The distortion principle for insurance pricing: properties, identification and robustness
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Abstract
Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia.
Keywords
Ambiguity Distortion premium Dual representation Premium principles Risk measures Wasserstein distance1 Introduction
The function of the insurance business is to carry the risk of a loss of the customer for a fixed amount, called the premium. The premium has to be larger than the expected loss, otherwise the insurance company faces ruin with probability one. The difference between the premium and the expectation is called the risk premium. There are several principles, from which an insurance premium is calculated on the basis of the loss distribution.
Let X be a (nonnegative) random loss variable. Traditionally, an insurance premium is a functional, \(\pi {:}\, \{ X\ge 0 \text { defined on } (\varOmega , \mathcal {F}, P) \} \rightarrow \mathbb {R}_{\ge 0}\). We will work with functionals that depend only on the distribution of the loss random variable (sometimes called lawinvariance or versionindependence property, Young 2014). If X has distribution function F we use the notation \(\pi (F)\) for the pertaining insurance premium, and \(\mathbb {E}(F)\) for the expectation of F. We use alternatively the notation \(\pi (F)\) or \(\pi (X)\), resp. \(\mathbb {E}(F)\) or \(\mathbb {E}(X)\) whenever it is more convenient. To the extent of the paper, a more specific notation is used for particular cases of the premium.
1.1 The distortion principle
1.2 Examples of distortion functions
 the power distortion with exponent s. If \(0<s< 1\),The premium is known as the proportional hazard transform (Wang 1995) and calculated as$$\begin{aligned} g^{(s)}(v)=v^{s},\quad h^{(s)}(v)=s(1v)^{s1}. \end{aligned}$$(3)If \(s\ge 1\), then we take$$\begin{aligned} \pi _{h^{(s)}}(F) = \int _0^\infty 1 F(x)^s \, dx = s\int _0^1 F^{1}(v)(1v)^{s1} \, dv. \end{aligned}$$(4)The premium is$$\begin{aligned} g^{(s)}(v)= 1 (1v)^s, \quad h^{(s)}(v) = s v^{s1}. \end{aligned}$$(5)If we consider integer exponent, the premium has a special representation.$$\begin{aligned} \pi _{h^{(s)}}(F) = \int _0^\infty 1 (1F(x))^s \, dx = s\int _0^1 F^{1}(v)v^{s1} \, dv. \end{aligned}$$(6)
Proposition 1
Proof
 the Wang distortion or Wang transform (Wang 2000)where \(\varPhi \) is the standard normal distribution and \(\phi \) its density.$$\begin{aligned} g(v)=\varPhi \left( \varPhi ^{1}(v)+\lambda \right) ,\qquad h(v)=\frac{\phi (\varPhi ^{1}(1v)+\lambda )}{\phi \left( \varPhi ^{1}(1v)\right) },\quad \lambda >0, \end{aligned}$$
 the \({\text {AV@R}}\) (average valueatrisk) distortion function and density arewhere \(0\le \alpha <1\). The pertaining premium has different names, such as conditional tail expectation (CTE), CV@R (conditional value at risk) or ES (expected shortfall) (Embrechts et al. 1997). The premium is$$\begin{aligned} g_\alpha (v)=\min \left\{ \frac{v}{1\alpha } ,1\right\} ,\qquad h_\alpha (v)=\frac{1}{1\alpha }\,\mathbb {1}_{v\ge \alpha }, \end{aligned}$$(7)$$\begin{aligned} \pi _{h_\alpha }(F)=\int _{0}^{\infty }\min \left\{ \frac{1F(x)}{1\alpha },1\right\} \,dx=\frac{1}{1\alpha }\int _{\alpha }^{1}F^{1}(v)\,dv. \end{aligned}$$(8)

piecewise constant distortion densities. The insurance industry uses also piecewise constant increasing distortion functions. For example, the following distortion function is used by a large reinsurer.
v  \(h\,(v)\)  v  \(h\,(v)\) 

[0,0.85]  0.8443  [0.988,0.992)  3.6462 
[0.85,0.947)  1.1731  [0.992,0.993)  4.0572 
[0.947,0.965)  1.4121  [0.993,0.996)  6.5378 
[0.965,0.975)  1.7335  [0.996,0.997)  12.7020 
[0.975,0.988)  2.4806  [0.997,1]  14.9436 
1.3 Certainty equivalence principle
1.4 The ambiguity principle
Remark 1
Remark 2
Recall the fundamental pricing formula of derivatives in financial markets states that the price can be obtained by taking the maximum of the discounted expected payoffs, where the maximum is taken over all probability measures, which make the discounted price of the underlying a martingale. This can be seen as an ambiguity price.
1.5 Combined models
2 The distortion premium and generalizations
The characterization and represestations of the distortion premium were studied exhaustively. Among some of the most classic contributions we mention the dual theory of Yaari (1987); and the characterization by axioms of this premium developed in Wang et al. (1997), where the power distortion for \(0<s<1\) is also characterized in a unique manner. A summary of other known representations and new generalization of this premium will be presented below. Recall that any mapping \(X \mapsto \pi (X)\) which is monotone, convex and fulfils translation equivariance^{4} is a risk measure. Furthermore, if \(\pi \) is also positively homogeneous, monotonic w.r.t. the first stochastic order and subadditive^{5}, then it is a coherent risk measure (Artzner et al. 1999). The distortion premium fulfils all these properties, therefore by the Fenchel–Moreau–Rockefellar theorem, it has a dual representation.
Theorem 1
Theorem 2
Remark 3
3 Continuity of the premium w.r.t. the Wasserstein distance
In this section we study sensitivity properties of the distortion premium respect to the underlying distribution. Some results in this section are related to those in Pichler (2013), Pflug and Pichler (2014) and Kiesel et al. (2016). Similar results of continuity for variability measures are studied in Furman et al. (2017). To start, we recall the notion of the Wasserstein distance.
Definition 1
Here the infimum is over all joint distributions of the pair (X, Y), such that the marginal distributions are P resp. \(\tilde{P}\), i.e. \(X\sim P\), \(Y \sim \tilde{P}\).
Proposition 2
Proof
See Pichler (2010). \(\square \)
Remark 4
The boundedness of h is ensured if g has a finite right hand side derivative at 0, and also if g has finite Lipschitz constant L, since \(\Vert h\Vert _\infty \le L\).
Proposition 2 can be easily generalized as follows.
Proposition 3
Proof
Example 1
 For the \({\text {AV@R}}\) distortion premium \( h_\alpha _\infty = \frac{1}{1\alpha }\), and therefore$$\begin{aligned}  \pi _{h_\alpha }(F)  \pi _{h_\alpha }(G)\le \frac{1}{1\alpha } \cdot WD_{1,d_1} (F,G) . \end{aligned}$$
 For the power distortion with \(s\ge 1\), \( h^{(s)}_\infty = s\), and therefore$$\begin{aligned}  \pi _{h^{(s)}}(F)  \pi _{h^{(s)}}(G)\le s\cdot WD_{1,d_1} (F,G) . \end{aligned}$$
The power distortion with \(0<s<1\) is not bounded. The next result is dedicated for this particular case.
Proposition 4
Proof
The next result is a direct consequence of Proposition 4.
Corollary 1
Corollary 2
Remark 5
Corollary 2 holds when the sequence of distributions are the empirical distributions \(\widehat{F}_n\) defined on an i.i.d. sample of size n, \((x_1, \ldots , x_n)\) from \(X\sim F\). If F has finite pmoments, then \(WD_{p,d_1} (\widehat{F}_n,F) \xrightarrow [n \rightarrow \infty ]{} 0\), hence \( \left \pi _h(\widehat{F}_n)  \pi _h(F) \right \xrightarrow [n \rightarrow \infty ]{} 0\). This result follows by applying Lemma 4.1 in Pflug and Pichler (2014).
Finally notice that, for continuity, the order of the Wasserstein distance r coincides with the number of finite moments of F.
3.1 Partial coverage
Theorem 3
Proof
For the XLinsurance, the Hölderconstant is a Lipschitz constant (\(\beta =1\)) and has the value 1.
From the previous Theorem we can conclude that, if two probabilities are close, then the image probabilities by a mapping T with the characteristics of Theorem 3, are close in Wasserstein distance as well. Theorem 3 isolates the argument also used in Theorem 3.31 in Pflug and Pichler (2014). Note that the underlying distances for the Wasserstein distances are the metrics of the respective spaces.
Corollary 3
We proceed now to study sensitivity properties of the distortion premium w.r.t. the distortion density.
4 Continuity of the premium w.r.t. the distortion density
Previously, we studied the mapping \(F \mapsto \pi _h(F)\) for fixed h. In this section, we consider and present properties of the mapping \(h \mapsto \pi _h(F)\) for fixed F. Different sensitivity properties w.r.t. the distortion parameters were studied in Gourieroux and Liu (2006).
Proposition 5
Proof
Use Hölder inequality and the result is direct. \(\square \)
We can conclude that, if \(h_1\) and \(h_2\) are close, then also the premium prices are close. However, h is always identifiable by the following Proposition.
Proposition 6
Proof
\(\square \)
Remark 6
Note the previous proposition is true if the family of distributions where the premium prices coincide contains all the Bernoulli variables. Compare also Theorem 2 in Wang et al. (1997).
Remark 7
5 Estimating the distortion density from observations

Power future markets A future contract fixes the price today for delivery of energy later. There is the risk of price changes between now and the delivery period. Thus, such a contract has the character of an insurance and the pricing principles apply, although the price is found in exchange markets (e.g. electricity future markets).

Exotic options While standard options are priced through a replication strategy argument, this argument does not apply for other types of options and these options have the character of insurance contracts. Pricing of such contracts is often done over the counter, but again the pricing principle is not revealed to the counterparty.

Credit derivatives Also these contracts carry the character of insurance and can be priced according to insurance price principles.
The goal of this section is to show how the distortion density h can be regained from the observations of the insurance prices, which would help us to shed more light on the price formation of contract counterparties. Notice that our aim is not to estimate the distortion premium prices from empirical data as is done in Gourieroux and Liu (2006) or Tsukahara (2013).
5.1 Estimation of the distortion density with a step function
5.2 Estimation of the distortion density with a cubic monotone spline
Any linear combination with positive scalars of the splines in (26) define a spline which is an increasing and positive function.
The estimations obtained by solving (\(P_1\)) and (\(P_2\)) are presented below.
AV@R distortion premium We consider particular cases of \(h_\alpha \) for \(\alpha =0.9, 0.95\). We estimate the distortion density for each of the cases, with two different step functions, corresponding to \(l=8, 10\) steps, and two different spline basis functions of dimensions \(l=8, 13\), respectively.
Power distortion premium For this case we consider\(h^{(s)}\) for \(s=0.8, 3\). We solve (\(P_1\)) and (\(P_2\)) with the same number of steps and number of spline basis functions as before.
Optimal values of the problems (\(P_1\)) and (\(P_2\)) for the \({\text {AV@R}}\)distortion and the power distortion
\({\text {AV@R}}\)  \(\alpha =0.9\)  \(\alpha =0.95\)  Power  \(s =0.8\)  \(s =3\) 

Step \(l=8\)  7.3248  107.1562  Step \(l=8\)  0.0012  1.1466e\({}\)04 
Step \(l=10\)  0  58.4835  Step \(l=10\)  0  5.1772e\({}\)05 
Spline \(l=8\)  0.0322  13.0785  Spline \(l=8\)  3.6976e\({}\)04  0 
Spline \(l=13\)  0.0154  0.0502  Spline \(l=13\)  1.3251e\({}\)04  0 
6 Ambiguity
In this section we combine the distortion premium with the ambiguity principle. Such an approach allows us to incorporate model uncertainty into the premium. Recall that, by setting the distortion density to \(h=1\), we would price just with the ambiguity principle. As was mentioned in Sect. 1, distances can be used to define ambiguity sets. Here, closed Wasserstein balls will serve as ambiguity sets. These sets will be centred at F, an initial distribution, that we refer to as our baseline model.
Definition 2
Remark 8
We can say more about the value and solution of (Pr) if we choose \(r=p\). We start with bounded distortion densities, i.e. for \(p=1\) and \(q=\infty \).
Proposition 7
 (i)
If h is unbounded, then (Pr) for \(r = 1\) is unbounded.
 (ii)If h is bounded with \(\sup _v h(v) = \Vert h\Vert _\infty \), then (Pr) is bounded for all \(r\ge 1\). If \(r=1\), the optimal value of (Pr) isWe interpret the additional term \(\epsilon \cdot \Vert h\Vert _\infty \) as the ambiguity premium. For the worst case distribution,$$\begin{aligned} \pi ^\epsilon _{h,1,d_1}(F) = \pi _h(F) + \epsilon \cdot \Vert h\Vert _\infty . \end{aligned}$$
 if \(h(v) = \Vert h\Vert _\infty \) for \(v \ge 1\eta \) and \(0<\eta \le 1\), then the supremum is attained at$$\begin{aligned} F_\eta ^*(x) = \left\{ \begin{array}{ll} F(x) &{} \quad x< F^{1}(1\eta ),\\ 1\eta &{} \quad F^{1}(1\eta )\le x < F^{1}(1\eta ) + \epsilon / \eta ,\\ F\left( x \epsilon / \eta \right) &{} \quad x \ge F^{1}(1\eta ) + \epsilon / \eta . \end{array} \right. \end{aligned}$$

Otherwise, the supremum is not attained, but can be approximated by the sequence \(F^*_{1/n}(x)\), \(\forall n\in \mathbb {N}\).

Proof
 (i)Given that h is increasing and unbounded, the increasing sequence \(K_n = h\left( 1 1/n \right) \), is such that \(\lim _{n\rightarrow \infty } K_n =\infty \). For all \(n\in \mathbb {N}\) we define a distribution \(G_n\) such that$$\begin{aligned} G_n^{1}(v) = F^{1}(v) + \epsilon \cdot n\, \mathbb {1}_{[11/n , 1]}. \end{aligned}$$
Remark 9
The solution \(F^*_\eta \) in Proposition 7 is not unique. Any distribution \(\tilde{F}_\eta \) such that \({\tilde{F}_\eta }^{1}(v) = F^{1}(v) + \frac{\epsilon }{\eta }\cdot k(v)\mathbb {1}_{[1\eta ,1]} \), with \(\frac{1}{\eta }\cdot k(v)\mathbb {1}_{[1\eta ,1]}\) a density on [0, 1], attains the supremum.
If h is unbounded we can characterize the solution of (Pr) as follows.
Proposition 8
Proof
Under some conditions on h we can also prove unboundness of (Pr) for \(r>p>1\) in the case where h is not in \(L^q\), where q is the conjugate of p, the finite moments of F.
Proposition 9
(Unboundness for \(\mathbf {r}>\mathbf {p}>\mathbf {1}\)) Let the baseline distribution F have finite pmoments and let \(h\notin \mathcal {L}^q\), for \(p,\, q\) conjugates and \(r,\,s\) conjugates with \(r>1\). If there exists \(s_1<s\) such that \(\int _0^1 h(v)^{s_1} \, dv =\infty \) and \(h\in \mathcal {L}^t\), for all \(t<s_1\), then (Pr) is unbounded for all \(r>p\).
Proof
Remark 10
7 Conclusions

the premium function \(F \mapsto \pi _h(F)\), i.e. the properties of \(\pi _h\) as a premium principle,

the direct function \(h \mapsto \pi _h(F)\), i.e. the dependency on the distortion density,

the inverse functions \(\pi _h(F) \mapsto h\).
Footnotes
 1.
The derivative of a concave function is a.e. defined, even if it is not differentiable everywhere.
 2.
\(F_1\) is first order stochastically larger than \(F_2\) if \(F_1(x)\le F_2(x)\) for all x.
 3.
The original notion of a utility function introduced by Neumann/Morgenstern was a concave monotonic U, such that the decision maker maximizes the expectation \(\mathbb {E}(U(Y))\) of a profit variable Y. A disutility function can be defined out of a utility function by setting \(V(u) =  U(u)\).
 4.
\(\pi \) has translation equivariance property, if \(\pi (X+c) = \pi (X) + c\), for \(c\in \mathbb {R}\).
 5.
A premium \(\pi \) is called subadditive, if \(\pi (X+Y) \le \pi (X) + \pi (Y)\). Subadditivity and positive homogeneity imply convexity.
Notes
Acknowledgements
Open access funding provided by University of Vienna.
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