Insurance and Forest Rotation Decisions Under Storm Risk

Abstract

Forests are often threatened by storms; and such a threat is likely to rise due to climate change. Using private forest insurance as a vehicle to fund resilience and adaptation emerge as a policy recommendation. Hence the forest owners would have the possibility to consider insurance when defining their forest management practices. In this context, we analyze the impact of the forest owner’s insurance decision on forest management under storm risk. We extend the Faustmann optimal rotation model under risk, first, considering the forest owner’s risk preferences, and second, integrating the decision of insurance. With this analytical model, we show that as the insurance coverage increases, the rotation length increases independently of the forest owner’s risk aversion. In addition, we identify some cases where it may be optimal for the forest owner to not adopt insurance contract. Finally, we prove that a public transfer, reducing the insurance premium, may encourage the forest owner to insure.

Appendices

The Faustmann Value

As we assume that the storm occurs independently of one another and randomly in time, we deduce the following expression for the land value:

$$\begin{aligned} J_F = E\left[\sum _{i=1}^{\infty } e^{-\delta ({\tau }_1+{\tau }_2+\cdots +{\tau }_i)} {\mathcal {Y}}_i\right] = \sum _{i=1}^{\infty } \prod _{j=1}^{i-1} E\left[e^{-\delta {\tau }_ j}]\cdot E[e^{-\delta {\tau }_i} {\mathcal {Y}}_i\right]= \frac{E[e^{-\delta \tau } {\mathcal {Y}}]}{1 - E[e^{-\delta \tau }]} \end{aligned}$$

(3)

From the definition of $\tau$ we deduce the expectation $\displaystyle E[e^{-\delta \tau }]$:

$$\begin{aligned} \displaystyle E[e^{-\delta \tau }]= \int _{t_L}^{T} e^{-\delta \tau } dF(\tau -t_L) + e^{-\delta T} (1-F(T-t_L))= \frac{\lambda + \delta e^{-(\delta +\lambda )(T-t_L)}}{\delta +\lambda } e^{-\delta t_L} \end{aligned}$$

We deduce the expected net economic return $\displaystyle E[e^{-\delta \tau }{\mathcal {Y}}]$ actualized at initial time:

$$\begin{aligned}&\frac{1-e^{-\delta {\underline{t}}}}{\delta } u(-{\mathcal {P}}) +\frac{e^{-\delta {\underline{t}}}-e^{-\delta t_L}}{\delta } u(h-{\mathcal {P}}) + \frac{e^{-\delta t_L}-e^{-(\delta +\lambda )T+\lambda t_L}}{\delta +\lambda } u(h-{\mathcal {P}}) \\&\quad + \int _{t_L}^T (E[u(V_1(\theta ,\tau )+ {\mathcal {I}}(\theta _\tau ,\tau )-c_1- C_n(\theta _\tau ,\tau ))] ) e^{-\delta \tau } dF(\tau -t_L) \\&\quad + u(V(T)-c_1) e^{-\delta T} (1-F(T-t_L)) \end{aligned}$$

and hence the result.

Proof of Proposition 1

To lighten the presentation, in some calculus, we omit the $\lambda , \delta$ dependencies in functions $a_1, b$ and denote $a_T$ (resp. $b_T$) the partial derivative of $a_1$ (resp. b) with respect to T.

From (1), differentiating $J_F$ with respect to the rotation age T gives the following first-order condition:

$$\begin{aligned} J_T = \left[ \frac{W_F(0,T) + \frac{a_1(\delta ,T)}{ \delta } (u(h-{\mathcal {P}}) -u(-{\mathcal {P}}))}{b(\delta ,T)} \right] _T= 0 \end{aligned}$$

We consider the case $h> {\mathcal {P}}$ (the other case is simpler). From the first-order condition $J_T = 0$ we deduce:

$$\begin{aligned} W_F(0,T)_Tb(T) - W_F(0,T) b_T(T) + (a_T(T) b(T) - a_1(T) b_T(T)) \frac{u(h-{\mathcal {P}}) - u({\mathcal {P}})}{\delta }=0 \end{aligned}$$

(4)

Differentiating $J_T=0$ with respect to ${\mathcal {P}}$ yields: $\displaystyle J_{TT} T_{{\mathcal {P}}} + J_{T{\mathcal {P}}}=0$. From $J_{TT} < 0$, $T_{{\mathcal {P}}}$ and $J_{T{\mathcal {P}}}$ have the same sign. $J_{T{\mathcal {P}}}$ is proportional (with the same sign) to the sum of:

$$\begin{aligned} A_{{\mathcal {P}}} = W_F(0,T)_{T{\mathcal {P}}} b(T) - W_F(0,T)_{{\mathcal {P}}}b_T(T) { \text{ and } (a_T(T) b(T) - a_1(T) b_T(T)) (u'(-{\mathcal {P}}) - u'(h-{\mathcal {P}}) )} \end{aligned}$$

$\displaystyle a_T(T) b(T) - a_1(T) b_T(T)$ and $u'(-{\mathcal {P}}) - u'(h-{\mathcal {P}})$ are positive. Let $\alpha (t)=E[1-\theta _t]$ the expected proportion of survival trees. Concerning $A_{{\mathcal {P}}}$, in a first step we consider a forest owner that is not risk averse for positive income :

$$\begin{aligned} W_F(0,T)_{{\mathcal {P}}} =&\frac{\lambda }{{\overline{P}}} \int _{t_L}^T (1-\alpha (\tau )) V(\tau ) e^{(\lambda +\delta )(T-\tau )}d\tau \\ W_F(0,T)_{T{\mathcal {P}}} =&\frac{\lambda }{{\overline{P}}} (1-\alpha (T)) V(T) + (\lambda +\delta ) W_F(0,T)_{{\mathcal {P}}} \end{aligned}$$

From $b_T(T) = \delta +(\delta +\lambda ) b(T)$ we deduce:

$$\begin{aligned} A_{{\mathcal {P}}} = &\frac{\lambda }{{\overline{P}}} (1-\alpha (T)) V(T) b(T) - \delta W_F(0,T)_{{\mathcal {P}}} \\ = &\frac{\lambda }{{\overline{P}}} [(1-\alpha (T)) V(T) b(T)-\delta \int _{t_L}^T (1-\alpha (\tau )) V(\tau ) e^{(\lambda +\delta )(T-\tau )}d\tau ] \end{aligned}$$

Let $\displaystyle \gamma (t) = \frac{\lambda }{{\overline{P}}} (1-\alpha (t)) V(t)$ then:

$$\begin{aligned} A_{{\mathcal {P}}} = \gamma (T) b(T)-\delta \int _{t_L}^T \gamma (\tau ) e^{(\lambda +\delta )(T-\tau )}d\tau \end{aligned}$$

Moreover, studying the behavior of $\displaystyle A_{{\mathcal {P}}}(t)= \gamma (t) b(t)-\delta \int _{t_L}^t \gamma (\tau ) e^{(\lambda +\delta )(t-\tau )}d\tau$, then:

$$\begin{aligned} A_{{\mathcal {P}}}'(t) = &\gamma '(t) b(t) + \gamma (t) b_T(t) - \delta \gamma (t) -(\lambda +\delta )(\gamma (t)b(t)- A_{{\mathcal {P}}}(t)) \\ = &\gamma '(t) b(t) + (\lambda +\delta ) A_{{\mathcal {P}}}(t) \end{aligned}$$

From $\alpha '(t) < 0$ we deduce $\gamma '(t) > 0$, moreover from $A_{{\mathcal {P}}}(0) =0$ we deduce that $A_{{\mathcal {P}}}(T) > 0$. In a second step, we consider a forest owner that is risk averse for positive income. In this case we have to consider the function $\gamma$ defined by: $\displaystyle \gamma (t) = \frac{\lambda }{{\overline{P}}} E[u'(V_1(\theta _t,t)+{\mathcal {I}}(\theta _t,t)-c_1- C_n(\theta _t,t)) \theta _t] V(t)$. Hence from the increasing of $\theta _t$ with respect to time t, we deduce that for a sufficiently weak risk averse forest owner for positive income, i.e. the absolute risk aversion is lower than the relative increasing of final revenue: $\displaystyle \frac{-u''}{u'} \le \frac{V'}{V}$, $\gamma$ is increasing and $A_{{\mathcal {P}}}(T) >0$ .

Proof of Proposition 2

We consider the derivative of the Faustmann Value $J_F$ with respect to insurance premium ${\mathcal {P}}$.

For a risk averse forest owner, we have:

$$\begin{aligned} \frac{\partial J_F}{\partial {\mathcal {P}}}&= -\frac{a_0(\delta ,T) u'(-{\mathcal {P}}) + a_1(\delta ,T) u'(h-{\mathcal {P}})}{\delta b(\delta ,T)} \\&\quad +\frac{1}{(1+l_f)(1+m) \delta _A} \frac{b(\delta _A,T)}{b(\delta ,T)} \frac{\int _{t_L}^T E[{u'(V_1(\theta ,\tau )+\mathcal I(\theta _\tau ,\tau )-c_1- C_n(\theta _\tau ,\tau ))} L(\theta ,\tau )] e^{(\lambda +\delta )(T-\tau )} d \tau }{\int _{t_L}^T E[L(\theta _\tau ,\tau )] e^{(\lambda +\delta _A)(T-\tau )} d \tau } \end{aligned}$$

From the concavity of the utility u, and using $-{\mathcal {P}}< h -{\mathcal {P}} < V_1(\theta ,\tau )+{\mathcal {I}}(\theta _\tau ,\tau )-c_1$ then:

$$\begin{aligned} \frac{\partial J_F}{\partial {\mathcal {P}}}&< \frac{1}{\delta } (-1 + \frac{1}{(1+l_f)(1+m)}\frac{\delta }{\delta _A} \frac{b(\delta _A,T)}{b(\delta ,T)} \frac{\int _{t_L}^T E[L(\theta _\tau ,\tau )] e^{(\lambda +\delta )(T-\tau )} d \tau }{\int _{t_L}^T E[L(\theta _\tau ,\tau )] e^{(\lambda +\delta _A)(T-\tau )} d \tau })u'(h-{\mathcal {P}}) \\&< \frac{1}{\delta } (-1 + \frac{1}{(1+l_f)(1+m)} \frac{\Gamma (\delta ,T)}{ \Gamma (\delta _A,T)})u'(h-{\mathcal {P}}) \end{aligned}$$

(i) If $\delta _A=\delta$ then $\displaystyle \frac{\partial J_F}{ \partial {\mathcal {P}}}< \frac{1}{\delta } (-1 + \frac{1}{ (1+l_f)(1+m)}) u'(h-{\mathcal {P}}) < 0$, hence the result.

(ii) With $\Gamma$ is decreasing with respect to $\delta$, if $\delta > \delta _A$ then $\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}} < 0$. But if $\delta$ is sufficiently lower than $\delta _A$, $\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}}$ may be equal to 0.

For a risk neutral forest owner, we have:

$$\begin{aligned} \frac{\partial J_F}{\partial {\mathcal {P}}}&= \frac{1}{\delta } \left(-1+ \frac{1}{(1+l_f)(1+m)} \frac{\Gamma (\delta ,T)}{\Gamma (\delta _A,T)}\right) \end{aligned}$$

(iii) If $\Gamma$ is decreasing with respect to $\delta$:

• if $\delta > \delta _A$ then $\displaystyle \frac{\partial J_F}{ \partial {\mathcal {P}}} <0$ and $\xi =0$.

• if $\delta$ is sufficiently lower than $\delta _A$ then $\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}}>0$ and $\xi =1$.

Similar results are obtained for $\Gamma$ increasing with respect to $\delta$ and the opposite relationships between $\delta$ and $\delta _A$.

Proof of Proposition 3

(i) For a risk averse forest owner, if $\delta _A=\delta$ then:

$$\begin{aligned} \frac{\partial J_F}{\partial {\mathcal {P}}} =&\frac{1}{ \delta } [- \frac{a_0(\delta ,T) u'(-{\mathcal {P}}) + a_1(\delta ,T) u'(h-{\mathcal {P}})}{b(\delta ,T)} \\&+\frac{\int _{t_L}^T E[u'(V_1(\theta ,\tau )+ {\mathcal {I}}(\theta _\tau ,\tau )-c_1- C_n(\theta _\tau ,\tau )) L(\theta ,\tau )] e^{(\lambda +\delta )(T-\tau )} d \tau }{(1-\gamma ) (1+l_f)(1+m) \int _{t_L}^T E[L(\theta _\tau ,\tau )] e^{(\lambda +\delta )(T-\tau )} d \tau }] \end{aligned}$$

$\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}}$ is positive for sufficiently large value of $\gamma$ and is negative for $\gamma =0$. Hence we deduce that for intermediate $\gamma$, $\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}}$ may be equal to 0, hence a non null premium ${\mathcal {P}}$ will be optimal. More precisely $\displaystyle \delta \frac{\partial J_F}{\partial {\mathcal {P}}} = -B({\mathcal {P}}) +\frac{C({\mathcal {P}})}{1-\gamma }$ with $\displaystyle B({\mathcal {P}}) = \frac{a_0(\delta ,T)u'(-{\mathcal {P}}) +a_1(\delta ,T) u'(h-{\mathcal {P}})}{b(\delta ,T)}$ and $\displaystyle C({\mathcal {P}}) =\frac{\int _{t_L}^T E[u'(V_1(\theta ,\tau )+{\mathcal {I}}(\theta _\tau ,\tau )-c_1- C_n(\theta _\tau ,\tau )) L(\theta ,\tau )] e^{(\lambda +\delta )(T-\tau )} d \tau }{(1+l_f)(1+m) \int _{t_L}^T E[L(\theta _\tau ,\tau )] e^{(\lambda +\delta )(T-\tau )} d \tau } \le B({\mathcal {P}})$.

We deduce that for $\gamma \in ]{{\underline{\gamma }}}=1-\max _{\mathcal P} \frac{C({\mathcal {P}})}{B({\mathcal {P}})}, {{\overline{\gamma }}}= 1-\min _{{\mathcal {P}}}\frac{C({\mathcal {P}})}{B({\mathcal {P}})}[$, it exists ${\mathcal {P}} >0$ such that $\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}}=0$.

For a risk neutral forest owner:

$$\begin{aligned} \frac{\partial J_F}{\partial {\mathcal {P}}}&= \frac{1}{\delta } \left(-1+ \frac{1}{ (1-\gamma )(1+l_f)(1+m)} \frac{\Gamma (\delta ,T)}{ \Gamma (\delta _A,T)}\right) \end{aligned}$$

(ii) If $\displaystyle \gamma < \gamma _0$ and $\Gamma$ decreasing with respect to $\delta$:

• if $\delta > \delta _A$ then $\displaystyle \frac{\partial J_F}{ \partial {\mathcal {P}}} <0$ and $\xi =0$.

• if $\delta$ is sufficiently lower than $\delta _A$ then $\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}}>0$ and $\xi =1$.

  (iii) If $\displaystyle \gamma > \gamma _0$ and $\Gamma$ decreasing with respect to $\delta$:

• if $\delta$ is sufficiently larger than $\delta _A$ then $\displaystyle \frac{\partial J_F}{\partial {\mathcal {P}}} <0$ and $\xi =0$.

• if $\delta < \delta _A$ then $\displaystyle \frac{\partial J_F}{ \partial {\mathcal {P}}}>0$ and $\xi =1$.