Transcript MOTAD and Focus Loss - University of Florida

Farm Portfolio Problem: Part II
Lecture XIII
MOTAD
 Hazell,
P.B.R. “A Linear Alternative to
Quadratic and Semivariance Programming
for Farm Planning Under Uncertainty.”
American Journal of Agricultural
Economics 53(1971):53-62.
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

Hazell’s approach is two fold. He first sets out
to develop review expected value/variance as a
good methodology under certain assumptions.
Then he raises two difficulties.
 The
first difficulty is the availability of code to solve
the quadratic programming problem implied by EV.
 The second problem is the estimation problem.
Specifically, the data required for EV are the mean
and the variance matrix.
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
The variance of a particular farming plan can be
expressed as
 1 s

x j xk 
(chj  g j )(chk  gk ) 



j 1 k 1
 s  1 h 1

n
n


1
 
 chj x j   g j x j 

s  1 h 1  j 1
j 1

s
n
n
2
2
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 Hazell
suggests replacing this objective
function with the mean absolute deviation
s
n


1
A    chj  g j x j
s h 1 j 1
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
Thus, instead of minimizing the variance of the
farm plan subject to an income constraint, you
can minimize the absolute deviation subject to
an income constraint. Another formulation for
this objective function is to let each observation
h be represented by a single row
n




yh   chj  g j x j
j 1
n
yh  yh   chj  g j x j
j 1
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min sA    yh  yh 
s
x
n
h 1


st  chj  g j x j  yh  yh  0
j 1
n
g x
j 1
j
j

j
 bi
n
a x
j 1
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s
min sA   yh
x

n
st
j 1
h 1

chj  g j x j  yh  0
n
g x
j 1
j
j

j
 bi
n
a x
j 1
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Table 1. Hazell’s Florida Farm
Obs.
Carrots
Celery
Cucumbers
Peppers
1
292
-128
420
579
2
179
560
187
639
3
114
648
366
379
4
247
544
249
924
5
426
182
322
5
6
259
850
159
569
Average
253
443
284
516
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Obs.
Carrots
Celery
Cucumbers
Peppers
1
39
-571
136
63
2
-74
117
-97
123
3
-139
205
82
-137
4
-6
101
-35
408
5
173
-261
38
-511
6
6
407
-125
53
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y1  y2  y3  y4  y5  y6
min
x
x1 
x2 
x3 

x4
25 x1  36 x2  27 x3  87 x4
 x1 
x2 
x3 
 10000

x4
39 x1  571 x2  136 x3  63 x4  y1
 74 x1  117 x2  97 x3  123 x4
 139 x1  205x2  82 x3  137 x4
0

 y2
 y3
 6 x1  101 x2  35 x3  408 x4
173 x1  261 x2  38 x3  511 x4
6 x1  407 x2  125 x3  53 x4
253 x1  443 x2  284 x3  516 x4
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 y4
 y5
 y6
0

0

0

0

0

0


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Focus-Loss
 Two


factors make Focus-Loss acceptable
First, like Hazell’s MOTAD, the Focus-Loss
problem is solvable using linear programming.
Second, Focus-Loss has a direct appeal in that
it focuses attention on survivability
 The
first step in the Focus-Loss
methodology is to define the maximum
allowable loss
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n
L  E ( z)  zc   E (c j ) x j  E ( F )  zc
j 1
L
- Maximum allowable loss
E(z) - Expected income for the firm
zc - Required cash income
E(cj) - Expected income from each crop, j
xj
- Level of the jth crop (activity)
E(F) - Expected level of fixed cost
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
Given this definition, the next step is to define
the maximum deficiencies or loss arising from
activity j.
rj  E (c j )  r
*
j
where rj* is the worst expected outcome. For
example, a crop failure may give an rj of -$100
which would represent your planting cost
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
Given this potential loss, the Focus-Loss
scenario is based on restricting the largest
expected loss to be above some stated level
L
rr x j 
k
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max 72 x1  53.4 x2  88.8 x3  200
x
x1 
x2 
x3
x1

x3
30 x1  20 x2  40 x3
5 x1  5 x2  8 x3
60 x1
44.5 x2
74 x3
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 12
8
 400
 80
1
 L 0
3
1
 L 0
3
1
 L0
3
16
 The
choice of k = 3 is somewhat arbitrary.
 Two points about the Focus-Loss


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Allowing L - the Focus-Loss solution is the
profit maximizing solution.
L can become large enough to make the linear
programming problem infeasible.
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
A Better Justification for k
 One
alternative for setting k results from the notion
that
r  j  t p  j
*
j
 Thus,
if we let tp be -1.96, the maximum loss would
be 1.96 j
L
 j t p  x j  k
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Direct Expected Utility
 We
have been discussing several
alternatives to utility maximization based on
efficiency criteria or ad hoc specifications
of risk aversion as in the case of focus-loss.
 One alternative is direct use of expected
utility.
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Table 2. Data for Direct Utility
Maximization
Corn
Soybeans
Wheat
Observation 1
176.24
94.81
97.09
Observation 2
232.93
114.39
120.18
Observation 3
273.01
144.50
108.75
Observation 4
221.59
114.32
87.48
Observation 5
-7.87
97.22
100.46
Observation 6
247.59
126.41
108.34
Observation 7
226.79
113.49
98.16
Observation 8
250.11
123.27
107.60
Observation 9
255.99
136.15
102.81
Observation 10
246.91
131.04
104.68
Average
212.33
119.56
103.56
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Parameterization of the
Expected Utility Model
 Total


acres do not exceed 1280.
Annual profit of $271,782.
Amortizing this amount into perpetuity using a
discount rate of 15% yields a total value of
$1,811,880.
 Assuming
the debt-to-asset position of the
farm is 60%, the value of the asset
represents equity of $724,752 and debt of
$1,087,130.
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 Assuming
an interest rate of 12.5% yields
an annual cash flow requirement of
$135,891 to cover the interest payments.
 Assuming a family living requirement of
$50,000 yields a minimum cash
requirement of $185,891.
Wi  176.24 x1  94.81 x 2  97.09 x 3  538861
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1 W1b 1 W2b
1 W10b


10 b 10 b
10 b
max
x
x1 
x2 
 1280
x3
 176.2 x1  938
. x2  97.1 x3
 232.9 x1  114.4 x2  120.2 x3
 W1
 W2
 538,861
 538,861

 246.9 x1  1310
. x2  104.7 x3
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 W10  538,861
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Table 4. Portfolio from
Expected Utility
r
x1
x2
x3


-0.001
929.12 350.88
0.00
239,231.70
75,923.56
-0.1
882.59 397.41
0.00
234,915.10
72,865.46
-1.0
532.69 747.31
0.00
202,455.70
50,130.77
-10.0
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4.93
0.00
284.59
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