Stop-Loss estimation via Single Loss Approximation (Poster)

Objective

We calculate the actuarial price and optimum retention point of a Stop-Loss reinsurance for the costs associated with high-cost illiness (HCI onwards) medical services, with emphasis on surgery services. We apply Single Loss Analysis Albrecher, Beirlant, & Teugels (2017). And we discuss the validity of the LogNormal assumption used in some works for the valuation of this kind of services.

Introduction

We assume the classical Cramer-Lundberg collective risk model, for the aggregate loss as \begin{align*} \label{ALD} S= \sum_{i=1}^N X_i \tag{1} \end{align*} where, $X_1,X_2,\ldots,X_N$ is a sequence of random variables iid $X_i\geq 0$, with cumulative distribution $F_X(x)=\mathbb{P}(X\le x)$ with support $[0,\infty)$, such that $F_{X}(x)<1$, $\forall x>0$, $F_{X}(0_+)=0$, and with $\mathbb{E}(X_{i})<\infty$. $N$ and $X_{i}$ are asumed mutually independent random variables $\forall i=1,2,\ldots$. (Bowers, Gerber, Hickman, Jones, & Nesbitt, 1997, p. 367).

The variable $N$ represents the number of surgery claims in one year of HCI, with a mass function $p_{n}(x) = \mathbb{P}(N=x)$. We employ several models included in R library gamlss from Rigby, Stasinopoulos, Heller, & Voudouris (2014), which include several Mixed Poisson distributions.

The variable $X$ represent the severities of individual surgeries in HCI. Their distribution is model by mixtures of light and heavy-tailed distributions, also called spliced distribution, defined as \begin{align*} F_X(x)=F_L(x)1_{\{x\leq u\}}+ F_H(x)1_{\{x>u\}} \tag{2} \end{align*}

where $1_{x\leq u}$ and $1_{x>u}$ are indicator variables, $u$ is a threshold, $F_L$ and $F_H$ are light and heavy-tailed distributions, respectively. A requisite is that $1-F_H$ is a regular varying function with index $-\alpha$, i.e, $1-F_H\in \mathcal{R}_{-\alpha}$.

The aggregate loss distributions $F_{S}(x)$ is defined as \begin{align*} F_{S}(x) & =\sum_{n=0}^{\infty}p_{n}F_{X}^{*n}(x) \tag{3} \end{align*}

where $p_{n}=\mathbb{P}(N=n)$ is the probability distribution of $N$ evaluated in $n$ and $F_{X}^{*n}(x)=\mathbb{P}(X_{1}+X_{2}+\ldots+X_{n}\le x)$ is the $n$-th convolution of $F_X$ with itself.

The calculation of the distribution and the quantil for $S$ is known a difficult task. Some authors have proposed algorithms and methods for their approximation. Some are described in Kaas, Goovaerts, Dhaene, & Denuit (2008, ch. 2-3), and Albrecher et al. (2017, ch. 6).

For this approximation are use for the calculation of the optimal retention point and Stop-Loss reinsurance premium, defined later, we use two alternatives proposed in the literature:

• The methodology Single Loss Approximation (SLA onwards).
• The assumption of the Black-Scholes methodology on which $S\sim LogNormal(\mu,\sigma^2)$

Frequency Model

Mixed Poisson Law

The Mixed Poisson Law is applied when the variance of the random variable $N$ is significantly greater than its expected value. We assume that the mass function of $N$, follows a Poisson Law with random parameter $\lambda$ (replaced by $\theta$), and mixing distribution function $U$ (also called risk structure function), such that \begin{align*} \mathbb{P}(N=x) = \int_{0}^{\infty} \frac{e^{-\theta t}(\theta t)^x}{x!}dU(\theta) \tag{4} \end{align*}

obtaining goodness of fit tests

Severity Model

Spliced distribution

Albrecher et al. (2017, p. 50) defines a m-component spliced distribution with a probability density function as \begin{align*} \begin{split} f_{X}(x) = \begin{cases} \pi_{1} \frac{f_{1}(x)}{F_{1}(c_{1})-F_{1}(c_{0})} & c_{0} < x \leq c_{1} \\ \pi_{2} \frac{f_{2}(x)}{F_{2}(c_{2})-F_{2}(c_{1})} & c_{1} < x \leq c_{2} \\ \quad \quad \quad \vdots & \quad \quad \ \vdots \\ \pi_{m} \frac{f_{m}(x)}{F_{m}(c_{m})-F_{m}(c_)}& c_ < x \leq c_{m} \end{cases} \end{split} \tag{5} \end{align*}

where $f_{i}(x)$ and $F_{i}(x)$ are the pdf and cdf of the $i$-th interval of a random variable $X$, respectively. $\pi_{i}>0$ are the weights of each of the categories, with $\sum_{i=1}^m \pi_{i} = 1$. $c_{k_i}$ are the intervals for which the random variable $X$ is defined in each category or also called union points.

obtaining goodness of fit tests

Single Loss Approximation

For the implementation of this approach, we start with the Böcker & Klüppelberg (2005) result, where the authors point out that, if $F_{X_k}$ could be classified within Subexponential class distributions, such that \begin{align*} \lim_{x\to\infty}\frac{1-F_{X}^{*n}(x)}{1-F_{X}(x)}=\lim_{x\to\infty}\frac{\bar{F}_{X}^{*n}(x)}{\bar{F}_{X}(x)}=n \tag{6} \end{align*} and assuming that $p_{n}$ satisfies \begin{align*} \sum_{n=0}^\infty (1+\varepsilon)^np_{n}<\infty \text{ for some } \varepsilon>0 \tag{7} \end{align*} then, it is said that $F_{S}$ is classify within the Subexponential class, when its tail behavior is given by \begin{align*} \label{BK} \bar{F}_{S}(x) \sim \mathbb{E}(N)\bar{F}_{X}(x); \quad \quad x\to\infty \tag{8} \end{align*}

Risk Measures

Value at Risk (VaR)

From the equation (\ref{BK}) Böcker & Klüppelberg (2005, p. 91), present the first approximation of the VaR by SLA method, for a level $\kappa$, with $0<\kappa<1$ and defined as \begin{align*} \label{VaR1} VaR_{S}(\kappa) = F_{X}^{-1}\left(1 - \frac{1-\kappa}{\mathbb{E}(N)})\right); \quad \quad \kappa\to1 \tag{9} \end{align*}

Although the equation (\ref{VaR1}) has a very attractive structure, in Böcker & Sprittulla (2006, p. 96), the authors point out that this equation must be used carefully because this approximation underestimate the VaR, since the aproximation does not take into account all loss events $X_i$ which contribute to the aggregate loss $S$.

In order to overcome such an inconvenience and give analytical support to the SLA method, from Albrecher et al. (2017, p. 199) we can obtain a corrected expression for VaR \begin{align*} \label{VaR2} VaR_S(\kappa)\approx F_{X}^{-1}\left(1 - \frac{1-\kappa}{\mathbb{E}(N)}\right)+ \mathbb{E}(X)\left(\frac{\mathbb{E}(N^2)}{\mathbb{E}(N)}-1\right); \quad \kappa\to1 \tag{10} \end{align*}

Expected Shortfall (ES/TVaR/AVaR/CVaR)

For the calculation of the ES, we use the basic definition of ES in Kaas et al. (2008, p. 129) \begin{align*} \label{VaR3} ES_{S}(\kappa)= \frac{1}{1-\kappa}\int_\kappa^1 VaR_{S}(\theta)\text{d}\theta \tag{11} \end{align*}

Stop-Loss Premium

Finally, from VaR and ES, it is possible to define the Stop-Loss premium by the identity [Kaas2008, p. 129] \begin{align*} \label{VaR4} \pi(\kappa)=(1-\kappa) \left[ES_{S}(\kappa) - VaR_{S}(\kappa)\right] \tag{12} \end{align*}

Reinsurance

If we define the aggregate costs $S$ as the equation (\ref{ALD}), for the validity period of a policy (usually one year). Then a *reinsurance contract for a *HCI **can be defined as \begin{align*} \label{RE1} S= & D + R \tag{13} \end{align*}

where $D$ represents the amount deductible or retained by the insurer and $R$ represents the amount paid by the reinsurer.

In order to define the reinsurance premium $\delta(\kappa)$, we assume the expected value principle, defined as \begin{align*} \label{RE2} \delta(M^*)=(1+\rho)\pi(M^*) \tag{14} \end{align*}

where $M^*=VaR_S(\rho*)$ is the optimal retention point, $\rho^*=\frac{1}{1+\rho}$ is the optimal condition for the retention, $\rho>0$ is defined as the reinsurer’s relative safety load factor, which can be interpreted as a risk premium rateand $\pi(.)$ is the Stop-Loss premium.

Optimum Retention Point Estimation

From the equations (\ref{VaR2}), (\ref{VaR3}), (\ref{VaR4}), (\ref{RE1}), (\ref{RE2}) and the adjusted distribution DEL for $N$ and W-GP for $X$, we calculate optimal retention points and optimal premiums for different levels of $\rho$.

Discussion of the validity of the LogNormal assumption for the aggregated costs

• To justify the LogNormal assumption for the aggregated cost $S= \sum_{i=1}^N X_i$ requires that a least the first two moments of $S$ exists, in order to specify a LogNormal distribution. And then calculate $VaR_S(\kappa)$, $ES_S(\kappa)$, and $\pi(\kappa)$.
• As LogNormal distribution is belongs to the Subexponential class of heavy-tailed distributions. The problem is, to guarantee that $S$ is LogNormal without any other kind of information. We do not know of any kind of conditions on severities $X_i$ which guarantee that $S$ is LogNormal.
• Based on this considerations and the results exposed, we consider that the assumption of LogNormal for the aggegated costs is not justified. As a conclusion several studies which expose valuation of the medical services in surgery or another HCI, based on Black-Scholes methodology, used for instance in (Chicaíza & Cabedo (2009), and Girón & Herrera (2015)), are lacking a more rigurous justification.

Conclusions

• The spliced distributions present a better adjustment in the adjustment of the severities than the other adjusted distributions.
• Although the goodness-of-fit tests may have higher values in their P-values for some distributions, it does not mean that these are the ones that offer a better adjustment. Therefore, it is always recommended to accompany these measures with a support graphic.
• For the calculation of the reinsurance premium, we assume the expected value principle, which depends on the reinsurer’s relative safety load factor $\rho$, but different principles can be used to calculate the optimal retention point.
• It is not correct to make assumptions of distributions, if you do not have a strict justification for making such assumptions.

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Chicaíza, L., & Cabedo, D. (2009). Using the Black-Scholes method for estimating high-cost illness insurance premiums in Colombia. Innovar, 19(33), 119–130.

Girón, L., & Herrera, F. (2015). Cálculo y comparación de la prima de un reaseguro de salud usando el modelo de opciones de Black-Scholes y el modelo actuarial. Revista de Economía Del Rosario. Julio-Diciembre, 18(2), 211–248. https://doi.org/10.12804/rev.econ.rosario.18.02.2015.03

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