## Abstract

Pricing tools for non-proportional reinsurance treaties often only provide layer prices, but no layer-independent collective risk model. There are, however, situations where such a layer-independent model is needed. Examples are large loss and catastrophe loss models for proportional reinsurance treaties. We show that the expected losses of a tower of reinsurance layers can always be matched using a piecewise Pareto distributed severity and provide an algorithm that can be used to convert layer information into a layer-independent collective risk model.

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## Acknowledgements

I would like to thank the anonymous referee for his constructive comments which helped to improve the presentation of the paper substantially.

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## Appendices

### Appendix 1: Technical details about the Pareto distribution

The purpose of this appendix is to define the Pareto alpha between two pieces of information (expected excess frequencies and/or expected losses of layers) more rigorously than in Sect. 3, and to prove existence and uniqueness. Moreover, we prove a technical lemma that is needed in Example 3.

Let \(0<t\le t_1<t_2\) and let \(f_1> f_2\) be the expected frequencies in excess of \(t_1\) and \(t_2\), respectively. Then we have

###
**Definition 5**

In this situation, we say that

is the *Pareto alpha between* \((t_1,f_1)\) *and* \((t_2,f_2)\).

Let \(\alpha \) be the Pareto alpha between \((t_1,f_1)\) and \((t_2,f_2)\) and let \(S=\sum _{n=1}^NX_n\) be a collective risk model with claim sizes \(X_n\sim {\text {Pareto}}(t,\alpha )\). Then the expected frequency in excess of \(t_i\) is given by \({\text {E}}(N)\cdot (1-F_{t,\alpha }(t_i))\), i.e. the model has the expected frequency \(f_1\) in excess of \(t_1\) if and only if the expected frequency in excess of \(t_2\) equals \(f_2\).

###
**Lemma 4**

*Let* \(0<t\le a< b<+\infty \). *Let* \(c:=b-a\) *and let* \(f>0\) *be the expected frequency in excess of* *t*. *Then the map*

*is a strictly decreasing homeomorphism. In case of an unlimited layer* \(+\infty \) *xs* *a* *we have a strictly decreasing homeomorphism*

###
*Proof*

We only consider the case of a limited layer. The proof for unlimited layers is similar. Since

is strictly decreasing for all \(x\in (a,b)\), the map

is strictly decreasing, too. We have

i.e. \(\Psi ^{(t,f)}_{c{\text {xs}}a}\) is surjective. Moreover, \(\Psi ^{(t,f)}_{c{\text {xs}}a}\) is continuous and consequently \(\Psi ^{(t,f)}_{c{\text {xs}}a}\) is a homeomorphism. \(\square \)

In the proof of Lemma 4 we have used the fact that a bijective, strictly monotonic and continuous function \(f:I\rightarrow J\subset \mathbb {R}\), where *I* is an open interval, is always a homeomorphism, i.e. the inverse \(f^{-1}\) is continuous as well. We will also use this fact in the proofs below.

###
**Definition 6**

Let \(0<t\le a<b\le +\infty \) and \(c:=b-a\). Let *e* denote the expected loss of the layer *c* xs *a* and let *f* be the expected frequency in excess of *t*. If \(0<e< f\cdot c\) then

is called the *Pareto alpha between* (*t*, *f*) *and* *c* *xs* *a*.

Let \(t \le t_1\le a\) and \(f_1>e/c\). Let \(\alpha \) be the Pareto alpha between \((t_1,f_1)\) and *c* xs *a*, and assume that \(S=\sum _{n=1}^N X_n\) is a collective risk model with \(X_n\sim {\text {Pareto}}(t,\alpha )\). Then the model has the expected frequency \(f_1\) in excess of \(t_1\), if and only if the expected loss of the layer *c* xs *a* equals *e*. Note that \(f_1>e/c\) only excludes the extreme case of a total loss model (cf. Sect. 2).

###
**Lemma 5**

*Let* \(0<t\le a<b\le +\infty \) and \(c:=b-a\). *Let* \(e>0\) *denote the expected loss of the layer* *c* *xs* *a* *and let* \(f_1>f_2>e/c\). *For* \(i\in \{1,2\}\) let \(\alpha _i\) *be the Pareto alpha between* \((t,f_i)\) *and* *c* *xs* *a*. *Then we have*

*and if* \(b<+\infty \), *then*

Let \(\sum _{n=1}^{N_i}X_{i,n}\), \(i\in \{1,2\}\) be collective risk models with \({\text {E}}(N_i)=f_i\) and \(X_{i,n}\sim {\text {Pareto}}(t,\alpha _i)\), which both match the expected loss of a layer *c* xs *a* with \(a\ge t\) and let \(f_1>f_2\). Then Lemma 5 states that the model with \(i=1\) has a greater expected layer entry frequency, whereas the model with \(i=2\) has a greater expected layer exit frequency (if \(b<+\infty \)). Lemma 5 is used in Example 3.

###
*Proof*

We have \(f_1\cdot I_{t,\alpha _2}(a,b)>f_2\cdot I_{t,\alpha _2}(a,b)=e=f_1\cdot I_{t,\alpha _1}(a,b)\). Applying Lemma 4 we conclude that \(\alpha _1>\alpha _2\). Since

there exits an \(x\in (a,b)\) such that \(f_1\cdot (1- F_{t,\alpha _1}(x)) \ge f_2\cdot (1- F_{t,\alpha _2}(x))\). Since \(\alpha _1>\alpha _2\), we have

For the case \(b<+\infty \), the proof of \( f_1\cdot (1- F_{t,\alpha _1}(b)) <f_2\cdot (1- F_{t,\alpha _2}(b)) \) is similar. \(\square \)

###
**Lemma 6**

*Let* \(0<t\le a_1<a_2\) *and let* \(b_1\le b_2\le +\infty \) *with* \( b_i>a_i\) *and* \(c_i:=b_i-a_i\). *If* \(b_2<+\infty \), *then the map*

*is well-defined (i.e. does not depend on* *t**) and is a strictly increasing homeomorphism. If *\(b_1<b_2=+\infty \), *then we have the (well-defined) strictly increasing homeomorphism*

*If* \(b_1=b_2=+\infty \), *then we have the (well-defined) strictly increasing homeomorphism*

###
*Proof*

The case \(b_1<b_2=+\infty \) follows easily from

which is a consequence of Lemma 1 in Sect. 3. The case \(b_1=b_2=+\infty \) follows directly from

which is also a consequence of Lemma 1. For the case \(b_2<+\infty \), which is slightly more difficult to prove, see Riegel [11]. \(\square \)

###
**Definition 7**

Let \(0<t\le a_1<a_2\) and let \(b_1\le b_2\le +\infty \) with \( b_i>a_i\) and \(c_i:=b_i-a_i\). Moreover, let \(e_1>0\) and \(e_2>0\) be the expected losses of the layers \(c_1\) xs \(a_1\) and \(c_2\) xs \(a_2\), respectively. If \(b_2<+\infty \), then we additionally assume that \(e_1/e_2>c_1/c_2\) and if \(b_1=b_2=+\infty \), then we require \(e_1/e_2>1\). Then

is called the *Pareto alpha between the layers* \(c_1\) *xs* \(a_1\) *and* \(c_2\) *xs* \(a_2\).

Let \(\alpha \) be the Pareto alpha between the layers \(c_1\) xs \(a_1\) and \(c_2\) xs \(a_2\) (with \(a_1<a_2\) and \(a_1+c_1 \le a_2+c_2\)) and assume that \(S=\sum _{n=1}^N X_n\) is a collective risk model with \(t\le a_1\) and \(X_n\sim {\text {Pareto}}(t,\alpha )\). Then the model matches the expected loss \(e_1\) for the layer \(c_1\) xs \(a_1\) if and only if it matches the expected loss \(e_2\) of the layer \(c_2\) xs \(a_2\). Note that the condition \(e_1/e_2>c_1/c_2\) in the case \(c_2<+\infty \) simply means that the risk rate on line of the lower layer \(c_1\) xs \(a_1\) is greater than the risk rate on line of the higher layer \(c_2\) xs \(a_2\) (cf. Sect. 2).

### Appendix 2: Proof of Proposition 1

Let *L* denote the Lévy metric, i.e.

for distribution functions *F* and *G*. Let *G* be a distribution function with \(G(0)=0\) and let \(\varepsilon > 0\). We show that there exist parameter vectors \(\mathbf {t}\) and \(\varvec{\alpha }\) such that \(L(F_{\mathbf {t},\varvec{\alpha }}, G) \le \varepsilon \). Choose \(0<\delta \le \varepsilon \) such that \(G(\delta )<1-\delta \). This is possible since *G* is right-continuous. Let

We define \(t_1:=\delta /2\), \(s_1:=1\) and for \(k=2,\ldots , n\)

For \(k=1,\ldots ,n-1\) let

and choose an arbitrary \(\alpha _n>0\). Let \(\mathbf {t}:=(t_1,\ldots ,t_{n})\), \(\varvec{\alpha }:=(\alpha _1,\ldots ,\alpha _n)\). Since \(\alpha _k\) is the Pareto alpha between \((t_k,s_k)\) and \((t_{k+1},s_{k+1})\) we then have

for \(k=0,\ldots ,n-2\). Moreover, we have \(F_{\mathbf {t},\varvec{\alpha }}(x),G(x)\in [1-\delta ,1]\) for \(x\ge (n-1)\cdot \delta \). This implies

For \(x\in [k\delta ,(k+1)\delta ]\) with \(k+1\le n-2\) we have

For \(x\in [(n-2)\delta ,(n-1)\delta ]\) we have

For \(x\ge (n-1)\delta \) we have

\(\square \)

### Appendix 3: Proof of Lemma 3

We have to show that it is possible to find an \(f_1>e_1/c_1\) such that Matching Algorithm 1 does not stop at an \(i<k\) in Step 2.

The case \(k=1\) is clear. For \(k\in \{2,3\}\) we only sketch the proof and leave the (simple) technical details to the reader. Let \(\alpha (f_i)\) denote the Pareto alpha between \((a_i,f_i)\) and the layer \(c_i\) xs \(a_i\). Then, for every \(i<k\),

is a strictly decreasing homeomorphism. If \(k=3\), then we choose \(f_2\), such that

(possible, since \(e_1/c_1>e_2/c_2>e_3/c_3\)). If \(k=2\), then we only require \(e_1/c_1>f_2>e_2/c_2\). Then Matching Algorithm 1 provides the requested result if we start with \(f_1:=\phi _1^{-1}(f_2)\). \(\square \)

### Appendix 4: Proof of Theorem 1

Either frequencies \(f_i\) with \(f_1 > e_1/c_1\) and \(e_{i-1}/c_{i-1}>f_i>e_{i}/c_{i}\) for \(i=2,\ldots ,k\) are given or they are calculated in Step 1 of Matching Algorithm 2. After Step 2 of the algorithm, where we define \(s_i:=f_i/f_1\) and \(l_i:=e_i/f_1\), the preconditions of the following proposition are fulfilled.

###
**Proposition 2**

*Consider a sequence of attachment points* \(0<a_1<\cdots <a_{k}\), *a sequence of excess probabilities* \(1=s_1>s_2>\cdots>s_{k}>0\) *and a sequence of loss expectations* \(l_1,\ldots ,l_k>0\) *for the layers* \(a_{i+1}-a_i\) *xs* \(a_i\) *(with* \(a_{k+1}:=+\infty \)), *such that*

*for* \(i=1,\ldots ,k-1\). *Let* \(n=2k-1\). *Then there exist parameters* \(\mathbf {t}=(t_1,\ldots ,t_n)\) *with* \(t_{2i-1}=a_i\) *and* \(\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n)\) *with* \(\alpha _i>0\) *such that*

*for* \(i=1,\ldots ,k\).

In the following proof of Proposition 2, it is explicitly shown how the parameter vectors \(\mathbf {t}\) and \(\varvec{\alpha }\) can be calculated inductively. Matching Algorithm 2 uses exactly the same approach for the calculation of the parameter vectors of the piecewise Pareto distribution. Therefore, the algorithm always leads to the desired collective risk model.

###
*Proof*

We use induction to prove this statement. In the base case \(k=1\) we can use \(t_1:=a_1\) and \(\alpha _1:=a_1/l_1+1\) (cf. Lemma 1). For the inductive step \(k\rightarrow k+1\) we assume now that the statement is true for a \(k\ge 1\). Then we have thresholds \(\mathbf {t}^{(k)}=(t_1,\ldots ,t_{2k-1})\) with \(t_{2i-1}=a_i\) and \(\varvec{\alpha }^{(k)}=(\alpha _1,\ldots ,\alpha _{2k-2},\alpha _{2k-1}^{(k)})\) such that

for \(i=1,\ldots ,k\). Let \(t_{2k+1}:=a_{k+1}\) and

We will show that there exist \(t_{2k}\in (a_k,a_{k+1})\) and \(\alpha _{2k-1},\alpha _{2k}> 0\) such that

for \(\mathbf {t}=(t_1,\ldots ,t_{2k+1})\) and \(\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _{2k+1})\). Due to \(1-F_{\mathbf {t},\varvec{\alpha }}(a_{k+1})=s_{k+1}\) and the definition of \(\alpha _{2k+1}\) we then also have

(cf. Lemma 1 in Sect. 3). For \(\tau \in (a_k,a_{k+1})\) and \(\alpha \ge 0\) we define

and

For \(\alpha _{2k-1}>0\) and \(\alpha _{2k}>0\) we have

i.e. (1) is equivalent to

If (3) is fulfilled then (2) is equivalent to

For fixed \(\tau \in (a_k,a_{k+1})\) the functions \(\alpha \mapsto \sigma ^{(k)}(\tau ,\alpha )\) and \(\alpha \mapsto \lambda ^{(k)}(\tau ,\alpha )\) are continuous and strictly decreasing and we have

Therefore, for a given \(t_{2k}\in (a_k,a_{k+1})\), the Eqs. (3) and (4) can be solved with \(\alpha _{2k-1}>0\) and \(\alpha _{2k}=\sigma ^{(k)}(t_{2k},\alpha _{2k-1})>0\) if and only if

The functions

are strictly increasing and we have

and

Let

and

We have \(\tau _u^{(k)}>a_k\), and if \(\tau _l^{(k)}>a_k\), then we have

Since \(\tau _l^{(k)}<a_{k+1}\), it results that \(\tau _l^{(k)}<\tau _u^{(k)}\). We select a \(t_{2k}\in (\tau _l^{(k)},\tau _u^{(k)})\) and obtain

i.e. Eqs. (3) and (4) can be solved with \(\alpha _{2k-1}>0\) such that \(\alpha _{2k}:=\sigma ^{(k)}(t_{2k},\alpha _{2k-1})>0\). \(\square \)

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Riegel, U. Matching tower information with piecewise Pareto.
*Eur. Actuar. J. * **8**, 437–460 (2018). https://doi.org/10.1007/s13385-018-0177-3

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DOI: https://doi.org/10.1007/s13385-018-0177-3