This project report examines the modeling and estimation of loss functions for crop insurance in India, addressing the agricultural sector's exposure to systemic and covariant risks. It details various approaches to crop insurance, such as yield index, weather index, and biomass index-based insurance, while providing theoretical analyses of loss functions and empirical studies based on yield data from Hanumangarh district. The report aims to develop a usable loss function to guide individual farmers in their insurance decisions based on production costs, indemnity, and climate conditions.

1. MODELING AND ESTIMATION OF LOSS
FUNCTION FOR CROP INSURANCE UNDER
INDIAN SCENARIO
(A MAJOR PROJECT REPORT)
ANIL KUMAR
M.Sc. Statistics
2011-2013
Under the Guidance of-
Dr. Vidyottama Jain, Assistant Professor
Department of Mathematics Central
University of Rajasthan
Dr. S. Dharmaraja, Associate Professor,
Department of Mathematics,
Indian Institute of Technology Delhi

2. Contents of the Project Report
Introduction
Approaches of Crop Insurance in India
Yield Index Based Crop Insurance
Weather Index Based Crop Insurance
Biomass (Crop Health) Index Based Crop Insurance
Multiple (Index) Trigger Index Based Crop Insurance
Previous and Current Crop Insurance schemes in India
Theoretical Analysis
Model of Loss Function in American Scenario
Model of Loss Function in Indian Scenario
Empirical Study
Data Sources
Data Generation
Data and Data Analysis
Conclusion
References

3. Introduction
Indian Agriculture sector is subject to a great many uncertainties.
Agriculture, particularly prone to systemic and co-variant risk
doesn’t easily lend itself to insurance.
Our main objective is to introduce simple but useable loss function
incorporating appropriate feature relevant to the area of interest.
The loss function for the individual farmer depends upon the
production of individual, area, cost of production. In addition, if
farmer is having any insurance policy, then loss function will
depend upon the premium (paid by the farmer for that insurance
policy) and indemnity (paid by the company if any loss occurs to
the farmer). The loss function is also expected to take account of
reserve creation together with climatic conditions and also for the
future investments of the company.

4. Approaches of Crop Insurance in India:
 Yield index based crop insurance.
 Weather index based crop insurance.
 Biomass index based crop insurance.
 Multiple (index) trigger crop insurance.
These product are based on the supply side.

5. Yield
index
based
insurance
• homogeneity
• Basis risk
Weather
index
based
insurance
• Weather
index
• Basis risk
Biomass
index
based
insurance
• Biomass
index
• Basis risk
Multiple
(index)
trigger
crop
APPROACHES OF
CROP INSURANCE
IN INDIA

6. YIELD INDEX BASED INSURANCE
The basic character of the yield index approach (area approach), is that
it sets up, for each area, an independent chance-system entirely
dependent on the annual average yields of the crop in that area and
avoids altogether any reference to individuals or groups of individuals in
the area not only while fixing the premium rate but also for assessment
of indemnity. This makes a crop insurance scheme based on this
approach a fair betting system in principle. But if the area is sufficiently
‘homogeneous’ to make the annual crop experience of a majority of the
farmers similar, it serves for them as crop insurance as well. Within
these limits, the scheme appears to be an operationally simple and
practically useful.
The basis risk in yield index based crop insurance is homogeneity
because the area is rarely homogeneous.

7. Indemnity is calculated based on the
formula:
here ,
b=Threshold yield.
a= Actual yield.
c= Sum Insured.
a)/b}.c,(b{indemnity  0max

8. WEATHER INDEX BASED CROP INSURANCE:-
The basic purpose of ‘weather index’ insurance is to estimate the
percentage deviation in crop output due to adverse deviations in weather
conditions. So company decide the premium and assessment of
indemnity basis on the weather index in low income countries where crop
insurance could not take off for various regions, including lack of
historical yield or loss data.
Basis risk in weather index crop insurance is lack of good density of
weather stations, and poor index design, non-availability of reliable and
quality weather data.

9.  Effect of Basis Risk & Poor Design of Weather
Index:-

10. BIOMASS INDEX BASED CROP INSURANCE
‘Biomass index’ based on satellite image is called Normalized Difference
Vegetative Index (NDVI) and it is used in India by AIC for estimating the crop
health (biomass) index and they find highest correlation between the
satellite images and final yield. So basis of this index the AIC decide the
premium rate and assessment of indemnity.
Basis risks are in this type of policy are high start up cost, requirement of all-
weather satellites, requirement frequent fly-overs, not suitable for crops
where the economic product is formed below the surface etc.
MULTIPLE (INDEX) TRIGGER CROP INSURANCE.
In this type of policies index is combination of these indexes and it can
bridge the ‘gap ’ in indemnity and lower the basis risk so behalf of this we
decide the premium rate and assessment of indemnity.

11. Previous and Present Crop insurance schemes
in India:
a) Program based on ‘individual’ approach (1972-1978)
b) Comprehensive Crop Insurance Scheme – CCIS (1985-1999)
c) Pilot Crop Insurance Scheme – PCIS (1979-1984)
d) National Agriculture Insurance Scheme –NAIS (1999):
 States and areas covered:
 Farmers covered
 Crops covered:
 Sum insured:
 Premium rates
 Premium subsidy:
 Loss assessment, levels of indemnity & threshold yield
 Sharing of risk
 The steps of agriculture insurance
 Estimation of crop yield
 Corpus fund
 Reinsurance cover:

12. Theoretical Analysis
We know that the loss associated with farmer is given by
Here
Model of Loss function for American Scenario:
The loss associated with individual farmer is given by
    indemnitypremiumrevenueproductionoftloss  cos




policyinsurancenohavefarmerif
policyinsurancehavefarmerif
,0
,1

     ][, *
,,,
1




dk
k
s
kikkiki
s
k
s
kk
K
k
k PZqRPYqCxf 

13. Here, Yk is the total yield of crop k in scenario s
And, is difference between the insured yield and the true yield
so loss function is given by
Here assume that farm grows k types of crops for each crop where k = 1, 2, , ,
,K. The possible planting dates for a crop k are indexed by dk. Scenarios
indexed by s = 1, 2, . . . ,N are historical records for each season. Crop-
insurance contracts are indexed by i = 1, 2,. . .,l crops on the allocates area qk
Where is the decision vector and is the random
vector.
s
kiZ ,
 s
dkki
dk
dk
s
ki yyXZ   *
,,

14. Variable Unit Description
$/ha Production cost of crop k per hectare.
$/ha Premium of the insurance policy i for
crop k per ha.
$/kg Market price of crop k per kg, scenario
s.
$/ha Price election of crop k, i.e. the
expected market price. In India, it is set
by Government of India as MSP.
Kg/ha Yield of crop k per ha for planting date
dk in scenario s.
Kg/ha Insured yield of crop k per ha by policy i.
Ha Number of hectares of land for crop k
with planting date dk
kC
kiR ,
s
kP
*
kp
s
dky
*
,kiy
dkX

15. Model of Loss Function in Indian Scenario:
Assumptions:
1. Farmer grow k type of crops.
2. The possible planting date and scenario are considered as a single.
3. Crop insurance contract are also single considered and level of indemnity
is medium(80%).
the loss function in Indian scenario is given by-
Here
So loss function for individual is given by-
Where is the decision vector and is the random
vector.
     i
kk
i
k
i
kkkik
i
k
i
k
i
k
K
k
ki qPZqYRPYqqCxf **
1
,


  
  %80** i
kk
i
k YYZ 
     i
kk
i
kk
i
kkkik
i
k
i
k
i
k
K
k
ki qPYYqYRPYqqCxf ***
1
*%80*,


  

16. Variable Unit Description
RS/ha Production cost of crop k per
hectare.
Ha Area allocated to kth crop for
ith individual farmer..
RS/kg . price per kg for crop k.
kg/ha Yield for crop k for ith farmer.
Unit less Premium rate in %.
Kg/ha Insured yield (moving
average of past 3 year
production data)
Kg/ha Indemnity kg/ha payable to
ith farmer for kth crop.
Indicator function Selection of insurance policy
for ith individual (indicator
function) where
if ith farmer selects policy
otherwise .
RS/kg Minimum Support Price
(MSP)
kC
i
kq
kP
i
kY
kR
*
ky
 
kiZ ,
i 1i
0i
*
kP

17. Empirical study
For empirical study we choose the Hanumangarh district located
northern western, Rajasthan. Farmer grow majorly Rice, Cotton
and Guar crops in Kharif season.
for our study we consider only single planting date and for a
crop.
 Data sources.
 Data generation.
 Data and data analysis.
 Data sources:
http://www.krishi.gov.in
http://www.eands.dacnet.nic.in
http://www.mandionline.gov.in

18.  Data generation:
The individual yield data is generated from truncated Normal
distribution. The cdf, mean and variance of truncated Normal
distribution is given by
Cdf
Mean
variance
)/(2
'
22
2/







e
    












2
/2/
22
)/(2/2
1)('
2222




ee
YVg
 


















 








1
y
yFg

19. We select data of crop yield of cotton, rice and guar of year 1994-1995 to
2007-2008 from Hanumangarh district are
DISTRICT: HANUMANGARH
YIELD DATA IN KG/HA
Year guar cotton Rice
1994-1995 540 1890 2009
1995-1996 372 2443 2264
1996-1997 1073 2298 2557
1997-1998 471 1249 1535
1998-1999 557 1239 1918
1999-2000 146 1863 2464
2000-2001 356 1782 2565
2001-2002 353 347 3048
2002-2003 47 392 1739
2003-2004 1140 2269 3213
2004-2005 146 2159 3677
2005-2006 726 2306 3860
2006-2007 265 2318 4639
2007-2008 870 2217 4194
Mean 504.4286 1769.429 2834.429
variance 107222.5 458560.5 862670.1

20. On the basic of these data, we calculate the mean and variance of
truncated normal distribution is given in table-
 Data and data analysis-
 Cost of production data for these three crops are given in table are-
MEAN AND STANDARD DEVIATION FOR TRUNCATED
NORMAL DISTRIBUTION
crop mean standard deviation
Cotton 1754.92 721.92
Rice -998.195 934.58
Guar 292.76 471.42
COST OF PRODUCTION DATA FOR 2008-
2009
Crop Cost(RS/ha)
Rice 34643
Cotton 20782.7
Gour 8483.54

21. • Minimum support price for these crops are-
• Market Price During 2008-2009
MARKET PRICE DURING 2008-2009
crop Price(RS/quintal)
cotton 4718
guar 2450
Rice 984.5
MINIMUM SUPPORT PRICE
crop price
cotton 2500
guar 1500
rice 880

22. • Moving average of past(just last three year from insured year) three
year yield data is obtained from yield production data are-
• Individual yield data, Insurance indicator function, Proportion of
crop in field data are considered as shown in Project Report.
Here,
Cost, Revenue, Premium and Indemnity for a particular crop of
individual farmer are calculated based on following-
MOVING AVERAGE OF PAST 3
YEAR YIELD DATA
Crop Yield
Guar 620.33
Cotton 2280.33
Rice 4231

23. cost = ,
Revenue = ,
Premium = and
Indemnity =
 Data Analysis
Using all data, we can calculate the following
 Loss for individual farmer for all crops (here we consider 3 crops).
 Loss for particular crop of individual farmer.
 Loss for the farmer, who have not insurance policy.
 Loss for farmers, who have insurance policy.

24. Loss for all categories are plotted in graphs (As Project Report).
By graph we can conclude that the result of loss function for various
crops it is shown that for some crop the loss is very low or negative
for all time. For some crop, the loss is positive and farmer faces loss.
In our data analysis, for rice, a large number of farmers face loss but
in case of cotton and gour the maximum numbers of farmers are in
profit.
In this project, we try to present a model to individual farmer’s loss
by representing loss function for Indian Scenario.
On the basic of these loss function farmer can take decision that
whether to go for insurance policy or not according to crop and
weather condition.

25. References:-
1. J. Liu, C. Men, V. E. Cabrera, S. Uryasev(2006), C. W. Fraisse: CVaR
Model for Optimizing Crop Insurance under Climate Variability: Research
report 2006-1.
2. Kolli N. Rao : Index Based Crop Insurance(2010): Agriculture and
Agricultural Science Procedia 1 (2010) 193–203.
3. Raju S.S., CHAND R.(2007): Progress and Problems in Agricultural
Insurance: Economic and Political Weekly, Vol. 42, No. 21 (May 26 - Jun. 1,
2007), pp. 1905-1908.
4. Modified National Agricultural Insurance Scheme (MNAIS) by AIC of India.
5. Uryasev Stan: Risk Management and Financial Engineering Lab, University
of Florida.
6. Sai T., Yulian W., and Xiaofeng H (2010): An Empirical Study of Agricultural
Insurance--Evidence from China: Agriculture and Agricultural Science
Procedia 1 (2010) 62–66.
7. NIAS scheme, AIC of India and Govt. of India.
8. Dandekar, V.M. [1976]: Crop Insurance in India, Economic and Political
Weekly, Vol. 11, No. 26, pp. A61-A80.