Lecture V

• Deriving the EV Frontier
– Let us begin with the traditional portfolio model. Assume that we want to minimize the variance associated with attaining a given level of income. To specify this problem we assume a variance matrix:

max ( ) x
 x '
 x st Ax b
 



924 41 .
.
.
458 52 .
.
.
.
.
.
.


135 22 .
.
.




.
.
1 0
.
.



 x
1 x x
3
2





.
x
4
.
x
1

.
x
2

.
x
3

.
x
4

.
x
1
 x
2
 x
3
 x
4



• In this initial formulation we find that the optimum solution is x which yields a variance of 228.25.
x







.
11613

.
10666


.
.
39508
59546





• GAMS Program
– Sets
– Tables
– Parameters
– Variables
– Equations
– Model Setup

• Starting with the basic model of portfolio choice:
x x
 x st x
 
Y
*

• Freund showed that the expected utility of a normally distributed gamble given negative exponential preferences could be written as
 
  x
 2
x x x x
 x

January-
June
July-
December
Period 1
Period 2
Period 3
1.199
.000
1.064
-2.064
-2.064
1.382
1.382
2.776
2.776
Production Capital
.484
.038
.020
.107
-1.504
-1.145
.000
.482
.000
.229
--1.229
60.0
60.0
24.0
12.0
0.0

Period 1
Period 2
Period 3
5.276
2.158
.000
Managerial Labor
4.836
.000
4.561
4.561
.000
.000
1.000
Unit Level of $100.0
1.000
1.000
.000
4.198
13.606
1.000
799.0
867.0
783.0

• The Variance matrix for the problem is
 



.
.

.

.
.

.
.
1124 64 .




– The maximization problem can then be written as: max x

1 1 1 1






 x
1 x
2 x
3 x
4








2





 x
1 x x
3 x
2
4













 x x x
3 x
1
2
4







.
x
1

.
x
2

.
x
3 x
3


.
482
0 x x
4
4


.
.


2 .
0 x
1

.
x
2

.
158
0 x x
1 x
1
1



.
.
.
484 x x x
2
2
2



.
038 x
3
0 x
4

.
x
1

.
020 x
2

.
107 x
3

.
229 x
4

.
.
x
1

.
x
2

.
x
3

.
x
4

.
.
x
1

.
x
2

0
0 x x
3
3



.
.
0 x
4

.
x
4

0 x
3

.
x
4

.
.

• Using 
= 1/1250.0 we obtain an optimal solution under risk of x




.





• The objective function for this optimum solution is 5,383.08. Putting
 equal to zero yields an objective function of 9,131.11 with a allocation of x




.





– Question: How does the current solution compare to the risk averse solution? Which crop makes the greatest gain? Which crop has the largest loss? Why?
– A second point is that although the objective function under risk aversion is 5,383.08, the expected income is 7207.24. What does this difference manifest?

• Quantifying Gains to Risk Diversification
Using Certainty Equivalence in a Mean-
Variance Model: An Application to Florida
Citrus
– The traditional formulation of the meanvariance rules begins with the negative exponential utility function:
 
  e
 

– Our discussion of Bussey indicated that this expected utility can be rewritten under normality as
   
  e
  2



– Hence, our tradition of maximizing z
 
( )
 2

– The implications of this objective function is actually much broader, however. Solving the negative exponential utility function for wealth ln




*
( )
*
( )


 
  e

 
W
* x
*
( )

*
( )
  1
 ln



*
( )
 
*
W x
  x
 2

– Other implications include the interpretation of the shadow values of the constraint as changes in certainty equivalence. For example, given the original specification of the objective function, the shadow values of the second land constraint is 34.73 and the shadow value of the first capital constraint is 93.98. These values are then the price of each input under uncertainty

• Moss, Charles B., Allen M. Featherstone, and Timothy G. Baker. “Agricultural Assets in an Efficient Multiperiod Investment
Portfolio.”
Agricultural Finance Review
49(1987): 82-94.

– Historically, ownership of agricultural assets has been dominated by farmer equity and debt capital.
• The implication of this form of ownership are increased variability in the return on equity to farmers
• A direct manifestation of the unwillingness of nonfarm investors to invest in agriculture can be seen in the unexplained premium on farm assets in the Capital Asset Pricing Model.

– This study examines whether autocorrelation in the returns on farm assets versus other assets may explain the discrepancy.
• Autocorrelation in farm returns refers to the tendency of increased returns to persist over time.
Mathematically:
 t

P
 t

1
  t

– Given this vector of returns, the problem is to design the expected value/variance problem for holding a given portfolio of assets over several periods. Mathematically, this produces two problems:
• Given the autoregressive structure of the problem, what is the expected return?
( I

AL )
  t

0
 t
I AL )

1

0

• A similar problem involves the variance matrix.
Using the autoregressive estimation above, the variance matrix for the investment can be written as
 










P '

T

1
(
P

P P
 
( P P '
 
P

P P '
 
P
T

2
(
PP


'
 
)

' '
 
'
 


' '
 
'
 
P
T

3





P
T

3
(
P
T
T

2


1
(
P
P i
T

1

' '





P P
( ' ) i

)

'
 
)





 i


0

Corporate Bonds
Common Stocks
Farm Assets
Government Bonds
Money
Prime Interest
Small Stocks
Treasury Bills
Mean
1.24
6.01
4.65
.55
-3.06
1.81
8.64
.22
Std. Dev.
6.71
19.29
3.87
6.71
7.14
20.74
29.83
2.00

1.0000
2.
.1899
-.0106
1.0000
.9925
.4487
.1671
.0654
.1833
.2208
.1426
.0723
.0026
1.0000
-.1023
-.2204
-.2053
1.0000
.4005
1.0000
.1568
.8711
.1443
.0296
.9041
-.0896
-.0017
-.1608
.1654
.9085
1.0000
-.1612
.9807
1.0000
-.1828
1.0000
1.0000
-.4939
-.9823
1.0000
.5543
1.0000
.9984
.8598
.8848
.9273
.5664
-.5033
-.4700
-.5825
-.2194
-.6234
-.9801
-.8265
-.9056
-.9041
-.6406
1.0000
.8650
.8865
.9177
.5686
1.0000
.8754
1.0000
.7268
.7684
.5543
.7978
1.0000
.7978
1.0000

1.0000
-.4939
-.9823
1.0000
.5543
1.0000
.9984
.8598
.8848
.9273
.5664
-.5033
-.4700
-.5825
-.2194
-.6234
-.9801
-.8265
-.9056
-.9041
-.6406
1.0000
.8650
.8865
.9177
.5686
1.0000
.8754
1.0000
.7268
.7684
.5543
.7978
1.0000
.7978
1.0000
1.0000
.05481
-.9854
1.0000
.0225
.9987
.0326
.8086
.8559
-.0909
-.2253
1.0000
-.9852
-.7793
-.8961
1.0000
.8197
.8649
1.0000
.7827
1.0000
.9570
.2720
-.9286
.9461
.6871
.7404
-.4795
-.5033
.374
-.4585
-.3222
-.0469
1.0000
-.4586
1.0000