Insurers and Insured Individuals Interaction as Basis for Persistent Agricultural Production Process

Likehood of losses occurring

We consider a conventional approach to deal with individual losses (Falin & Falin, 2004):

$X=I.Y,$(1)

Where $I$ – indicator function of losses occurring, $Y$ – insurance compensation after losses occurring. The mean insurance compensation is mathematical expectation of random variable $X$:

$m=EX=P\left\{I=1\right\}\bullet EY$.

When crops insuring, it is proposed in (Baskakov, Кrylova, Selivanova & et al, 2016) to consider a drop in crop yield by share $\alpha$ exceeding its mean value in the particular area over the last five years as criterion of losses occurring $\left(I=1\right)$ is:

$\frac{\underset{K=i-5}{\overset{i-1}{\sum }}{U}_{K^{-5\bullet U_i}}^{}}{\underset{K=i-5}{\overset{i-1}{\sum }}{U}_K}>\alpha ,$ (2)

Where $U_i$ – crop yields in year i.

We note that crop yield $U_i$ depends on a lot of factors, some of them are accidental (rainfall, average daily temperature over the period of vegetation), although there are also determined factors (grade of quality, amount of used fertilizers etc.) (Posypanov, Dolgodvorov & Zherukov, 2016). Therefore, a crop yield can be viewed as a function dependent on determined and stochastic factors:

$U=f\left(A,B\right)$,

Where $A$ – a set of determined factors influencing on the crop yield, $B$– a set of stochastic factors effecting on the crop yield (Tashchiyan, Sushko & Grichin, 2015).

Assuming linearity of function $f$ and sufficient quantity of factors in set $B$, we can suggest in terms of the central limit theorem that random variable $U$ has normal distribution with the function of density:

$p_u\left(x\right)=\frac{1}{\sqrt{2\pi \sigma }}{e}^{-\frac{{\left(x-a\right)}^2}{2{\sigma }^2}}$,(3)

Where $a$ – mathematical expectation $U,\sigma$– standard deviation $U$.

Relying on the above-mentioned, $U$ is modeled according to distribution (3). The probability of event $I=1$ is calculated as (2). $$P\left\{I=1\right\}=P\left\{\left(1-\alpha \right)\underset{K=i-5}{\overset{i-1}{\sum }}{U}_K>5U_i\right\}=P\left\{\left(1-\alpha \right)\underset{K=i-5}{\overset{i-1}{\sum }}{U}_K-5U_i>0\right\}.$$(4)

Random variable $\xi =\left(1-\alpha \right)\underset{K=i-5}{\overset{i-1}{\sum }}{U}_k-5U_i$ has normal distribution with mathematical expectation $E_{\xi }=\left(1-\alpha \right)\cdot 5EU-5EU=-5\alpha EU$ and standard deviation ${\sigma }_{\xi }=\sqrt{{\left(1-\alpha \right)}^2\cdot 25{\sigma }_U^2+25{\sigma }_U^2}=5{\sigma }_U\cdot \sqrt{1+{\left(1-\alpha \right)}^2}$. Taking into account the result of (4), we obtain:

$P\left\{\left(1-\alpha \right)\underset{K=i-5}{\overset{i-1}{\sum }}{U}_K-5U_i>0\right\}=\mathrm{0,5}-\Phi \left(\frac{\alpha EU}{\sigma U\sqrt{1+(1+\alpha {)}^2}}\right),$(5)

Where $\mathrm{\Phi }\left(x\right)=\frac{1}{\sqrt{2\pi }}\underset{0}{\overset{x}{\int }}{e}^{-\frac{t^2}{2}}dt$ – Laplace’s function.

We point out that relation $\frac{{\sigma }_U}{EU}$ is variation coefficient $V$ of the crop yield. Hence, result (5) for $\alpha =\mathrm{0,3}$ is written as $P\left\{I=1\right\}=0.5-\mathrm{\Phi }\left(\frac{0.3}{V\sqrt{2.69}}\right)$.

Distribution of individual losses

We study the model of losses distribution. We proceed from the assumption that in case of an insured event losses are calculated as follows (Baskakov, Кrylova, Selivanova & et al, 2016; Lizunkov, Marchuk & Podzorova, 2015; Malushko, 2015; Kindaev & Moiseev, 2016):

$$Y=\frac{P}{10}\cdot S\cdot \left(\frac{\underset{K=i-5}{\overset{i-1}{\sum }}{U}_K}{5}-U_i\right),$$(6)

where $P$– mean farmers’ price per 1000 kilograms of a crop formed in the subject of the Russian Federation over a year before year i, when an insurance contract was signed according to the data of the Federal State Statistics Service; $S$– area under crop. Provided that $P$ and $S$ are determined, distribution of the value described by formula (4) is calculated as random variable $\frac{\underset{K=i-5}{\overset{i-1}{\sum }}{U}_K}{5}-U_i$. As all $U_i$ are similar independent random variables with distribution (3), $\frac{\underset{K=i-5}{\overset{i-1}{\sum }}{U}_K}{5}-U_i$ has also normal distribution with parameters $a=\frac{1}{5}\underset{K=i-5}{\overset{i-1}{\sum }}E{U}_k-E{U}_i=0$ and $\sigma =\sqrt{\frac{1}{25}\underset{K=i-5}{\overset{i-1}{\sum }}D{U}_k+DU\_i}={\sigma }_D\sqrt{\frac{\theta }{5}}$. The following parameters of normal distribution are obtained for $Y:a=0$, $\sigma \sqrt{\frac{6}{5}}\cdot \frac{P\cdot S}{10}\cdot {\sigma }_U$. The distribution function of random variable $X$ is written as equality (1):

$\begin{array}{c}{F}_X\left(x\right)=\mathrm{P}\{X\le x\}=\mathrm{P}\{I\cdot Y\le x\}=\mathrm{P}\{I\cdot Y\le x\mid I=0\}\cdot \mathrm{P}\{I=0\}+\mathrm{P}\left\{I\cdot Y\le x∣I=1\right\}\cdot \mathrm{P}\left\{I=1\right\}=\mathrm{P}\left\{0\le x\right\}\cdot \mathrm{P}\left\{I=0\right\}+\mathrm{P}\left\{Y\le x∣I=1\right\}\cdot \mathrm{P}\left\{I=1\right\}.\end{array}$(7)

Let us consider result (7) for diverse $x$:

$F_X\left(x\right)=0\cdot P\left\{I=0\right\}+P\left\{Y\le x|I=1\right\}\cdot P\left\{I=1\right\}=0$ for $x<0$.

For $x\ge 0$ we obtain:

$$F_X\left(x\right)=1\cdot P\left\{I=0\right\}+P\left\{Y\le x|I=1\right\}\cdot P\left\{I=1\right\}=\mathrm{0,75}+\left(\mathrm{0,5}-\mathrm{\Phi }\left(\frac{10\sqrt{5}\cdot x}{P\cdot S\cdot {\sigma }_U\sqrt{6}}\right)\right)\cdot \mathrm{\Phi }\left(\frac{\alpha }{V\sqrt{1+{\left(1+\alpha \right)}^2}}\right)+\mathrm{0,5}\mathrm{\Phi }\left(\frac{10\sqrt{5}\cdot x}{P\cdot S\cdot {\sigma }_U\sqrt{6}}\right).$$

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When calculating the distribution function of random variable $X$, the distribution function of random variable $Y$ was used mainly:

$$F_Y\left(x\right)=\mathrm{0,5}+\mathrm{\Phi }\left(\frac{10\sqrt{5}\cdot x}{\sqrt{6}\cdot P\cdot S\cdot {\sigma }_U}\right)$$

and the conditional distribution function of random $Y$ is:

$$F_Y\left(x|I=1\right)=\left\{\begin{array}{c}0,x\le 0\\ \mathrm{0,5}+\mathrm{\Phi }\left(\frac{10\sqrt{5}\cdot x}{\sqrt{6}\cdot P\cdot S\cdot {\sigma }_U}\right),x>0\end{array}\right.$$.

The distribution function of individual losses is plotted in Figure 1 for the parameters: $P=1$ thousand rubles, $S=1000$ hectares, ${\sigma }_U=\mathrm{5,25}$ centner/hectare $V=\mathrm{0,290}$. The parameters of crop yield ale in line with those of oats in Bashmakovo district, Pensa region (Suzdalova, Politsinskaya & Sushko, 2015; Malushko, 2016).