A Better Approach to
Calculate Approved Yield Indexing
Dr. Myles J. Watts
Professor, Montana State University
Agricultural Economics & Economics Department
Economic Consultant, Watts and Associates, Inc.
Crop Insurance
• A risk management tool for producers to
alleviate financial stress from:
– Low yields.
– Unexpected price declines.
2
Concerns
• Producers ability to obtain meaningful
crop insurance is eroded after a series of
poor yielding years.
• After unusually good or bad years, rates
are actuarially less sound.
• Technology improvements resulting in
increasing yield are ignored by current
system.
3
Current Method of Calculating
Indemnity Trigger
• Approved yield = simple average of 4-10
years of producer supplied yield history.
• (Approved yield)*(Coverage level) =
Indemnity trigger.
• If yield outcome falls below the indemnity
trigger, an indemnity is paid.
4
Current Method
• Pros
–Easy to administer.
–Easy to understand.
5
Current Method
• Cons
– Does not account for technological increases
• Biased.
– Small sample size
• Efficiency of approve yield estimates.
– Series of unusually low yielding years will
dramatically lower approved yield, making an
indemnity payment less likely.
– Series of high yields increases probability of
an indemnity payment.
6
U.S. Corn Historic Yields
160
140
120
100
80
60
40
20
0
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
Cor n P l ant ed Y i el d
1972
1976
1980
1984
1988
1992
1996
2000
2004
T r end
7
U.S. Wheat Historic Yields
45
40
35
30
25
20
15
10
5
0
1918
1922
1926
1930
1934
1938
1942
1946
1950
1954
1958
1962
US Wht pl yl d
1966
1970
1974
1978
1982
1986
1990
1994
1998
2002
2006
T r end
8
Wheat Yield, Petroleum County, Montana
40
35
30
25
20
15
10
5
0
1924
1932
1940
Planted Yield
1948
1956
Trend
1964
10 year simple average
1972
1980
5 year simple average
1988
2005 yield
1996
2004
9
Objectives
• Develop an approved yield as accurate as
possible point estimate of expected yield.
– Reduce bias of a simple average.
– Increased efficiency over a simple average.
– Reduce adverse selection and moral hazard.
– Administratively feasible.
10
Proposed Alternative – Indexing
• Longer term (e.g. > 50 years) regional data
(NASS) sets used along with producer actual
production history.
• Method overview
– Statistically estimate trend line from long term
regional data to forecast expected regional yield.
– Calculate average of farm level and regional series
for given time period.
– Difference between two is added (subtracted) to
(from) expected regional yield to calculate the
producers approved yield.
11
Detailed Discussion
• Let
TI  number of farm yield observations
Tr  number of regional yield observations
b  annual increase (slope) of yield trend line
 2  farm level variance
 e2  variance of trend line residuals or errors
 2f   2   e2
 farm level variance beyond regional variance
12
Bias
• Simple average of current method is
biased because of technology. Bias is
TI  1
b
.
2
• Indexed yield predicted from linear
regression has no bias.
13
Efficiency
2
2

• Variance of simple average = Y 
.
TI
• Variance of indexed yield has two
components which are orthogonal and
additive.
2

– Variance of regional expected yield ( r̂ ) in
year Tr  1 =


 1 T  1  t 2 
4  Tr  2
r
2
2
2


 rˆ   e
 Tr
 e 
.
Tr Tr  1
2 
 Tr
t  t  


t 1


14
Efficiency cont.
– Variance of difference between farm &
regional average yield =
 
2
f

2
f
TI
.
Therefore, variance of Indexed yield is

2
Indexed
4  Tr  2 
     

.
Tr Tr  1 TI
2
rˆ
2
f
2
e
2
f
15
Efficiency Gain
• Large number of observations at regional level
increases the efficiency of indexing.
• The efficiency of the simple average and indexing is
the same when the number of observations satisfies
Tr Tr  1
TI 
.
4  Tr  2
• The Indexed yield will provide a more efficient
estimate of the approved yield if the length of the
regional data series is greater than approximately
four times the length of the farm data series.
16
Illustration of Efficiency
Let
 2  900  farm yield total variance
 e2  450  regional variance
 2f  450  farm level added variance
Length of
Simple
Farm Data Average
(years)
Variance
4
225
6
150
8
113
10
90
20
210
172
154
142
Length of Regional Time Series (years)
30
40
50
60
70
Indexed Yield Variance
176
159
150
143
139
138
122
112
106
101
120
103
94
87
83
108
92
82
76
71
80
135
98
79
68
• For approved yields to be efficient (stable) and unbiased,
the distribution of approved yields must be concentrated
around the expected yield (small variance).
17
Critical Component Indexing
• Estimation of yield trend lines.
Most General Form:
2

t
1
Yˆt   o 

3  t
•
4
 ‘s are parameters to be statistically
estimated.
18
Estimating Trend Lines
• Form is flexible.
For example:
IF
THEN
 2  1,  4  0
Linear
4  0
Exponential
2  4
Sigmoid
19
Figure 3
Illustration of the Effect of
2
400
Y  15  20t  2
350
300
 2  1.25
Expected Yield
250
2  1
200
150
 2  .5
100
2  0
50
 2  1
0
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Time
20
Figure 4
Illustration of the Effect of
40
2
20t  2
Y  15 
100  t  2
35
2  4
30
2  8
Expected Yield
25
2  2
2  1
20
15
10
5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Time
21
Figure 5
Illustration of the Effect of
40
3
20t 4
Y  15 
3  t 4
35
30
 3  10
 3  100
 3  1000
Expected Yield
25
20
15
10
5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Time
22
Figure 6
Illustration of the Effect of
90
4
20t 4
Y  15 
100  t  4
80
70
Expected Yield
60
 4  3.5
50
4  4
40
30
4  6
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Time
23
Model Selection
• Model chosen by:
– Standard statistical tests such as F-test.
– Visual inspection of graphs.
• Model choice experience
– Have estimated 100’s of trend lines for crop yields in
the US and other countries.
Selected Form
Linear ( 2  1,  4  0)
Proportion of Time
Selected
80%
( 2   4 )
15%
No Trend ( 2  0)
5%
Sigmoid
24
Wheat Yield, Petroleum County, Montana
40
35
30
25
20
15
10
5
0
1924
1932
1940
Planted Yield
1948
1956
Trend
1964
10 year simple average
1972
1980
5 year simple average
1988
2005 yield
1996
2004
25
Petroleum County, Montana
• Estimated trend line equation is
9.656
14.106
t
Yˆt  8.487 
.
9.656
988392  t
• Scale t
year  1918
t
.
10
• Expected regional indexed yield 2005 =
22.6.
26
Adjustment to Indexed Method
• Indexed Farm Yield = Indexed
Regional Yield = (Farm Average –
Regional Average).
27
Regional (county) and Hypothetical
Farm Yields for Petroleum County
Year
Example Farm Yield
Regional Yield
1995
30.00
20.50
1996
35.00
26.00
1997
40.00
32.80
1998
30.00
36.30
1999
30.00
28.60
2000
15.00
9.10
2001
0.00
7.50
2002
10.00
5.60
2003
26.00
21.40
2004
25.00
21.40
28
Petroleum County Simple Farm
Average and Indexed Yield
Number Of
Years*
Farm
Average
Regional
Average
Difference
of
Averages
4
15.25
13.98
1.28
23.88
6
17.67
15.60
2.07
24.67
8
22.00
20.34
1.66
24.26
10
24.10
20.92
3.18
25.78
Indexed
Yield
*Most Recent Years
29