Two-sided distributions with applications in insurance loss modeling

Abstract

A framework of two-sided densities is presented for asymmetric continuous distributions consisting of two branches each with its own generating density. The framework supports the construction of distributions with positive support and a specified mode. Examples thereof shall be constructed using the beta and Burr Type XII distributions for their left and right branch densities. The examples are parameterized via left and right branch scale parameters, a mode parameter and two parameters determining the heaviness of its right tail. Keeping one of the tail parameters fixed, a procedure solving for their parameters is presented given a lower and upper quantile, a mode and a conditional-value-at-risk, popular in risk management of insurance losses. While valuable on its own right, that solution may be used as a starting point for a maximum likelihood routine. The estimation of the parameters is demonstrated using the classical insurance Danish fire loss data set and a French business loss interruption data set. Both data sets are publicly available. Developed models compare favorably with prior models fitted to the Danish fire loss data in the literature.

Appendices

Appendix A: The Burr Type XII distribution

Let $X\sim Burr(n,\tau ,\nu )$ (Burr 1942) with pdf

$$\begin{aligned} f_X(x\vert n,\tau ,\nu )=\frac{\nu \tau }{n}\frac{(y/n)^{\tau -1}}{[1+(y/n)^\tau ]^{ \nu +1}},\nu ,\tau>0,n>0. \end{aligned}$$

(A1)

From (A1) one obtains the cdf

$$\begin{aligned} F_X(x\vert n,\tau ,\nu )=1- \left[ \frac{1}{1+(y/n)^\tau } \right] ^\nu , \end{aligned}$$

(A2)

and from (A2) the quantile function

$$\begin{aligned} F^{-1}_X(p\vert n,\tau ,\nu )=n \left[ (1-p)^{-1/\nu }-1 \right] ^{1/\tau }. \end{aligned}$$

(A3)

After som algebraic manipulations, one derives with (A1) that

$$\begin{aligned} E [X^k\vert n,\tau ,\nu ]=\frac{kn^k}{\tau }\mathbb {B}(k/\tau ,\nu -k/\tau ),\, \text {provided }\nu >k/\tau . \end{aligned}$$

(A4)

Finally, from (A1) one obtains

$$\begin{aligned} CVaR_p[X]=E \left[ X\vert X>x_p \right] =x_p+\frac{n}{\tau }\frac{\mathbb {B}(\nu -1/\tau ,1/ \tau )}{1-p}I_{(1-p)^{1/\nu }}(\nu -1/\tau ,1/\tau ), \end{aligned}$$

(A5)

where $x_p$ is given by (A3). Thus $CVaR_p[X]$ (A5) is finite for $\nu >1/\tau$ and infinite otherwise.

Appendix B: The generalized log-Moyal distribution

Let $X\sim GlogM(\theta ,\nu )$, i.e. generalized log-Moyal distributed (Bhati and Ravi 2018), with pdf

$$\begin{aligned} f_X(x\vert \theta ,\nu )=\frac{1}{\sqrt{2\pi }}\frac{1}{\sigma x}(\theta /x)^\nu exp \left[ -\frac{1}{2} \left\{ (\theta /x)^\nu \right\} ^2 \right] . \end{aligned}$$

(B6)

One obtains from (B6) the cdf

$$\begin{aligned} F_X(x\vert \theta ,\nu )=2\times \left\{ 1-\Phi \left[ (\theta /x)^\nu \right] \right\} , \end{aligned}$$

(B7)

where $\Phi (\cdot )$ is the standard normal cdf. From (B7) one derives the quantile function

$$\begin{aligned} F_X^{-1}(p\vert \theta ,\nu )=\theta \left[ \Phi ^{-1} \left( 1-\frac{p}{2} \right) \,\right] ^{-1/\nu }. \end{aligned}$$

(B8)

After some algebraic manipulations, one obtains with (B6) for the raw moments

$$\begin{aligned} E[X^k\vert \theta ,\nu ]=\frac{\theta ^k\Gamma \left[ \frac{1}{2} \left( 1-\frac{k}{\nu } \right) \right] }{\sqrt{ \pi 2^{k/\nu }}}, \end{aligned}$$

(B9)

which are finite for $\nu >k$ and infinite otherwise. Finally, from (B6) one derives

$$\begin{aligned} CVaR_p(X)=\frac{1}{1-p}\frac{\theta \Gamma \left[ \frac{1}{2} \left( 1-\frac{1}{\nu } \right) \right] }{ \sqrt{\pi 2^{1/\nu }}}F_V\left\{ \frac{1}{2} \left[ \Phi ^{-1} \left( 1-\frac{p}{2} \right) \, \right] ^2 \left| \frac{1}{2} \left( 1-\frac{1}{\nu } \right) ,1 \right. \right\} , \end{aligned}$$

(B10)

where $V\sim Gamma(\alpha ,\beta )$, $\alpha =\frac{1}{2}(1-\frac{1}{\nu })$, $\beta =1$ with pdf

$$\begin{aligned} f_V(v\vert \alpha ,\beta )=\frac{\beta ^\alpha }{\Gamma (\alpha )}v^{\alpha -1}exp(-\beta v). \end{aligned}$$

(B11)

Thus, $CVaR_p(X)$ (B10) is finite for $\nu >1$ and infinite otherwise.

Appendix C: The generalized inverse gamma distribution

Let $X\sim Gamma(n,\theta )$ with pdf (B11). Let $Y=X^{-1/\nu }$. From (B11) one obtains the pdf

$$\begin{aligned} f_Y(y\vert n,\theta ,\nu )=\frac{\nu \theta ^n}{\Gamma (n)}y^{-\nu n-1}e^{-\theta y^{-\nu }}. \end{aligned}$$

(C12)

The pdf (C12) of the rv Y is generalized inverse gamma $GIG(n,\theta ,\nu )$ (Mead 2015). For the cdf one derives

$$\begin{aligned} Pr(Y\le y)=1-Pr(X\le y^{-\nu })=1-F_X(y^{-\nu }\vert n,\theta ), \end{aligned}$$

(C13)

and from (C13) the quantile function

$$\begin{aligned} F^{-1}_Y(p\vert n,\theta ,\nu )= \left[ F_X^{-1}(1-p\vert n,\theta ) \right] ^{-1/\nu }. \end{aligned}$$

(C14)

The raw moments for Y follow directly from the raw moments of $X\sim Gamma(n,\theta )$ as

$$\begin{aligned} E \left[ Y^k\vert \theta ,n \right] =E \left[ X^{-k/\nu }\vert \theta ,n \right] =\frac{\theta ^{k/\nu }\Gamma (n-k/\nu )}{\Gamma (n)}. \end{aligned}$$

(C15)

Hence, raw moments (C17) exist for $\nu > k/n$ and are infinite otherwise. To derive $CVaR_p(Y)=E[Y\vert Y>y_p]$ one sets with (C12)

$$\begin{aligned} \begin{aligned} \int _{y_p}^\infty uf_Y(u\vert n,\theta ,\nu )du&=\int _{y_p}^\infty \frac{\nu \theta ^n}{\Gamma (n)}(u^{-\nu })^ne^{-\theta u^{-\nu }}du =\int _0^{y^{-\nu }_p}\frac{\theta ^n}{\Gamma (n)}v^{n-1/\nu -1}e^{-\theta v}dv \\&=\theta ^{1/\nu }\frac{\Gamma (n-1/\nu )}{\Gamma (n)}F_W \left[ F_X^{-1} \left( 1-p\vert n,\theta \right) \vert n-1/\nu ,\theta \right] . \end{aligned} \end{aligned}$$

(C16)

where in (C16), $W\sim Gamma(n-1/\nu ,\theta )$, $X\sim G(n,\theta )$. Finally, from (C16)

$$\begin{aligned} CVaR_p(Y)=\frac{\theta ^{1/\nu }}{1-p}\frac{\Gamma (n-1/\nu )}{ \Gamma (n)}F_W \left[ F_X^{-1} \left( 1-p\vert n,\theta \right) \vert n-1/\nu ,\theta \right] . \end{aligned}$$

(C17)

Thus, $CVaR_p(Y)$ (C17) is finite for $\nu >1/n$ and infinite otherwise.