Transcript Methods of Risk Analysis for Farms: Mean/Variance Models

The Farm Portfolio Problem:
Part I
Lecture V
An Empirical Model of MeanVariance
• 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 f ( x )  x '  x
x
st
Ax  b
 924.41 458.52 202.22 135.22


.
452.99 72.55 
 458.52 76129

202.22 452.99 49011
.
109.09


 135.22 72.25 109.09 284.17
Ax  b
 x1 
 
.
11366
.
6.298 8.014  x2   7.0
 8119

    
.
10
.
10
.
10
.  x3  10
. 
 10
 
 x4 
8119
.
x1  11366
.
x2  6.298 x3  8.014 x4  7.000
x1 
x2 
x3 
x4  1000
.
• In this initial formulation we find that the
optimum solution is x which yields a
variance of 228.25.
 .11613 


 .10666
x
.39508 


 .59546 
Parts of the GAMS Program
• GAMS Program
–
–
–
–
–
–
Sets
Tables
Parameters
Variables
Equations
Model Setup
• Starting with the basic model of portfolio
choice:
min 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

EU [ x ]   ( x )  2  ( x )

2
max x '   x x'  x
x
JanuaryJune
JulyDecember
1.199
1.382
2.776
.000
60.0
.000
1.382
2.776
.482
60.0
Production Capital
Period 1
1.064
.484
.038
.000
24.0
Period 2
-2.064
.020
.107
.229
12.0
Period 3
-2.064
-1.504
-1.145
--1.229
0.0
Managerial Labor
Period 1
5.276
4.836
.000
.000
799.0
Period 2
2.158
4.561
.000
4.198
867.0
Period 3
.000
4.561
.000
13.606
783.0
Unit Level of $100.0
1.000
1.000
1.000
1.000
• The Variance matrix for the problem is
.
 688.73  182.05
 7304.69 90389


62016
.
 47114
.
110.43 


1124.64 750.69 


3689.53 

– The maximization problem can then be written
as:

 x1 
 x1   x1 
 
   
 x2    x2   x 2 
max 1 1 1 1    2    
x
x3
x3
x3
 
   
 x4 
 x4   x 4 
1199
.
x1  1382
.
x2  2.776 x3 
0 x4  60.0
0 x1  1382
.
x2  2.776 x3  .482 x4  60.0
1064
.
x1  .484 x2  .038 x3  . 0 x4  24.0
 2.064 x 1  .020 x2  .107 x3  .229 x4  12.0
 2.046 x1  1504
.
x2  1145
.
x3  1229
.
x4  0.0
5.276 x1  4.836 x2 
0 x3 
0 x4  799.0
2.158 x1  4.561 x2 
0 x3  4.198 x4  867.0
0 x1  4.146 x2 
0 x3  13.607 x4  787.0
• Using  = 1/1250.0 we obtain an optimal
solution under risk of
 10.29


 26.76
x
2.68 


 32.35
• 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
 22.14


 0.00 
x
1162
. 


 57.55
– 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 MeanVariance Model: An Application to Florida
Citrus
– The traditional formulation of the meanvariance rules begins with the negative
exponential utility function:
U W ( x)   e
 W ( x )
– Our discussion of Bussey indicated that this
expected utility can be rewritten under
normality as
EU W  x    e

    ( x )  2  2 ( x ) 
– Hence, our tradition of maximizing

z   ( x)  2  ( x)
2
– The implications of this objective function is
actually much broader, however. Solving the
negative exponential utility function for wealth
U W ( x )    e
*

 W * ( x )
ln  U W * ( x )    W * ( x )


W * ( x )   1  ln  U W * ( x )

W ( x)   ( x)  2  2 ( x)
*

– 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) 0
1
• A similar problem involves the variance matrix.
Using the autoregressive estimation above, the
variance matrix for the investment can be written as


P

PP  
 P '
 P ' P '
( PP') P'





T 1
( PP' )( P' ) T  2
 ( P ' )

PP
P( PP')
PPP' P' PP'

( PPP' P' PP' )( P' ) T  3



T 3
 P ( PPP' P' PP')




T 1


P i ( P') i



i 0


P T 1
P T  2 ( PP  )
Mean
Corporate Bonds
Common Stocks
Farm Assets
Government Bonds
Money
Prime Interest
Small Stocks
Treasury Bills
Std. Dev.
1.24
6.01
4.65
.55
-3.06
1.81
8.64
.22
6.71
19.29
3.87
6.71
7.14
20.74
29.83
2.00
1.0000
1.0000
2.
.1899
1.0000
-.4939
1.0000
-.0106
.2208
1.0000
.9925
.1426
-.1023
1.0000
-.9823
.5543
1.0000
.9984
-.5033
-.9801
1.0000
.4487
.0723
-.2204
.4005
1.0000
.8598
-.4700
-.8265
.8650
1.0000
.1671
.0026
-.2053
.1568
.9041
1.0000
.8848
-.5825
-.9056
.8865
.8754
1.0000
.0654
.8711
.1443
.0296
-.0896
-.1612
1.0000
.9273
-.2194
-.9041
.9177
.7268
.7684
1.0000
.1833
-.0017
-.1608
.1654
.9085
.9807
-.1828
1.0000
.5664
-.6234
-.6406
.5686
.5543
.7978
.7978
1.0000
1.0000
1.0000
-.4939
1.0000
.05481
1.0000
-.9823
.5543
1.0000
-.9854
.0225
1.0000
.9984
-.5033
-.9801
1.0000
.9987
.0326
-.9852
1.0000
.8598
-.4700
-.8265
.8650
1.0000
.8086
-.0909
-.7793
.8197
1.0000
.8848
-.5825
-.9056
.8865
.8754
1.0000
.9273
-.2194
-.9041
.9177
.7268
.7684
1.0000
.5664
-.6234
-.6406
.5686
.5543
.7978
.7978
1.0000
.8559
-.2253
-.8961
.8649
.7827
1.0000
.9570
.2720
-.9286
.9461
.6871
.7404
1.0000
-.4795
-.5033
.374
-.4585
-.3222
-.0469
-.4586
1.0000