STOCHASTIC

Consider the following four utility functions:
r
-1.00
-0.10
0.00
0.0395
0.05
0.0605
0.10
0.20
0.30
1.10
tfifr)
-1.00
-0.10
0.00
0.0395
0.05
0.0605
0.10
0.20
0.30
1.10
u2(r)
-1.00
-0.082
0.00
0.0313
0.0395
0.0477
0.078
0.152
0.222
0.640
u,(r)
-1.00
-0.19
0.00
0.0806
0.1025
0.1246
0.21
0.44
0.69
3.41
«*(r)
-1.00
-0.82
0.00
0.0313
0.0395
0.0477
0.078
0.152
0.280
1.20
(b) Graph these utility functions. Discuss their properties.
(c) Compute the expected utility of each asset for the various utility functions and rank
the corresponding assets. [Hint: Note that Ui(r) = r, U2(r) = 0.8r—0.2r 2 , and
l/3(r) = 2r + r 2 .]
Exercise Source Notes
Portions of Exercise 3 were adapted from Ziemba et al. (1974); Exercise 4 was adapted
from Penner (1964); portions of Exercises 7 and 8 were adapted from Samuelson (1967);
portions of Exercise 9 were adapted from Press (1972); the idea for Exercise 10 was suggested
by Professor J. Ohlson; Exercise 11 was adapted from Hester (1967); portions of Exercise 12
were adapted from R ichter (1960) and Ziemba (1974); portions of Exercise 13 were adapted
from Sharpe (1970); some results similar to those in Exercise 14 were independently derived
in Merton (1972); and Exercise 27 was adapted from Tobin (1965).
COMPUTATIONAL AND REVIEW EXERCISES 341