Rainfall Insurance and Basis Risk Professor Tobacman, Li Zheng

Overview Definition 1 Definition 2 Reality check Moving Forward: Empirical Studies

Basis Risk Occurs when Payout from indexed insurance Actual losses experienced do not match.

Definition 1 I defined the basis risk facing farmers as: 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘= 𝜎 𝑀 𝑖 2 | 𝑢𝑛𝑑𝑒𝑟 𝑛 ∗ 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝜎 𝑀 𝑖 2 | 𝑢𝑛𝑑𝑒𝑟 0 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝑀 𝑖 is the income of the 𝑖 𝑡ℎ farmer / state

Definition 1 (Continued) 𝑦 𝑖 = 𝜇 𝑖 + 𝑓 𝑤 +𝑔 ∙ + 𝜀 𝑖 𝜀 𝑖 ~ 𝑁(0, 𝜎 𝜀 𝑖 2 ) 𝐼 𝑤 = 𝑃𝑎𝑦𝑜𝑢𝑡 𝑓𝑟𝑜𝑚 𝑜𝑛𝑒 𝑝𝑜𝑙𝑖𝑐𝑦 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 𝑞=𝐸 𝐼 𝑤 (fair insurance) Variable w here represents the index or indices in question

Definition 1 (Continued) Under 𝑛 insurance policies, 𝑀 𝑖 =𝑝𝐴 𝑦 𝑖 + 𝑛 𝐼 𝑤 − 𝑞 𝐸 𝑀 𝑖 =𝑝𝐴 𝜇 𝑖 𝑉𝑎𝑟 𝑀 𝑖 = 𝐴 2 𝑝 2 𝜎 𝑦 𝑖 2 + 𝑛 2 𝜎 𝐼(𝑤) 2 + 2𝐴𝑛𝑝 𝐶𝑜𝑣( 𝑦 𝑖 , 𝐼 𝑤 ) Here A is the area under farming and p is the price per unit yield. It is expected that 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑓 <0.

Definition 1 (Continued) With CARA Utility, 𝑈 𝑀 𝑖 =𝐸 𝑀 𝑖 − 1 2 ∅𝑉𝑎𝑟 𝑀 𝑖 = 𝑝𝐴 𝜇 𝑖 − 1 2 ∅ 𝐴 2 𝑝 2 𝜎 𝑦 𝑖 2 + 𝑛 2 𝜎 𝐼(𝑤) 2 + 2𝐴𝑛𝑝 𝐶𝑜𝑣( 𝑦 𝑖 , 𝐼 𝑤 ) ∅ is the coefficient of risk aversion

Definition 1 (Continued) Maximizing utility with respect to 𝑛 : 𝜕(𝑈( 𝑀 𝑖 ) 𝜕𝑛 =0− 1 2 ∅ 0+ 2 𝑛 ∗ 𝜎 𝐼(𝑤) 2 + 2𝐴𝑝 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑤 =0 𝜕 2 (𝑈( 𝑀 𝑖 ) 𝜕 𝑛 2 =2 𝜎 𝐼(𝑤) 2 >0 ∀ 𝑛 ϵ 𝐙 0 + 𝑛 ∗ = − 𝐴𝑝 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑤 𝜎 𝐼 𝑤 2 For now we are assuming ability to purchase non-discrete amounts of insurance

Definition 1 (Continued) Under this definition and the expression for 𝑛 ∗ : 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘= 𝜎 𝑀 𝑖 2 | 𝑢𝑛𝑑𝑒𝑟 𝑛 ∗ 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝜎 𝑀 𝑖 2 | 𝑢𝑛𝑑𝑒𝑟 0 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 = 𝐴 2 𝑝 2 𝜎 𝑦 𝑖 2 + (𝑛 ∗ ) 2 𝜎 𝐼 𝑤 2 + 2𝐴 𝑛 ∗ 𝑝 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑤 𝐴 2 𝑝 2 𝜎 𝑦 𝑖 2 =1+ 𝐴 2 𝑝 2 𝜎 𝐼(𝑤) 2 [ 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑤 ] 2 𝐴 2 𝑝 2 𝜎 𝑦 𝑖 2 [ 𝜎 𝐼 𝑤 2 ] 2 + −2 𝐴 2 𝑝 2 [ 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑤 ] 2 𝐴 2 𝑝 2 𝜎 𝑦 𝑖 2 𝜎 𝐼(𝑤) 2 =1− 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑤 𝜎 𝑦 𝑖 𝜎 𝐼(𝑤) 2 =1− 𝜌 𝑦 𝑖 , 𝐼(𝑤) 2

Definition 1 (Continued) Given this specification of basis risk, one has to maximize 𝜌 𝑦 𝑖 , 𝐼(𝑤) 2 to minimize basis risk. If we assume that rainfall is independent of all other explanatory variables, we will arrive at one obvious candidate for the functional form of 𝐼 𝑤 : 𝐼 𝑤 =𝑎𝑓 𝑤 +𝑏 . I will now show that this functional form is indeed one of the candidates that minimizes this definition of basis risk under independence of rainfall.

Definition 1 (Continued) Under independence of rainfall: 𝜌 𝑦 𝑖 , 𝐼(𝑤) 2 = 𝐶𝑜𝑣 𝑦 𝑖 , 𝐼 𝑤 𝜎 𝑦 𝑖 𝜎 𝐼(𝑤) 2 = 𝐶𝑜𝑣 𝑓(𝑤), 𝐼 𝑤 𝜎 𝑦 𝑖 𝜎 𝐼(𝑤) 2 = 𝑎𝐶𝑜𝑣 𝑓(𝑤), 𝑓(𝑤) 𝑎 𝜎 𝑦 𝑖 𝜎 𝑓(𝑤) 2 = 𝜎 𝑓(𝑤) 2 𝜎 𝑦 𝑖 𝜎 𝑓(𝑤) 2 = 𝜎 𝑓(𝑤) 𝜎 𝑦 𝑖 2 I claim that 𝜎 𝑓(𝑤) 𝜎 𝑦 𝑖 2 is the maximum value of 𝜌 𝑦 𝑖 , 𝐼(𝑤) 2 .

Definition 1 (Continued) Prove by Contradiction:  If ∃ 𝐶𝑜𝑣 𝑓 𝑤 , 𝐼 𝑤 𝜎 𝑦 𝑖 𝜎 𝐼 𝑤 2 , 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡: 𝐶𝑜𝑣 𝑓 𝑤 , 𝐼 𝑤 𝜎 𝑦 𝑖 𝜎 𝐼 𝑤 2 > 𝜎 𝑓 𝑤 𝜎 𝑦 𝑖 2 𝐶𝑜𝑣 𝑓 𝑤 , 𝐼 𝑤 𝜎 𝐼 𝑤 2 > 𝜎 𝑓 𝑤 2 𝐶𝑜𝑣 𝑓 𝑤 , 𝐼 𝑤 2 > 𝜎 𝑓(𝑤) 2 𝜎 𝐼(𝑤) 2 But this violates the Cauchy-Schwarz Inequality.

Definition 1 - Assumptions The above assumes: No interactions between rainfall and other explanatory variables in the model if 𝐼 𝑤 =𝑎𝑓 𝑤 +𝑏 is one of the basis risk minimizing insurance models. Using the definition alone allows for interactions between all explanatory variables. CARA utility functions Ability to purchase non-discrete amounts of insurance policies

Definition 1 – Criticisms Criticisms of the above definition: Too blunt: Does not isolate basis risk due to rainfall. This definition is a measure of risk due to uncompensated exposure to agriculture as a whole, not due to uncompensated exposure due to rainfall. Too pessimistic: Due to its bluntness, it will likely result in overstated levels of basis risk No clear minimum. It is not clear what the benchmark level of basis risk is, and no clear target to work towards. 0 is unrealistic.

Definition 1 – Summary What we learnt: Setting 𝐼 𝑤 =𝑎𝑓 𝑤 +𝑏 is one useful functional form that could minimize basis risk.

Definition 2 To increase precision, I defined basis risk as: 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘= 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑖𝑛𝑐𝑜𝑚𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑢𝑛𝑑𝑒𝑟 𝑛 ∗ 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑖𝑛𝑐𝑜𝑚𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑢𝑛𝑑𝑒𝑟 0 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 = 𝜎 𝑀 𝑖 𝑑𝑢𝑒 𝑡𝑜 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑎𝑛𝑑 𝑒𝑟𝑟𝑜𝑟 2 | 𝑢𝑛𝑑𝑒𝑟 𝑛 ∗ 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝜎 𝑀 𝑖 𝑑𝑢𝑒 𝑡𝑜 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑎𝑛𝑑 𝑒𝑟𝑟𝑜𝑟 2 | 𝑢𝑛𝑑𝑒𝑟 0 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 With this definition, basis risk measures risk from unknown and hence uncompensated exposure due to rainfall in agriculture.

Definition 2 (Continued) Similarly, we define yield and insurance: 𝑦 𝑖 = 𝜇 𝑖 + 𝑓 𝑤 +𝑔 ∙ + 𝜀 𝑖 𝜀 𝑖 ~ 𝑁(0, 𝜎 𝜀 𝑖 2 ) 𝐼 𝑤 = 𝑃𝑎𝑦𝑜𝑢𝑡 𝑓𝑟𝑜𝑚 𝑜𝑛𝑒 𝑝𝑜𝑙𝑖𝑐𝑦 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 𝑞=𝐸 𝐼 𝑤 We now strictly assume that 𝑔 ∙ 𝑎𝑛𝑑 𝑓 𝑤 are independent. Other than that there are no restrictions on the functional form of the yield model.

Definition 2 (Continued) Intuitively, we expect that all known relationships between rainfall and yield would have been factored into the ideal insurance policy (i.e. compensated). Thus we suspect that: 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘= 𝜎 𝜀 𝑖 2 𝜎 𝑓(𝑤) 2 + 𝜎 𝜀 𝑖 2 Indeed, using similar methods and assumptions as above we arrive at our intuition.

Definition 2 (Continued) To increase precision, we only consider income due to contribution of rainfall and unknown factors (error terms). Denote this income as 𝑀 𝑅𝑖 , and the corresponding yield 𝑦 𝑅𝑖 Under 𝑛 insurance policies, 𝑦 𝑅𝑖 = 𝜇 𝑖 + 𝑓 𝑤 +𝑔 ∙ + 𝜀 𝑖 =𝑓 𝑤 + 𝜀 𝑖 𝑀 𝑅𝑖 =𝑝𝐴 𝑦 𝑅𝑖 + 𝑛 𝐼 𝑤 − 𝑞 𝐸 𝑀 𝑅𝑖 =𝑝𝐴𝐸[𝑓 𝑤 ] 𝑉𝑎𝑟 𝑀 𝑅𝑖 = 𝐴 2 𝑝 2 𝜎 𝑦 𝑅𝑖 2 + 𝑛 2 𝜎 𝐼(𝑤) 2 + 2𝐴𝑛𝑝 𝐶𝑜𝑣( 𝑦 𝑅𝑖 , 𝐼 𝑤 )

Definition 2 (Continued) With CARA preferences: 𝑈 𝑀 𝑅𝑖 =𝐸 𝑀 𝑅𝑖 − 1 2 ∅𝑉𝑎𝑟 𝑀 𝑅𝑖 = 𝑝𝐴 𝛽 𝑖 𝐸[𝑓 𝑤 ] − 1 2 ∅ 𝐴 2 𝑝 2 𝜎 𝑦 𝑅𝑖 2 + 𝑛 2 𝜎 𝐼(𝑤) 2 + 2𝐴𝑛𝑝 𝐶𝑜𝑣( 𝑦 𝑅𝑖 , 𝐼 𝑤 )

Definition 2 (Continued) Maximizing utility by solving for 𝑛 ∗ : 𝜕(𝑈( 𝑀 𝑅𝑖 ) 𝜕𝑛 =0− 1 2 ∅ 0+ 2 𝑛 ∗ 𝜎 𝐼(𝑤) 2 + 2𝐴𝑝 𝐶𝑜𝑣 𝑦 𝑅𝑖 , 𝐼 𝑤 =0 𝜕 2 (𝑈( 𝑀 𝑅𝑖 ) 𝜕 𝑛 2 =2 𝜎 𝐼(𝑤) 2 >0 ∀ 𝑛 ϵ 𝐙 0 + 𝑛 ∗ = − 𝐴𝑝 𝐶𝑜𝑣 𝑦 𝑅𝑖 , 𝐼 𝑤 𝜎 𝐼(𝑤) 2 Again, we ignore discretized purchase requirements in reality.

Definition 2 (Continued) Under this definition and the expression for 𝑛 ∗ : 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘= 𝜎 𝑀 𝑅𝑖 2 | 𝑢𝑛𝑑𝑒𝑟 𝑛 ∗ 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝜎 𝑀 𝑅𝑖 2 | 𝑢𝑛𝑑𝑒𝑟 0 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 = 𝐴 2 𝑝 2 𝜎 𝑦 𝑅𝑖 2 + (𝑛 ∗ ) 2 𝜎 𝐼 𝑤 2 + 2𝐴 𝑛 ∗ 𝑝 𝐶𝑜𝑣 𝑦 𝑅𝑖 , 𝐼 𝑤 𝐴 2 𝑝 2 𝜎 𝑦 𝑅𝑖 2 =1+ 𝐴 2 𝑝 2 𝜎 𝐼(𝑤) 2 [ 𝐶𝑜𝑣 𝑦 𝑅𝑖 , 𝐼 𝑤 ] 2 𝐴 2 𝑝 2 𝜎 𝑦 𝑅𝑖 2 [ 𝜎 𝐼 𝑤 2 ] 2 + −2 𝐴 2 𝑝 2 [ 𝐶𝑜𝑣 𝑦 𝑅𝑖 , 𝐼 𝑤 ] 2 𝐴 2 𝑝 2 𝜎 𝑦 𝑅𝑖 2 𝜎 𝐼(𝑤) 2 =1− 𝐶𝑜𝑣 𝑦 𝑅𝑖 , 𝐼 𝑤 𝜎 𝑦 𝑅𝑖 𝜎 𝐼(𝑤) 2 =1− 𝜌 𝑦 𝑅𝑖 , 𝐼(𝑤) 2

Definition 2 (Continued) It is not unreasonable to assume that all known relationships are modeled into the insurance. Taking the clue from the first definition, we set : 𝐼 𝑤 =𝑎𝑓 𝑤 +𝑏. 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘= 1− 𝐶𝑜𝑣 𝑦 𝑅𝑖 , 𝐼 𝑤 𝜎 𝑦 𝑅𝑖 𝜎 𝐼 𝑤 2 =1− 𝐶𝑜𝑣 𝑓 𝑤 + 𝜀 𝑖 , 𝑎𝑓 𝑤 +𝑏 𝜎 𝑓 𝑤 + 𝜀 𝑖 𝜎 𝑎𝑓 𝑤 +𝑏 2 = 1− 𝑎𝐶𝑜𝑣 𝑓 𝑤 , 𝑓 𝑤 𝑎 𝜎 𝑓 𝑤 2 + 𝜎 𝜀 𝑖 2 𝜎 𝑓 𝑤 2 =1− 𝜎 𝑓 𝑤 2 𝜎 𝑓 𝑤 2 + 𝜎 𝜀 𝑖 2 𝜎 𝑓 𝑤 2 = 1− 𝜎 𝑓 𝑤 𝜎 𝑓 𝑤 2 + 𝜎 𝜀 𝑖 2 2 =1− 𝜎 𝑓 𝑤 2 𝜎 𝑓 𝑤 2 + 𝜎 𝜀 𝑖 2 = 𝜎 𝜀 𝑖 2 𝜎 𝑓(𝑤) 2 + 𝜎 𝜀 𝑖 2 Hence we see that we are back with our intuition.

Definition 2 - Assumptions The above assumes: No interactions between rainfall and other explanatory variables in the model. CARA utility functions Ability to purchase non-discrete amounts of insurance policies Although subject to the stricter assumptions as the first definition, we now know that the minimum basis risk is 0.

Comparison of the 2 Definitions   Definition 1 Definition 2 Pros Probably more relevant to farmers Clear benchmark: 0 Cons No clear benchmark Ignores overall impact on income

Reality Check If we had defined basis risk to be risk from understood exposure to rainfall, we expect that an insurance model that incorporates this understood exposure to reduce such rainfall risks to zero. 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑖𝑛𝑐𝑜𝑚𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑢𝑛𝑑𝑒𝑟𝑠𝑡𝑜𝑜𝑑 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑢𝑛𝑑𝑒𝑟 𝑛 ∗ 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑖𝑛𝑐𝑜𝑚𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑢𝑛𝑑𝑒𝑟𝑠𝑡𝑜𝑜𝑑 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑢𝑛𝑑𝑒𝑟 0 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 = 𝜎 𝑀 𝑖 𝑑𝑢𝑒 𝑡𝑜 𝑢𝑛𝑑𝑒𝑟𝑠𝑡𝑜𝑜𝑑 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 2 | 𝑢𝑛𝑑𝑒𝑟 𝑛 ∗ 𝑖𝑛𝑠𝑢𝑟𝑎𝑛𝑐𝑒 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠 𝜎 𝑀 𝑖 𝑑𝑢𝑒 𝑡𝑜 𝑢𝑛𝑑𝑒𝑟𝑠𝑡𝑜𝑜𝑑 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 2 | 𝑢𝑛𝑑𝑒𝑟 0 𝑝𝑜𝑙𝑖𝑐𝑖𝑒𝑠

Reality Check Using similar methods as above, defining the income due to understood rainfall as 𝑦 𝑅𝑖 =𝑓 𝑤 , we arrive at: 𝐵𝑎𝑠𝑖𝑠 𝑅𝑖𝑠𝑘= 1− 𝐶𝑜𝑣 𝑓 𝑤 , 𝐼 𝑤 𝜎 𝑓 𝑤 𝜎 𝐼 𝑤 2 =0 when we set : 𝐼 𝑤 =𝑎𝑓 𝑤 +𝑏.

Moving Forward – Empirics A. Measure Historical basis risk using Definitions 1 and 2 Need data on rainfall and yields APHRODITE data set for rainfall data “India Agricultural and Climate Data Set” (IACDS) prepared by Apurva Sanghi, K.S. Kavi Kumar, and James W. McKinsey, Jr. * Yield modeling With and without interactions Begin with basic forms, then try parametric modeling *The IACDS provides information on agricultural yields and prices for 20 major and minor crops, for each of 270 districts in India, from 1956 to 1987. Based on this information, an area-weighted average revenue per unit area was calculated as a proxy for farmer’s revenue per unit area.

Moving Forward – Empirics B. Model Historical Basis Risk using current insurance models Start with unitary period model (definitions 1 and 2) Allow for non CARA utility and test robustness of current definitions. Using the growth phase policies from the term sheets, estimate historical basis risk under Ideal 𝑛 ∗ Discretized ceiling / floor of 𝑛 ∗ Once theory arrives incorporate multiple phases Compare with basis risk minimizing model of insurance

Moving Forward – Empirics C. Model Historical Basis Risk using new insurance models Allow for assumption of non-optimal 𝑛 ∗ due to demand side factors, test robustness of definitions with respect to 𝑛 ∗ Allow for non CARA utility

Summary Stats for Data Statistics / States State 1 State 2 State 3 Mean Rainfall   Variance of Rainfall State Mean / Country Mean State Variance / Country Variance Mean Monsoon Rainfall (Area weighted) Mean Monsoon Rainfall (Population weighted) Variance of Monsoon Rainfall Monsoon Start Period Monsoon End Period Major Crops Average Yield (area weighted)

Models with no Covariates and no Interactions   Model 1 Model 2 Model 3 Model 4 f(w) Linear Quadratic Parametric g(x) Omitted h(w,x) R^2 Basis Risk Definition 1 – n=0 Basis Risk Definition 2 – n=0 Historic Insurance Specification Basis Risk Definition 1 – continuous n* Basis Risk Definition 2 – continuous n* Basis Risk Definition 1 –discrete n* Basis Risk Definition 2 – discrete n* Alternative Insurance Specification 1 phases 3 phases

Models with Covariates and no Interactions   Model 1 Model 2 Model 3 Model 4 f(w) Linear Quadratic g(x) h(w,x) Omitted R^2 Basis Risk Definition 1 – n=0 Basis Risk Definition 2 – n=0 Historic Insurance Specification Basis Risk Definition 1 – continuous n* Basis Risk Definition 2 – continuous n* Basis Risk Definition 1 –discrete n* Basis Risk Definition 2 – discrete n* Alternative Insurance Specification 1 phases 3 phases

Models with Covariate and Interactions   Model 1 Model 2 Model 3 Model 4 f(w) Linear Quadratic g(x) h(w,x) R^2 Basis Risk Definition 1 – n=0 Basis Risk Definition 2 – n=0 Historic Insurance Specification Basis Risk Definition 1 – continuous n* Basis Risk Definition 2 – continuous n* Basis Risk Definition 1 –discrete n* Basis Risk Definition 2 – discrete n* Alternative Insurance Specification 1 phases 3 phases

Summary 2 Definitions of basis risk Moving Forward: Empirics

Acknowledgments Professor Tobacman for his unending patience, understanding, guidance and support. Professor Cole, Daniel Stein and many more with providence of key data