

      

  







     

       

     

     

     

     

1/ 1/

a 2 a 2 a 0



      

  

  

a



    

         

  

f0A

ff0A

ff0

r1YR

ar1YR

ar1YU















Hence

            

ff0f0Aff0 rr

ar1YUr1YRrr

ar1YU 







Since

0rr

~f

for this case,

(i)

              

fff0f0Afff0 rr

ar1YUr1YRrr

ar1YU 







Case 2:

~

    

         

  

.r1YR

ar1YR

ar1YU

f0A

ff0A

ff0















Thus,

1 – π

5.4. Let

BA R

and R

represent the gross returns on the two assets ;

 

    

 

     

 

  

BABABA R

UEa1R

UaER

Ua1R

aUER

a1R

aUE 

     

BA R

UER

UE 

since the utility function is concave and gross asset returns are identically distributed.

This result means that the investor is going to invest in both securities – it is never

optimal in this situation to invest only in one of the two assets. He thus chooses to

diversify. Moreover if the investor cares only about the first two moments he will invest

equal amounts in the assets to minimize variance. To show this note that the expected

5.5

1Y0

payouts =

riskless asset: 1

risky asset:











)b1)(b)x1(x(''U)1( 1

11 

)b1)(b)x1(x(''U)1()a1)(a)x1(x(''U

)a)x1(x(''U)x1)(a1()a)x1(x('U

111111 







as the numerator is positive and the denominator negative.

We would expect that an increase in the probability of the unfavorable risky asset state a,

5.7 a.

xzx(EUmax 2211

x,x 21



02211 Yxpxp .t.s 

Since we assume (maintained assumption) U’( )>0,

02211 Yxpxp 

, and

220

xpY

x



The problem may thus be written :

























221

220

xpY

EUmax

The necessary and sufficient F.O.C. (under the customary assumption) is :

xpY

'EU 1

2221

220 





































b. Suppose

bea)y(U 



; the above equation becomes :

xpY

expAbE 1

2221

220 

























































equivalently,

exp

expAbE 1

2221

10 



















































The first term which contains

can be eliminated from the equation. The intuition for

5.8. The problem with linear mean variance utility is

 

)max( x

sxsEsy





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978-0123865496 Chapter 4 Solution Manual

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978-0123865496 Chapter 4 Solution Manual

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