Comparison of Different Water Supply Risk Management Tools for Irrigators: Option Contracts and Insurance

Abstract

Irrigators must cope with the risk of not having enough water to meet crop demands. There are different tools for managing this risk, including water market mechanisms and insurance. Given the choice, farmers will opt for the tool that offers the greatest positive change in expected utility. This paper presents a theoretical assessment of farmers’ expected utility for two different water option contracts and a drought insurance policy. We analyze the conditions that determine farmers’ preferences for these instruments and perform a numerical application to a water-stressed Spanish region. Results show that farmers’ willingness to pay for the considered risk management tools are greater than the preliminary estimates of these instruments costs. This suggests that option contracts and insurance may help farmers manage water supply availability risks.

Appendices

Appendix 1: Farmers’ Expected Utility with No Risk Management Tool

$$\begin{aligned} EU_0 \left( {\widetilde{\uppi }} \right)= & {} \mathop \int \nolimits _0^{\overline{w}} U\left( {\widetilde{\pi } _0 }\right) f\left( w \right) dw\nonumber \\= & {} \mathop \int \nolimits _0^{\overline{w}}\left[ {1-e^{-r\left( {a+bw} \right) }} \right] f\left( w \right) dw\nonumber \\= & {} \mathop \int \nolimits _0^{\overline{w}} f\left( w \right) dw - \mathop \int \nolimits _0^{\overline{w}} e^{-ra} e^{-rbw} f\left( w \right) dw \nonumber \\= & {} 1-e^{-ra}\mathop \int \nolimits _0^{\overline{w}} e^{-rbw }f\left( w \right) dw \nonumber \\= & {} 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right) \end{aligned}$$

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Appendix 2: Expected Utility and Risk Premium with Option Contract (a)

$$\begin{aligned} EU_{opt_a } \left( {\widetilde{\pi } } \right)= & {} Z{\mathop {\int }\nolimits _{0}^{w_{g}}} \left[ {1-e^{-r\left( {a+bw_g -P_{opt_a } -P_e \left( {w_g -w} \right) } \right) }} \right] f\left( w \right) dw\nonumber \\&\quad +\,\left( {1-Z} \right) {\mathop {\int }\nolimits _{0}^{w_{g}}} \left[ {1-e^{-r\left( {a+bw-P_{opt_a } } \right) }} \right] f\left( w \right) dw\nonumber \\&\quad +\,{\mathop {\int }\nolimits _{w_{g}}^{\bar{w}}} \left[ {1-e^{-r\left( {a+bw-P_{opt_a } } \right) }} \right] f\left( w \right) dw\nonumber \\= & {} Z\gamma - Ze^{{ - r(a + bw_{g} - P_{{opt_{a} }} - P_{e} w_{g} )}} {\textit{LIMGF}}_{w} ( - rP_{e} )\nonumber \\&\quad +\,\gamma - e^{{ - r(a - P_{{opt_{a} }} )}} {\textit{LIMGF}}_{w} ( - rb) - Z\gamma \nonumber \\&\quad +\,Ze^{{ - r(a - P_{{opt_{a} }} )}} {\textit{LIMGF}}_{w} ( - rb) + (1 - \gamma ) - e^{{ - r(a - P_{{opt_{a} }} )}} {\textit{UIMGF}}_{w} ( - rb) \nonumber \\= & {} 1 - e^{{ - r(a - P_{{opt_{a} }} )}} ({\textit{UIMGF}}_{w} ( - rb) + {\textit{LIMGF}}_{w} ( - rb)) \nonumber \\&\quad -\,Ze^{{ - r(a + bw_{g} - P_{{opt_{a} }} - P_{e} w_{g} )}} {\textit{LIMGF}}_{w} (-rP_{e} )\nonumber \\&\quad +\,Ze^{{ - r(a - P_{{opt_{a} }} )}} {\textit{LIMGF}}_{w} (- rb) =1-e^{-r\left( {a-P_{opt_a } } \right) }{\textit{MGF}}_w \left( {-rb} \right) \nonumber \\&\quad -\,Ze^{-r\left( {a+bw_g -P_{opt_a } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) \nonumber \\&\quad +\,Ze^{-r\left( {a-P_{opt_a } } \right) }{\textit{LIMGF}}_w \left( {-rb} \right) \end{aligned}$$

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$$\begin{aligned} EU_0 \left( {\widetilde{\pi } } \right)= & {} EU_{opt_a } \left( {\widetilde{\pi } } \right) \\ 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} 1-e^{-r\left( {a-R_{opt_a } } \right) }{\textit{MGF}}_w \left( {-rb} \right) \end{aligned}$$

$$\begin{aligned}&\quad -Ze^{-r\left( {a+bw_g -R_{opt_a } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) +\,Ze^{-r\left( {a-R_{opt_a } } \right) }{\textit{LIMGF}}_w \left( {-rb} \right) \nonumber \\&\quad -\,e^{-ra}{\textit{MGF}}_w \left( {-rb} \right) = -e^{-ra}e^{rR_{opt_a } }\left[ {\textit{MGF}}_w \left( {-rb} \right) +Ze^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) \right] \nonumber \\&\quad -\,{\textit{ZLIMGF}}_w \left( {-rb} \right) \nonumber \\&R_{opt_a } = \frac{1}{r} ln\left( {\frac{{\textit{MGF}}_w \left( {{-}rb} \right) }{\left( {1{-}Z} \right) {\textit{MGF}}_w \left( {{-}rb} \right) +\hbox {Z}\left[ {e^{{-}r\left( {b{-}P_e } \right) w_g }{\textit{LIMGF}}_w \left( {{-}rP_e } \right) +{\textit{UIMGF}}_w \left( {{-}rb} \right) } \right] }} \right) \nonumber \\ \end{aligned}$$

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Appendix 3: Upper Incomplete Moment Generation Function (UIMGF)

We consider that variable $\widetilde{w}$ follows a gamma distribution $f(w)$:

$$\begin{aligned} f\left( w \right)= & {} \frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }w^{\alpha -1}e^{-\lambda w}\nonumber \\ {\textit{UIMGF}}_w \left( {-rb} \right)= & {} \mathop \int \nolimits _{w_g }^{\overline{w}} e^{-rbw}\frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }w^{\alpha -1}e^{-\lambda w} dw= \left[ {-\frac{\lambda ^{\alpha }w^{\alpha }E_{1-\alpha } \left( {\left( {\lambda +rb} \right) w} \right) }{{\Gamma }\left( \alpha \right) }} \right] _{w_g }^{\overline{w}} \end{aligned}$$

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$E$ is an exponential integral function.

$$\begin{aligned} E_n \left( z \right)= & {} z^{n-1}{\Gamma }\left( {1-n,z} \right) \\ E_{1-\alpha } \left( {\left( {\lambda +rb} \right) w} \right)= & {} \left( {\left( {\lambda +rb} \right) w} \right) ^{-\alpha }{\Gamma }\left( {\alpha ,\left( {\lambda +rb} \right) w} \right) \end{aligned}$$

So, the expression of ${{\textit{UIMGF}}}_{w}(-{rb})$ is

$$\begin{aligned} {\textit{UIMGF}}_w \left( {-rb} \right)= & {} -\frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }\left[ {w^{\alpha }\left( {w\left( {\lambda +rb} \right) } \right) ^{-\alpha }{\Gamma }\left( {\alpha ,\left( {\lambda +rb} \right) w} \right) } \right] _{w_g }^{\overline{w}}\nonumber \\= & {} -\frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }\left[ {\left( {\lambda +rb} \right) ^{-\alpha }{\Gamma }\left( {\alpha ,\left( {\lambda +br} \right) w_g } \right) } \right] _{w_g }^{\overline{w}}\nonumber \\= & {} \frac{\lambda ^{\alpha }}{\left( {\lambda +rb} \right) ^{\alpha }}\left[ {\frac{-{\Gamma }\left( {\alpha ,\left( {\lambda +rb} \right) w_g } \right) }{{\Gamma }\left( \alpha \right) }} \right] _{w_g }^{\overline{w}}\nonumber \\= & {} {\textit{MGF}}_w \left( {-rb} \right) \left( {\left[ {Q\left( {\alpha ,\left( {\lambda +rb} \right) w_g } \right) } \right] -\left[ {Q\left( {\alpha ,\left( {\lambda +rb} \right) {\overline{w}}} \right) } \right] } \right) \end{aligned}$$

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$Q$(.) is the regularized gamma function, whose domain is [0,1].

Appendix 4: Expected Utility and Risk Premium with Option Contract (b)

If we assume that irrigators will always exercise the option at the maturity date when their water allotment is below $w_{g}$, their profit function is

$$\begin{aligned} \widetilde{\pi } _{opt_b } \left( w \right)= & {} a+b\widetilde{w}-P_{opt_b } \quad if\quad \widetilde{w}\ge w_g\nonumber \\ \widetilde{\pi } _{opt_b } \left( w \right)= & {} a+b\hbox {w}_g -P_{opt_b } -P_e \left( {w_g -\widetilde{w}} \right) \quad if \quad \widetilde{w}<w_g \end{aligned}$$

$$\begin{aligned} EU_{opt_b } \left( {\widetilde{\pi } } \right)= & {} \mathop \int \nolimits _0^{w_g } \left[ {1-e^{-r\left( {a+bw_g -P_{opt_b } -P_e \left( {w_g -w} \right) } \right) }} \right] f\left( w \right) dw\nonumber \\&+\,\mathop \int \nolimits _{w_g }^{\overline{w}} \left[ {1-e^{-r\left( {a+bw-P_{opt_b } } \right) }} \right] f\left( w \right) dw\nonumber \\= & {} \gamma -e^{-r\left( {a+bw_g -P_{opt_b } -P_e w_g } \right) }\mathop \int \nolimits _0^{w_g } e^{-rP_e w}f\left( w \right) dw\nonumber \\&+\,\left( {1-\gamma } \right) -e^{-r\left( {a-P_{opt_b } } \right) }\mathop \int \nolimits _{w_g }^{\overline{w}} e^{-rbw} f\left( w \right) dw\nonumber \\= & {} 1-e^{-r\left( {a+bw_g -P_{opt_b } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) -e^{-r\left( {a-P_{opt_b } } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ \end{aligned}$$

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$$\begin{aligned} EU_0 \left( {\widetilde{\pi } } \right)= & {} EU_{opt_b } \left( {\widetilde{\pi } } \right) \nonumber \\ 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} 1-e^{-r\left( {a+bw_g -R_{opt_b } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) \nonumber \\&\quad -e^{-r\left( {a-R_{opt_b } } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ -e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} -e^{-r\left( {a+bw_g -R_{opt_b } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) \nonumber \\&\quad -e^{-r\left( {a-R_{opt_b } } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ {\textit{MGF}}_w \left( {-rb} \right)= & {} e^{rR_{opt_b } }\left[ {e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) +{\textit{UIMGF}}_w \left( {-rb} \right) } \right] \nonumber \\ R_{opt_b }= & {} \frac{1}{r}\ln \left( {\frac{{\textit{MGF}}_w \left( {-rb} \right) }{e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) +{\textit{UIMGF}}_w \left( {-rb} \right) }} \right) \end{aligned}$$

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Appendix 5: Expected Utility and Risk Premium with Insurance

The farmers’ profit function in this case is

$$\begin{aligned} \widetilde{\pi } _{ins} \left( w \right)= & {} a+b\widetilde{w} -P_{ins}\quad if\quad \widetilde{w}\ge w_g\\ \widetilde{\pi } _{ins} \left( w \right)= & {} {a+b\hbox {w}_{g}} -{P}_{ins} \quad if \quad \widetilde{w}<w_g \end{aligned}$$

$$\begin{aligned} EU_{ins} \left( {\widetilde{\pi } } \right)= & {} \mathop \int \nolimits _0^{w_g } 1-e^{-r\left( {a+bw_g -P_{ins} } \right) }f\left( w \right) dw\nonumber \\&+\,\mathop \int \nolimits _{w_g }^{\overline{w}} 1-e^{-r\left( {a+bw-P_{ins} } \right) }f\left( w \right) dw\nonumber \\= & {} \gamma -\gamma e^{-r\left( {a+bw_g -P_{ins} } \right) }+\left( {1-\gamma } \right) -e^{-r\left( {a-P_{ins} } \right) }{\textit{UIMGF}}_w \left( {-rb}\right) \nonumber \\= & {} 1-\gamma e^{-r\left( {a+bw_g -P_{ins} } \right) }-e^{-r\left( {a-P_{ins} } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \end{aligned}$$

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$$\begin{aligned} EU_0 \left( {\widetilde{\pi } } \right)= & {} EU_{ins} \left( {\widetilde{\pi } } \right) \nonumber \\ 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} 1-\gamma e^{-r\left( {a+bw_g -R_{ins} } \right) }-e^{-r\left( {a-R_{ins} } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ R_{ins}= & {} \frac{1}{ r} ln \left( {\frac{{\textit{MGF}}_w \left( {-rb} \right) }{\gamma e^{-rbw_g }+ {\textit{UIMGF}}_w \left( {-rb} \right) }} \right) \end{aligned}$$

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Appendix 6: Comparison of Instruments

1.1 Comparison Between the Two Option Contracts (a) and (b)

We first compare the risk premiums and then assess the conditions that make one instrument more attractive to the farmer than the other. $R_{opt_b}$ will be greater than $R_{opt_a}$ for all cases. Intuitively, the conclusion is the same, as option contract (b) offers more guarantees than contract (a), and the probability of the farmer being able to purchase the optioned volume at the maturity date is greater.

If $R_{opt_b } >R_{opt_a }$, then

$$\begin{aligned}&e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) + {\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\&\quad <\left( {1-Z} \right) {\textit{MGF}}_w \left( {-rb} \right) +\hbox {Z }\left[ {e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) +{\textit{UIMGF}}_w \left( {-rb} \right) } \right] \nonumber \\&e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) + {\textit{UIMGF}}_w \left( {-rb} \right) < {\textit{MGF}}_w \left( {-rb} \right) \end{aligned}$$

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For $R_{opt_b }$ to be positive, the above expression must hold (as the numerator of the logarithm on the right side of the expression has to be greater than the denominator on the left side).

We calculate the conditions that determine whether farmers will take out an option contract or insurance thus:

$$\begin{aligned} P_{opt_b } - P_{opt_a } <\frac{1}{r} ln\left( {\frac{D_{opt_a } }{D_{opt_b } }} \right) \end{aligned}$$

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If $P_{opt_b } <P_{opt_a } +\frac{1}{r} ln\left( {\frac{D_{opt_a } }{D_{opt_b } }} \right)$, they would choose the option contract (b); and if $P_{opt_b } >P_{opt_a } +\frac{1}{r }\ln \left( {\frac{D_{opt_a } }{D_{opt_b } }} \right)$, they would purchase the option contract (a) $(D_{opt_a }$ is always higher than $D_{opt_b } )$. The farmer would be indifferent to the two if $P_{opt_b } =P_{opt_a } +\frac{1}{r} ln\left( {\frac{D_{opt_a } }{D_{opt_b } }} \right)$.