*Article* **Unit-Linked Tontine: Utility-Based Design, Pricing and Performance †**

**An Chen <sup>1</sup> , Thai Nguyen 2,\* and Thorsten Sehner <sup>1</sup>**


**Abstract:** Due to the low demand for conventional annuities, alternative retirement products are sought. Quite recently, tontines have been frequently brought up as a promising option in this respect. Inspired by unit-linked life insurance and retirement products, we introduce unit-linked tontines in this article, where the tontine payoffs are directly linked to the development of the underlying financial market. More specifically, we consider two different tontine payoff structures differing in the (non-)inclusion of guaranteed payments. We first price the unit-linked tontines by using the risk-neutral pricing approach. Consequently, we study the attractiveness of these products for a utility-maximizing policyholder and compare them with non-unit-linked tontines. Our numerical analysis sheds light on the design challenges and gives explanations why similar products might not be widely adopted already.

**Keywords:** unit-linked tontine; product design; risk neutral pricing; utility optimization; utility performance

**JEL Classification:** G13; G22

#### **1. Introduction**

Unit-linked insurance policies belong to the most frequently concluded contracts in the life insurance sector; for example, more than 50% of the UK life (re)insurance gross written premiums were attributed to the index- and unit-linked insurance field in 2019 according to Statista (2020b). Among other attractive features, higher return expectations, flexibility, design possibilities and tax advantages (see, e.g., Schiereck et al. 2020) certainly play a driving role in the attractiveness of these policies. Interesting subject areas related to unit-linked insurance contracts, such as variable annuities, include pricing and valuation from the insurer's or the customers' perspective (see, e.g., Aase and Persson 1994; Ekern and Persson 1996; Gatzert et al. 2011), hedging strategies (see, e.g., Møller 1998), impact of stochastic interest rates (see, e.g., Schrager and Pelsser 2004) or guarantee components (see, e.g., Ledlie et al. 2008). In this paper, inspired by variable annuities, we design and investigate a new type of tontine that is directly linked to the developments in the financial market.

Yet, why is it even reasonable to consider tontines when dealing with old-age provision? From a theoretical point of view, actuarially fairly priced annuities should actually be regarded favorably by rational customers (see, e.g., Peijnenburg et al. 2016; Yaari 1965). However, annuitization rates are rather low in reality (see, e.g., Hu and Scott 2007). This adverse phenomenon known as the annuity puzzle (see, e.g., Ramsay and Oguledo 2018) is hitting conventional annuities. Moreover, due to low interest rate environments and tightening solvency regulations, it is hard to expect that annuitization rates will go up any

**Citation:** Chen, An, Thai Nguyen, and Thorsten Sehner. 2022. Unit-Linked Tontine: Utility-Based Design, Pricing and Performance. *Risks* 10: 78. https://doi.org/ 10.3390/risks10040078

Academic Editors: Ermanno Pitacco and Annamaria Olivieri

Received: 27 January 2022 Accepted: 29 March 2022 Published: 7 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

time soon. Therefore, alternative retirement products are naturally searched by insurers and customers, which brings up tontines as an option. Due to the backdrop of the demographic change (see, e.g., Margaras 2019), the so-called tontine retirement investment has become more and more important (see, e.g., Milevsky and Salisbury 2015; Sabin 2010). A main characteristic of tontines is that, in contrast to annuities, longevity risk is borne, to a great extent, by the pooled policyholders themselves. Hence, tontines are normally cheaper and, thus, potentially more attractive. Further discussions on practicalities, qualitative regulatory, technological and risk management issues associated with a tontine product can be found in Milevsky et al. (2018); Winter and Planchet (2021).

Let us briefly mention some of the recent literature that has addressed relevant topics related to tontine products. A general and historical view on tontines, as well as their possible applications for retirement income planning, is given in Milevsky (2015). Specific forms for the tontine payout structure are discussed in, e.g., Milevsky and Salisbury (2015). The question regarding how the tontine principle can be used to create tontine pensions for employees is studied in Forman and Sabin (2015). In Gemmo et al. (2020), investment possibilities in both tontines and traditional financial assets are investigated. Fairness issues when considering heterogeneous cohorts are considered in, e.g., Chen et al. (2020); Denuit (2019); Donnelly et al. (2014); Milevsky and Salisbury (2016); Sabin (2010). Bernhardt and Donnelly (2019) study the inclusion of bequest motives in tontine products. Recently, research on reasonable ways to combine tontines and annuities has been more extensively explored, see, e.g., Chen and Rach (2019); Chen et al. (2019, 2020); Weinert and Gründl (2020). However, to the best of our knowledge, the idea to consider a tontine as a unit-linked product has not yet been considered in detail in the literature.

In this article, inspired by unit-linked life and retirement insurance products, we introduce unit-linked tontines (see Sehner (2021)). We analyze the pricing and attractiveness of such products where two concrete unit-linked tontine payoffs are considered. We base our product model on the tontine concept applied in, e.g., Milevsky and Salisbury (2015), where the deterministic payout function is replaced by a stochastic payout process that depends on the developments in the financial market. In the specific setting, one tontine payoff is designed to coincide with the pure value of a portfolio following a certain investment strategy in the financial market, while the other one includes guaranteed payments, such that the policyholders participate in high portfolio values, but are secured in bad market scenarios. We rely on the risk-neutral pricing approach to determine the premiums required to buy the corresponding unit-linked tontines. In order to highlight the potential of our unit-linked tontine variant, we conduct an expected utility analysis that is commonly used in such a context (see, e.g., Mitchell 2002; Yaari 1965). More specifically, we first search for the optimal investment strategy that maximizes the expected utility of the policyholder for a given unit-linked tontine variant. We then numerically compare the maximum expected utilities of the two variants. Our comparison also takes two traditional tontine alternatives without unit-linked payments into account, namely the optimal and the natural traditional tontine.

The main observations and results, which can be drawn from our numerical analysis, are as follows: The unit-linked tontine may perform better than the traditional tontine alternatives if the following circumstances are present: First, the initial number of pooled individuals is either very low or high. Second, the expected return of the tradable risky asset is high or its volatility is low, which leads to a higher market price of risk, working naturally in favor of the unit-linked tontine. Third, the policyholder's risk aversion or subjective discount rate is low. The additional financial risk component in the unit-linked tontine and the steady increase of the expected payment of the unit-linked tontine over time are respectively responsible for this. For our baseline parametrization, the certainty equivalent induced by the variant, whose payout process is defined by the pure portfolio value, is, for instance, about 8% higher than the one belonging to the optimal traditional tontine and about 11% higher than the one belonging to the natural traditional tontine. As the unit-linked tontine can be more successful among customers than the traditional

counterpart, it seems reasonable to further study it. We further observe that, if the pure portfolio value stipulates its payout process, the unit-linked tontine may yield a higher utility level than in the case where it includes guaranteed payments. For our baseline parametrization, the corresponding certainty equivalent is, for instance, about 27% higher. Nevertheless, the latter case might be attractive, especially to customers who consider additional guarantee components important. In particular, its performance approaches that of the superior variant if the expected return of the risky asset decreases or if the volatility of the risky asset or the policyholder's risk aversion increases.

The remainder of this article is organized as follows: Section 2 introduces the model setting including the general nature of the unit-linked tontine product and the underlying financial and mortality risks. In Section 3, we derive the pricing formulas not only for the general payment structure, but also for both concrete variants of the unit-linked tontine. In Section 4, we discuss the solution of the utility optimization problem for our two particular unit-linked payment designs. In Section 5, we conduct the numerical study and present its outputs. Section 6 concludes the article. Some additional mathematical derivations can be found in the Appendices A–D.

#### **2. Model Setting**

#### *2.1. Unit-Linked Tontine Product*

In order to model the unit-linked tontine product, we employ the tontine concept presented in, e.g., Milevsky and Salisbury (2015), and modify it according to our purposes. Threfore, the idea behind the tontine type established in Milevsky and Salisbury (2015), to which we also refer as the traditional tontine, is shortly reviewed here first. Initially, i.e., at time 0, the buyer of such a tontine pays a single premium to the providing life insurance company. After the insurer has issued tontines to *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> individuals at time 0, they are grouped together into a pool. For simplicity, it is assumed that these *n* individuals, who are also referred to as policyholders or participants, are homogeneous, i.e., they are all of the same age *x* ≥ 0 at time 0 and of the same gender (which implies that they all have the same mortality rate). As time goes by, the insurance company disburses contractually predetermined payments to living participants. Specifically, a living individual holding one of the traditional tontine contracts receives at time *t* ≥ 0, in the first place, a specific amount of money determined by the so-called tontine payout function denoted by *d<sup>t</sup>* , which is deterministic and initially stipulated. What is more, contingent on being alive at time *t*, there is the possibility that she obtains more than *d<sup>t</sup>* due to the fact that the theoretical payments to the dead participants, if existent, are distributed among the survivors in the pool. Owing to the homogeneity between the participants, this extra payment is given by (*n*−*Nt*)*d<sup>t</sup> Nt* , where the random variable *N<sup>t</sup>* denotes the stochastic number of participants alive at time *t*. Overall, we can summarize the total payment that is disbursed to the considered traditional tontine holder at time *t*, given that she is alive, in the following expression:

$$\mathbb{E}\left(\frac{(n-N\_t)d\_t}{N\_t} + d\_t\right) \mathbb{1}\_{\{\mathbb{f}\_x > t\}} = \frac{nd\_t}{N\_t} \mathbb{1}\_{\{\mathbb{f}\_x > t\}'} \tag{1}$$

where the random variable *ζ<sup>x</sup>* represents the stochastic remaining lifetime of the individual aged *x* at time 0. As there are no death benefits, it is clear that the policyholder's payments proportionally increase if more individuals in the pool pass away. Note that throughout the following sections, we always assume that the payments of the insurer to a tontine holder are continuously disbursed.

When considering the unit-linked tontine product, we focus on payments stemming from the purchase of this tontine that are explicitly linked to the financial market. In this way, the participants directly partake in the developments in the financial market. Our corresponding product model is adopted, to a great extent, from the traditional tontines described above. The only difference is that the deterministic tontine payout function *d<sup>t</sup>* is replaced by the so-called tontine payout process denoted by the stochastic process Ψ*<sup>t</sup>* . This process depends on the performance of the financial market and, hence, makes the tontine a unit-linked product. Apart from that, the role of Ψ*<sup>t</sup>* stays the same as the one of *d<sup>t</sup>* . Note that in this article, we study two specified variants for Ψ*<sup>t</sup>* that are introduced in Section 3.2. On the whole, similar to (1), the total payment being disbursed to a unit-linked tontine holder at time *t* and described by the stochastic process *D<sup>t</sup>* is given by

$$D\_t = \frac{n\Psi\_t}{N\_t} \mathbb{1}\_{\{\mathbb{\zeta}\_x > t\}}. \tag{2}$$

#### *2.2. Financial Market and Mortality Risk*

For the examination of the unit-linked tontine introduced in Section 2.1, we need to model the financial market. Hereinafter, we always consider the financial market in continuous time that consists of one risky and one risk-free asset. We assume that there are no transaction costs or liquidity risk when trading the assets in the market. Following the well-known Black–Scholes model (see Black and Scholes 1973), the stochastic value of the risky asset at time *t*, denoted by *S<sup>t</sup>* , is described by the following geometric Brownian motion:<sup>1</sup>

$$\mathbf{dS}\_{l} = \mu \mathbf{S}\_{l} \mathbf{d}t + \sigma \mathbf{S}\_{l} \mathbf{d} \mathbf{W}\_{l\prime} \quad \text{S}\_{0} > \mathbf{0},\tag{3}$$

where *W* is a standard Brownian motion. The dynamics of the risk-free asset is given by

$$\mathbf{d}B\_{l} = rB\_{l}\mathbf{d}t, \quad B\_{0} = 1,\tag{4}$$

where *r* is the risk-free interest rate. The three parameters *µ*, *σ* and *r* are constant over time in our setting and *µ* > *r* is assumed. Note that possible dividend payments existing in the described financial market are neglected in our framework.

Let *V<sup>t</sup>* be the value of a portfolio at time *t* that is generated by the investments of the insurer in the financial market. We assume that the fraction of the portfolio invested in the risky asset at time *t* is described by the *deterministic* trading strategy *πt*∈ [0, 1]. This means that neither short selling of the risky portfolio nor leverage is allowed. The remaining fraction (1 − *πt*) is invested in the risk-free asset. By the self-financing property, the dynamics of *V<sup>t</sup>* under P is given by

$$\mathbf{d}V\_{\mathbf{f}} = \pi\_{\mathbf{f}} \frac{V\_{\mathbf{f}}}{S\_{\mathbf{f}}} \mathbf{d}S\_{\mathbf{f}} + (1 - \pi\_{\mathbf{f}}) \frac{V\_{\mathbf{f}}}{B\_{\mathbf{f}}} \mathbf{d}B\_{\mathbf{f}} = (r + \pi\_{\mathbf{f}}(\mu - r))V\_{\mathbf{f}} \mathbf{d}t + \sigma \pi\_{\mathbf{f}} V\_{\mathbf{f}} \mathbf{d}W\_{\mathbf{f}\iota} \quad V\_{0} > 0. \tag{5}$$

It can be shown that the explicit solution of the stochastic differential Equation (5) is given by

$$\mathbf{V}\_{t} = V\_{0}\mathbf{e}^{rt+(\mu-r)}\int\_{0}^{t} \pi\_{s}\mathbf{ds} - \frac{\mathbf{e}^{2}}{2}\int\_{0}^{t} \pi\_{s}^{2}\mathbf{ds} + \sigma\int\_{0}^{t} \pi\_{s}\mathbf{dW}\_{s} \tag{6}$$

Besides the financial risk, mortality risk is also contained in the unit-linked tontine. It stems from two sources, namely the unsystematic mortality risk and the systematic mortality risk (see, e.g., Dahl et al. 2008). The unsystematic mortality risk arises from the randomness of deaths in the pool with a known mortality law. This risk is diversifiable, i.e., it disperses if the size of the pool grows. In contrast, the systematic mortality risk is not diversifiable, even if the pool size is large, as it results from overarching changes in the underlying mortality intensity. For the traditional tontines (with mortality risk exclusively) and an infinite pool size, all the mortality risk is shared by the policyholders. With a finite pool size, the insurer only has the risk generated by the death time of the last survivor, at which the insurer stops its payment. Additionally, in unit-linked tontines, there is financial market risk. Depending on the risk management strategies the insurer chooses, the insurer might still retain some financial market risk.

To model the mortality risk, we use the following framework: The probability (under P) that the considered individual survives the next *t* years from time 0 on, at which she is *x* years old, is denoted by *<sup>t</sup> p<sup>x</sup>* ∈ (0, 1]. To include the above-mentioned systematic mortality

risk component, we, similar to, e.g., Lin and Cox (2005), allow for a mortality shock that is represented by a random variable denoted by *e*. We assume that *e* has a density function denoted by *f<sup>e</sup>* and that its moment-generating function denoted by *M<sup>e</sup>* exists. The shocked survival curve is then given by *<sup>t</sup> p<sup>x</sup>* 1−*e* . We set the range of *e* to (−∞, 1), so that *<sup>t</sup> p<sup>x</sup>* <sup>1</sup>−*<sup>e</sup>* <sup>∈</sup> (0, 1] is preserved. If no mortality shock is existent, simply let *<sup>e</sup>* <sup>=</sup> <sup>0</sup> a.s. We remark that the latest insurance solvency regulations require insurers to test their balance sheets against various stress-test scenarios. For instance, in the Canadian solvency regulation, a 10–20% decrease of mortality rates (depending on the type of annuity is assumed for a longevity shock). The U.S. regulation assumes a stress on mortality improvement between 16–40% (depending on the age). This results in lower mortality rates between 0.7–6%. In Solvency II, which is implemented for insurance undertakings in the EU, a longevity shock is defined as a decrease of annual death probabilities by 20%. The simple model we have chosen reflects the spirit of these realistic regulation frameworks.

For the random variable *N<sup>t</sup>* , which is affected by mortality risk, we can obtain the following distribution under P when conditioning on the survival of the considered policyholder and on *e*:

$$(N\_t - 1 \vert \zeta\_x > t, \varepsilon) \stackrel{\mathbb{P}}{\sim} \text{Bin}\left(n - 1, \iota\_t p\_x^{1 - \varepsilon}\right),\tag{7}$$

where we use the assumption that the lifetimes of the participants are stochastically independent under P.

Following the main stream of unit-linked insurance products (e.g., Aase and Persson 1994; Bacinello et al. 2018; Bernhardt and Donnelly 2019; Briys and de Varenne 1994), we suppose that *W*, constituting the financial risk, is stochastically independent of (*ζx*, *N*, *e*) under P. Note that this requirement does usually not pose a restriction as the development of the value of the risky asset and the chances of survival do generally not interact. We remark that the independence assumption of actuarial and financial risk in the real world may be quite reasonable in many situations. Recent research however finds that shocks in stock market wealth might have an impact on mortality. For example, Giulietti et al. (2020) provide evidence that daily fluctuations in the stock market have important effects on fatal car accidents. Schwandt (2018) demonstrates that stock wealth shocks that lead to losses in the wealth of stock-holding retirees affect the health of retirees in the US. In our paper, the independence assumption of these risks allows us to analyze the pricing problem and individual welfare of the unit-linked tontine in a semi-explicit way.

Let G = {G*t*}*t*≥<sup>0</sup> be the filtration generated by the Brownian motion *W* and denote the natural filtration with respect to *<sup>ζ</sup>x*, *<sup>N</sup>* and *<sup>e</sup>* by <sup>H</sup> <sup>=</sup> {H*t*}*t*≥<sup>0</sup> . The resulting progressively enlarged filtration is given by <sup>F</sup> <sup>=</sup> {F*t*}*t*≥<sup>0</sup> , whose element F*<sup>t</sup>* = G*<sup>t</sup>* ∨ H*<sup>t</sup>* contains all relevant information revealed until time *t*.

#### **3. Pricing**

In this section, we aim at pricing the unit-linked tontine product established in Section 2.1, i.e., we determine the single initial premium denoted by *P*<sup>0</sup> > 0 that needs to be paid by a policyholder to the insurance company. As we employ the standard risk-neutral pricing approach to find *P*0, we have to clarify how a risk-neutral probability measure denoted by Q is chosen when mortality risk is also taken into account. First, it is clear that, due to the dependence of *D* on the survival of the policyholder and the other participants, the market, in which the unit-linked tontine is traded, is incomplete. Thus, a risk-neutral probability measure is not unique and, hence, there is, in general, also no unique price *P*0. For a concrete choice of Q, we assume that the insurer considers the financial risk and the mortality risk separately when determining Q, whereby the stochastic independence of these two risk categories is also supposed under Q. Further discussions about the independence property between financial and actuarial risks in the Pand the Q-worlds can be found in, e.g., Dhaene et al. (2013), where the authors investigate

the conditions under which it is possible (or not) to transfer the independence assumption from the physical measure P to the risk-neutral pricing measure Q.

Regarding the financial risk that is captured by the filtration G, we expect the insurer to use the risk-neutral probability measure, which, if we restrict ourselves to G, exists and is unique due to the completeness of the financial market described in Section 2.2. Note that the explicit solution for *V<sup>t</sup>* , which is under P given in (6), changes accordingly under Q to

$$dV\_t = V\_0 e^{rt - \frac{\sigma^2}{2} \int\_0^t \pi\_s^2 \text{ds} + \sigma \int\_0^t \pi\_s \text{dW}\_s^{\mathbb{Q}}}\,\tag{8}$$

where *W* Q *t* is a standard Brownian motion under Q.

*t*≥0 Following Chen and Rach (2019), we assume that the choice of Q on H for pricing purposes depends on the nature of the overall insurance business of the life insurance company. If a large product range is offered, there may already be some natural hedges between the products and, thus, the insurer would be faced with less mortality risk than in the case in which it solely concentrates on one specific product field. We assume that the insurer only trades tontine products and that, also due to the resulting higher mortality risk exposure, the insurer is prudent when charging premiums, i.e., safety loadings are to be included in some way. If *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* <sup>∈</sup> (0, 1] denotes the survival probability under <sup>Q</sup>, and since tontines belong to the retirement product type, a possibility to reflect the insurer's pricing prudence is to require that

$$
\mu\_t \widetilde{p}\_{\mathbf{x}} \ge \, \_t p\_{\mathbf{x} \boldsymbol{\nu}} \tag{9}
$$

and that the mortality shock *e* follows the same distribution under Q as under P. Given these requirements, to which we stick in the following, the (shocked) survival curve under Q runs at a higher level than the one under P, which leads to the inclusion of implicit safety loadings in premiums. If the insurer increases *<sup>t</sup> <sup>p</sup>*e*x*, the company is more conservative about pricing. The choice of the magnitude of *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* usually depends on the pool size *<sup>n</sup>* since, as already pointed out in Section 2.2, the unsystematic mortality risk becomes less relevant if *n* grows. Therefore, *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* normally attains a rather low value if the pool size is large. As it is determined that changing the probability measure from P to Q does not have an impact on the distribution type of the random variable *N<sup>t</sup>* , we simply replace <sup>P</sup> by <sup>Q</sup> and *<sup>t</sup> <sup>p</sup><sup>x</sup>* by *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* in (7) when specifying the distribution of *N<sup>t</sup>* under Q. The stochastic independence of the remaining lifetimes of the participants is preserved under Q accordingly.

Having clarified the risk-neutral probability measure Q, we discuss the pricing of the unit-linked tontine in the following. We will start with a general tontine payout process Ψ*<sup>t</sup>* and then continue by examining specified alternatives for it. We always assume that the rates of convergence of *<sup>t</sup> <sup>p</sup><sup>x</sup>* and *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* towards 0 if *<sup>t</sup>* goes to infinity exceed the rates of convergence or divergence of the other time-dependent quantities in order to guarantee that all improper integrals with respect to *t* necessary throughout the subsequent sections exist.<sup>2</sup>

#### *3.1. General Payment Structure*

First, let Ψ*<sup>t</sup>* be a general tontine payout process. Next, the single initial premium *P*<sup>0</sup> can be calculated via the risk-neutral pricing approach as

$$\begin{split} P\_0 &= E\_{\mathbb{Q}} \left[ \int\_0^\infty e^{-rt} D\_l \mathbf{d}t \bigg| \mathcal{F}\_0 \right] = E\_{\mathbb{Q}} \left[ \int\_0^\infty e^{-rt} \frac{n \Psi\_t}{N\_l} \mathbb{1}\_{\{\zeta\_x > t\}} \mathbf{d}t \right] \\ &= n \int\_0^\infty e^{-rt} E\_{\mathbb{Q}} [\Psi\_t] E\_{\mathbb{Q}} \left[ \frac{\mathbb{1}\_{\{\zeta\_x > t\}}}{N\_l} \right] \mathbf{d}t, \end{split} \tag{10}$$

where the stochastic independence between *W*<sup>Q</sup> and (*ζx*, *N*, *e*) is applied in the last step. The latter expected value in (10) is given by

$$E\_{\mathbb{Q}}\left[\frac{\mathbb{1}\_{\{\xi\_{x}>t\}}}{N\_{t}}\right] = \frac{1}{n}I\_{t\prime} \tag{11}$$

where

$$I\_l = \int\_{-\infty}^{1} \left( 1 - \left( 1 - \, \widetilde{p}\_x \, ^{1-z} \right)'' \right) f\_\varepsilon(z) \, \mathrm{d}z. \tag{12}$$

The detailed derivation of (11) is reported in Appendix A. Consequently, we obtain the following general pricing formula:

$$P\_0 = \int\_0^\infty e^{-rt} I\_t E\_\mathbb{Q}[\Psi\_t] \mathbf{d}t. \tag{13}$$

#### *3.2. Specified Payment Structures*

In the following, we consider two specified variants for the tontine payout process Ψ*<sup>t</sup>* , which can be interesting to examine and may have potential for tontine product design. We determine the single premiums that need to be contributed by the individual if she wants to buy the corresponding unit-linked tontine.

As our focus is on payments with a direct linkage to the financial market, i.e., to the developments of the risky and of the risk-free asset, hereafter, we assume that for payout purposes, the insurer creates a tontine payment account Ψ whose value can be amounted to the portfolio given in (8). We assess the following cases on how to potentially define Ψ*<sup>t</sup>* :

(A) Let us first consider the case where the tontine payout process is equal to the portfolio value *V* explicitly given in (8), i.e.

$$
\Psi\_t = V\_t.\tag{14}
$$

This means that the tontine payout process at time *t* simply complies with a money stock amounting to *V<sup>t</sup>* . To generate this amount, the insurance company can invest in the risky and the risk-free asset according to the trading strategy applied in the corresponding portfolio. By the choice given in (14), the full potential of the financial market will be passed on to the customers within a tontine framework. By (2), the total tontine payment to the policyholder at time *t* in this case is given by *nV<sup>t</sup> Nt* <sup>1</sup>{*ζx*>*t*} .

(B) Second, inspired by participating life insurance policies with guaranteed payments (see, e.g., Briys and de Varenne 1994), we stipulate

$$\Psi\_{\rm I} = \mathcal{G}\_{\rm I} + \mathfrak{a} \left( V\_{\rm I} - \mathcal{G}\_{\rm I} \right)^{+},\tag{15}$$

where *G<sup>t</sup>* > 0 denotes the guaranteed payment at time *t* and *α* ∈ (0, 1] is the constant participation rate, and where (*V<sup>t</sup>* − *Gt*) <sup>+</sup> <sup>=</sup> max{*V<sup>t</sup>* <sup>−</sup> *<sup>G</sup><sup>t</sup>* , 0}. Thus, the tontine payout process coincides here with a predetermined payment function represented by *G<sup>t</sup>* as long as the financial market performs poorly, i.e., *V<sup>t</sup>* is low, so that *V<sup>t</sup>* ≤ *G<sup>t</sup>* holds. On the contrary, if the financial market performs well, i.e., *V<sup>t</sup>* is high, so that *V<sup>t</sup>* > *G<sup>t</sup>* , an additional participation in the positive difference *V<sup>t</sup>* − *G<sup>t</sup>* at the rate *α* is included. Employing the choice given in (15) can satisfy customers, who appreciate additional guarantee components smoothing uncertain payout structures. By (2), the total tontine payment to the policyholder at time *<sup>t</sup>* in this case is given by *<sup>n</sup>*(*Gt*+*α*(*Vt*−*Gt*) + ) *Nt* <sup>1</sup>{*ζx*>*t*} . The tontine pricing in Cases A and B is summarized in the following two propositions:

**Proposition 1** (Case A)**.** *If* Ψ*<sup>t</sup> is defined as in* (14)*, the single initial premium of the resulting version of the unit-linked tontine product is given by*

$$P\_0 = V\_0 \int\_0^\infty I\_l \,\mathrm{d}t.\tag{16}$$

**Proof.** With the aid of the general pricing formula given in (13) and by using

$$E\_{\mathbb{Q}}[\mathbf{Y}\_{t}] = E\_{\mathbb{Q}}[V\_{t}] = V\_{0}e^{rt}$$

due to the fact that the discounted portfolio value process is a Q-martingale, we obtain (16).

**Proposition 2** (Case B)**.** *If* Ψ*<sup>t</sup> is defined as in* (15)*, the single initial premium of the resulting version of the unit-linked tontine product is given by*

$$P\_0 = \int\_0^\infty e^{-rt} I\_t\left(\mathcal{G}\_t + \mathfrak{a}\left(V\_0 e^{rt} \Phi\left(\tilde{d}\_t\right) - \mathcal{G}\_t \Phi\left(\hat{d}\_t\right)\right)\right) \mathrm{d}t,\tag{17}$$

*where* Φ *is the distribution function of the standard normal distribution and the functions d*e*<sup>t</sup> and d*b*<sup>t</sup> are given by*

$$\tilde{d}\_{t} = \frac{\ln\left(\frac{V\_{0}}{G\_{t}}\right) + rt + \frac{\sigma^{2}}{2}\int\_{0}^{t} \pi\_{s}^{2} \mathrm{ds}}{\sigma\sqrt{\int\_{0}^{t} \pi\_{s}^{2} \mathrm{ds}}} \quad \text{and} \quad \hat{d}\_{t} = \tilde{d}\_{t} - \sigma\sqrt{\int\_{0}^{t} \pi\_{s}^{2} \mathrm{ds}}.\tag{18}$$

**Proof.** The proof of Proposition 2 is reported in Appendix B.1.

**Remark 1.** *From the insurer's perspective, managing such unit-linked products would require the insurer to pay transaction costs that are linked to hedging activities against fluctuations of the risky asset in the financial market and of the mortality development. For instance, in Case B, if we ignore the mortality risk, the insurer has to hedge against selling a guaranteed amount plus the call option, which by put-call parity is equivalent to selling the portfolio value plus the put option. In bad market scenarios when the risk asset price goes down, more hedging activities would be needed; hence, it is true that the relative transaction price will be higher if the tail is longer. A thorough analysis that includes transaction costs is interesting and left for future research. In the real-world implementation, these transaction costs do impact the product design. We remark that in the presence of transaction costs, hedging and pricing are no longer valid in the classical Black and Scholes model. In such contexts, Leland's increasing volatility method, as per Leland (1985), would be helpful for compensating transaction costs and an approximately complete replication can be expected by using the delta strategy calculated from a modified Black–Scholes equation with an appropriate modified volatility. This prescription is based on the idea that the presence of transaction costs implies an extra fee, which is necessary for the option seller in the replication problem, i.e., options become more expensive in the presence of transaction costs.*

#### **4. Utility Optimization**

In the following, we conduct a utility maximization analysis to find out which of the two unit-linked tontine variants suggested in Section 3.2 is more preferable to an individual investor. To this end, for a given unit-linked tontine variant, we search for the optimal investment strategy that maximizes the discounted expected utility of the policyholder in this section. We numerically compare the utility optimums of the two variants with each other and with those of the traditional tontine alternatives without unit-linked payments in Section 5.

Subsequently, we always assume that the policyholder's utility function, denoted by *u*, is of constant relative risk aversion:

*<sup>u</sup>*(*c*) = *<sup>c</sup>* 1−*γ* 1 − *γ* , (19)

where *c* > 0 represents the consumable input and *γ* > 0 adhering to *γ* 6= 1 is the measure of the policyholder's relative risk aversion. This choice is one of the most frequently used utility functions to capture the preferences of individuals (see, e.g., Levy 1994; Sharpe 2017).<sup>3</sup> In the design problems below, we assume that expectations are not subjective.

#### *4.1. General Payment Structure*

For a general tontine payout process Ψ*<sup>t</sup>* , the objective of the optimization problem, i.e., the discounted expected utility, can be formulated and transformed as follows:

$$E\_{\mathbb{P}}\left[\int\_{0}^{\infty} e^{-\rho t} \mu\left(\frac{n\Psi\_{t}}{N\_{t}}\right) \mathbb{1}\_{\{\zeta\_{x}>t\}} \mathrm{d}t\right] = \frac{n^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\rho t} \kappa\_{t} E\_{\mathbb{P}}\left[\Psi\_{t}^{1-\gamma}\right] \mathrm{d}t,\tag{20}$$

where *ρ* is the constant subjective discount rate of the individual and

$$\begin{split} \kappa\_{l} &= E\_{\mathbb{P}} \left[ \frac{\mathbb{1}\_{\{\xi\_{\mathbf{x}} > 1\}}}{N\_{l}^{1-\gamma}} \right] \\ &= \sum\_{k=0}^{n-1} \frac{1}{(k+1)^{1-\gamma}} \binom{n-1}{k} \int\_{-\infty}^{1} \binom{p\_{\mathbf{x}} - z}{}^{k+1} \left(1 - {}\_{l}p\_{\mathbf{x}}\, ^{1-z}\right)^{n-1-k} f\_{\mathbf{c}}(z) \, \mathrm{d}z. \end{split} \tag{21}$$

The formulation of the discounted expected utility in (20) arises from translating the formula in (10) into the utility framework, while its transformation results from applying the power utility function given in (19) and similar calculation techniques as before. Since the individual has to provide a single initial premium out of her available initial wealth, denoted by *v* > 0, to buy the tontine product, the pricing formula found in Section 3, where the general version is given in (13) and the specified ones in (16) and (17), naturally forms the budget constraint in the optimization problem. The decision variables in the optimization problem are typically appropriate quantities occurring in the tontine payout process Ψ*<sup>t</sup>* . This means that we eventually search for the optimal specific form of Ψ*<sup>t</sup>* , which determines the tontine disbursements in such a way that the policyholder is endowed with the highest utility level possible. The general representative maximization problem overall is given by:

#### **Problem 1.**

$$\begin{aligned} \max\_{\left(\Psi\_{t}\right)\_{t\geq 0}} & \frac{n^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\rho t} \kappa\_{t} E\_{\mathbb{P}} \left[\Psi\_{t}^{1-\gamma}\right] \mathbf{d}t, \\\text{s.t. } & v = P\_{0} = \int\_{0}^{\infty} e^{-rt} I\_{t} E\_{\mathbb{Q}} [\Psi\_{t}] \mathbf{d}t. \end{aligned}$$

Note that, strictly speaking, we shall put *v* ≥ *P*<sup>0</sup> in the budget constraint. However, as is typically done in this kind of optimization problem, the budget constraint is binding in the optimal solution due to the steadily positive slope of *u*, such that we start immediately with equality in the constraint.

#### *4.2. Specified Payment Structures*

Now, we consider the particular unit-linked payment designs from Section 3.2 specifying the tontine payout process Ψ*<sup>t</sup>* in two different ways and modify Problem 1 accordingly. The emerging optimization problems are then, if possible, solved analytically. Concerning the fractions invested in the risky and the risk-free asset, we henceforth assume that they stay constant over time and are non-negative and bounded from above by 1, i.e., *π<sup>t</sup>* = *π* ∈ [0, 1] for all *t*. Note that these assumptions do not actually pose a strict restriction: By their invariability, the fractions can also be regarded as the perpetual average percentages which determine the long-term mean composition of the portfolio. By generally forbidding short selling, we account for the fact that bans on short selling (can) indeed exist, as in the case in Europe in March 2020 during the coronavirus pandemic showed (see, e.g., Smith 2020). Applying a constant *π* simplifies the equations in (5), (6), (8) and (18), accordingly.

**Case A:** Recall that we assume for Case A that Ψ*<sup>t</sup>* = *V<sup>t</sup>* holds. Therefore, it is reasonable to choose *π* and *V*<sup>0</sup> (note that *V*<sup>0</sup> is not *ν*, the initial wealth) as the decision variables in the corresponding optimization problem. In other words, we look for the optimal portfolio parameter combination, namely for the fraction invested in the risky asset and the initial investment amount that is supposed to be determined in such a way that the policyholder comes off best.<sup>4</sup> The appropriate maximization problem derived from Problem 1 and (16) is given by

**Problem 2** (Case A-bounded investment strategy)**.**

$$\begin{aligned} \max\_{(\Psi\_t)\_{t\geq 0}} & \frac{n^{1-\gamma}}{1-\gamma} \int\_0^\infty e^{-\rho t} \kappa\_t E\_\mathbb{P} \left[ \Psi\_t^{1-\gamma} \right] \mathbf{d}t, \\\text{s.t. } & v = P\_0 = \int\_0^\infty e^{-rt} I\_t E\_\mathbb{Q} \left[ \Psi\_t \right] \mathbf{d}t. \end{aligned}$$

The objective of Problem 2 results from employing *E*<sup>P</sup> h *V* 1−*γ t* i = *V*0*e r*+(*µ*−*r*)*π*− *γσ*2 <sup>2</sup> *π* 2 *t* 1−*γ* .

As it is possible to solve this problem analytically, we summarize the related optimizing quantities in a proposition:

**Proposition 3.** *The optimal values π* ∗*A and V*∗ 0 *A for π and V*<sup>0</sup> *solving Problem 2 are given by*

$$
\pi^{\*A} = \frac{\mu - r}{\gamma \sigma^2} \mathbb{1}\_{\left\{ \mu - r \le \gamma \sigma^2 \right\}} + \mathbb{1}\_{\left\{ \mu - r > \gamma \sigma^2 \right\}} \quad \text{and} \quad V\_0^{\*A} = \frac{v}{\int\_0^\infty I\_l \mathbf{d}t}. \tag{22}
$$

**Proof.** The proof of Proposition 3 is reported in Appendix B.2.

We observe that the optimal value for the trading strategy in Proposition 3 coincides with Merton's fraction if *<sup>µ</sup>* <sup>−</sup> *<sup>r</sup>* <sup>≤</sup> *γσ*<sup>2</sup> (see Merton 1969).

**Case B:** As we assume for Case B that Ψ*<sup>t</sup>* = *G<sup>t</sup>* + *α*(*V<sup>t</sup>* − *Gt*) + holds, it is sensible to again choose *π* and *V*<sup>0</sup> as the decision variables in the corresponding optimization problem.<sup>5</sup> By means of Problem 1 and (17), the maximization problem for Case B can be formulated as follows:

**Problem 3** (Case B-bounded investment strategy)**.**

$$\begin{split} \max\_{(\boldsymbol{\pi},\boldsymbol{V}\_{0}) \in [0,1] \times (0,\infty)} & \frac{n^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\rho t} \boldsymbol{\kappa}\_{t} \\ \quad \cdot \boldsymbol{G}\_{t}^{1-\gamma} \left( \boldsymbol{\Phi} \left( \overline{\boldsymbol{d}}\_{t} \right) + \int\_{0}^{\infty} \left( 1 + a \left( e^{\sigma \pi \sqrt{t} \boldsymbol{y}} - 1 \right) \right)^{1-\gamma} \boldsymbol{\phi} \left( \boldsymbol{y} + \overline{\boldsymbol{d}}\_{t} \right) \mathbf{d} \boldsymbol{y} \right) \mathbf{d}t \\ \text{s.t. } \boldsymbol{v} = \boldsymbol{P}\_{0} = \int\_{0}^{\infty} e^{-\boldsymbol{r}t} I\_{t} \left( \boldsymbol{G}\_{t} + a \left( V\_{0} \boldsymbol{e}^{\boldsymbol{r}t} \boldsymbol{\Phi} \left( \widetilde{\boldsymbol{d}}\_{t} \right) - \boldsymbol{G}\_{t} \boldsymbol{\Phi} \left( \widetilde{\boldsymbol{d}}\_{t} \right) \right) \right) \mathbf{d}t. \end{split}$$

The objective of Problem 3 results from employing similar calculation techniques as before, which, inter alia, leads to

$$\begin{split} &E\_{\mathbb{P}}\left[\left(\mathcal{G}\_{l}+\mathfrak{a}\left(V\_{l}-\mathcal{G}\_{l}\right)^{+}\right)^{1-\gamma}\right] \\ = &\mathcal{G}\_{l}^{1-\gamma}\Phi\left(\overline{d}\_{l}\right)+\int\_{\overline{d}\_{l}}^{\infty}\left(\mathcal{G}\_{l}+\mathfrak{a}\left(V\_{0}e^{\tau t+(\mu-r)\pi t-\frac{\sigma^{2}}{2}\pi^{2}t+\sigma\pi\sqrt{l}z}-\mathcal{G}\_{l}\right)\right)^{1-\gamma}\phi(z)dz \\ = &\mathcal{G}\_{l}^{1-\gamma}\left(\Phi\left(\overline{d}\_{l}\right)+\int\_{0}^{\infty}\left(1+\mathfrak{a}\left(e^{\sigma\pi\sqrt{l}y}-1\right)\right)^{1-\gamma}\phi\left(y+\overline{d}\_{l}\right)dy\right), \end{split} \tag{23}$$

where *φ* is the density of the standard normal distribution. Further, the substitution *y* = *z* − *d<sup>t</sup>* is applied in the third line and the function *d<sup>t</sup>* is given by

$$\overline{d}\_{l} = \frac{\ln\left(\frac{G\_{l}}{V\_{0}}\right) - rt - (\mu - r)\pi t + \frac{\sigma^{2}}{2}\pi^{2}t}{\sigma\pi\sqrt{t}}.\tag{24}$$

If we try to solve this problem by using the method of Lagrange multipliers (see, e.g., Bertsekas 2014), the corresponding Lagrange function L(*π*, *V*0, *λ*), where *λ* is the Lagrange multiplier, is defined as

$$\begin{split} \mathcal{L}(\boldsymbol{\pi}, \boldsymbol{V}\_{0}, \boldsymbol{\lambda}) &= \frac{n^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\rho t} \kappa\_{t} \mathbb{G}\_{t}^{1-\gamma} \left( \boldsymbol{\Phi} \left( \tilde{\boldsymbol{d}}\_{t} \right) + \int\_{0}^{\infty} \left( 1 + a \left( e^{\boldsymbol{\sigma} \boldsymbol{\pi} \sqrt{t} \boldsymbol{y}} - 1 \right) \right)^{1-\gamma} \boldsymbol{\Phi} \left( \boldsymbol{y} + \tilde{\boldsymbol{d}}\_{t} \right) \mathbf{d} \boldsymbol{y} \right) \mathrm{d}t \\ &+ \lambda \left( \boldsymbol{\sigma} - \int\_{0}^{\infty} e^{-\boldsymbol{\sigma} t} \boldsymbol{I}\_{t} \left( \boldsymbol{G}\_{t} + a \left( V\_{0} e^{\boldsymbol{\sigma} t} \boldsymbol{\Phi} \left( \tilde{\boldsymbol{d}}\_{t} \right) - \boldsymbol{G}\_{t} \boldsymbol{\Phi} \left( \tilde{\boldsymbol{d}}\_{t} \right) \right) \right) \mathrm{d}t \right) . \end{split} \tag{25}$$

The first-order condition with respect to *π* is given as

$$\frac{\partial}{\partial \pi} \mathcal{L}(\pi, V\_0, \lambda) = \frac{n^{1-\gamma}}{1-\gamma} \int\_0^\infty e^{-\rho t} \kappa\_t G\_t^{1-\gamma} \left(\phi\left(\overline{d}\_t\right) \frac{1}{\pi} \widetilde{d}\_t + \int\_0^\infty \left(1 + a\left(e^{\sigma \pi \sqrt{\lambda} y} - 1\right)\right)^{-\gamma} \phi\left(y + \overline{d}\_t\right) \right. \\ \left. + \left. \left(y + \overline{d}\_t\right) \right|\_{\mathcal{L}} - \int\_0^\infty \left(y + \overline{d}\_t\right) \left(y + \overline{d}\_t\right) \frac{1}{\pi} \widetilde{d}\_t \right) \mathrm{d}y \right) dt \tag{26}$$
 
$$ -\lambda a V\_0 \sigma \int\_0^\infty I\_t \phi\left(\widetilde{d}\_t\right) \sqrt{t} \mathrm{d}t = 0.$$

The first-order condition with respect to *V*<sup>0</sup> is given as

$$\begin{split} \frac{\partial}{\partial V\_{0}} \mathcal{L}(\boldsymbol{\pi}, \boldsymbol{V}\_{0}, \boldsymbol{\lambda}) &= \frac{n^{1-\gamma}}{(1-\gamma)\boldsymbol{V}\_{0}\boldsymbol{\sigma}\boldsymbol{\pi}} \int\_{0}^{\infty} e^{-\rho t} \kappa\_{t} G\_{t}^{1-\gamma} \frac{1}{\sqrt{t}} \left( \int\_{0}^{\infty} \left(1 + a\left(e^{\boldsymbol{\sigma}\boldsymbol{\pi}\sqrt{t}\boldsymbol{y}} - 1\right)\right)^{1-\gamma} \\ & \cdot \phi\left(\boldsymbol{y} + \overline{d}\_{t}\right) \left(\boldsymbol{y} + \overline{d}\_{t}\right) \mathrm{d}y - \phi\left(\overline{d}\_{t}\right)\right) \mathrm{d}t - \lambda a \int\_{0}^{\infty} \boldsymbol{I}\_{t} \Phi\left(\widetilde{d}\_{t}\right) \mathrm{d}t = 0, \end{split} \tag{27}$$

and the one with respect to *λ* naturally coincides with the budget constraint:

$$v = \int\_0^\infty e^{-rt} I\_l\left(\mathcal{G}\_l + a\left(V\_0 e^{rt} \Phi\left(\widetilde{d}\_l\right) - \mathcal{G}\_l \Phi\left(\widehat{d}\_l\right)\right)\right) \mathrm{d}t. \tag{28}$$

From (26) and (27), the following equation must hold true:

$$\begin{split} &\int\_{0}^{\infty} e^{-\rho t} \kappa\_{l} G\_{l}^{1-\gamma} \left( \phi \left( \overline{d}\_{l} \right) \widetilde{d}\_{l} + \int\_{0}^{\infty} \left( 1 + a \left( e^{\rho \pi \sqrt{t}y} - 1 \right) \right)^{-\gamma} \phi \left( y + \overline{d}\_{l} \right) \right. \\ &\left. \cdot \left( (1-\gamma) a e^{\rho \pi \sqrt{t}y} \sigma \pi \sqrt{t} y - \left( 1 + a \left( e^{\rho \pi \sqrt{t}y} - 1 \right) \right) \left( y + \overline{d}\_{l} \right) \widetilde{d}\_{l} \right) \mathrm{d}y \right) \mathrm{d}t \int\_{0}^{\infty} I\_{l} \Phi \left( \widetilde{d}\_{l} \right) \mathrm{d}t \\ &= \int\_{0}^{\infty} e^{-\rho t} \kappa\_{l} G\_{l}^{1-\gamma} \frac{1}{\sqrt{t}} \left( \int\_{0}^{\infty} \left( 1 + a \left( e^{\rho \pi \sqrt{t}y} - 1 \right) \right)^{1-\gamma} \Phi \left( y + \overline{d}\_{l} \right) \left( y + \overline{d}\_{l} \right) \mathrm{d}y - \phi \left( \overline{d}\_{l} \right) \right) \mathrm{d}t \\ &\cdot \int\_{0}^{\infty} I\_{l} \Phi \left( \widetilde{d}\_{l} \right) \sqrt{t} \mathrm{d}t. \end{split} \tag{29}$$

For the calculations in (26) and (27), the following identities are applied: *<sup>∂</sup>d<sup>t</sup> ∂π* <sup>=</sup> <sup>1</sup> *π d*e*t* , *∂d*b *t ∂π* = *∂d*e *t ∂π* − *σ* √ *t* and

$$V\_0 e^{rt} \phi \left(\widetilde{d}\_l\right) - G\_l \phi \left(\widehat{d}\_l\right) = 0. \tag{30}$$

The detailed derivation of (30) is reported in Appendix A. The solution of the system of Equations (28) and (29) (when it exists) provides the optimal values for *π* and *V*<sup>0</sup> in Case B. However, due to the complexity of this system of equations, we are unable to find explicit formulas for the solution of Problem 3. Therefore, in what follows, we numerically solve Problem 3 to find the optimal values *π*g∗<sup>B</sup> and *V*g<sup>∗</sup> 0 B .

#### **5. Numerical Analysis**

In this section, we aim at discovering distinct characteristics of the introduced unitlinked tontine product by means of numerical studies. For these studies, concrete assumptions about definite numbers for the various appearing parameters and about other modeling implementations need to be made initially. Subsequently, the specific main objective is to compare, in terms of the utility of a policyholder, our two different variants for the unit-linked tontine product established and priced in Section 3.2, and optimized in Section 4.2 within several sensitivity analyses. Additionally, we seek to integrate the traditional tontine with non-unit-linked payments into this comparison. Thereby, we are able to indicate whether the individual, in the analyzed instances, prefers that the tontine payment is linked to the financial market.

#### *5.1. Setup*

First, let us set up the overall framework with the different assumptions for our numerical studies. We start with the determination of the modeling of the shocked survival curves *<sup>t</sup> p<sup>x</sup>* <sup>1</sup>−*<sup>e</sup>* and *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e* , respectively. We initially specify the survival probabilities *<sup>t</sup> p<sup>x</sup>* and *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* as

$$\mathbf{p}\_t \mathbf{p}\_x = e^{-\int\_0^t \mathbf{m}\_{x+\tau} \mathbf{d}\tau} = e^{\frac{\mathbf{r} - \mathbf{r}\_2}{\mathcal{S}\mathbf{l}} \left(1 - e^{\frac{\mathbf{r}}{\mathcal{S}\mathbf{l}}}\right)} \quad \text{and} \quad \prescript{}{t}{\mathbf{p}}\_x = e^{-\int\_0^t \tilde{\mathbf{m}}\_{x+\tau} \mathbf{d}\tau} = e^{\frac{\mathbf{r} - \mathbf{r}\_2}{\mathcal{S}\mathbf{l}} \left(1 - e^{\frac{\mathbf{r}}{\mathcal{S}\mathbf{l}}}\right)} , \tag{31}$$

where

$$m\_{\mathbf{x}+\mathbf{\tau}} = \frac{1}{\mathcal{S}\_1} e^{\frac{\mathbf{x}+\mathbf{\tau}-\underline{\mathbf{x}}\mathbf{\tau}}{\mathcal{S}\_1}} \quad \text{and} \quad \tilde{m}\_{\mathbf{x}+\mathbf{\tau}} = \frac{1}{\mathcal{S}\_1} e^{\frac{\mathbf{x}+\mathbf{\tau}-\bar{\mathbf{x}}\mathbf{\tau}}{\mathcal{S}\_1}},\tag{32}$$

are the individual's forces of mortality at the age of *x* + *τ* with *τ* ≥ 0 following the Gompertz law of mortality (see, e.g., Milevsky and Salisbury 2015) under P and Q, respectively. We refer to *<sup>g</sup>*<sup>1</sup> <sup>&</sup>gt; 0, *<sup>g</sup>*<sup>2</sup> <sup>&</sup>gt; <sup>0</sup> and *<sup>g</sup>*e<sup>2</sup> <sup>&</sup>gt; <sup>0</sup> as the first Gompertz parameter describing the dispersion and the second Gompertz parameters describing the modal ages at death. Note that we assume that *g*<sup>1</sup> remains the same under both probability measures P and Q, and that *<sup>g</sup>*e<sup>2</sup> <sup>≥</sup> *<sup>g</sup>*2, so that (9) is fulfilled. For the mortality shock *<sup>e</sup>*, following Chen et al. (2019), we assume its distribution to be truncated normal:

$$
\epsilon \sim \mathcal{N}\_{( - \infty, 1)} \left( \eta\_1, \eta\_2^2 \right) . \tag{33}
$$

where *η*<sup>1</sup> and *η* 2 2 are the mean and the variance parameter of the normal distribution truncated on the interval (−∞, 1), respectively. Table 1 summarizes the assumed baseline values for the relevant parameters and their corresponding ranges for *n*, *µ*, *σ*, *γ* and *ρ*, used in our sensitivity analyses.

When choosing the parameter values given in Table 1, we include the following considerations:



**Table 1.** Specification of relevant parameters for numerical studies.

We also need to introduce a practicable choice in Case B for the guaranteed payment *G<sup>t</sup>* , which has not been specified so far. Since we aim at taking account of the circumstance that the individual's attitude towards the guaranteed payment can change if she gets older, we choose

*G<sup>t</sup>* = *Geδ<sup>t</sup>* , (34)

where *G* > 0 is the prescribed constant initial guarantee amount and *δ* the guarantee growth rate. By this stipulation, we can consider different situations, such as the case in which the liquidity needs of the policyholder increase with age, which can be modeled by choosing a positive *δ*. If it is required that *G<sup>t</sup>* is time-independent, i.e., a constant over time, simply let *δ* = 0. We choose *G* in such a way that the value of the guaranteed payments at time 0 corresponds to a fraction, say *g* ∈ (0, 1), of the total premium. Relying on (13), which represents the described correspondence if *P*<sup>0</sup> = *v* is multiplied by *g* and Ψ*<sup>t</sup>* is replaced by *G<sup>t</sup>* , we obtain

$$G = \frac{gv}{\int\_0^\infty e^{-(r-\delta)t} I\_t \mathbf{d}t}.\tag{35}$$

For the three case-related parameters *α*, *δ* and *g*, we summarize their assumed baseline values in Table 2. For the sensitivity analyses below, the corresponding ranges of *δ* and *g* are also presented in Table 2.


**Table 2.** Specification of relevant parameters related to Case B for numerical studies.

The following considerations are taken into account when choosing the parameter values given in Table 2:


#### *5.2. Comparison*

The main questions we intend to answer in this numerical analysis are as follows:


These questions will be answered in Section 5.2.2, where we present our numerical results and sensitivity analyses based on the assumptions made in Section 5.1. In preparation for this, a short overview of the necessary details on the traditional tontine is given and the precise comparison approach is explained in Section 5.2.1.

#### 5.2.1. Traditional Tontine and Comparison Approach

Recall that the traditional tontines established in Milevsky and Salisbury (2015) are introduced in Section 2.1, where its total payment is given in (1). In order to buy a traditional tontine, we assume that the individual also spends her available initial wealth *v* to pay the single initial premium charged for it. By replacing the tontine payout process Ψ*<sup>t</sup>* in (13) by the tontine payout function *d<sup>t</sup>* , this premium can be calculated via

$$P\_0 = \int\_0^\infty e^{-rt} I\_l d\_l \mathbf{d}t.\tag{36}$$

We consider two different variants of specific forms of *d<sup>t</sup>* , one rather theoretical and one rather practical. The first one, which is also examined in, e.g., Chen et al. (2019), arises directly from the maximization of the discounted expected utility associated with the purchase of the traditional tontine. In the corresponding optimization problem, *d<sup>t</sup>* is naturally chosen as the decision variable. Details on this problem and its solution that is given by the optimal version *d* ∗ *t* for *d<sup>t</sup>* are reviewed in Appendix C. We refer to the resulting product as the *optimal traditional tontine*. For the second specific form of *d<sup>t</sup>* , we use one of the so-called natural tontines proposed by Milevsky and Salisbury (2015). This more practicable payout function is given by

$$d\_l = E\_{\mathbb{Q}}\left[\mathbbm{1}\_{\{\zeta\_x > t\}}\right]d = {}\_t\tilde{p}\_x E\_{\mathbb{Q}}\left[e^{-\ln({}\_t\tilde{p}\_x)\cdot\varepsilon}\right]d = {}\_t\tilde{p}\_x M\_{\varepsilon}(-\ln({}\_t\tilde{p}\_x))d,\tag{37}$$

where *d* > 0 is constant over time and determined by plugging (37) in the budget constraint *v* = *P*0, where *P*<sup>0</sup> is given in (36):

$$d^\* = \frac{v}{\int\_0^\infty e^{-rt} I\_{t^\*} \tilde{p}\_x M\_\varepsilon (-\ln(\_t \tilde{p}\_x)) \,\mathrm{d}t}.\tag{38}$$

Note that by applying (37), the total payment to the living traditional tontine holder is actually also constant over time if deaths in the pool occur as expected. We refer to the product resulting from (37) and (38) as the *natural traditional tontine*.

For the comparison, we look at the (maximized) discounted expected utilities arising from the optimal findings in Section 4.2 and from above that the individual attains when acquiring the respective tontine product alternatives. They are denoted by EU∗<sup>A</sup> and EU∗<sup>B</sup> in case of the two unit-linked tontines from Case A and Case B, respectively, by EU∗OT in case of the optimal traditional tontine and by EU∗NT in case of the natural traditional tontine. For the sake of completeness, an overview of the formulas for the different (maximized) discounted expected utilities is given in Appendix D. <sup>7</sup> The reason why such a direct comparison approach is valid within our framework is that the individual spends the same initial wealth *v* for every product variant. Therefore, since the purchase costs for the policyholder are always identical, she rationally prefers the tontine that provides her with the highest utility. To make our comparison results easier to interpret, we do not straightforwardly consider the different (maximized) discounted expected utility levels, but the corresponding certainty equivalents, which are the safe amounts that make the individual indifferent between obtaining them and the optimal uncertain total payments of the tontine products. These certainty equivalents, which are denoted by CE∗*<sup>j</sup>* with *<sup>j</sup>* <sup>∈</sup> {A, B, OT, NT} marking the respective product variant, are thus calculated by using the same concept as in (20) and the quantities EU∗*<sup>j</sup>* for equating:

$$\begin{aligned} \mathrm{E}\_{\mathbb{P}}\left[\int\_{0}^{\infty} e^{-\rho t} u \left(\mathrm{CE}^{\*j}\right) \mathbf{1}\_{\{\zeta\_{x} > t\}} \mathrm{d}t\right] &= \mathrm{EU}^{\*j} \\\\ \Leftrightarrow \mathrm{CE}^{\*j} &= \left( (1 - \gamma) \mathrm{EU}^{\*j} \left( \int\_{0}^{\infty} e^{-\rho t} \int\_{-\infty}^{1} p\_{x}^{1 - z} f\_{\varepsilon}(z) \mathrm{d}z \mathrm{d}t \right)^{-1} \right)^{\frac{1}{1 - \gamma}} . \end{aligned} \tag{39}$$

As EU∗*<sup>j</sup>* is strictly increasing in CE∗*<sup>j</sup>* , comparing the (maximized) discounted expected utilities is equivalent to comparing the certainty equivalents.

#### 5.2.2. Numerical Results and Sensitivity Analyses

In Table 3, we show the first numerical findings, namely the ones for CE∗*<sup>j</sup>* , that emerge from applying the baseline parameter values given in Tables 1 and 2 (for Case B).<sup>8</sup>


**Table 3.** Certainty equivalents of different tontines with baseline parameter values.

Comparing the certainty equivalents reported in in Table 3 shows that the policyholder is in the best position as long as she holds the unit-linked tontine designed in Case A. When comparing only the unit-linked tontine variants, it is more beneficial for the individual if the tontine payout process does not include an additional guaranteed payment as in Case B, but rather simply complies with the entire portfolio value that arises entirely out of optimally investing in the financial market. The unit-linked tontine variant from Case B actually performs worse than the traditional tontine, where even the more practicable version, the natural traditional tontine, surpasses it by far, i.e., CE∗NT CE∗<sup>B</sup> . Do the previous observations also hold if certain parameter values change?

In Figures 1 and 2, we show the numerical comparison findings that emerge from applying the parameter ranges given in Table 1. In particular, we present the resulting curves for CE∗*<sup>j</sup>* if the parameters *n*, *µ*, *σ*, *γ* and *ρ* vary, respectively.<sup>9</sup>

**Figure 1.** Effects of *n* (**a**), *µ* (**b**) and *σ* (**c**) on certainty equivalents.

**Figure 2.** Effects of *γ* (**a**) and *ρ* (**b**) on certainty equivalents.

Two main observations can be universally drawn from Figures 1 and 2:


As already pointed out by Chen et al. (2021) (Theorem 5.2), the impact of the pool size on the attractiveness of a tontine is not monotonically increasing. In their context, they compare tontines with annuities and the critical pool size determines the preference ordering between annuities and tontines. After the pool size reaches a certain magnitude, tontines will become, for instance, more attractive than conventional annuities. They observe that this number is rather small for a conventional tontine case. Now, in our unit-linked products, this number seems rather large, shown in Figure 1a to be larger than 200, beyond which the attractiveness of the unit-linked tontine products increases in the pool size.

In order to get a better understanding of the findings derived from Table 3 and of the above-mentioned observations based on Figures 1 and 2, we show in Figure 3 the means and 0.01-/0.99-quantiles under P of the optimal total payments for Cases A and B and the traditional tontine with respect to age. For the generation of the graphs, we assume that the considered policyholder is always alive and that the parameters attain their baseline values given in Tables 1 and 2 (for Case B).<sup>10</sup>

By comparing Case A and Case B by means of Figure 3, the effect of the guaranteed payment picked up in Case B becomes clear: In Figure 3c, we notice that the 0.01-quantile curve for Case B is almost always significantly above the one for Case A. This implies

that the inclusion of the guaranteed payment prevents the policyholder in Case B from receiving a very low total payment in bad market scenarios. Yet, at the same time, the guaranteed payment also limits a possible positive development of the total payment in good market scenarios, which is, however, completely exploited by the unit-linked tontine from Case A. This is recognizable by the 0.99-quantile curves displayed in Figure 3a,b. Since the scale of the 0.99-quantiles, especially in Figure 3b, is much larger than the one of the 0.01-quantiles in Figure 3c, the dominance of Case A in good market scenarios clearly outperforms the dominance of Case B in bad market scenarios. Hence, as visible in Figure 3a,b, the mean of the total payment for Case A is consistently higher than the one for Case B. Due to the fact that the power utility function is strictly increasing, we can infer from this finding that CE∗<sup>A</sup> > CE∗<sup>B</sup> holds, as observed above for the given parameters. Moreover, we, particularly in Figure 3a,c, see that the curves for the traditional tontine, which is represented here by the optimal version, can be above or below the ones for the unit-linked tontine. That is why the policyholder prefers the traditional tontine to the unitlinked tontine in some instances, while in others she does not, as apparent from Figures 1 and 2. The partial dominance of the traditional tontine explicitly shown in Figure 3 suffices to beat the performance of the unit-linked tontine from Case B, but not the one from Case A. This can be observed from Table 3, where all parameters also attain their baseline values.

(**b**) Means (black) and 0.99-quantiles (grey).

**Figure 3.** Means and 0.99-quantiles at earlier retirement ages (**a**) and at more advanced retirement ages (**b**), and 0.01-quantiles (**c**) of optimal total payments for Cases A and B and the optimal traditional tontine (OT) depending on age, assuming that the policyholder is always alive and the parameters attain baseline values.

In the following, the impacts of the parameters *n*, *µ*, *σ*, *γ* and *ρ* and, eventually of the varying parameters *δ* and *g*, being only related to Case B, will be discussed in detail.

#### **Sensitivity Analyses Regarding** *n*

In Figure 1a, we notice right away the converse behavior of CE∗<sup>A</sup> and CE∗<sup>B</sup> with regard to CE∗OT and CE∗NT as long as the initial number *n* of participants in the pool ranges within relatively small values. Especially when an extremely small pool takes in a very few new participants, the policyholder's benefit drops sharply in case of the unit-linked tontine, whereas it rises quickly for the traditional tontine. From around *n* = 250 on, the courses of the curves belonging to the unit-linked tontine switch to an upward movement, which becomes even steeper than the one for the traditional tontine. In summary, a purchase decision in favor of the unit-linked tontine is wise if the pool size is either very small or large.

In order to get a better understanding of the recorded observations, we let *n* vary again and study the resulting optimal values *V* ∗ 0 <sup>A</sup> and *V*g<sup>∗</sup> 0 B for the initial investment amount in Figure 4.

**Figure 4.** Effect of *n* on optimal values for the initial investment amount.

We observe very similar curve shapes for *V* ∗ 0 <sup>A</sup> and *V*g<sup>∗</sup> 0 B in Figure 4 compared to the ones for CE∗<sup>A</sup> and CE∗<sup>B</sup> in Figure 1a, namely the strong decline in *n* at the beginning, which quickly lessens and, from around *n* = 250 on, turns into an increase. Consequently, it seems that the behavior of the initial investment amount for a varying pool size causes the performance development of the unit-linked tontine described above. If we exemplarily consider the formula in (22) for *V* ∗ 0 <sup>A</sup>, only the initial decrease appears plausible at first glance. However, when we recall that lower (implicit) safety loadings included in the premiums can be chosen if *<sup>n</sup>* grows by reducing *<sup>t</sup> <sup>p</sup>*e*x*, just like we do, it is comprehensible why the decrease can be slowed down and possibly even be reversed at some point. Below, we provide more interpretations to the impact of the pool size *n*:


#### **Sensitivity Analyses Regarding** *µ* **and** *σ*

If the policyholder chooses the unit-linked tontine, we observe in Figure 1b,c that her utility enhances more and more as long as the drift rate *µ* of the risky asset increases and its volatility *σ* decreases, respectively. This is because the risky asset, in which investments are made within the framework of the unit-linked tontine, is clearly more profitable if its return grows and its risk reduces, as can be seen, for example, from a higher Sharpe ratio *<sup>µ</sup>*−*<sup>r</sup> σ* , which eventually is naturally also more beneficial to the policyholder. As the certainty equivalents associated with the traditional tontine are apparently not affected by a varying *µ* or *σ* due to its payout's independence of the financial market, there is a certain level at which the performance of the risky asset is so good that the traditional tontine is no longer preferred.

#### **Sensitivity Analyses Regarding** *γ* **and** *ρ*

In Figure 2a, we find that CE∗*<sup>j</sup>* declines for all *j* for higher values of *γ*, which means that each tontine variant gets less interesting for the individual when she becomes more risk-averse. This is because the risk inherent in the tontines is borne, to a great extent, by all participants in the pool, and the payments to the policyholder are, hence, uncertain to some extent. If the policyholder embraces less of this risk, i.e., she is more risk-averse and prefers more stable payments, her personal benefit is, thus, smaller. However, the curves displaying CE∗<sup>A</sup> and CE∗<sup>B</sup> exhibit (partly much) steeper slopes than those for CE∗OT and CE∗NT due to the fact that the unit-linked tontine alternatives contain more risk, namely not only the mortality risk but also the financial risk component. Therefore, if the policyholder tolerates more risk, i.e., she is less risk-averse (*γ* decreases) and prefers riskier payments, the unit-linked tontine is definitely the better choice. In Figure 2b, it can be observed that when the subjective discount rate *ρ* grows, the personal utilities induced by buying the examined tontines constantly diminish. The only exception is the optimal traditional tontine that regains some attractiveness for higher values of *ρ* in consequence of the specific structure of *d* ∗ *t* , which is explicitly given in Appendix C. Since a higher subjective discount rate means that the individual tends to consume more at earlier retirement ages, the decreases of CE∗<sup>A</sup> and CE∗<sup>B</sup> in *ρ* are explainable by the steady increases of the means of the total unit-linked tontine payments over time, as this is exemplarily illustrated in Figure 3a,b. In these two figures, we also observe that the magnitudes of the two mean curves for the unit-linked tontine variants are a lot greater compared to the traditional tontine. This gives a reason for the steeper slopes of the curves displaying CE∗<sup>A</sup> and CE∗<sup>B</sup> in Figure 2b.

#### **Sensitivity Analyses Regarding** *δ* **and** *g*

When considering the choice for *G<sup>t</sup>* as introduced in (34) and (35) for Case B, we are especially interested in the impact of the guarantee growth rate *δ* and the guaranteed premium fraction *g* on the policyholder's tontine product preference. To analyze this, we look at the resulting curves for CE∗<sup>B</sup> , CE∗OT and CE∗NT depicted in Figure 5, where the ranges given in Table 2 are applied.

**Figure 5.** Effects of *δ* (**a**) and *g* (**b**) on certainty equivalents.

Both graphs of Figure 5 demonstrate a similar curve progression for CE∗<sup>B</sup> . In particular, the resulting certainty equivalents are negatively proportional to *δ* and *g*. However, as the payout of the traditional tontine does not depend on *δ* and *g*, neither CE∗OT nor CE∗NT changes. As a consequence, it is possible that the policyholder benefits more from the unit-linked tontine designed in Case B than from the traditional tontine if the guaranteed payment is low enough. On the other hand, a high guaranteed component in the unit-linked tontine may adversely affect the performance of the product due to stronger limitations on possible investment gains.

#### **6. Conclusions**

In the present article, we propose unit-linked tontine products that combine the tontine concept with the idea underlying unit-linked insurance policies, i.e., to tie payouts to the developments in the financial market. We examine a general payment structure of the product and analyze two specified payment structures. The two risk types contained in the unit-linked product are the financial risk stemming from the risky asset existing in the financial market and the mortality risk, for which we actually also incorporate the systematic part in our model. The premium required to buy the unit-linked tontine is determined in a risk-neutral pricing framework. Further, we study the optimal expected utility of an individual purchasing the unit-linked tontine by adjusting the payment structure. In our numerical comparison and sensitivity analyses, we contrast the policyholder's benefits arising out of the two optimized unit-linked tontine variants, as well as the optimal and the natural traditional tontine. In particular, we find that there exist circumstances in which the unit-linked tontine endows the policyholder with a higher utility level than the traditional tontine, emphasizing the potential of the suggested unit-linked tontines. More precisely, under our numerical setting with power utility functions, the unit-linked tontines might be a potential choice for the policyholder when the expected return of the risky asset is high or if the volatility of the risky asset, the policyholder's risk aversion or her subjective discount rate is low. Moreover, we observe that if its payout process is stipulated by the pure financial market portfolio value, the unit-linked tontine consistently makes the policyholder better off than in the case where it includes guaranteed payments. However, its performance approaches more and more that of the superior variant if the expected return of the risky asset decreases or if the volatility of the risky asset or the policyholder's risk aversion increases. Furthermore, when comparing the case with guaranteed payments with the traditional tontine with no financial market component, this case can nevertheless be attractive, especially to customers who consider additional guarantee elements important. Our findings would give reason to further study this new type of product in more realistic settings that take practical aspects into account, for instance, how the provider hedges the mortality and financial market risks related to the unit-linked tontines and what the net loss of the provider is. A thorough analysis of the hedging perspective requires a more dynamic framework and will be left for future research.

**Author Contributions:** Conceptualization, A.C. and T.N.; Methodology, A.C., T.N. and T.S.; Software, T.S.; Writing—original draft, T.S.; Writing—review & editing, A.C., T.N. and T.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** Thai Nguyen acknowledges the support of the Natural Sciences and Engineering Research Council of Canada [RGPIN-2021-02594].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Detailed Derivations**

The equality in (11) holds for all *t* as

*E*<sup>Q</sup> <sup>1</sup>{*ζx*>*t*} *Nt* = *E*<sup>Q</sup> *t p*e*x* <sup>1</sup>−*eE*<sup>Q</sup> 1 *Nt ζ<sup>x</sup>* > *t*, *e* <sup>=</sup> *<sup>E</sup>*<sup>Q</sup> *t p*e*x* <sup>1</sup>−*eE*<sup>Q</sup> 1 *N<sup>t</sup>* − 1 + 1 *ζ<sup>x</sup>* > *t*, *e* = *E*<sup>Q</sup> " *t p*e*x* 1−*e n*−1 ∑ *k*=0 1 *k* + 1 *n* − 1 *k t p*e*x* 1−*e k* <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n*−1−*<sup>k</sup>* # = *E*<sup>Q</sup> " *n*−1 ∑ *k*=0 *n n* (*n* − 1)! (*k* + 1)!(*n* − 1 − *k*)! *t p*e*x* 1−*e k*+1 <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n*−1−*<sup>k</sup>* # = 1 *n E*<sup>Q</sup> " *n*−1 ∑ *k*=0 *n k* + 1 *t p*e*x* 1−*e k*+1 <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n*−1−*<sup>k</sup>* # = 1 *n E*<sup>Q</sup> " *n* ∑ *k*=1 *n k t p*e*x* 1−*e k* <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n*−*<sup>k</sup>* # = 1 *n E*<sup>Q</sup> " *n* ∑ *k*=1 *n k t p*e*x* 1−*e k* <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n*−*<sup>k</sup>* + *n* 0 *t p*e*x* 1−*e* 0 <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n* − *n* 0 *t p*e*x* 1−*e* 0 <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n* = 1 *n E*<sup>Q</sup> " *n* ∑ *k*=0 *n k t p*e*x* 1−*e k* <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n*−*<sup>k</sup>* − <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n* # = 1 *n E*<sup>Q</sup> h<sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* <sup>1</sup>−*<sup>e</sup>* <sup>+</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n* − <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n* i = 1 *n E*<sup>Q</sup> h 1 − <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*e n* i = 1 *n* Z 1 −∞ 1 − <sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>p</sup>*e*<sup>x</sup>* 1−*z n fe*(*z*)d*z* = 1 *n It* .

The equality in (30) holds for all *t* as, with *π* denoting the ratio of a circle's circumference to its diameter,

$$\begin{split} V\_{0}e^{rt}\phi\left(\hat{d}\_{l}\right) - G\_{l}\phi\left(\hat{d}\_{l}\right) &= \frac{1}{\sqrt{2\pi}} \Biggl( e^{\ln(V\_{0}) + rt - \frac{\tilde{d}\_{l}^{2}}{2}} - e^{\ln(G\_{l}) - \frac{\tilde{d}^{2}}{2}} \Biggr) \\ &= \frac{1}{\sqrt{2\pi}} \Biggl( e^{\ln(V\_{0}) + rt - \frac{\tilde{d}\_{l}^{2}}{2}} - e^{\ln(G\_{l}) - \frac{\tilde{d}\_{l}^{2} - 2\tilde{d}\_{l}\pi\pi\sqrt{t + \sigma^{2}\pi^{2}\_{l}}}}{2} \Biggr) \\ &= \frac{1}{\sqrt{2\pi}}e^{-\frac{\tilde{d}\_{l}^{2}}{2}} \underbrace{\left( e^{\ln(V\_{0}) + rt} - e^{\ln(G\_{l}) + \ln\left(\frac{V\_{0}}{C\_{l}}\right) + rt + \frac{\sigma^{2}}{2}\pi^{2}t - \frac{\sigma^{2}}{2}\pi^{2}t} \right)}\_{=0} = 0. \end{split}$$

#### **Appendix B. Proofs**

*Appendix B.1. Proposition 2*

**Proof.** With the aid of the general pricing formula given in (13), we obtain the claim since the equality in (17) holds as

$$E\_{\mathbb{Q}}[\Psi\_{t}] = E\_{\mathbb{Q}}\Big[G\_{t} + \mathfrak{a}(V\_{t} - G\_{t})^{+}\Big] = G\_{t} + \mathfrak{a}E\_{\mathbb{Q}}\Big[\left(V\_{t} - G\_{t}\right)^{+}\Big]\_{\mathcal{A}}$$

with

$$\begin{split} E\_{\mathbb{Q}}\left[\left(V\_{\mathbb{f}}-G\_{\mathbb{f}}\right)^{+}\right] &= E\_{\mathbb{Q}}\left[\left(V\_{0}e^{rt}-\frac{e^{2}}{2}\int\_{0}^{t}\pi\_{s}^{2}ds+\sigma\sqrt{\int\_{0}^{t}\pi\_{s}^{2}ds}-\operatorname{G}\_{\mathbb{f}}\right)\mathbbm{1}\_{\{Z>\beta\_{t}\}}\right] \\ &= V\_{0}e^{rt}\int\_{\hat{\mathbb{f}}\_{\mathbb{f}\_{t}}^{\infty}}^{\infty}e^{-\frac{\sigma^{2}}{2}\int\_{0}^{t}\pi\_{s}^{2}ds+\sigma\sqrt{\int\_{0}^{t}\pi\_{s}^{2}ds}}\varPhi(z)\operatorname{d}z-\operatorname{G}\_{\mathbb{f}}\int\_{\hat{\mathbb{f}}\_{\mathbb{f}\_{t}}^{\infty}}^{\infty}\varPhi(z)\operatorname{d}z \\ &= V\_{0}e^{rt}\int\_{\hat{\mathbb{f}}\_{\mathbb{f}\_{t}}^{\infty}}^{\infty}\varPhi(z)\operatorname{d}z-\operatorname{G}\_{\mathbb{f}}\varPhi(-\beta\_{t}) \\ &= V\_{0}e^{rt}\varPhi\left(\sigma\sqrt{\int\_{0}^{t}\pi\_{s}^{2}ds}-\beta\_{t}\right)-\operatorname{G}\_{\mathbb{f}}\varPhi(-\beta\_{t})=V\_{0}e^{rt}\varPhi\left(\tilde{d}\_{t}\right)-\operatorname{G}\_{\mathbb{f}}\varPhi\left(\hat{d}\_{t}\right). \end{split}$$

where

$$\beta\_t = \frac{\ln\left(\frac{G\_t}{V\_0}\right) - rt + \frac{\sigma^2}{2} \int\_0^t \pi\_s^2 \mathrm{d}s}{\sigma \sqrt{\int\_0^t \pi\_s^2 \mathrm{d}s}}.$$

#### *Appendix B.2. Proposition 3*

**Proof.** As the budget constraint in Problem 2 depends only on *V*<sup>0</sup> and not on *π*, the optimal value for *V*<sup>0</sup> is already completely determined by this constraint, so that we immediately obtain

$$V\_0^{\*A} = \frac{v}{\int\_0^\infty I\_t \mathbf{d}t} \prime$$

which is obviously positive, so that we also stick to the condition that *V*<sup>0</sup> > 0. Consequently, the budget constraint is entirely taken care of by *V* ∗ 0 <sup>A</sup> and, thus, the determination of the optimal value of the trading strategy *π* can be done by simply maximizing the objective of Problem 2 with respect to *π*. To this end, we realize the shape of the objective as a function of *π* by considering the corresponding derivative:

$$\begin{split} &\frac{\partial}{\partial\pi} \left( \frac{\left( nV\_{0}^{\ast \mathbf{A}} \right)^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\rho t} \kappa\_{\ell t} e^{(1-\gamma)\left(r+(\mu-r)\pi-\frac{\gamma\sigma^{2}}{2}\pi^{2}\right)t} \mathbf{d}t \right) \\ &= \frac{\left(nV\_{0}^{\ast \mathbf{A}}\right)^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\rho t} \kappa\_{\ell t} e^{(1-\gamma)\left(r+(\mu-r)\pi-\frac{\gamma\sigma^{2}}{2}\pi^{2}\right)t} (1-\gamma) \left(\mu-r-\gamma\sigma^{2}\pi\right)t \mathbf{d}t \\ &= \underbrace{\left(nV\_{0}^{\ast \mathbf{A}}\right)^{1-\gamma}}\_{>0} \left(\mu-r-\gamma\sigma^{2}\pi\right) \underbrace{\int\_{0}^{\infty} e^{-\rho t} \kappa\_{\ell t} e^{(1-\gamma)\left(r+(\mu-r)\pi-\frac{\gamma\sigma^{2}}{2}\pi^{2}\right)t} t \mathbf{d}t \end{split}$$

The identified derivative is positive (negative), i.e., the objective is strictly increasing (decreasing) in *π*, if

$$\left(\mu - r - \gamma \sigma^2 \pi \begin{array}{c} \geq \\ \leq \end{array}\right) \mathfrak{0} \Leftrightarrow \pi \stackrel{<}{\left(>\right)} \frac{\mu - r}{\gamma \sigma^2} \cdot 1$$

Since we also need to adhere to the condition that *π* ∈ [0, 1], it is clear that, as long as *<sup>µ</sup>* <sup>−</sup> *<sup>r</sup>* <sup>≤</sup> *γσ*<sup>2</sup> , the optimal value for *π* is given by *π* <sup>∗</sup><sup>A</sup> = *µ*−*r γσ*<sup>2</sup> . Otherwise, if *<sup>µ</sup>* <sup>−</sup> *<sup>r</sup>* <sup>&</sup>gt; *γσ*<sup>2</sup> , it is *π* <sup>∗</sup><sup>A</sup> = 1. Overall, we find

$$
\pi^{\*\mathcal{A}} = \frac{\mu - r}{\gamma \sigma^2} \mathbf{1}\_{\left\{ \mu - r \le \gamma \sigma^2 \right\}} + \mathbf{1}\_{\left\{ \mu - r > \gamma \sigma^2 \right\}}.
$$

#### **Appendix C. Review of Optimization Problem for Traditional Tontine**

In the style of Problem 1, the maximization problem for the traditional tontine with the decision variable *d<sup>t</sup>* can be, by using (36) and replacing Ψ*<sup>t</sup>* by *d<sup>t</sup>* , formulated as follows:

$$\begin{aligned} \max\_{(d\_t)\_{t\geq 0}} & \frac{n^{1-\gamma}}{1-\gamma} \int\_0^\infty e^{-\rho t} \kappa\_t d\_t^{1-\gamma} \mathbf{d}t, \\ \text{s.t. } & v = P\_0 = \int\_0^\infty e^{-rt} I\_t d\_t \mathbf{d}t. \end{aligned}$$

By applying the techniques in Chen et al. (2019), it can be shown that the optimal solution is given by

$$d\_t^\* = \left(\frac{\lambda^\* e^{-rt} I\_t}{n^{1-\gamma} e^{-\rho t} \kappa\_t}\right)^{-\frac{1}{\gamma}}.$$

where the optimal Lagrange multiplier is given by

$$
\lambda^\* = v^{-\gamma} \left( \int\_0^\infty \frac{\left(e^{-rt}I\_t\right)^{1-\frac{1}{\gamma}}}{\left(n^{1-\gamma}e^{-\rho t}\kappa\_t\right)^{-\frac{1}{\gamma}}} \mathrm{d}t \right)^\gamma.
$$

#### **Appendix D. Overview of Formulas for (Maximized) Discounted Expected Utilities**

The formulas for the different (maximized) discounted expected utilities EU∗*<sup>j</sup>* with *j* ∈ {A, B, OT, NT} that are applied for the comparison are listed in the following overview:

$$\begin{split} & \mathrm{EU}^{\*\mathsf{A}} = \frac{\left(\boldsymbol{n}\_{0}^{\mathsf{T}\mathsf{A}}\right)^{\mathsf{1}-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\boldsymbol{\rho}t} \kappa\_{t} e^{(1-\gamma)\left(r+(\boldsymbol{\mu}-\boldsymbol{r})\boldsymbol{\pi}^{\*\mathsf{A}}-\frac{\boldsymbol{\mu}^{2}}{2}\left(\boldsymbol{\pi}^{\*\mathsf{A}}\right)^{2}\right)} \mathrm{d}t, \\ & \mathrm{EU}^{\*\mathsf{B}} = \frac{\left(\boldsymbol{n}^{1-\gamma}\right)}{1-\gamma} \int\_{0}^{\infty} e^{-\boldsymbol{\rho}t} \kappa\_{t} G\_{t}^{1-\gamma} \left(\boldsymbol{\Phi}\left(\overline{d}\_{t}^{\*}\right) + \int\_{0}^{\infty} \left(1+a\left(e^{\boldsymbol{\rho}\boldsymbol{\pi}^{\*\mathsf{B}}\boldsymbol{\sqrt}}\boldsymbol{\tilde{\nu}}-1\right)\right)^{1-\gamma} \boldsymbol{\phi}\left(\boldsymbol{y}+\overline{d}\_{t}^{\*}\right) \mathrm{d}t \right) \\ & \mathrm{EU}^{\*\mathsf{OT}} = \frac{n^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\boldsymbol{\rho}t} \kappa\_{t}(d\_{t}^{\*})^{1-\gamma} \mathrm{d}t, \\ & \mathrm{EU}^{\*\mathsf{NT}} = \frac{n^{1-\gamma}}{1-\gamma} \int\_{0}^{\infty} e^{-\boldsymbol{\rho}t} \kappa\_{t}(t\tilde{\boldsymbol{\rho}}\_{\mathcal{X}}M\_{t}(-\ln(t\_{\tilde{\boldsymbol{\rho}}\_{\mathcal{X}}}))d^{\*})^{1-\gamma} \mathrm{d}t, \end{split}$$

where *d* ∗ *t* is given as in (24), but with *π* replaced by *π*g∗<sup>B</sup> and *V*<sup>0</sup> replaced by *V*g<sup>∗</sup> 0 B .

#### **Notes**

<sup>1</sup> For simplicity, we have assumed log-normal risky asset dynamics, which, as well documented, may not be very realistic. It would be interesting to look at the unit-linked tontine design problem in more general settings where the asset volatility is random when fat-tailed returns and volatility clustering are taken into account (see, e.g., Cont and Tankov 2004; Fouque et al. 2000). The continuity assumption of the stock price is relaxed in order to capture sudden and unpredictable market changes (see, e.g., Cont and Tankov 2004). Also, for such long-term investment problems, it would be more realistic to incorporate interest rate fluctuations (see, e.g., Hull and White 1990; Vasicek 1977).


for Case B, by *<sup>n</sup>*(*Gt*+*α*(*<sup>V</sup>* ∗ *<sup>t</sup>* −*Gt*) + ) *Nt* , where *V* ∗ *<sup>t</sup>* = *V*g<sup>∗</sup> 0 B *e rt*+(*µ*−*r*)*π*g∗<sup>B</sup> *<sup>t</sup>*<sup>−</sup> *<sup>σ</sup>* 2 2 *π*g∗<sup>B</sup> 2 *t*+*σπ*g∗B*W<sup>t</sup>* , and, for the optimal traditional tontine, by *nd*<sup>∗</sup> *t Nt* . Note that the computation of all depicted quantities is done numerically, where we divide the relevant time line running from *t* = 0 to *t* = 35 by a constant discretization step size of 0.025, which means that we overall analyze 1401 points, and simulate each occurring random variable 450,000 times.

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### *Article* **Equivalent Risk Indicators: VaR, TCE, and Beyond**

**Silvia Faroni 1,2 , Olivier Le Courtois 1,\* and Krzysztof Ostaszewski <sup>3</sup>**


**Abstract:** While a lot of research concentrates on the respective merits of VaR and TCE, which are the two most classic risk indicators used by financial institutions, little has been written on the equivalence between such indicators. Further, TCE, despite its merits, may not be the most accurate indicator to take into account the nature of probability distribution tails. In this paper, we introduce a new risk indicator that extends TCE to take into account higher-order risks. We compare the quantiles of this indicator to the quantiles of VaR in a simple Pareto framework, and then in a generalized Pareto framework. We also examine equivalence results between the quantiles of high-order TCEs.

**Keywords:** VaR; TCE; extended TCE; insurance regulation; risk measurement

#### **1. Introduction**

In the second half of the twentieth century, developed market economies had undergone an inflationary period after World War II, followed by a disinflationary period that started in the early 1980s accompanied by less regulation and greater reliance on market forces. These developments resulted in improved economic performance, but also increased volatility and market pressure on financial institutions following traditional fixed capital standards. In order to address these developments, regulators have moved towards risk-based capital requirements for financial institutions, while also allowing gradual relaxation of certain rules of governance of financial institutions and replacing them with a principle-based approach.

The risk measures imposed by regulators on insurance companies share a common feature: they are all related to the behavior of tails of the probability distribution of a firm's financial results. The reason is that the regulatory purpose of capital requirements is to make capital available for absorbing losses occurring in extreme events, i.e., events of large financial losses, which could bring about insolvency. Risk-based capital requirements are a natural outgrowth of traditional prudential regulation aiming at preserving the solvency of private financial institutions.

Hence, safety capital should be computed by looking at the extreme risks that can impact financial institutions in general and insurance enterprises in particular. However, regulators differ in their specification of the tail indicator that they recommend. Some regulators impose the use of quantiles of the distribution (value at risk, or VaR), while other regulators impose the use of partial moments (tail conditional expectation or TCE, or an equivalent measure of expected shortfall). Furthermore, regulators also differ in their choice of time horizon. Finally, they also differ in their choice of confidence level (i.e., probability value for the quantile, or for the conditional expectation).

Comité Européen des Assurances and Mercer Oliver Wyman Limited (2005) and CEA (2007) provide a comparison of different regulatory regimes for capital requirements. The most consequential regulation of risk-based capital is the European Union's Solvency II. Solvency II imposes risk-based capital requirements computed using the VaR as a risk measure over a one-year period and with a confidence level of 99.5%. When Solvency II

**Citation:** Faroni, Silvia, Olivier Le Courtois, and Krzysztof Ostaszewski. 2022. Equivalent Risk Indicators: VaR, TCE, and Beyond. *Risks* 10: 142. https://doi.org/10.3390/ risks10080142

Academic Editors: Ermanno Pitacco and Annamaria Olivieri

Received: 31 May 2022 Accepted: 18 July 2022 Published: 22 July 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

was being designed, it was to some degree modeled on the Basel II banking regulation, also using VaR as a risk measure for capital requirement purposes, for market risk. The credit crisis of 2008 in many ways exposed the weaknesses of VaR, and a new system of capital regulation for banking, Basel III (see Basel Committee on Banking Supervision's documents from 2019 and 2022), has been developed and implemented since (see also Gatzert and Wesker (2011) for a comparison of Solvency II and Basel regulations).

Solvency II is not only the main regulatory law for prudential regulation of insurance enterprises in the European Union, but it has become a model for risk-based capital requirements worldwide. The United States is one major exception to this trend, as the U.S. model regulation of insurance firms' capital preceded Solvency II, and it is designed around a formula provided by the regulatory body, the National Association of Insurance Commissioners (NAIC) (see also National Association of Insurance Commissioners (2007) on ORSA perspectives).

The Canadian regulation of insurance capital differs from that of the European Union, and it is based on risk assessment of the firm in the context of certain extreme events. Canada's Office of Supervision of Financial Institutions (OSFI) risk assessment process begins with an evaluation of the inherent risk within each significant activity of an insurer and the quality of risk management applied to mitigate these risks (see Canada Office of Supervision of Financial Institutions 2022). After considering this information, OSFI determines the level of net risk and direction (i.e., whether it is decreasing, stable, or increasing) of the rating for each significant activity. The net risks of the significant activities are combined, by considering their relative importance, to arrive at the overall net risk (ONR) of the insurer. Furthermore, OSFI provides capital requirement guidelines, which must be then included in insurers' own risk and solvency assessment.

In the cases of both the United States and Canada, we see a significant regulatory involvement in the supervision of risk-based capital. All regulators are, of course, involved in this process, but the approach of the European Union and, notably, Switzerland, is more principle-based than rule-driven. The Swiss regulation of risk-based capital for insurers includes a capital standard, and stress-testing of certain extreme scenarios. The capital standard is based on the expected shortfall, or, equivalently, tail conditional expectation (TCE).

In this work, we focus on two regulations: Solvency II and the Swiss Solvency Test (SST). The Swiss Solvency Test, implemented in 2004, preceded Solvency II, but in 2015 the European Union recognized the SST as the first regime to be fully equivalent to Solvency II. Solvency II imposes a capital requirement computed using value at risk (VaR) as a risk measure over a 1-year period and with a confidence level of 99.5%, whereas SST uses TCE with a confidence level of 99% over a 1-year period.

VaR and TCE are the most classic examples of risk measures (see for instance Linsmeier and Pearson 2000; Klugman et al. 2012; Acerbi et al. 2001). A risk measure is a mapping from the random variable representing risk exposure to the set of real numbers. It can be interpreted as the amount of capital required to protect against adverse outcomes of a given risk. The paper by Artzner et al. (1999) introduced the concept of coherence of risk measures and has been very influential in the further development of risk measurement. A coherent risk measure is defined by the following four properties: subadditivity, monotonicity, positive homogeneity, and translation invariance. The properties of risk measures in the context of insurance are discussed by Wang and Zitikis (2021). Acerbi and Tasche (2002) noted that TCE is a coherent risk measure, while VaR is not (see also Society of Actuaries (2000) on TCE).

Rostek (2010) provides an interesting alternative to evaluations of risk, a model of preferences, in which, given beliefs about uncertain outcomes, an individual evaluates an action by a quantile of the induced distribution. Fadina et al. (2021) designed a unified axiomatic framework for risk evaluation principles, which quantifies jointly a loss random variable and a set of plausible probabilities. They called such an evaluation principle a generalized risk measure.

Fuchs et al. (2017) show that a notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval. In this general setting, they prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. They also present a particularly short proof of the subadditivity of the expected shortfall risk measure. Denuit et al. (2006) provide a comprehensive review or modeling risk in incomplete markets, with emphasis on insurance risks, expanding on and combining in a comprehensive review the existing literature on quantitative risk management.

Li and Wang (2022) noted that the Basel Committee on Banking Supervision proposed the shift from the 99% value at risk (VaR) to the 97.5% expected shortfall (ES) for internal models in market risk assessment (see Basel Committee on Banking Supervision 2019, 2022). Inspired by that development, Li and Wang introduced a new distributional index, the probability equivalence level of VaR and ES (PELVE), which identifies the balancing point for the equivalence between VaR and ES. PELVE has desirable theoretical properties, and it distinguishes empirically heavy-tailed distributions from light-tailed ones.

Barczy et al. (2022) generalized and further developed the PELVE measure and applied this indicator to a high-order TCE, which is distinct from the extended TCE introduced in the present paper. Fiori and Rosazza Gianin (2021) construct another generalization by constructing an indicator that relates monotone risk measures.

Our paper was developed independently from this stream of papers and develops related comparisons of VaR and TCE quantiles, but without introducing an intermediate indicator such as PELVE. A key contribution of our paper is the introduction of a new risk indicator that extends TCE to take into account higher-order risks. We also provide comparisons of this new indicator with more classic risk indicators.

More deeply, the goal of our paper is to understand how regulators choose such or such risk indicators for solvency computations. Thus, our goal is to understand the implicit utility function of insurance regulators and to understand how equivalent risk valuation systems can be put in place, but we do so without introducing any type of utility function. It is, of course, entirely possible that the regulations put in place are not fully a result of a certain intent, but rather of a political process so that what we determine may not be the output of an actual utility function of a specific regulator. However, making the implicit functioning of actual regulations explicit should be a valuable contribution in assuring that regulations function in an effective and efficient manner.

The paper makes use of two probabilistic assumptions, where we assume that claims can be either Pareto or generalized Pareto (GPD) distributed. We make these assumptions for two main reasons: they allow us to very conveniently derive readable results and they are consistent with what is observed for claims with heavily distributed tails. The reader interested in exploring situations where claims could be associated with semi-heavy tails can be referred for instance to Le Courtois (2018) or Le Courtois and Walter (2014).

The paper is organized as follows. In Section 2, we provide a general overview of the two main risk measures, VaR and TCE, and briefly discuss equivalence results that relate to the confidence levels of VaR and TCE. In Section 3, we introduce a new and generalized tail indicator, and we discuss the relation between the quantiles of this indicator and the quantiles of VaR, in a simple Pareto framework, and then in a generalized Pareto framework. Section 4 examines equivalence results between the quantiles of generalized indicators.

#### **2. VaR and TCE**

In this section, we recall key preparatory elements on VaR and TCE. We first recall classic definitions. Then, we examine the link between VaR and TCE. We conclude with an illustration of this link. These elements will be useful in the next section, where the core contribution of the paper is developed.

#### *2.1. Definitions*

The most used risk measure is value at risk (VaR), which is the expected worst loss over a given horizon at a given confidence level. Linsmeier and Pearson (2000) define VaR as the loss that is expected to be exceeded with a probability of only (1 − *α*) percent during the next holding period *T*. The role of regulators is to choose the value of the confidence level *α* and of the horizon *T*.

From a mathematical viewpoint, suppose that *FX<sup>T</sup>* (*x*) represents the distribution function of outcomes over a fixed period of time *T* of a portfolio of risks. An adverse outcome is a loss and, in this case, positive values of the random variable *X<sup>T</sup>* are losses. The VaR of the random variable is the *α* percentile of the distribution of *XT*, denoted by

$$\mathbf{V}\mathbf{a}\mathbf{R}\_{\mathfrak{a}}(X\_T) = F\_{X\_T}^{-1}(\mathfrak{a}).$$

According to the report of the National Association of Insurance Commissioners (2007), tail conditional expectation (TCE)—or conditional tail expectation (CTE)—measures the amount of risk within the tail of a distribution of outcomes, expressed as the probabilityweighted average of the outcomes beyond a chosen point in the distribution.

In the report produced by the CEA (2007), TCE measures the average losses over the defined threshold (typically set as the VaR at a given confidence level *α*). In other words, TCE is a conditional mean value, given that the loss exceeds the (1 − *α*) percentile. It is also often called tail value at risk (TVaR) or expected shortfall (ES). A broader analysis of TCE and its properties can be found in Society of Actuaries (2000). From a mathematical viewpoint, TCE is defined as follows:

$$\text{TCE}\_{\mathfrak{A}}(X\_T) = \mathbb{E}[X\_T \mid X\_T \ge \text{VaR}\_{\mathfrak{A}}(X\_T)].$$

Furthermore, if the random variable is continuous, we can write:

$$\text{TCE}\_{\mathfrak{A}}(X\_T) = \frac{1}{1-\mathfrak{a}} \int\_{\mathfrak{a}}^{1} \text{VaR}\_{\mathfrak{U}}(X\_T) \, d\mathfrak{u}.$$

#### *2.2. Relation between VaR and TCE Quantiles*

If we compute VaR and TCE using the same quantile, TCE will be always higher than VaR, by construction. However, the quantiles for VaR and TCE are usually chosen to be different by regulators. We study which relation should exist between these two quantiles. Specifically, we examine how it is possible to find *c* and *q* such that VaR*<sup>q</sup>* = TCE*c*, where *q* > *c*. We conduct our analysis when risks follow a generalized Pareto distribution.

Let *X<sup>T</sup>* be a random variable that follows a generalized Pareto distribution with three parameters: location *µ*, scale *σ*, and shape *ξ*. The cumulative distribution function of *X<sup>T</sup>* admits the Jenkinson–von Mises representation, which can be expressed as follows:

$$F(X\_T) = 1 - \left(1 + \frac{\xi(\varkappa - \mu)}{\sigma}\right)^{-\frac{1}{\xi}},\tag{1}$$

for *ξ* 6= 0.

In this situation, we can show that the VaR can be computed as follows:

$$\text{VaR}\_q(X\_T) = \mu + \left( (1 - q)^{-\tilde{\xi}} - 1 \right) \cdot \frac{\sigma}{\tilde{\xi}} \, ^\prime \tag{2}$$

while the tail conditional expectation admits the following expression:

$$\text{TCE}\_{\mathfrak{c}}(X\_T) = \mu + \frac{\sigma}{\tilde{\xi}} \left( \frac{(1-\mathfrak{c})^{-\tilde{\xi}}}{1-\tilde{\xi}} - 1 \right).$$

To solve VaR*q*(*XT*) = TCE*c*(*XT*), we first rewrite TCE*c*(*XT*) as a function of VaR*c*(*XT*):

$$\text{TCE}\_{\mathfrak{c}}(X\_T) = \frac{1}{1 - \mathfrak{f}} \cdot \text{VaR}\_{\mathfrak{c}}(X\_T) + \frac{\sigma - \mathfrak{f}\mu}{1 - \mathfrak{f}}.$$

Thus, TCE*c*(*XT*) = VaR*q*(*XT*) is equivalent to

$$\frac{1}{1-\tilde{\xi}} \cdot \text{VaR}\_{\mathfrak{c}}(X\_T) + \frac{\sigma - \tilde{\xi}\mu}{1-\tilde{\xi}} - \text{VaR}\_{\mathfrak{q}}(X\_T) = 0. \tag{3}$$

Using the above equality, we can relate the quantities *c* and *q* as follows.

**Theorem 1.** *In the generalized Pareto framework, the quantile of TCE and the quantile of VaR obey the following relationship when the two risk indicators are equal:*

$$c = 1 - (1 - \mathfrak{f})^{-\frac{1}{\mathfrak{g}}} \cdot (1 - q) \tag{4}$$

*where* 0 < *ξ* < 1*.*

#### **Proof.** See Appendix A.

Note that Equation (4) depends on *ξ*, but not on *µ* or *σ*. Also note that if *ξ* = <sup>1</sup> *α* , the result obtained using the generalized Pareto distribution boils down to an identical result in the subcase of Pareto Type I distributions.

#### *2.3. Illustration*

Next, we illustrate Theorem 1 in Figure 1, where we plot the TCE quantile as a function of its equivalent VaR quantile. We plot this relation for different values of the market risk parameter *ξ*, where we let *ξ* take values between 0.01 and 0.99. We recall that a higher value of *ξ* is equivalent to an increased presence of extreme risks in the phenomenon under study.

**Figure 1.** TCE quantile as a function of VaR quantile.

We can see that when there is more risk on the market, that is, when *ξ* → 1, the VaR quantile has to be a number close to 1, whereas the TCE quantile can take a broad range of values between 0.9 and 1.

This feature implies that in the presence of a lot of extreme risks, there is a large variability in the choice of the TCE quantile, making it a difficult value to choose by regulators. Conversely, in the presence of a lot of extreme risks, the VaR quantile is easy to set, where the regulator only needs to choose a sufficiently high value, as is observed in the case of the Solvency II regulation.

However, this feature is not observed when *ξ* is small, where a lot of admissible values can be taken by both the VaR and TCE quantiles, and where these two quantiles vary linearly.

Figure 1 also illustrates that a lot of the solutions to Theorem 1 are consistent with the Solvency II and the Swiss insurance regulations. However, the figure also illustrates that the two regulations are inconsistent. Indeed, it is practically impossible to find a pair of reasonable values of the VaR and TCE quantiles that is consistent with both regulations.

#### **3. A New High-Order TCE Indicator**

In this section, we introduce a new generalized TCE indicator, which is a conditional higher-order moment of the probability distribution under study. We compute this indicator when losses follow a Type I Pareto distribution and we derive equivalence relations with value at risk. We also conduct a similar study when losses follow a generalized Pareto distribution.

#### *3.1. Definition*

By analogy with higher-order moments, which are key characteristics of probability distributions, we construct a higher-order measure of risk, which is a TCE at order m. We denote this indicator by TCE(*m*) and we define it as follows:

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \mathbb{E}[X\_T^m \mid X\_T \ge \text{VaR}\_{\mathfrak{c}}(X\_T)].\tag{5}$$

As an illustration, TCE(2) *c* is a conditional non-central second-order moment, where the condition is that losses exceed the <sup>1</sup> <sup>−</sup> *<sup>c</sup>* percentile. Further, TCE(1) *c* is the standard tail conditional expectation indicator.

Because

$$\mathbb{E}[\mathbf{X}\_T^m \mid \mathbf{X}\_T \ge \text{VaR}\_c(\mathbf{X}\_T)] = \frac{\mathbb{E}\left[\mathbf{X}\_T^m \: \mathbb{1}\_{\mathbf{X}\_T \ge \text{VaR}\_c(\mathbf{X}\_T)}\right]}{\Pr(\mathbf{X}\_T \ge \text{VaR}\_c(\mathbf{X}\_T))},$$

we can rewrite the extended TCE indicator as follows:

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{1}{1-\mathfrak{c}} \operatorname{\mathbb{E}}\left[X\_T^m \, \mathbb{1}\_{X\_T \ge \text{VaR}\_{\mathfrak{c}}(X\_T)}\right],$$

so that

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{1}{1-\mathfrak{c}} \int\_{\text{VaR}\_{\mathfrak{c}}(X\_T)}^{+\infty} \mathfrak{x}^m \, dF(\mathfrak{x}) \, . \tag{6}$$

where *c* = *F*(VaR*c*(*XT*)).

Let us change variables as follows: *F*(*x*) = *s*, *x* = *F* −1 (*s*) = VaR*s*(*XT*), and *ds* = *dF*(*x*). We readily obtain a third equivalent representation of the extended TCE indicator:

$$\mathrm{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{1}{1-\mathfrak{c}} \int\_{\mathfrak{c}}^{1} \left(\mathrm{VaR}\_{\mathfrak{s}}(X\_T)\right)^{m} ds.\tag{7}$$

Note that another extended TCE indicator Ξ (*m*) can be found in the risk management literature (see for instance Barczy et al. (2022)). This indicator is defined by:

$$\mathbb{E}^{(m)} = \frac{m}{1-c} \int\_{\mathcal{c}}^{1} \left(\frac{s-c}{1-c}\right)^{m-1} \text{VaR}\_{\mathbf{s}}(X\_T) \, ds = \frac{m}{1-c} \int\_{\mathcal{c}}^{1} \left(\frac{s-c}{1-c}\right)^{m-1} F^{-1}(s) \, ds \dots$$

If we again change variables as follows: *F*(*x*) = *s*, *x* = *F* −1 (*s*) = VaR*s*(*XT*), and *ds* = *dF*(*x*), we obtain:

$$\Xi^{(m)} = \frac{m}{1-c} \int\_{\text{VaR}\_{\text{c}}(X\_T)}^{+\infty} \left( \frac{\mathcal{F}(\mathfrak{x}) - c}{1-c} \right)^{m-1} \ge dF(\mathfrak{x}).$$

All of these expressions are distinct from Equations (5)–(7) and confirm that Ξ (*m*) cannot be interpreted as a conditional higher-order moment, contrary to the indicator examined in this paper.

#### *3.2. Pareto Distributed Losses*

Let us now assume that losses follow a Type I Pareto distribution, whose probability density function is represented by:

$$f\_{X\_T}(\mathbf{x}) = \frac{\mathfrak{a}\theta^{\mathfrak{a}}}{\mathfrak{x}^{\mathfrak{a}+1}} \,\tag{8}$$

where *α* is the shape parameter and *θ* the scale parameter.

We apply the definition in Equation (6) to compute TCE(*m*) when losses follow a Type I Pareto distribution. We write:

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{1}{1-\mathfrak{c}} \int\_{\text{VaR}\_{\mathfrak{c}}(X\_T)}^{+\infty} \mathfrak{x}^m \, \frac{\mathfrak{a}\theta^\alpha}{\mathfrak{x}^{\alpha+1}} \, d\mathfrak{x}.$$

This expression can be developed in closed form as a function of value at risk, as shown in the next theorem.

**Theorem 2.** *When losses are Pareto distributed, the extended TCE indicator admits the following expression.*

$$T\mathbb{C}E\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{\mathfrak{a}}{\mathfrak{a} - m} \left( VaR\_{\mathfrak{c}}(X\_T) \right)^m \tag{9}$$

*when α* > *m.*

**Proof.** See Appendix A.

From this theorem, we deduce that VaR*q*(*XT*) = TCE*c*(*XT*) is equivalent to

$$\frac{\mathfrak{a}}{\mathfrak{a}-\mathfrak{m}} \cdot (\mathsf{VaR}\_{\mathfrak{c}}(X\_T))^m - \mathsf{VaR}\_{\mathfrak{q}}(X\_T) = 0. \tag{10}$$

This equation allows us to find the relation between the extended TCE and the VaR quantiles. Indeed, we obtain:

**Theorem 3.** *In the Pareto framework, the extended TCE quantile (c) and the VaR quantile (q) obey the following relationship when the two risk indicators are equal:*

$$c = 1 - \left(\frac{\left(\alpha - m\right)\theta^{1-m}}{a}\right)^{-\frac{a}{m}} (1 - q)^{\frac{1}{m}}.\tag{11}$$

**Proof.** See Appendix A.

For consistency with the generalized Pareto approach, we replace the shape parameter *α* of the classic Pareto distribution with *ξ* = <sup>1</sup> *α* , which can also be interpreted as a shape parameter.

Thus, Equation (11) can be rewritten as follows:

$$\mathcal{L} = 1 - \left( (1 - m\xi) \,\,\theta^{1 - m} \right)^{-\frac{1}{m\xi}} (1 - q)^{\frac{1}{m}} \,\, \, \, \tag{12}$$

where 0 < *ξ* < <sup>1</sup> *m* .

It appears that Equation (12) generalizes the result of Theorem 1. Indeed, Equation (12) reduces to Equation (4) when *m* = 1.

Figure 2 shows the extended TCE quantile as a function of the VaR quantile, for different values of the shape parameter *ξ*. In the left panel of Figure 2, the scale parameter is equal to *θ* = 1, while the scale parameter is equal to *θ* = 5 in the right panel of this figure. Both panels are plotted assuming that *m* = 2.

**Figure 2.** Extended TCE quantile as a function of VaR quantile. (**Left panel**): *m* = 2 and *θ* = 1. (**Right panel**): *m* = 2 and *θ* = 5.

As in the case of Figure 1, we see that a higher value of the shape parameter leads to more intricate situations, where the VaR quantile takes a value close to one, while the extended TCE quantile can take a broad range of values. However, Figure 2 also shows that a higher value of the scale parameter leads to even more intricate situations. Thus, when value at risk is not able to distinguish between extreme risk situations, a more sophisticated indicator such as the extended TCE indicator is able to produce such a distinction.

Figure 3 is constructed in a similar way as Figure 2, but now both panels are plotted assuming that *m* = 3. By comparing Figures 2 and 3, we see that the curves are pushed to the right for higher values of *m*, all other parameters being equal. Thus, varying the value of *m* allows us to construct risk indicators that are more or less sensitive to the presence of extreme risks.

**Figure 3.** Extended TCE quantile as a function of VaR quantile. (**Left panel**): *m* = 3 and *θ* = 1. (**Right panel**): *m* = 3 and *θ* = 5.

#### *3.3. GPD Losses*

Let us now generalize the previous study to the case where the loss random variable *X* follows a generalized Pareto distribution. In accordance with the cdf shown in Equation (1), we rely on the following probability density function:

$$f\_{X\_T}(\mathbf{x}) = \frac{1}{\sigma} \left( 1 + \frac{\xi(\mathbf{x} - \boldsymbol{\mu})}{\sigma} \right)^{\left( -\frac{1}{\xi} - 1 \right)}\Big|\tag{13}$$

which generalizes Equation (8) by introducing a location parameter *µ*. The scale parameter is now denoted by *σ* and the shape parameter is now *ξ* 6= 0. For information on the estimation of tail parameters, see for instance Hill (1975) or Hosking and Wallis (1987).

We obtain a quasi-closed-form formula for the extended TCE indicator in this setting. Indeed, we have:

**Theorem 4.** *In the GPD case, the extended TCE indicator can be computed as follows:*

$$\begin{split} \operatorname{TCE}\_{c}^{(m)}(X\_{\mathrm{T}}) &= \frac{1}{1-c} \cdot \int\_{\mathrm{VaR}\_{c}(X\_{\mathrm{T}})}^{+\infty} x^{m} \frac{1}{\sigma} \left( 1 + \frac{\xi(x-\mu)}{\sigma} \right)^{\left(-\frac{1}{\xi}-1\right)} dx \\ &= \frac{1}{1-c} \left[ -(-1)^{-\left(m-\frac{1}{\xi}+1\right)} \frac{1}{\overline{\xi}} \frac{\Gamma\left(-m+\frac{1}{\xi}\right) \Gamma(m+1)}{\Gamma\left(1+\frac{1}{\xi}\right)} \left( \frac{\mu\xi-\sigma}{\sigma} \right)^{-\frac{1}{\xi}} \left( \frac{\mu\xi-\sigma}{\overline{\xi}} \right)^{m} \\ &- (1-c) \frac{\left(\mathrm{VaR}\_{c}(X\_{\mathrm{T}})\right)^{m+1}}{(m+1)\cdot(\sigma-\mu\xi)} \,\_2F\_{1}\left(1, m-\frac{1}{\xi}+1; m+2; \frac{\mathrm{VaR}\_{c}(X\_{\mathrm{T}})\,\xi}{\mu\xi-\sigma} \right) \right], \end{split} \tag{14}$$

*where* <sup>2</sup>*F*1(·, ·; ·; ·) *is the hypergeometric function,* <sup>Γ</sup>(·) *is the gamma function,* <sup>0</sup> <sup>&</sup>lt; *<sup>ξ</sup>* <sup>&</sup>lt; <sup>1</sup> *m , and VaRc*(*XT*) = *µ* + *<sup>σ</sup> ξ* (1 − *c*) <sup>−</sup>*<sup>ξ</sup>* <sup>−</sup> <sup>1</sup> *.*

**Proof.** See Appendix A.

To solve TCE(*m*) *<sup>c</sup>* (*XT*) = VaR*q*(*XT*), we can equivalently solve:

$$\frac{1}{1-c}\left[-(-1)^{-\left(m-\frac{1}{\xi}+1\right)}\frac{1}{\tilde{\xi}}\frac{\Gamma\left(-m+\frac{1}{\xi}\right)\Gamma(m+1)}{\Gamma\left(1+\frac{1}{\xi}\right)}\left(\frac{\mu\xi-\sigma}{\sigma}\right)^{-\frac{1}{\xi}}\left(\frac{\mu\xi-\sigma}{\xi}\right)^{m}\right]$$

$$-\frac{(1-c)}{(m+1)}\frac{(\mathrm{VaR}\_{\mathrm{c}}(X\_{T}))^{m+1}}{\left(m+1\right)}\,\_2F\_1\left(1,m-\frac{1}{\xi}+1;m+2;\frac{\mathrm{VaR}\_{\mathrm{c}}(X\_{T})}{\mu\xi-\sigma}\right)\right]=\mathrm{VaR}\_{\mathrm{d}}(X\_{T}).\tag{15}$$

To numerically solve Equation (15), the following three conditions must be met:


Although Equation (15) does not admit closed-form solutions, we numerically solve it to show the relation between the TCE(*m*) quantile and the VaR quantile, as a function of the order *m* and of the three generalized Pareto distribution parameters *ξ*, *µ*, and *σ*.

We start by plotting in Figure 4 the relation between the TCE(*m*) and the VaR quantiles when *m* = 2 and *σ* = 0.1. The left panel of the figure presents the situation where *µ* = −0.05, while the right panel presents the situation where *µ* = 0.05.

**Figure 4.** Extended TCE quantile as a function of VaR quantile. (**Left panel**): *m* = 2, *σ* = 0.1, and *µ* = −0.05. (**Right panel**): *m* = 2, *σ* = 0.1, and *µ* = +0.05.

From Figure 4, we see that the location parameter *µ* has a sharp impact on quantile dependences. When *µ* is high, the relation between indicator quantiles becomes nearly linear, which is an ideal situation from a risk management viewpoint.

Next, we plot in Figure 5 the relation between the TCE(*m*) and the VaR quantiles when *m* = 2 and *µ* = 0. The left panel of the figure presents the situation where *σ* = 0.1, while the right panel presents the situation where *σ* = 0.4. From the comparison of the two panels of the figure, we see that the ideal situation where the relation between the indicator quantiles is quasi-linear occurs for small values of the scale parameter *σ*.

**Figure 5.** Extended TCE quantile as a function of VaR quantile. (**Left panel**): *m* = 2, *σ* = 0.1, and *µ* = 0. (**Right panel**): *m* = 2, *σ* = 0.4, and *µ* = 0.

Finally, we plot in Figure 6 the relation between the TCE(*m*) and the VaR quantiles when *m* = 3, *µ* = 0. We set *σ* = 0.1 in the left panel and *σ* = 0.4 in the right panel. Thus, Figure 6 presents the same comparison as Figure 5, but for a larger value of the order parameter *m*.

**Figure 6.** Extended TCE quantile as a function of VaR quantile. (**Left panel**): *m* = 3, *σ* = 0.1, and *µ* = 0. (**Right panel**): *m* = 3, *σ* = 0.4, and *µ* = 0.

From the comparison of Figures 5 and 6, we see that higher values of the order parameter *m* lead to more intricate situations from a management viewpoint. Thus, the presence of extreme risks in the system being considered, and their taking into account via higher-order conditional moments, makes risk management more complicated in the sense that choosing an indicator quantile becomes a more critical and sensitive decision.

#### **4. Equivalence between High-Order Indicators**

In this section, we study the relation between the quantiles *q* (*m*) and *q* (*n*) of distinct extended tail conditional expectation indicators, where each indicator is associated with a different order *m* or *n*. Namely, we examine the situation where:

$$\mathrm{TCE}^{(m)}\_{q^{(\mathfrak{m})}}(X\_T) = \mathrm{TCE}^{(n)}\_{q^{(\mathfrak{n})}}(X\_T). \tag{16}$$

We also study the sub-case where TCE is compared with a high-order TCE, that is, we study the quantiles *q* and *q* (*m*) that satisfy:

$$\text{TCE}\_q(X\_T) = \text{TCE}\_{q^{(m)}}^{(m)}(X\_T). \tag{17}$$

#### *4.1. Pareto Distributed Losses*

If we model losses as random variables that follow a classic type I Pareto distribution, we can use Equation (9) to compute TCE(*m*) *q* (*m*) (*XT*) where *α* > *m*. Thus, we can compare two higher-order TCEs as follows:

$$\frac{\mathfrak{a}}{\mathfrak{a}-m} \left( \text{VaR}\_{q^{(\mathfrak{m})}} (X\_T) \right)^m = \frac{\mathfrak{a}}{\mathfrak{a}-n} \left( \text{VaR}\_{q^{(\mathfrak{n})}} (X\_T) \right)^m \mathfrak{a}$$

where

$$\text{VaR}\_{q^{(m)}}(X\_T) = \theta \left(1 - q^{(m)}\right)^{-\frac{1}{\alpha}}.$$

We obtain:

$$\frac{\mathfrak{a}}{\mathfrak{a}-m} \, \theta^{\mathfrak{m}} \left(1 - q^{(m)}\right)^{-\frac{m}{\mathfrak{a}}} = \frac{\mathfrak{a}}{\mathfrak{a}-n} \, \theta^{\mathfrak{m}} \left(1 - q^{(n)}\right)^{-\frac{n}{\mathfrak{a}}}.$$

which leads us to:

$$q^{(n)} = 1 - \left(\frac{\alpha - n}{\alpha - m} \theta^{m-n}\right)^{-\frac{\alpha}{n}} \left(1 - q^{(m)}\right)^{-\frac{m}{n}} \tag{18}$$

where *α* > *m*, *α* > *n*, and *θ* > 0.

If we denote *ξ* = <sup>1</sup> *α* , we can rewrite Equation (18) as in the following proposition.

**Proposition 1.** *When losses follow a classic Pareto distribution, the quantiles of high-order TCEs that solve Equation* (16) *can be related as follows:*

$$q^{(n)} = 1 - \left(\frac{\frac{1}{\xi} - m}{\frac{1}{\xi} - n}\right)^{\frac{1}{\xi n}} \theta^{\frac{n-m}{\xi n}} \left(1 - q^{(m)}\right)^{-\frac{m}{n}},\tag{19}$$

*where* 0 < *ξ* < <sup>1</sup> *m ,* 0 < *ξ* < <sup>1</sup> *n , and θ* > 0*.*

We now illustrate this proposition.

Figure 7 plots the relation between the quantiles *q* (*m*) and *q* (*n*) when *m* = 5 and *n* = 2. The left panel of the figure shows the situation where *θ* = 1, while the right panel of the figure shows the situation where *θ* = 2.

**Figure 7.** Extended TCE quantile at order *n* = 2 as a function of extended TCE quantile at order *m* = 5. (**Left panel**): *θ* = 1. (**Right panel**): *θ* = 2.

The left panel of Figure 7 can be interpreted as follows. The relation between the high-order quantiles is countermonotonic contrary to the relation between the TCE quantile and the VaR quantile, for instance. This means that a high value of *q* (*m*) corresponds to a small value of *q* (*n*) , and conversely.

This feature is a consequence of the fact that high-order TCEs concentrate on different parts of probability tails. Thus, the figure shows us that a manager that reduces high-order extreme risks at a given order, say *m*, is not simultaneously reducing high-order extreme risks at another order, say *n*. The right panel of the figure tells us that this aspect is even more pronounced for higher values of *θ*.

We now come to the specific case where *n* = 1, that is to the study of the relation between TCE and a higher-order TCE:

$$\frac{\alpha}{\alpha - 1} \left( \text{VaR}\_q(X\_T) \right)^1 = \frac{\alpha}{\alpha - m} \left( \text{VaR}\_{q^{(m)}}(X\_T) \right)^m.$$

Equation (18) becomes

$$q = 1 - \left(\frac{\alpha - m}{(\alpha - 1)\theta^{m-1}}\right)^{\alpha} \left(1 - q^{(m)}\right)^{-m}.$$

when *α* > *m* and *θ* > 0. Similarly, Equation (19) becomes

$$q = 1 - \left(\frac{\frac{1}{\frac{7}{5}} - m}{\left(\frac{1}{\frac{7}{5}} - 1\right) \theta^{m-1}}\right)^{\frac{1}{5}} \left(1 - q^{(m)}\right)^{-m}$$

when 0 < *ξ* < <sup>1</sup> *m* and *θ* > 0.

We show in Figure 8 the relation between the TCE quantile and the high-order TCE quantile when *m* = 2. The left panel of the figure shows the situation where *θ* = 1, while the right panel of the figure shows the situation where *θ* = 2.

**Figure 8.** TCE quantile as a function of extended TCE quantile at order 2. (**Left panel**): *θ* = 1. (**Right panel**): *θ* = 2.

Figure 8 confirms the results of Figure 7. Reducing risks using TCE does not necessarily reduce risks as measured by a high-order TCE, and conversely. Again, this effect is more pronounced for higher values of *θ*.

#### *4.2. GPD Losses*

Let us now come to the more general situation where losses are modeled using a generalized Pareto distribution. Our goal is to solve Equation (16) when the extended TCE indicator TCE(*m*) *q* (*m*) is given by Equation (14). Thus, to derive the relation between *q* (*m*) and *q* (*n*) , we numerically solve:

$$\frac{1}{1-q^{(m)}}\left[-(-1)^{-\left(m-\frac{1}{\xi}+1\right)}\frac{1}{\frac{\Gamma}{\xi}}\frac{\Gamma\left(-m+\frac{1}{\xi}\right)\Gamma(m+1)}{\Gamma\left(1+\frac{1}{\xi}\right)}\left(\frac{\mu\xi-\sigma}{\sigma}\right)^{-\frac{1}{\xi}}\left(\frac{\mu\xi-\sigma}{\xi}\right)^{m}\right]$$

$$-\left(1-q^{(m)}\right)\frac{\left(\mathrm{VaR}\_{q^{(m)}}(X\_{\Gamma})\right)^{m+1}}{(m+1)\cdot(\sigma-\mu\xi)}\,\_2F\_1\left(1,m-\frac{1}{\xi}+1;m+2;\frac{\mathrm{VaR}\_{q^{(m)}}(X\_{\Gamma})}{\mu\xi-\sigma}\right)\right]$$

$$=\frac{1}{1-q^{(n)}}\left[-(-1)^{-\left(n-\frac{1}{\xi}+1\right)}\frac{1}{\xi}\frac{\Gamma\left(-n+\frac{1}{\xi}\right)\Gamma(n+1)}{\Gamma\left(1+\frac{1}{\xi}\right)}\left(\frac{\mu\xi-\sigma}{\sigma}\right)^{-\frac{1}{\xi}}\left(\frac{\mu\xi-\sigma}{\xi}\right)^{n}\right]$$

$$-\left(1-q^{(n)}\right)\frac{\left(\mathrm{VaR}\_{q^{(n)}}(X\_{\Gamma})\right)^{n+1}}{(n+1)\cdot(\sigma-\mu\xi)}\,\_2F\_1\left(1,n-\frac{1}{\xi}+1;n+2;\frac{\mathrm{VaR}\_{q^{(n)}}(X\_{\Gamma})\,\xi}{\mu\xi-\sigma}\right)\right]\tag{20}$$

Figure 9 plots the relation between the quantiles *q* (*m*) and *q* (*n*) when *m* = 5, *n* = 2, and *σ* = 0.1. The left panel of the figure shows the situation where *µ* = −0.05, while the right panel of the figure shows the situation where *µ* = 0.05.

**Figure 9.** Extended TCE quantile at order *n* = 2 as a function of extended TCE quantile at order *m* = 5 with *σ* = 0.1. (**Left panel**): *µ* = −0.05. (**Right panel**): *µ* = +0.05.

From Figure 9, we deduce that the link between the high-order TCE quantiles is linear when *σ* = 0.1, so this parameter of the GPD distribution is not problematic. By comparing the two panels of the figure, we see that the parameter *µ* has little effect on the curves linking the high-order TCE quantiles.

Figure 10 plots the relation between the quantiles *q* (*m*) and *q* (*n*) when *m* = 5, *n* = 2, and *µ* = 0. The left panel of the figure shows the situation where *σ* = 0.1, while the right panel of the figure shows the situation where *σ* = 0.4.

**Figure 10.** Extended TCE quantile at order *n* = 2 as a function of extended TCE quantile at order *m* = 5 with *µ* = 0. (**Left panel**): *σ* = 0.1. (**Right panel**): *σ* = 0.4.

Figure 10 shows us that high values of *σ* can yield problematic links between the high-order TCE quantiles, hinting at probability tails that are quantified differently by distinct high-order TCE indicators.

We now come to the specific case where *n* = 1, that is, to the study of the relation between TCE and a higher-order TCE. In that case also, the solutions are numerically obtained by solving Equation (20).

Figure 11 plots the relation between the quantiles *q* (*m*) and *q* when *m* = 2 and *σ* = 0.1. The left panel of the figure shows the situation where *µ* = −0.05, while the right panel of the figure shows the situation where *µ* = 0.05.

**Figure 11.** TCE quantile as a function of extended TCE quantile at order *m* = 2 with *σ* = 0.1. (**Left panel**): *µ* = −0.05. (**Right panel**): *µ* = +0.05.

Figure 11 tells us that the link between the second order TCE quantile and the TCE quantile is close to linear when *σ* = 0.1, so that, again, this parameter of the GPD distribution is not problematic when it is not set to a high value.

By comparing the two panels of Figure 11, we see, as in Figure 9, that large variations of the parameter *µ* have a quite limited impact on the position of the curves relating a high-order TCE quantile to the TCE quantile.

Figure 12 plots the relation between the quantiles *q* (*m*) and *q* when *m* = 2 and *µ* = 0. The left panel of the figure shows the situation where *σ* = 0.1, while the right panel of the figure shows the situation where *σ* = 0.4.

**Figure 12.** TCE quantile as a function of extended TCE quantile at order *m* = 2 with *µ* = 0. (**Left panel**): *σ* = 0.1. (**Right panel**): *σ* = 0.4.

Figure 12 confirms the conclusion of Figure 10. Specifically, high values of *σ* can yield problematic links between a high-order TCE quantile and the TCE quantile.

#### **5. Conclusions**

We end this paper with a brief illustration based on actual data. We use a fire insurance claim data set, labeled "beaonre" within the R package CASdatasets. This dataset includes 1823 observations of fire insurance claims from the year 1997. We transform this dataset of claim costs into a dataset of reimbursements, from which we can compute VaR, TCE, and TCE(*m*) . To compare VaR or TCE with a high-order TCE indicator, we need to adjust the latter quantity in terms of scale. For instance, we may want to solve VaR*q*(*XT*) = TCE(*m*) *<sup>c</sup>* (*XT*) 1 *m* . While exact solutions to this equation do not exist, we can still deduce a relation between VaR and TCE(*m*) .

We show in Figure 13 the link between VaR and the high-order TCE indicator (computed with *m* = 2) in the case of fire data. This figure is consistent with the theoretical results shown, for instance, in Figure 6. Thus, Figure 13 confirms, in passing, the relevance of the GPD assumption. Further illustration with actual data is out of the scope of the present paper but could be a matter of an interesting extension.

**Figure 13.** Extended TCE quantile, with *m* = 2, as a function of VaR.

To conclude, we introduce in this paper a new risk indicator that is a high-order TCE risk measure. We compare the quantiles of this indicator to the quantiles of VaR in a simple Pareto framework, and then in a generalized Pareto framework. We also examine equivalence results between the quantiles of high-order TCEs. By doing so, we aim at illustrating the interplay between implicit choices of risk measures by regulators and the characteristics of probability distribution tails.

Among the possible theoretical extensions of our paper, one could cite the verification that the high-order indicator that we introduce is indeed a coherent risk measure. While this is out of the scope of the present paper, a separate document is in the process of being written on this aspect. See for instance Krokhmal (2007) or Barbosa and Ferreira (2004) for references on the coherence of related indicators. Another possible extension of our paper could consist in examining the relation between high-order TCEs when the probability distribution admits tails that are not modeled using the generalized Pareto distribution, but using, for instance, the semi-heavy tails of infinitely divisible probability distributions. Finally, it could be interesting to examine the stability of high-order TCE indicators (see for instance the discussion in Le Courtois et al. (2020) on the cross-stability of second and fourth-order moments).

**Author Contributions:** All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

*Appendix A.1. Proof of Theorem 1*

Our goal is to solve Equation (3):

$$\frac{1}{1-\tilde{\xi}} \text{VaR}\_{\mathfrak{c}} + \frac{\sigma - \tilde{\xi}\mu}{1-\tilde{\xi}} - \text{VaR}\_{\mathfrak{q}} = 0.$$

We replace VaR with its expression given in Equation (2):

$$\frac{1}{1-\tilde{\xi}}\left(\left((1-c)^{-\tilde{\xi}}-1\right)\cdot\frac{\sigma}{\tilde{\xi}}+\mu\right)+\frac{\sigma-\tilde{\xi}\mu}{1-\tilde{\xi}}-\left(\left((1-q)^{-\tilde{\xi}}-1\right)\cdot\frac{\sigma}{\tilde{\xi}}+\mu\right)=0.\tilde{\xi}$$

We have:

*σ ξ*

$$\frac{1}{1-\tilde{\xi}} \left( \left( (1-c)^{-\tilde{\xi}} - 1 \right) \cdot \frac{\sigma}{\tilde{\xi}} \right) + \frac{\mu}{1-\tilde{\xi}} + \frac{\sigma - \tilde{\xi}\mu}{1-\tilde{\xi}} - \left( \left( (1-q)^{-\tilde{\xi}} - 1 \right) \cdot \frac{\sigma}{\tilde{\xi}} \right) - \mu = 0, \tilde{\xi}$$

or

$$\left( \frac{1}{1 - \xi} \left( (1 - c)^{-\xi} - 1 \right) - \left( (1 - q)^{-\xi} - 1 \right) \right) = \frac{\mu - \xi \mu - \mu - \sigma + \xi \mu}{1 - \xi}$$

,

.

so that

$$\frac{1}{1-\xi} \left( (1-c)^{-\xi} - 1 \right) - \left( (1-q)^{-\xi} - 1 \right) = -\frac{\sigma}{1-\tilde{\xi}} \frac{\tilde{\xi}}{\sigma}$$

Next, we write:

$$\frac{1}{1-\xi} \cdot (1-c)^{-\xi} - (1-q)^{-\xi} - \frac{\xi}{1-\xi} = -\frac{\xi}{1-\xi}$$

and

$$\frac{1}{1-\xi} \cdot (1-c)^{-\xi} = (1-q)^{-\xi}.$$

Finally, we obtain:

$$1 - c = \left( (1 - \xi)(1 - q)^{-\frac{\xi}{\xi}} \right)^{-\frac{1}{\xi}}.$$

which can also be reformulated as follows:

$$\mathcal{c} = 1 - (1 - \xi)^{-\frac{1}{\xi}} \cdot (1 - q) \cdot$$

*Appendix A.2. Proof of Theorem 2*

Our goal is to solve

$$\mathrm{TCE}\_{\mathsf{c}}^{(m)}(X\_T) = \frac{1}{1-\mathsf{c}} \cdot \int\_{\mathrm{vaR}\_{\mathsf{c}}(X\_T)}^{+\infty} \mathsf{x}^m f\_{\mathbf{X}\_T}(\mathbf{x}) d\mathbf{x} \mathsf{x}$$

where *fX<sup>T</sup>* (*x*) = *αθ<sup>α</sup> x α*+1 . We have:

$$\begin{aligned} \text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) &= \frac{1}{1-\mathfrak{c}} \cdot \int\_{\text{VaR}\_{\mathfrak{c}}(X\_T)}^{+\infty} \mathfrak{x}^m \frac{a\theta^\alpha}{\mathfrak{x}^{\alpha+1}} d\mathfrak{x}\_\prime \\ &= \frac{1}{1-\mathfrak{c}} \left[ -\frac{a\theta^\alpha}{\mathfrak{a}-m} \cdot \mathfrak{x}^{m-\mathfrak{a}} \right]\_{\text{VaR}\_{\mathfrak{c}}(X\_T)}^{+\infty} \end{aligned}$$

.

The quantity lim *<sup>x</sup>*→+<sup>∞</sup> <sup>−</sup> *αθ<sup>α</sup> α*−*m* · *x m*−*α* converges to zero only if *m* − *α* < 0. Assuming that *α* > *m*, we obtain:

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{1}{1-c} \frac{\mathfrak{a}\theta^{\alpha}}{\mathfrak{a}-m} \cdot \left(\text{VaR}\_{\mathfrak{c}}(X\_T)\right)^{m-\alpha}.$$

Next, we write:

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{1}{1 - c} \frac{a\theta^{\mathfrak{a}}}{a - m} (\text{VaR}\_{\mathfrak{c}}(X\_T))^{-a} \cdot (\text{VaR}\_{\mathfrak{c}}(X\_T))^m \mu$$

and we replace VaR with its expression:

$$\text{VaR}\_{\mathfrak{c}}(X\_T) = \theta(1-\mathfrak{c})^{-\frac{1}{\mathfrak{a}}}\mathfrak{c}$$

to obtain:

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{1}{1 - c} \frac{\mathfrak{a}\theta^a}{\mathfrak{a} - m} \left(\theta \cdot (1 - c)^{-\frac{1}{a}}\right)^{-a} \cdot \left(\text{VaR}\_{\mathfrak{c}}(X\_T)\right)^m.$$

Finally, we have:

$$\text{TCE}\_{\mathfrak{c}}^{(m)}(X\_T) = \frac{\mathfrak{a}}{\mathfrak{a} - m} \cdot \left( \text{VaR}\_{\mathfrak{c}}(X\_T) \right)^m \text{s}$$

which is our result.

#### *Appendix A.3. Proof of Theorem 3*

Our goal is to solve Equation (10) when VaR*c*(*XT*) = *θ*(1 − *c*) − 1 *<sup>α</sup>* , so when *X<sup>T</sup>* follows a Pareto Type I distribution. Replacing VaR with its expression, we can rewrite Equation (10) as follows:

$$\frac{\mathfrak{a}}{\mathfrak{a}-\mathfrak{m}} \cdot \left(\theta(1-c)^{-\frac{1}{\mathfrak{a}}}\right)^{m} - \left(\theta(1-q)^{-\frac{1}{\mathfrak{a}}}\right) = 0.$$

Next, we write:

$$\frac{\mathfrak{a}}{\mathfrak{a}-m} \cdot \theta^m (1-\mathfrak{c})^{-\frac{m}{\mathfrak{a}}} - \theta (1-q)^{-\frac{1}{\mathfrak{a}}} = 0,$$

so that

$$c = 1 - \left(\frac{\alpha - m}{\alpha \theta^m} \left. \theta \left(1 - q\right)^{-\frac{1}{\alpha}} \right)^{-\frac{\theta}{m}}\right)$$

which can readily be rewritten as Equation (11).

*Appendix A.4. Proof of Theorem 4*

The aim of this appendix is to demonstrate that the integral:

$$\mathrm{TCE}^{(m)}(X\_T) = \frac{1}{1-c} \int\_{\mathrm{VaR}\_{\mathrm{c}}(X\_T)}^{+\infty} x^m \frac{1}{\sigma} \left(1 + \frac{\tilde{\xi}(\varkappa - \mu)}{\sigma}\right)^{-\frac{1}{\xi}-1} d\varkappa$$

can be computed to provide the result in Equation (14).

Using classic results on special functions (see for instance Lebedev (1972)), we rewrite the integral as follows:

$$\text{TCE}^{(m)}(X\_T) = \frac{1}{1-c} \left[ \frac{\mathbf{x}^{m+1}}{(m+1)\cdot(\sigma-\mu\_\odot^x)} \left( \frac{\sigma+\xi(x-\mu)}{\sigma} \right)^{-\frac{1}{\xi}} \right]$$

$$\times \,\_2F\_1 \left( 1, m-\frac{1}{\xi}+1; m+2; \frac{\mathbf{x}\_\xi^x}{\mu\_\xi^x-\sigma} \right) \Big|\_{\text{VaR}\_\odot(X\_T)}^{+\infty}. \tag{A1}$$

To compute the limit when *x* tends to infinity of the quantity *J* defined by:

$$J = \frac{\mathbf{x}^{m+1}}{(m+1)\cdot(\sigma-\mu\xi)} \left(\frac{\sigma+\xi(\mathbf{x}-\mu)}{\sigma}\right)^{-\frac{1}{\xi}} \,\_2F\_1\left(1, m-\frac{1}{\xi}+1; m+2; \frac{\mathbf{x}\xi}{\mu\xi-\sigma}\right) J$$

we rewrite the hypergeometric function using a linear transformation:

$$\begin{split} \;\_2F\_1\left(1, m-\frac{1}{\xi}+1; m+2; \frac{\mathbf{x}\_{\xi}^{\mathbf{x}}}{\mu\xi-\sigma}\right) \\ = \;\_2F\_1\left(m-\frac{1}{\xi}\right)\Gamma(m+2) \\ = \;\_1F\_1\left(m-\frac{1}{\xi}+1\right)\Gamma(m+1) \left(-\frac{\mathbf{x}\_{\xi}^{\mathbf{x}}}{\mu\xi-\sigma}\right)^{-1} \;\_2F\_1\left(1, -m; -m+\frac{1}{\xi}+1; \frac{\mu\xi-\sigma}{\mathbf{x}\_{\xi}^{\mathbf{x}}}\right) \\ + \;\_2F\_1\left(-m+\frac{1}{\xi}\right)\Gamma(m+2) \left(-\frac{\mathbf{x}\_{\xi}^{\mathbf{x}}}{\mu\xi-\sigma}\right)^{-\left(m-\frac{1}{\xi}+1\right)} \;\_2F\_1\left(m-\frac{1}{\xi}+1, -\frac{1}{\xi}; m-\frac{1}{\xi}+1; \frac{\mu\xi-\sigma}{\mathbf{x}\_{\xi}^{\mathbf{x}}}\right). \end{split} \tag{A2}$$

Thus, *J* can be rewritten as follows:

$$\begin{split} J &= K \, \mathrm{x}^{\mathrm{m}} \left( 1 + \frac{\widetilde{\mathsf{z}}(\mathsf{x} - \mu)}{\sigma} \right)^{-\frac{1}{\xi}} \, \_2F\_1 \left( 1, -m; -m + \frac{1}{\widetilde{\mathsf{z}}} + 1; \frac{\mu \widetilde{\mathsf{z}} - \sigma}{\mathrm{x} \widetilde{\mathsf{z}}} \right) \\ &+ L \, \mathrm{x}^{\frac{1}{\xi}} \left( 1 + \frac{\widetilde{\mathsf{z}}(\mathsf{x} - \mu)}{\sigma} \right)^{-\frac{1}{\xi}} \, \_2F\_1 \left( m - \frac{1}{\widetilde{\xi}} + 1, -\frac{1}{\widetilde{\xi}}; m - \frac{1}{\widetilde{\xi}} + 1; \frac{\mu \widetilde{\mathsf{z}} - \sigma}{\mathrm{x} \widetilde{\mathsf{z}}} \right), \end{split}$$

where *K* and *L* are functions of the parameters that are independent of *x*. Specifically,

$$K = \frac{\Gamma\left(m - \frac{1}{\xi}\right)}{\Gamma\left(m - \frac{1}{\xi} + 1\right)\xi}$$

and

$$L = \frac{\Gamma\left(-m + \frac{1}{\xi}\right)\Gamma(m+1)}{\Gamma\left(1 + \frac{1}{\xi}\right)} \frac{1}{(\sigma - \mu\xi)} \left(\frac{\mathfrak{z}}{\sigma - \mu\xi}\right)^{-\left(m - \frac{1}{\xi} + 1\right)}.$$

Then, we use the fact that

$$\begin{aligned} \lim\_{\chi \to +\infty} & \,\_2F\_1 \left( 1, -m; -m + \frac{1}{\xi} + 1; \frac{\mu \mathfrak{F} - \sigma}{\mathfrak{x} \mathfrak{F}} \right) = \\ & \lim\_{\chi \to \infty} \left[ \left( \frac{\mu \mathfrak{F} - \sigma}{\mathfrak{x} \mathfrak{F}} \right)^0 + \frac{-m}{-m + \frac{1}{\xi} + 1} \frac{\mu \mathfrak{F} - \sigma}{\mathfrak{x} \mathfrak{F}} + \cdots \right] = 1 \end{aligned}$$

and

$$\begin{aligned} &\lim\_{\mathbf{x}\to+\infty} \,\_2F\_1\left(m-\frac{1}{\xi}+1, -\frac{1}{\xi}; m-\frac{1}{\xi}+1; \frac{\mu\xi-\sigma}{\varkappa\xi}\right) = \\ &= \lim\_{\mathbf{x}\to\infty} \left[ \left(\frac{\mu\xi-\sigma}{\varkappa\xi}\right)^0 + \frac{-\frac{1}{\xi}\left(m-\frac{1}{\xi}+1\right)}{m-\frac{1}{\xi}+1} \frac{\mu\xi-\sigma}{\varkappa\xi} + \cdots \right] = 1, \end{aligned}$$

and also

$$\lim\_{\mathbf{x}\to+\infty} \mathbf{x}^m \left( 1 + \frac{\xi(\mathbf{x}-\boldsymbol{\mu})}{\sigma} \right)^{-\frac{1}{\xi}} = \mathbf{0}$$

and

$$\lim\_{\varepsilon \to +\infty} \mathbf{x}^{\frac{1}{\xi}} \left( 1 + \frac{\xi(\varkappa - \mu)}{\sigma} \right)^{-\frac{1}{\xi}} = \left( \frac{\xi}{\sigma} \right)^{-\frac{1}{\xi}},$$

to show that

$$\lim\_{x \to \infty} f = L \left( \frac{\xi}{\sigma} \right)^{-\frac{1}{\xi}}.$$

when *ξ* < <sup>1</sup> *m*

.

A few elementary operations allow us to write that

$$\lim\_{\lambda \to +\infty} \frac{\mathfrak{x}^{m+1}}{(m+1)\cdot(\sigma-\mu\xi)} \left(\frac{\sigma+\xi(\mathfrak{x}-\mu)}{\sigma}\right)^{-\frac{1}{\xi}} \,\_2F\_1\left(1, m-\frac{1}{\xi}+1; m+2; \frac{\mathfrak{x}\mathfrak{x}}{\mu\xi-\sigma}\right)$$

$$\widetilde{\mathfrak{x}} = -(-1)^{-\left(m-\frac{1}{\xi}+1\right)} \frac{1}{\widetilde{\xi}} \frac{\Gamma\left(-m+\frac{1}{\xi}\right)\Gamma(m+1)}{\Gamma\left(1+\frac{1}{\xi}\right)} \left(\frac{\mu\xi-\sigma}{\sigma}\right)^{-\frac{1}{\xi}} \left(\frac{\mu\xi-\sigma}{\xi}\right)^{m} \tag{A3}$$

when *ξ* < <sup>1</sup> *m* .

Finally, we compute the value of the primitive in Equation (A1) when *x* is equal to VaR*c*(*XT*). This quantity is equal to

$$-\frac{\left(\text{VaR}\_{\text{c}}(X\_{\text{T}})\right)^{m+1}}{(m+1)\cdot(\sigma-\mu\tilde{\text{y}})} \left(\frac{\sigma+\xi(\text{VaR}\_{\text{c}}(X\_{\text{T}})-\mu)}{\sigma}\right)^{-\frac{1}{\tilde{\xi}}}{}\_{2}F\_{1}\left(1,m-\frac{1}{\tilde{\xi}}+1;m+2;\frac{\text{VaR}\_{\text{c}}(X\_{\text{T}})}{\mu\tilde{\text{y}}-\sigma}\right). \tag{A4}$$

We input Equations (A3) and (A4) into Equation (A1) and we derive Equation (14), which is our result.

#### **References**

Acerbi, Carlo, and Dirk Tasche. 2002. On the coherence of expected shortfall. *Journal of Banking & Finance* 26: 1487–503.


Hill, Bruce. 1975. A simple general approach to inference about the tail of a distribution. *Annals of Statistics* 3: 1163–74. [CrossRef]

