Some Comparative Statics for
Evaluating the Performance of
the U.S. Crop Insurance Program
by
Octavio A Ramirez
and
J. Scott Shonkwiler
Objectives

Develop an analytical framework that can
be used to:
Assess how changes in the current RMA ratesetting protocol could affect producer
participation and program cost
Evaluate the merits of alternative premium
estimation methods and other potential
strategies for improving actuarial performance
and other important features of the program
Compute elasticities and predict the impact of
changes in program “parameters” (e.g. PSR)
on key measures of performance (e.g. PPR)
Analytical Framework

Suppose that the decision rule for
producer i to purchase insurance is:
RPPxPPEi>(1–GSR)xIPEi
where RRP>1 and 0≤GSR≤1
Since neither the producer nor the insurer
knows what the Actuarially Fair Premium
(AFP), PPEi and IPEi have to be treated as
random variables
Analytical Framework
Specifically, let:
PPEi = AFPi + ui such that ui ~ N(μ1, σ12)
IPEi = AFPi + νi such that νi ~ N(μ2, σ22)
 This specification allows for bias in the
premium estimates when μ1 and/or μ2 are
not equal to zero and makes the degree of
uncertainty in the estimates explicit by
introducing random components with
variances of σ12 and σ22

Analytical Framework

We also allow for the possibility that the
producer and insurer estimates are
correlated, i.e.:
Cov(PPEi,IPEi) = Cov(ui, νi) = σ12

Then, the probability that producer (i) will
participate in the program is given by:
Pr[RPPxPPEi>(1–GSR)xIPEi] =
Pr[αPPEi –IPEi>0] = Pr[αui–νi>(1–α)AFPi]
where α=RPP/(1–GSR)
Analytical Framework

Then note that:
Pr[αui–νi>(1–α)AFPi] =
Φ((αμ1–μ2+(α–1)AFPi)/(Var(αui–νi))1/2)=
Φ(μ3/σ3), where:
μ3=αμ1–μ2+(α–1)AFPi
σ23=α2σ21+σ22–2ασ12
and Φ denotes the standard normal
cumulative distribution function
Analytical Framework

The question then is how the probability of
participation is affected by changes in:
Bias in the producer premium estimate (μ1)
Bias in the insurer premium estimate (μ2)
Error in the producer premium estimate (σ1)
Error in the insurer premium estimate (σ2)
Correlation between the producer and the
insurer premium estimates (ρ12)
The risk protection premium factor (RPP)
The rate at which the government subsidizes
the estimated premiums (GSR)
Analytical Framework
Thus we are interested in the partial
derivatives of Pr[αPPEi –IPEi>0] = Φ(μ3/σ3)
with respect to all these variables
 The results needed to obtain these
derivatives are:

 ∂Φ(μ3/σ3)/∂μ3 = (1/σ3)ϕ(μ3/σ3)
 ∂Φ(μ3/σ3)/∂σ3 = – (μ3/σ32)ϕ(μ3/σ3)
where μ3 and σ3 are as previously defined
and ϕ is a standard normal density function
Conclusions from the Derivatives
An increasing bias in the producer premium
estimate will increase the probability of
participation
 An increasing bias in the insurer premium
estimate will decrease the probability of
producer participation
 The absolute impact of an increased
producer bias is always equal to or higher
(likely twice as much) than the effect of a
higher insurer bias

Conclusions from the Derivatives
A decrease in the variability of either of the
premium estimates is likely to increase the
probability of participation
 However, it is possible for a decrease in
variability to have a negative effect on the
probability of participation
 An equal change in producer error has a
much larger impact on the probability of
participation than a change in insurer error

Conclusions from the Derivatives
A higher correlation between the producer
and the insurer premium estimates makes it
more likely for a producer to purchase crop
insurance
 When there is a strong correlation between
the producer and the insurer premium
estimates and the subsidy levels are very
high, it is actually possible for a further
increase in the GSR (or the RPP) to have a
detrimental effect on participation

Other Interesting Results

We also compute the expected value (E[.])
and the variance (Var[.]) of the effective peracre subsidy (PAS) received by a participating
producer :
E[PASp]=AFPp–(1–GSR)E[IPEp]
E[IPEp] = AFPp+μ2+δσ2ω
Var[PASp]=(1–GSR)2Var[IPEp]
Var[IPEp] = σ22(1–δ2ω(ω+μ3/σ3))
where δ=(αρ12σ1σ2–σ22)/σ2σ3 and
ω=φ(μ3/σ3)/Φ(μ3/σ3)
Other Results

And the percentage of the indemnities to be
paid to a participating producer that is
expected to be funded by the government
(PFG):
E[PFGp]=1– (1–GSR)E[IPEp]/AFPp
where again E[IPEp] = AFPp+μ2+δσ2ω
Illinois Corn Examples
AFPs corresponding to various mean-standard
deviation combinations assuming normally
distributed yields and a 75% coverage level
MEAN
STD
AFP
RMAEST
PROQUO
175.0
32.5
6.7
17.41
7.83
155.0
40.0
17.31
17.31
7.79
135.0
47.5
33.33
17.42
7.84
165.0
35.0
10.26
17.42
7.84
155.0
40.0
17.31
17.31
7.79
145.0
45.0
26.78
17.24
7.76
165.0
37.5
12.87
17.42
7.84
155.0
40.0
17.31
17.31
7.79
145.0
42.5
23.25
17.24
7.76
Scenario 1
RPP=1.10
CL=75%
GSR=0.55
μ1=0
σ1=AFP/4
μ2=17.3-AFP
σ2=0.05
ρ12=0
AFP
POP
7
9
11
13
15
17
19
21
23
25
27
29
0.481
0.803
0.923
0.966
0.983
0.990
0.994
0.996
0.997
0.998
0.998
0.999
PPR=
E[PASp] S[PASp]
-0.789
1.211
3.211
5.211
7.211
9.211
11.211
13.211
15.211
17.211
19.211
21.211
0.955
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.023
PFG=
PFG
-0.113
0.135
0.292
0.401
0.481
0.542
0.590
0.629
0.661
0.688
0.712
0.731
0.506
Figure 1: Expected Per Acre Subsidy
for Different Levels of Risk
25
22
19
16
Subsidy 13
($/acre)
10
7
4
1
-2
7
9
11 13 15 17 19 21 23 25 27 29 31 33
AFP ($/acre)
Elasticities
AFP
FREQ
∂popµ1
∂popµ2
εpopσ1
εpopσ2
εpopρ12
εpopGSR
εpopRPP
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
0.025
0.028
0.032
0.035
0.039
0.043
0.046
0.050
0.054
0.057
0.061
0.061
0.057
0.054
0.050
0.046
0.043
0.039
0.035
0.032
0.028
0.025
0.021
0.017
0.014
0.010
SL/EL=
0.228
0.179
0.123
0.081
0.053
0.035
0.023
0.016
0.011
0.008
0.006
0.005
0.004
0.003
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.019
-0.093
-0.073
-0.050
-0.033
-0.022
-0.014
-0.010
-0.007
-0.005
-0.003
-0.003
-0.002
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.008
0.038
-0.244
-0.294
-0.268
-0.223
-0.180
-0.143
-0.115
-0.092
-0.075
-0.062
-0.051
-0.043
-0.037
-0.032
-0.027
-0.024
-0.021
-0.019
-0.017
-0.015
-0.014
-0.013
-0.011
-0.011
-0.010
-0.075
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
4.093
2.295
1.328
0.795
0.493
0.316
0.210
0.143
0.101
0.073
0.054
0.041
0.031
0.025
0.020
0.016
0.013
0.011
0.009
0.008
0.007
0.006
0.005
0.004
0.004
0.003
0.219
3.349
1.878
1.087
0.651
0.403
0.259
0.171
0.117
0.082
0.060
0.044
0.033
0.026
0.020
0.016
0.013
0.011
0.009
0.007
0.006
0.005
0.005
0.004
0.004
0.003
0.003
0.180
Cost Analysis
Overall Expected Per-Acre Subsidy (OEPAS) and
Average Dollar Cost Measure (ADCM) for
different GSR/PPR/PFG combinations
GSR
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
PPR
0.810
0.844
0.876
0.906
0.933
0.955
0.972
0.992
PFG
0.276
0.316
0.359
0.405
0.454
0.506
0.560
0.671
OEPAS
6.335
7.093
7.882
8.699
9.541
10.403
11.281
12.166
ADCM
5.130
5.987
6.908
7.883
8.898
9.936
10.970
12.070
Scenario 2
RPP=1.10
CL=75%
GSR=0.325
μ1=0
σ1=AFP/4
μ2=0
σ2=AFP/3
ρ12=0.50
ADCM=5.52
AFP
POP
7
9
11
13
15
17
19
21
23
25
27
29
0.953
0.953
0.953
0.953
0.953
0.953
0.953
0.953
0.953
0.953
0.953
0.953
PPR=
E[PASp] S[PASp]
2.331
2.997
3.663
4.329
4.995
5.661
6.327
6.993
7.659
8.325
8.991
9.657
0.953
1.558
2.003
2.448
2.893
3.338
3.783
4.228
4.673
5.118
5.563
6.009
6.454
PFG=
PFG
0.333
0.333
0.333
0.333
0.333
0.333
0.333
0.333
0.333
0.333
0.333
0.333
0.317
Scenario 1 vs 2
AFP
POP
E[PASp] S[PASp]
PFG
POP
E[PASp]
S[PASp]
PFG
7
0.481
-0.789
0.023
-0.113
0.953
2.331
1.558
0.333
9
0.803
1.211
0.023
0.135
0.953
2.997
2.003
0.333
11
0.923
3.211
0.023
0.292
0.953
3.663
2.448
0.333
13
0.966
5.211
0.023
0.401
0.953
4.329
2.893
0.333
15
0.983
7.211
0.023
0.481
0.953
4.995
3.338
0.333
17
0.990
9.211
0.023
0.542
0.953
5.661
3.783
0.333
19
0.994
11.211
0.023
0.590
0.953
6.327
4.228
0.333
21
0.996
13.211
0.023
0.629
0.953
6.993
4.673
0.333
23
0.997
15.211
0.023
0.661
0.953
7.659
5.118
0.333
25
0.998
17.211
0.023
0.688
0.953
8.325
5.563
0.333
27
0.998
19.211
0.023
0.712
0.953
8.991
6.009
0.333
29
0.999
21.211
0.023
0.731
0.953
9.657
6.454
0.333
31
0.999
23.211
0.023
0.749
0.953
10.323
6.899
0.333
PPR=
0.955
PFG=
0.506
PPR=
0.953
PFG=
0.317
Sorry for running out of time!