As mentioned above, annuities are usually viewed as a very conservative product by individuals, as they represent an inflexible and illiquid asset, they imply an irreversible decision at issue, they do not satisfy bequest needs or they meet bequest preferences only partially, if a death benefit is included (see Pitacco 2016). Innovation can be pursued in various respects. Making the benefit amount contingent on mortality/longevity is undoubtedly an important innovation. In particular, it favours the adoption of not excessively prudential assumptions about future mortality, otherwise necessary given that annuities extend over a very long time-horizon. Bequest preferences can be met by introducing death benefits. While this is customary in annuities (as mentioned above), with death benefits payable up to some (not very high) age, Bernhardt and Donnelly (2019) introduce bequest in pooled arrangements. Several opportunities for innovation can be obtained when combining different benefits, such as the just mentioned case of annuities and death benefits. Particularly interesting in this regard is the combination of different forms of annuities; Chen et al. (2019, 2020) and Chen and Rach (2019) consider the case of fixed annuity benefits combined with mortality/longevity linked-benefits.

Another line of innovation can be developed in respect of the structure of the fees. It is traditional for life annuities to define and charge the premium loading at issue, when the individual transfers their money to the provider. Such a pricing structure concurs to the inflexibility of the product, as on one hand pricing assumptions are fully chosen at issue, on the other the provider is required to pre-define the management over time of the upfront loadings. Conversely, periodic fees can pave the way for greater product flexibility, for example making it easier to switch to different forms of benefits after issue, or making it possible to revise the pricing assumptions. In this latter regard, we note that while the possibility of revising pricing assumptions can be viewed with suspicion by individuals, any change will not necessarily be to the advantage of the provider (rather, it may be favourable to the individual); in principle, any change in pricing assumptions will be limited by the policy conditions underwritten at issue. Periodic fees are a natural choice when periodic premiums are paid, i.e., usually in the case of endowments. In a mortality/longevity linking framework, Hanbali et al. (2019) focus in particular on the case of pure endowments, by assuming level premiums (subject to possible adjustments in time). Periodic fees are common in variable annuities, independent of the time-profile of the premium stream. Here, guarantees are priced by charging a periodic fee to the policy account value, with the possibility for the policyholder to waive the guarantee at any time (except for agreed time windows), with a consequent interruption of the relevant fee charge; for a general overview, see (Bacinello et al. 2011). In a recent contribution by (Chen et al. 2021) periodic fees are introduced in a tontine scheme, and are expressed as a proportion of the benefit amount; the size of the fee that makes the individual indifferent between choosing a fixed-benefit or a tontine annuity is in particular examined, identifying this way the range of acceptable fee levels in tontine arrangements.

In this paper, we consider mortality/longevity-linked annuities, that include guarantees in the form of barriers for the benefit amount. We further develop Olivieri and Pitacco (2020a, 2020b), by introducing periodic fees instead of an upfront fee at issue. Periodic fees are charged to the policy fund value and their level is assessed based on the losses and profits retained by the provider, which in turn depend on the linking coefficient and the barriers for the benefit amount. The metric we use to assess the required fees is the business value for the provider; this quantity, defined as the present value of future profits net of the cost of capital, is suitable as a joint summary of the losses and profits retained by the provider. We show that the periodic fee identifies a discount factor to be used for pricing and reserving. We then define the individual reserve, and split it into two components, one covering the value of future benefits and one the value of future fees; such an information can be useful in some applications.

The remainder of the paper is organized as follows. In Section 2, we describe the model, in particular the mortality/longevity-linking annuity benefits examined in the paper (Section 2.1), the structure of periodic fees and the corresponding discount factor (Section 2.2), the individual reserve and its components (Section 2.3), the business value to the provider (Section 2.4); finally, we discuss how to assess the required fees, based on an assessment of the business value (Section 2.5). Some numerical findings are illustrated in Section 3, more specifically in Section 3.3, after having sketched the stochastic mortality model adopted (Section 3.1) and provided details about the arrangements analysed in the numerical implementation (Section 3.2). Finally, Section 4 concludes the paper, with some final comments.

### **2. Model Setup**

### *2.1. Mortality/Longevity-Linked Annuity Benefits*

We consider a discrete-time annuity immediate in arrears, i.e., with payments at the end of the year. For simplicity, one cohort only is addressed, homogeneous in all respects. The entry time is 0 and the entry age is *x*. In this paper, we focus on mortality/longevity risk only, while we disregard other risks. For this reason, a deterministic financial setting is adopted.

We follow the general linking structure described in (Olivieri and Pitacco 2020a), to which we refer for a detailed discussion about the rationale and actuarial technique backing the annuity benefit adjustment. We consider the following two alternative mortality/ longevity-linked annuity benefits:

$$b\_t = b\_{t-1} \cdot \frac{p\_{x+t-1}(0)}{\hat{p\_{x+t-1}}} \; ; \tag{1}$$

$$b\_t = b\_0 \cdot \frac{1 + a\_{x+t}(0)}{1 + a\_{x+t}(t)} \, . \tag{2}$$

where *b*0 is chosen at time 0 in both cases. The meaning of the quantities in Equations (1) and (2) is as follows: *px*+*t*−1(0) denotes the annual survival probability at age *x* + *t* − 1 provided by a best-estimate mortality assumption (i.e., life table) at time 0; *px*+*t*−1 represents the proportion of survivors (or longevity index) observed in a chosen population; *ax*+*t*(0), *ax*+*t*(*t*) denote the actuarial value at age *x* + *t* of a unitary discrete-time annuity in arrears, based on the best-estimate mortality assumptions (namely, life tables), respectively, at time 0 and *t*. An interpretation about the two adjustments coefficients follows below.

In Equation (1), which we will call linking by means of the survival probability, the benefit amount is adjusted based on the comparison between the survival rate realized in a given population, *px*+*t*−1, and a benchmark, *px*+*t*−1(0), chosen at time 0. An increase (decrease) of the benefit amount follows from *px*+*t*−1(0) > *px*+*t*−1 (*px*+*t*−1(0) < *px*+*t*−1), i.e., in the case of higher realized mortality (longevity) than predicted by the benchmark. We note that, for an annuity business, a profit (loss) is typically reported by the provider in the case of higher mortality (longevity); thus, the benefit adjustment in (1) serves to mitigate such a profit (loss), partially transferring it to the individuals. The strength of the mitigation effect depends, however, on the population in which *px*+*t*−1 is measured (as well as on other policy conditions, such as any barriers set for the benefit amount).

As far as the population is concerned, the choice is between the provider's pool or a reference population. While the provider's pool usually shows a small size and is thus subject to major random fluctuations, large reference populations are more appropriate, so that deviations between *px*+*t*−1 and *px*+*t*−1(0) can be mainly attributed to an unanticipated underlying mortality/longevity trend. Random fluctuations represent a traditional risk for insurers (and annuity providers), and the relating risk management actions should be arranged using the well-known pooling arguments. Conversely, mortality/longevitylinking benefits are recommended to cope with unanticipated mortality/longevity trends. In the following, we then assume that *px*+*t*−1 is measured on a large population. We finally note that any possible difference between the mortality/longevity in the provider's pool and the reference population may be due not only to random fluctuations, but also to different underlying trends. A basis risk follows for the provider, which we do not address in this paper.

It is interesting to note that rule (1) is equivalent to the following:

$$b\_t = b\_0 \cdot \frac{{}\_t p\_x(0)}{{}\_t \hat{p}\_x} \, , \tag{3}$$

where *t px*(0) is the survival probability from age *x* to age *x* + *t* provided by the bestestimate mortality assumption at time 0, while *t px* is the proportion of survivors from age *x* to age *x* + *t* in the reference population. Equation (3) shows an important feature, which does not emerge explicitly from Equation (1): the adjustment coefficient should be applied to the benefit amount defined at the time the benchmark is referred to, namely time 0 in this case. In other words, there must be a consistency between the time-frame involved by the adjustment coefficient and the reference time of the quantity to which such a coefficient is applied.

Coming to the benefit adjustment in Equation (2), which we will call linking by means of the actuarial value of the annuity, we first note that *ax*+*t*(*h*) = ∑*ω*−(*x*+*t*) *s*=1 (1 + *i*)−*s* · *s px*+*t*(*h*), where *i* is a chosen discount rate, *s px*+*t*(*h*) the survival probability (from age *x* + *t* to age *x* + *t* + *s*) based on best-estimate assumptions at time *h*, and *ω* the maximum attainable age (that we assume to be deterministic). The ratio 1+*ax*+*t*(0) 1+*ax*+*t*(*t*) then involves a comparison between possibly different mortality assumptions (or life tables) adopted to assess the actuarial value of a unitary annuity, at time 0 and time *t*. The actuarial value in the numerator, in particular, is a benchmark. If higher (lower) mortality is forecasted at time *t*, then *ax*+*t*(0) > *ax*+*t*(*t*) (*ax*+*t*(0) < *ax*+*t*(*t*)), implying an increase (reduction) of the benefit amount. Updated mortality/longevity assumptions in the actuarial value of the annuity can result in a profit (loss) for the provider, which is (partially) mitigated by the benefit adjustment. We note that, in line with what commented above for the linking by means of the survival probabilities, having set the benchmark at time 0, the benefit adjustment is applied to the benefit amount at the same time.

Both in the linking by means of the survival probability and by means of the actuarial value of the annuity, explicit guarantees can be introduced, for example, by setting bounds for the benefit amount or the benefit adjustment. Furthermore, it is also reasonable to accept a maximum age to apply the benefit adjustment (say, age 95), in order to avoid that the individual is exposed to the risk of benefit reductions at very advanced ages.

### *2.2. Policy Fund and Periodic Fees*

In this section, we describe the main setting for periodic fees. As with other annuity products, each individual only pays an initial capital *S* at entry time 0, and then will cash the annuity benefit annually, until death. Unlike the usual setting for annuities, where the premium loading is charged entirely to the initial capital at time 0, we assume that periodic fees are charged to the policy fund at the beginning of each year. This is similar to what happens in variable annuities. In our discussion, we disregard expenses; the fee (whether it is obtained as a single initial or a periodic loading) is justified by the logic of the safety loading, i.e., by the fact that the arrangement incorporates guarantees whose cost is charged to the individual. When expenses are also addressed, periodic expenses can be added to the periodic fee, while an upfront loading to cover initial expenses could be included.

In this paper, we are mainly concerned with identifying appropriate pricing and reserving logics when fees are periodic. Any possible revision after issue of the fee level requires criteria that we do not address in this paper. In what follows, we assume that the periodic fee level is chosen at time 0, and kept fixed over the whole policy duration; some comments about the possible revision of the fee level after issue will be made later, in Section 3.3.

As stated in Section 2.1, in this paper we consider a pool consisting of one cohort only, homogeneous in all respects (entry age, risk class, benefit amount). With *Nx*+*t* we denote the number of individuals in the pool at time *t* (age *x* + *t*). At time 0, *Nx* = *nx* known, while *Nx*+*t* is random because of the mortality in the cohort.

Let *At* denote the individual fund (or policy fund) for a policy in-force at time *t*. No death benefit is paid out by the provider, so that upon death the policy fund is released to the pool (as a form of mortality credit). The dynamics of the policy fund is then described by the following balance equation:

$$A\_t \cdot N\_{\mathbf{x}+t} = A\_{t-1} \cdot N\_{\mathbf{x}+t-1} \cdot (1 - \xi) \cdot (1 + i) - b\_t \cdot N\_{\mathbf{x}+t} \tag{4}$$

where: *ξ* is the proportional premium loading (or fee) that is charged to each policy fund at the beginning of each year and *i* is the (deterministic) return on investments. At time 0, *A*0 = *S*.

The policy fund is random because of the mortality in the pool and the path of the benefit amount; in a more general setting, random financial returns can also be addressed.

Solving backwards Equation (4) (note that *Aω*−*x* = 0), we find:

$$A\_0 = \sum\_{s=1}^{\omega \cdot - \chi} b\_s \cdot \left( (1 - \xi) \cdot (1 + i) \right)^{-s} \cdot \frac{N\_{\chi + s}}{n\_{\chi}} \,. \tag{5}$$

Equation (5) shows that the periodic fee *ξ* identifies a discount factor, to be used for pricing. However, in order to quantify *ξ*, we still need a valuation principle. While we know that *A*0 = *S*, the expression on the right hand side of Equation (5) is still random, as both the sequences *bs* and *Nx*+*s* are random. Furthermore, there are two unknowns in Equation (5), namely *b*0 and *ξ*. We further discuss this problem in Section 2.5, after having introduced some quantities. We point out, however, that the approach we suggest to assess the periodic fee will not directly make use of Equation (5); the discussion just developed allows us to say that the periodic fee identifies a discount factor to be used for pricing purposes (as well as for reserving; see Section 2.3).

We note that, if policy conditions admit, the periodic fee could be updated after issue first extending Equation (5) and then implementing an appropriate valuation principle. As already mentioned, in this paper we do not develop the discussion in this respect.

### *2.3. Individual Reserve and Components*

In order to define the individual reserve, we make the following comments. As is well-known, the individual reserve corresponds to the best-estimate value of liabilities plus a risk margin. We assume that the risk margin can be measured through the periodic fee, which (as noted in Section 2.2) identifies a discount factor. We further assume that, to be consistent, the best-estimate assumptions should be those defined when the periodic fee was assessed, i.e., at time 0 in this paper. Consider a policy in-force at time *t*. The individual reserve is defined as follows:

$$V\_t = b\_t \cdot \sum\_{s=1}^{\omega - (x+t)} \left( (1 - \xi) \cdot (1 + i) \right)^{-s} \cdot \,\_s p\_{x+t}(0) \,. \tag{6}$$

It is useful to note that no future update of the benefit amount is explicitly considered in the definition of the individual reserve, as the periodic fee already includes in the reserve an allowance for the future benefit adjustments.

For shortness, we denote the sum in Equation (6) as *ax*+*t*(0; *ξ*), where the symbol *ξ* refers to the fact that discounting is based also on the periodic fee (conversely, the notation *ax*+*t*(0) will be kept to denote the expression commented in Section 2.1, i.e., when discounting is based on the interest rate only).

It could be useful to split the individual reserve into components. A possible splitting is the following:

$$V\_t = b\_t \cdot a\_{x+t}(0; \underline{\mathfrak{z}}) = b\_t \cdot a\_{x+t}(0) + b\_t \cdot \left(a\_{x+t}(0; \underline{\mathfrak{z}}) - a\_{x+t}(0)\right). \tag{7}$$

The first quantity, *V*[ben] *t* = *bt* · *ax*+*t*(0) is the best-estimate value of future benefits, while the second, *V*[fee] *t* = *bt* · (*ax*+*t*(0; *ξ*) − *ax*+*t*(0)), can be interpreted as the part of the individual reserve accounting for fees. This decomposition could be useful first to identify the risk margin included in the individual reserve. It could be further useful in the case of a switch to a different benefit structure, as the provider could only use *V*[ben] *t* to determine the new benefit level or, in the case of revision of the fee, the revision itself could be limited to the component *V*[fee] *t* . These aspects are not further developed in this paper.

### *2.4. Pool Fund, Present Value of Future Benefits, Present Value of Future Profits and Business Value*

We now address quantities defined in the provider's perspective. Each individual in the pool pays the initial capital *S* at time 0, and will cash the annual amount *bt* at time *t*, *t* = 1, 2, ... , until death. The following quantity describes what we call the pool fund, i.e., the remaining money at time *t* of the total amount of initial capital, net of the benefits paid so far and including interest on investments:

$$F\_t = F\_{t-1} \cdot (1+i) - b\_t \cdot \mathcal{N}\_{x+t\ \prime} \tag{8}$$

with *F*0 = *S* · *nx*. Periodic fees do not enter the assessment of the pool fund; indeed, as individuals transfer money to the pool only at time 0, periodic fees do not originate a real periodic cash flow. Rather, they consist in a loading which is charged annually to the policy funds of the survivors, as expressed by Equation (4).

The difference between the pool fund and the pool reserve, namely

*SPt* = *Ft* − *Vt* · *Nx*+*t* , (9)

expresses the surplus cumulated by the provider in the time-interval (0, *t*). If we consider the surplus cumulated over the whole pool duration (0, *ω* − *x*), we find:

$$SP\_{\omega-x} = \left. S \cdot n\_x \cdot (1+i)^{\omega-x} - \sum\_{s=1}^{\omega-x} b\_s \cdot (1+i)^{\omega-x-s} \cdot N\_{x+s} \tag{10}$$

$$\mathbf{u} = \begin{array}{c} n\_{\mathbf{x}} \cdot (1+i)^{\omega - x} \cdot \left( S - \sum\_{s=1}^{\omega - x} b\_{\mathbf{s}} \cdot (1+i)^{-s} \cdot \frac{N\_{\mathbf{x}+\mathbf{s}}}{n\_{\mathbf{x}}} \right) \end{array} \tag{11}$$

where: PVFB0 = ∑*ω*−*x s*=1 *bs* · (1 + *i*)−*s* · *Nx*+*s nx* represents the Present Value of Future Benefits (PVFB) at time 0, expressed per policy issued. The quantity PVFP0 = *S* − PVFB0 then represents the total profit, usually called Present Value of Future Profits (PVFP) at time 0, expressed per policy issued.

The PVFB and the PVFP are quantities of great interest for the assessment of the business value. Their definition can easily be extended to times after issue. If *nx*+*t* is the number of policies in-force (i.e., in the portfolio) at time *t*, then

$$\text{PVFB}\_{l} = \sum\_{s=1}^{\omega - (x+t)} b\_{t+s} \cdot (1+i)^{-s} \cdot \frac{N\_{x+t+s}}{n\_{x+t}} \tag{12}$$

expresses the PVFB at time *t* per policy in-force at that time, while PVFP at time *t* can be assessed as follows:

$$\text{PVFP}\_{t} = V\_{t} - \text{PVFB}\_{t} \,. \tag{13}$$

Alternative valuation assumptions can be adopted in the assessment of PVFB, PVFP, in particular with regard to mortality. The proportions *Nx*+*t*+*s nx*+*t* in (12) lead to an entity-specific assessment , as the size of the total amount of future payments to be made by the provider (namely, the amounts *bt*+*s* · *Nx*+*t*+*s*) is measured with the number of survivors in the pool, which in their turn results from the mortality in the pool itself. Entity-specific assessments are convenient, for example, to perform a realistic valuation of provider's liabilities. The adoption of the proportion of survivors in the reference population, *s px*+*t*, in place of *Nx*+*t*+*s nx*+*t* is to be preferred in market-consistent assessments, such as those involved by pricing issues. In this case, mortality/longevity risks to which the provider is exposed (e.g., because of a small pool size or a pool composition affected by adverse-selection), but could be offset by appropriate market transactions, should be excluded from the valuation. Since we are discussing the setting of appropriate fees, in this paper we follow a market-consistent logic, and we disregard risks specific to the provider. In particular, we do not consider different mortality situations between the provider's pool and the reference population, so that we accept *Nx*+*t*+*s nx*+*t* = *s px*+*t* at any age.

For further details about the assessment of PVFB and PVFP in more general situations, we refer to Olivieri and Pitacco (2020a, 2020b).

We now assess the business value for the provider, which is the part of PVFP net of the cost of capital that the provider is required to hold to manage the business (for references and an application to traditional annuities, see Blackburn et al. 2017). We assess the cost of capital in a market-consistent style, i.e., as frictional costs net of the value of the limited liability put option. The limited liability put option takes a positive value when, because of the possible depletion of capital, the provider may fail to meet (at least in part) its obligations. We assume that the provider can always obtain the extra capital necessary to fulfil all obligations, in the case the amount initially allocated to the pool becomes insufficient. In this case, the limited liability put option takes a null value. Frictional costs (which arise mainly because of agency costs) are usually assessed as a proportion *ρ* of the capital held on top of the pool reserve. We assume that such a capital is in the amount required by regulation; we denote with RC*t* the capital required at time *t* per policy in-force at that time. The annual frictional cost per policy in-force at the beginning of the year is then:

$$\text{FC}\_t = \rho \cdot \text{RC}\_{t-1}.\tag{14}$$

The present value of frictional costs at time *t*, per policy in-force, is obtained as follows:

$$\text{PVFC}\_{t} = \sum\_{s=1}^{\omega - (x+t)} \text{FC}\_{t+s} \cdot (1+i)^{-s} \cdot \,\_s\widetilde{p}\_{x+t} \,. \tag{15}$$

We assess RC*t* following the Solvency 2 principles, which require an amount of capital so to avoid default with 99.5% probability, allowing only for risks non-diversifiable on the market. A long-term time horizon in which to assess possible defaults seems the most logical choice in respect of longevity risk, due to its long-term nature. We obtain the required capital at time *t* per policy in-force from the following condition:

$$\Pr(\text{RC}\_{l} + V\_{t} < \text{PVFB}\_{t}) = 0.005\,\text{.}\tag{16}$$

Finally, we define the business value at time *t*, per policy in-force, as follows:

$$\text{BV}\_{t} = \text{PVFP}\_{t} - \text{PVFC}\_{t} \,. \tag{17}$$

### *2.5. Setting the Periodic Fee*

As is well-known (see, for example, Duffie 2001), the market price of a security is given by the present value of its expected cash flows, where the present value is assessed with the risk-free rate and the expected value is obtained with a suitably risk-adjusted probability measure. When the market is incomplete, as is the case for example of insurance and pension markets, there are infinite suitable probability measures, among which the provider has to choose one to price the annuity contract. This approach apparently contrasts with the traditional insurance pricing model, which first employs best-estimate assumptions, and then adds an implicit or explicit safety loading. The safety loading, in particular, represents the expected profit to the provider and its size should be justified by the risks taken by the provider itself. What is common to both approaches is that the expected profit to the provider is 0 under the chosen valuation assumptions; however, a reward for the retained risks is included, either through the risk-adjustment of the probabilities or the safety loading.

Mortality/longevity-linked annuities certainly require an innovative pricing approach, as they imply a new concept of longevity guarantee, as we have commented in Sections 1 and 2.1. It could be convenient to match somehow market principles with the traditional model. To this aim, we note that of the various quantities described so far, there is one explicitly affected both by expected profits and the risks borne by the provider, which is the business value. Risks, in particular, affect the business value via frictional costs, as they are proportional to the required capital, whose size in turn depends on the potential losses

reported by the provider, apart from the largest losses as identified by the accepted default probability.

We extend the 0 expected profit principle to the business value. Adopting the periodic fee structure described in Section 2.2, at time 0 we assess the periodic fee *ξ* so that

$$\mathbb{E}[\mathbb{B}\mathbf{V}\_0] = \mathbf{0}.\tag{18}$$

Such an equation can be solved once a (stochastic) mortality model describing the mortality/longevity of the reference population has been chosen. In Section 3.1, we briefly describe the model that we have adopted in the numerical implementation.

We now address some computational issues. First we note that Equation (18) reasonably requires stochastic simulation, although this depends on the mortality model (however, one can hardly count on closed formulae). Second, as already noted (see Section 2.2), given *S*, there are two unknowns in the relevant equations, namely *b*0 and *ξ*, which among other things are related, as the higher is *ξ*, the lower is *b*0, and vice versa. Whatever is the approach adopted (market-based, traditional or the one we are discussing), there is one degree of freedom, in respect of which we proceed as follows. First we set *ξ* = 0 and assess *b*0 solving Equation (6), for *t* = 0 (having *V*0 = *S*, at that time). We denote such an initial benefit amount as *b*∗ 0 ; in practice, *b*∗ 0 = *S ax*(0). Then we assess the expected business value, under *b*∗ 0 ; we use the notation BV∗ 0 = E[BV0; *b*∗ 0 ]; if such a quantity is = 0 (reasonably, it takes a negative value), we adjust (typically, we reduce) the initial benefit amount, so to reach a value 0 for the business value. In practice, we set *V*[fee] 0 = −BV∗ 0 and we find *b*0 = *S*−*V*[fee] 0 *ax*(0) (see Equation (7)). Finally, we find the periodic fee *ξ* by solving at time *t* = 0 Equation (6) (which has now only one unknown) in respect of *ξ* (here, we can use a routine for the internal rate of return). We point out that once *b*0 and *ξ* have been set, we cannot exclude that E[BV0] = 0. This is because the components of BV0 are not necessarily proportional to *b*0 and usually they are not proportional to *ξ*. Equation (18) is used to set the fee in a consistent way in the various situations (in particular, working with different mortality/longevity-linking coefficients), but (as we will see in the numerical implementation in Section 3) the fee thus determined can still entail value creation for the provider.

### **3. Results**

### *3.1. Mortality Model*

A stochastic mortality model is required to simulate the survival rates realized in the reference population and in the pool, as well as to obtain updated best-estimate assumptions at every time, consistently with the simulated experience. After the seminal paper by (Lee and Carter 1992), a very prolific research has developed on the stochastic modelling of mortality. Several models are described in the literature; most of them are suitable to obtain accurate projections at the initial time, but can present computational complexity when processing future best-estimate assumptions. Here we adopt a model discussed in (Olivieri and Pitacco 2009), which fits the evaluation needs mentioned above, and is computationally tractable. Below, we recall the main features of the model, referring to (Olivieri and Pitacco 2009) for details.

We refer to a given cohort and denote with *qx*+*t* = 1 − *px*+*t* the random mortality rate at age *x* + *t*. We define *qx*+*t* as follows:

$$
\widetilde{q}\_{x+t} = q\_{x+t}(0) \cdot Z\_{x+t} \tag{19}
$$

where: *qx*+*t*(0) = 1 − *px*+*t*(0) is the mortality rate based on best-estimate assumptions at time 0; *Zx*+*t* is a (positive) random coefficient (ensuring 0 ≤ *qx*+*t* ≤ 1) which measures the deviation of the observed mortality rate compared to the best-estimate one at time 0. We assume:

$$Z\_{x+t} \sim \text{Gamma}(\alpha\_{x+t}, \beta\_{x+t}) \; ; \tag{20}$$

a Gamma distribution also follows for *qx*+*t*, with parameters obtained from those in (20). If the Poisson distribution is accepted for the annual number of deaths reported by the cohort, conditional on a given value for the mortality rate, then the unconditional distribution of the annual number of deaths is described by a negative binomial (or Poisson-Gamma) law. We point out that while the Poisson distribution describes random fluctuations in mortality, the Gamma (i.e., the coefficient *Zx*+*t*, whatever is its probability distribution) describes aggregate deviations.

The parameters of (20) are first chosen at time 0 , when the cohort starts being observed; thus, *αx*+*t* = *α*¯ 0 and *βx*+*t* = *β*¯ 0 at time 0, where *α*¯ 0, *β*¯ 0 are given values. Then, year after year the parameters are updated, through an inferential procedure based on the information carried by the observed annual number of deaths. In particular, after *h* years of observation, once the numbers of deaths *dx*, *dx*+1, ... , *dx*+*h*−1 and the number of survivors *nx*, *nx*+1 = *nx* − *dx*, ... , *nx*+*h*−1 = *nx*+*h*−2 − *dx*+*h*−2 have been reported, the initial *α*¯ 0, *β*¯ 0 are replaced with the following values: *α*¯ *h* = *α*¯ 0 + *dx* + *dx*+1 + ··· + *dx*+*h*−1, *β*¯ *h* = *β*¯ 0 + *nx* · *qx*(0) + *nx*+1 · *qx*+1(0) + ··· + *nx*+*h*−1 · *qx*+*h*−1(0). Updating the parameters *αx*+*t*, *βx*+*t* introduces an implicit correlation among the coefficients *Zx*+*t*'s, which is something one expects when an underlying longevity trend drives mortality. Nevertheless, fluctuations in the number of deaths in the opposite direction in respect of the prevailing trend are still admitted at any time. We point out that updating the parameters of the probability distribution of *Zx*+*t* allows to update the current best-estimate assumptions, as well as the projection of the future number of survivors in the cohort. Details of the inferential procedure are described in (Olivieri and Pitacco 2009).

In the numerical implementation, we consider a cohort initial age *x* = 65. We set *α*¯ 0 = *β*¯ 0, so to have E0[*Zx*+*t*] = 1, E0[*qx*+*t*] = *qx*+*t*(0) (we specify in the subscript of the symbol E the time at which the expected value is assessed, thus meaning that the state of information about mortality is specified at that time). Based on the information gained from the observed number of deaths up to time *h*, the best-estimate mortality rate is reassessed as *qx*+*t*(*h*) = *α*¯ *h β*¯ *h* · *qx*+*t*(0), where *α*¯ *h β*¯ *h* - 1, depending the realized mortality path. The best-estimate mortality rates *qx*+*t*(0) are modelled through a Gompertz law with parameters as in (Bacinello et al. 2018). The remaining expected lifetime at age 65 is almost 20 years at time 0; to avoid distortions from major random fluctuations at the highest ages, the cohort is examined up to age 100 (any payments beyond that age is disregarded). The values *α*¯ 0 = *β*¯ 0 = 100 *α*¯ 0 = *β*¯ 0 = 1000 are alternatively adopted; if *α*¯ 0 = *β*¯ 0 = 100, the coefficient of variation of *Zx*+*t* at time 0 is 0.1, while it is 0.0316 if *α*¯ 0 = *β*¯ 0 = 1000. Given the meaning of *Zx*+*t*, the former choice of *α*¯ 0, *β*¯ 0 then depicts a situation with a higher dispersion in aggregate mortality than the latter choice. To distinguish the two alternative scenarios, shortly we will refer to the choice *α*¯ 0 = *β*¯ 0 = 100 as to a scenario with major aggregate deviations; the term moderate aggregate deviations will be used instead to refer to the choice *α*¯ 0 = *β*¯ 0 = 1000 (it is useful to stress that the adjectives 'moderate' and 'major' do not have an absolute meaning here, but they are used in comparative terms between the two situations).

### *3.2. Benefit Arrangements*

We examine the following arrangements:

Arrangement 1 represents a standard case, to which arrangements with a mortality/longevity linking can be compared. For arrangements 2 and 3, barriers to the benefit

amount are adopted, both on an annual and a global basis. Year by year, we require 0.9 · *bt*−1 ≤ *bt* ≤ 1.1 · *bt*−1, so to avoid strong variations in the benefit amount from one year to the next. In respect of the possible total variation of the benefit amount we consider two cases, implying a different proportion of the risk sharing between the provider and the individual. Specifically, we require alternatively:

Finally, we admit that the benefit amount can be updated up to age 95; beyond that age, the benefit amount keeps flat, at the latest updated level.

As mentioned earlier, we disregard financial risk and we adopt a deterministic setting in this regard. Our aim is to make a comparison of the periodic fees and other quantities for the alternative linking solutions. These comparisons are not affected by the interest rate level, when it is deterministic. Thus, we simplify and take *i* = 0, supported in this choice by the low level of interest rates in recent years.

### *3.3. Implementation and Discussion*

In the implementation discussed in this section, the main purpose is to compare the periodic fees required for alternative annuity designs. Some of the quantities described in Section 2 are additionally quoted, in particular the components of the individual reserve, the present value of future profits and the business value. Rather than the absolute values we have obtained for the various quantities, it is the comparison between them that is significant. While absolute values are affected by the various choices made, in particular with regard to the mortality model and the interest rate, their comparison allows a better understanding of what the various linking mechanisms imply, in respect of alternative choices. All assessments have been developed simulating the number of survivors in the reference population; the same proportion of survivors than those of the reference population has been adopted for the pool. The information gained from the mortality observed in the reference population has also been used to update the best-estimate mortality assumptions after issue. The initial capital paid by each individual is *S* = 100 monetary units.

Table 1 quotes the periodic fee, assessed as described in Section 2.5, for alternative arrangements. The proportion *ρ* of frictional costs has been set to 2%, following market practice (see Blackburn et al. 2017). Tables 2 and 3 list the expected value, the 0.01- and the 0.99-quantiles of the benefit amounts for some times *t* and for the various arrangements. It is convenient to analyse these three tables jointly, as the time-profile of the benefit amounts helps understanding the size of the fee, while the latter explains the differences among the initial benefit amounts *b*0 under the different arrangements.

The arrangement with fixed benefits can be used as a reference case. It requires the highest fees, due to the absence of any form of risk sharing between the provider and the individual. In comparison, mortality/longevity-linking arrangements require lower fees, as they imply, in particular, possible reductions of the benefit amount. When interpreting the results, it should be remembered that when the benefit is linked to a mortality/longevity experience, either by means of the survival probability or the actuarial value of the annuity, the benefit is allowed both to decrease (and this implies a sharing of losses due to higher longevity) and increase (in this case, profits due to higher mortality are shared). The size of the fee is affected by the extent of the possible reduction in losses to the provider, but also by that of profits.

Overall, it seems that the fee is much affected by the loss sharing effect. This emerges, for example, when comparing the cases (a) and (b) for the linking by means of the survival probability; indeed, narrower barriers for the benefit amount imply in particular a reduced participation to possible losses.

The importance of the magnitude of possible losses as a key to interpreting the level of the required fee also emerges when comparing (for any given arrangement) a scenario with moderate deviations in aggregate mortality to one with major deviations. Higher fees are required under the latter scenario. In this regard, it is useful to note that deviations are admitted in terms both of higher and lower mortality rates. Under a scenario with major deviations, higher fees are required; this suggests that the fee itself is in particular affected by the possibility to share the possible losses occurring because of lower mortality rates.

When comparing the fees required for arrangement where benefits are linked by means of the survival probability in the different cases, a univocal behaviour emerges. The same does not happen for arrangements where benefits are linked by means of the actuarial value of the annuity; in this latter case, it seems that the trade-off between profits and losses assumes a different balance depending on the parameters accepted for the benefit barriers and the mortality scenario. For example, *ξ* takes the same value in a scenario of moderate aggregate deviations, independent of the barriers we have tested for the benefit amount (clearly, a different choice for such barriers could result in a different value for the required fee). From Table 2 we see also that the expected benefit amount and the 0.01- and 0.99-quantiles coincide (apart from roundings) in the two cases (a) and (b).

In principle, the coefficient linking the benefits to the survival rate implies a different time-profile of the adjustments when compared to the coefficient linking the benefit to the actuarial value of the annuity. This is shown, for example, by the quantiles of the benefit amounts quoted in Tables 2 and 3. For a given mortality scenario, the coefficient linking the benefits to the actuarial value of the annuity implies a larger benefit adjustment in earlier times than the coefficient linking the benefit to the survival probability; on the other hand, lower adjustments are then required later in time by the former coefficient. This effect can be explained by the time-horizon referred to by the quantities in the adjustment coefficient; if we compare (3) with (2), we can realize that the actuarial value of the annuity refers to a longer time-horizon than the survival probability in the early years of the annuity (i.e., when *t* is small); the opposite happens when *t* is high. The time-profile of the adjustments has an impact on the size of the cash flows, and then on the proportion of the total profits and losses retained by the provider, as well as on the fee.

**Table 1.** Periodic fee *ξ* (to be charged each year to the policy fund value).

**Table 2.** Benefit amount *bt* (expected value and 0.01- and 0.99-quantiles) for selected times. Moderate aggregate deviations.

**Table 3.** Benefit amount *bt* (expected value and 0.01- and 0.99-quantiles) for selected times. Major aggregate deviations.

Table 4 quotes the initial fee *π* equivalent to the periodic fees in Table 1, obtained from the following condition:

$$S = b\_0 \cdot a\_x(0) \cdot (1 + \pi) \,. \tag{21}$$

We note that Equation (21) represents the standard approach to premium loading for annuities. As for the periodic fee, a valuation principle is required, or an explicit choice of *π*, given that Equation (21) has two unknowns (namely, *b*0 and *π*). Here, we assess *π* consistently with what performed for *ξ*, i.e., solving Equation (18).

The fees quoted in Table 4 can be compared as discussed for Table 1. When comparing Table 4 with Table 1, we obtain an assessment of the overall loading implied by a given periodic fee. Among the advantages of periodic fees when compared to an upfront loading, we mention the fact that their structure is similar to that of other products, in particular financial investments; then, periodic fees can represent a solution in which individuals are more confident, being more familiar with such a pricing structure. Furthermore, as already mentioned, the application of periodic fees makes it easier to change the guarantee at some point in time, since in the event of a switch from a benefit structure to another the provider can stop applying the current fee and determine the new level according to the new underwritten guarantee, similarly to what happens (for example) in variable annuities. We also note that periodic fees may allow a revision of the pricing basis after issue, if justified by the emerging scenario and if admitted by policy conditions. In this respect, it is necessary to predefine appropriate triggers identifying situations where a revision of the fee is justified. Triggers could, for example, be related to a mortality/longevity index, or a measure of value to the provider, or the default probability of the provider. This topic deserves a specific research, and is not further developed in this paper.

**Table 4.** Equivalent initial fee *π* (to be charged to the initial capital).

Table 5 lists the individual reserve *Vt* and its components, namely *V*[ben] *t* and *V*[fee] *t* , for a sample of times and a sample of linking arrangements. First we note that at time 0 the proportion *V*[fee] *t Vt* corresponds to the equivalent initial fee *π*; then, such a proportion decreases in time, as it is quite natural, given that the time-horizon of the obligation of the provider gradually reduces. In the table we only include some arrangements and only a scenario of moderate aggregate deviations, as other situations suggest similar comments.

**Table 5.** Individual reserve *Vt* and components *V*[ben] *t* , *V*[fee] *t* , at selected times *t*. Moderate aggregate deviations.

In Table 6, we quote the expected value of the Present Value of Future Profits and the Business Value at time 0, per policy issued, for the various arrangements examined so far. We first note that the magnitude of E[PVFP0] is in line with that of the overall loading (see Table 4). Indeed, a large part of the profit is originated by the loading. As is well-known, large loadings impact negatively on the demand; while PVFP0 measures the profit per policy issued, the total profit gained by the provider also depends on the pool size. This should not be disregarded when performing a profit test of the business with the purpose of identifying a cost-effective solution that may prove attractive to the individual.

As to the business value, first we note that it is not 0. As commented in Section 2.5, condition (18) is a notional reference, which is useful to set the fee consistently in different situations; not necessarily such a condition leads to a situation of a 0 expected value for the business value, as it emerges from Table 6. In such a table, the business value is quoted as a proportion of the present value of future profits. We see that such a proportion is different, depending both on the arrangement and the scenario. In view of practical implementations, the fee obtained under condition (18) could be taken as the minimum acceptable fee for the provider. An additional loading could be suggested by further assessments; for example, instead of condition (18), reference to the tail of BV0 could be made, by setting an accepted level for the probability of incurring into a negative business value, or setting a target value for the expected business value. In any case, clearly, the loading must prove to be acceptable for individuals. Investigating this aspect is outside the scope of this paper.

**Table 6.** Present Value of Future Profits and Business Value (expected values), at time 0.

We finally note that the annual profits, the present value of future profits and the business value can be affected by basis risk in mortality, i.e., by a different mortality in the pool than in the reference population; however, such an aspect is not included in the assessments summarized in this section.

### **4. Conclusions**

In this paper, we investigate annuity designs in which the benefit amount is adjusted in time in relation to a given mortality/longevity experience, compared to a chosen benchmark. Such designs imply a new definition of the longevity guarantee, which deserves attention, given that individuals prove to be dissatisfied with traditional annuities, but need to obtain longevity protection in the private market.

In this paper, in particular, we are concerned with a pricing structure which is innovative for an annuity product. Instead of the traditional upfront single loading, we consider periodic fees, which seem more versatile to introduce opportunities of flexibility into the product. We consider a periodic fee charged year by year to the policy fund value, and we show that this identifies a discount factor, to be used for pricing and reserving. Trying to match traditional with market pricing rules, we assess the periodic fee using a condition expressed in terms of business value. This way, the fees incorporate an allowance for both the expected profit and the risk retained by the provider.

Future steps in the research may concern an assessment of individual preferences in respect of the alternative linking solutions. The business value for the provider could be further examined, by addressing the demand function, as well as the limited liability put option. The implications of switching between alternative linking rules or guarantees require a specific study, as well as the possibility of updating the periodic fee after issue. Addressing a pool consisting of multiple cohorts or heterogeneous in other respects is also significant, in particular to detect possible smoothing effects if the linking coefficient accounts for the mortality experienced over different cohorts. An explicit pricing of the guarantees is a topic to further develop. In this regard, modelling guarantees as financial options offers the possibility to test pricing models developed in that field. Matching the mortality/longevity with a financial linking should also be considered, as most annuities in the market are participating in respect of the return on investments.

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

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the paper, as they are entirely simulated on the assumptions stated.

**Acknowledgments:** The author wish to thank the anonymous referees for their constructive comments and suggestions.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**

### *Article* **Disruption of Life Insurance Profitability in the Aftermath of the COVID-19 Pandemic**

**Maria Carannante 1,\*,†, Valeria D'Amato 1,†, Paola Fersini 2,†, Salvatore Forte 3,† and Giuseppe Melisi 4,†**

**Abstract:** Life insurance profitability depends on reliable mortality risk projections and pricing. While the COVID-19 pandemic has caused disruptions around the world, this is a temporary mortality shock likely to dissipate. In this paper, we investigate the long-run impact of COVID-19 on life insurance profitability. Due to the long-run dynamics of the mortality characterised by a decreasing effect of the COVID-19 mortality acceleration, we suggest proactive mortality risk management by implementing prompt premium adjustments, in order to increase the resilience of the business.

**Keywords:** SCR; profitability; annuity; mortality projections

### **1. Introduction**

Aside from the social and health consequences of COVID-19, the pandemic has led to economic and market shocks. Interest rates and equity markets have declined, credit spreads have widened, and volatility has increased. The additional volatility in global markets affecting the value of equity, fixed investments, and low interest rate income has led to the need to implement unconventional monetary policy measures, such as negative rates, large asset purchase programmes, forward guidance, and targeted liquidity provision measures (ECB 2021). Likewise, the impact of COVID-19 on the insurance industry risks is becoming severe.

Insurance companies are required to investigate the potential disruption across the business caused by the pandemic. Indeed, the pandemic is likely to disrupt investments, finance, capital, underwriting, claims, and actuarial functions in several business areas. Over the next few months, due to the increasing uncertainty around new business and underwriting, the appetite for new insurance products may decline, as consumers face increasing temporary or permanent unemployment, potential loss of income, and general market volatility. Cash flow expectations over the next years also depend on global equity markets that have seen reduced investment returns. "The insurance sector must deal with challenging market conditions and maintain operations, while at the same time protecting employees and policyholders" (EIOPA 2020a). The decline in asset liquidity and the increase in overdue liabilities may cause a decrease in assets relative to liabilities (EIOPA 2020b).

According to Karlsson (2020), the pandemic may have seriously affected the operation of European insurance companies, by representing a serious threat for the solvency stability. Understanding how the COVID-19 pandemic has affected insurance companies is crucial especially in light of the "double-hit" scenario characterised by a resurgence of the virus,

**Citation:** Carannante, Maria, Valeria D'Amato, Paola Fersini, Salvatore Forte, and Giuseppe Melisi. 2022. Disruption of Life Insurance Profitability in the Aftermath of the COVID-19 Pandemic. *Risks* 10: 40. https://doi.org/10.3390/ risks10020040

Academic Editors: Anna Rita Bacinello and Nadine Gatzert

Received: 8 November 2021 Accepted: 3 February 2022 Published: 11 February 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/).

reported by previous stress test exercises by Moody's Analytics (2020). The insurance industry's profitability is linked to operational and financial management both of them suffering the effect of the pandemic. The future financial cash flows could be affected by the uncertainty and pessimism due to the pandemic, the spillover effect of the overall decline in the market, leading to investors' herd behaviour to negative abnormal returns. Conversely, in Farooq et al. (2021), the authors take also into account a possible opposite effect of the COVID-19 outbreak of the increasing demand for insurance contracts and premiums. From the operational management point of view, insurers are responding to the widening pandemic on multiple fronts as health insurance, non-life and life offices, some classes of business being most exposed to coronavirus and adversely impacted. Some business classes are more exposed to the COVID-19 outbreak than the others. The portfolio concentration of higher risk business classes seriously threatens the insurance companies, by suggesting more well-diversified portfolios.

In particular, the health insurance premiums continued to grow steadily after the outbreak as pointed out in Wang et al. (2020) and Nguyen and Vo (2020). In particular, in correspondence to the profound shock to the health care systems due to the surge of COVID-19, major commercial health insurance companies increased operating income from decreased care utilization: for instance UnitedHealth Group, CVS Health Care Benefits Segment, Anthem, and Humana all saw operating earnings over 200% of their 2019 amount, much of which has been attributed to delays in routine care (Bryan and Tsai 2021).

Focusing exclusively on aspects related to non-life insurance, the insurers tried to adapt the policies to the new challenges exposed by the crisis in response to the COVID-19, specifically by providing the business interruption (BI) insurance, the crisis having been reaffirmed the importance of business continuity planning. With regard to property and casualty (P&C), the impact on business and coverage has been profound, estimated at USD 80 to USD 100 billion in the case of business interruption (BI) coverage, a critical area of concern under COVID-19 (Marsh 2021). In general, according to an interesting study by Gründl et al. (2020), the insurance industry alone will not be able to provide sufficient coverage for business interruption losses like those occurring during the COVID-19 crisis, as the markup of a hypothetical insurance contract in the top 20% of the realised price markups of NatCat insurance would lead an expected shortfall of the loss distribution which is about 100 times higher. In the automobile insurance field, reductions in driving and accident claims led to premium refunds early during the pandemic by causing welldocumented premium changes (Scism 2020).

The uncertain mortality and morbidity events related to COVID-19 are also affecting the life insurance and annuity business. Mortality improvements over the past several years have been muted, likely to continue mainly as a combined effect of the temporary mortality shock due to the pandemic. Indeed, the debate is ongoing on how temporarily stressed mortality rates change post-COVID-19 mortality rates (Andresson and Lindholm 2021) and the mortality term structure (Milesky 2021; Spiegelhalter 2020).

The scarce literature on the topic enlightens that life insurance companies have been forced to significantly adjust life insurance premiums or offerings to account for the increased mortality risk (Pułanska 2021).

Harris et al. (2021) suggest minimal observable premium adjustments through February 2021. They find evidence that premiums raised "for unhealthy older smokers, and policies offered to individuals age 75 and above were differentially removed from the market". Overall, small adjustments in the life business offering correspond to increases in mortality risk perceived from insurers as modest in the short run, by implicitly assuming no effects in the long-run perspective. To the best of our knowledge, in light of the mortality stress temporariness, the academic literature has not extensively focused on the possible changes in profitability margins for life insurance companies.

The novelty of our research properly consists in examining the long-run impact of COVID-19 on life insurance profitability. We suggest connecting the profitability analysis to the temporary excess of deaths due to the COVID-19 which will be softened in the long run by the structural improvements of longevity projections (Carannante et al. 2021b).

Actuarial assumptions and forecasting are crucial for an effective mortality risk management strategy and preserving the expected cash flows over the coming years. Understanding the impact of future structural improvement scenarios, as well as increased short-term mortality combined with heightened attention to social and health care improvements in the longer term will allow life insurance offices to maintain profitability. In other words, proactive mortality risk management, which requires revising mortality assumptions to make timely decisions in reserves and forecasting, will enable the insurance industry to build resilience and tackle the immediate challenge of positioning the business for the future. The remainder of the paper is structured as follows. In Section 2, we introduce the issue of profitability, define profit resilience in life insurance, and how to quantify it with particular reference to annuity contracts. Section 3 details the numerical applications, focusing on the mortality, financial, and cash flow aspects. Section 4 concludes.

### **2. Profit Resilience in Annuity**

We analyse the expected profit of a variable immediate annuity contract.

The general actuarial model used for the evaluation of the insurance contract and to estimate the future cash flows belongs to the life insurance methodologies, which represent the actuarial practice in many countries, according to a time-discrete approach, which, see Olivieri and Pitacco (2015).

The contract under consideration is an immediate single premium annuity with a revalued instalment for an individual of age *x* at time 0 in which the contract is underwritten. Obviously, since it is an immediate annuity, the single premium is the only possible alternative. In this case, in order to implement the profit-sharing mechanism that prevails in the Italian market, we implement an actuarial model with cliquet guarantees with annual returns recognised to policyholders depending only on the most recent performance of an investment portfolio. The contract valuation can the be reduced to that of a sequence of one-year forward-start options, see Bacinello (2001, 2003a, 2003b).

To assess the effects depending on age, we consider policyholders aged 20, 40, and 60. In the case of an annuity, the instalment is constant and the pure premium for an individual at age *x* is given by:

$$P\_{\mathbf{x}} = R \cdot \sum\_{t=1}^{\omega - x - 1} \frac{l\_{\mathbf{x} + t}}{l\_{\mathbf{x}}} \cdot (1 + i)^{-t} \tag{1}$$

where:

*Px* is the pure premium based on the first-order mortality basis table;

*R* is the constant instalment paid by the insurance company during the policyholder's life with a value agreed at contract time;

*lx* is the number of policyholders at age *x* deduced by the first-order mortality basis table used to compute the pure premium;

*i* is the technical rate;

*ω* is the extreme age, thus *ω* − 1 is the last age for a policyholder and *lω* = 0.

Since we consider a variable annuity, the pure premium is defined as:

$$P\_{\mathbf{x}} = \,^r R \mathbf{0} \cdot \sum\_{t=1}^{\omega - x - 1} \frac{I\_{\mathbf{x} + t}}{I\_{\mathbf{x}}} \cdot \left(\mathbf{1} + i\right)^{-t} \tag{2}$$

where:

*R*0 is the first instalment defined at contract time.

The following instalments are variable based on segregated fund returns with the following formula:

$$R\_t = R\_{t-1} \cdot (1 + r\_t) \tag{3}$$

where:

*Rt* is the instalment at time *t* if the policyholder is alive;

*Rt*−1 is the instalment at time *t*−1 if the policyholder is alive;

*rt* is the downgraded rate of return used to vary the rate based on the segregated fund return rate using the following formula:

$$r\_t = \max\left(\frac{g\_t - i - mt}{1 + i}, \arg\right) \tag{4}$$

where:

*gt* is the segregated fund return rate for the period (*t* − 1, *t*) recognised at time *t*; *mt* is the rate retained by the insurance company on the segregated fund return; *mg* is the minimum guaranteed rate of the segregated fund.

Once the pure premium and method of variation of the instalment are determined, the expenses loaded premium at age *x* can be calculated:

$$PT\_x = \frac{P\_x \cdot (1 + \pi)}{(1 - \beta)}\tag{5}$$

where:

*α* is the loading rate of the annuity payment;

*β* is the loading rate of the administrative costs.

The expected profit is defined as:

$$E(\mathcal{U})\_{x,k} = PT\_{x,k} - BE\_{x,k} - Co\mathcal{C}\_{x,k} \tag{6}$$

where:

*E*(*U*)*x*,*k* is the present expected profit at time *k* when the contract is purchased by an individual at age *x*;

*PTx*,*k* is the expenses loaded premium at time *k* for a policyholder at age *x*; *BEx*,*k* is the best estimate of the contract liability a time *k* for a policyholder at age *x* according to Solvency II principles, by considering the financial options and guarantees to include in the insurance contract;

*CoCx*,*k* is the cost of capital due to the allocation of the capital requirement under Solvency II for a contract sold a time *k* for a policyholder at age *x*.

Cost of capital, *CoCx*,*k*, is determined according to Solvency II requirements:

$$CoC\_{x,k} = \partial \cdot \sum\_{l=1}^{m} \frac{\mathbb{C} \cdot SCR\_{x,k+l-1}}{\left(1 + i\_{rf}(k, k+l)\right)^{l}} \tag{7}$$

where:

*∂* is the cost of capital rate increase;

*ir f*(*k*, *k* + *l*) is the risk-free rate for the time horizon (*k*, *k* + *l*);

*C* is the cost of capital rate, that is, the unrealised extra-return compared to the risk-free rate; *SCRx*,*k*+*l*−1 is the solvency capital requirement for the time horizon *k* + *l* − 1 and a policyholder of age *x*;

*m* is the number of years when the risk expires in terms of capital requirements.

To determine *CoCx*,*k*, we consider the EIOPA (2014) standard formula with particular reference to the market, longevity, expense, and operational risks. To note is that in determining *RORAC*, the overall *SCR* is considered, while in determining *CoC* only nonhedgeable risks are considered.

Furthermore, important to note is that *BEx*,*k* is equal to the present expected value of the liability if considering a reliable technical basis, depending on the mortality table. We assume that the realistic projected mortality table is obtained using the stochastic mortality model considering the scenario without the effects of the COVID-19 pandemic, and the scenario with an acceleration of mortality due to COVID-19 (Carannante et al. 2021a, 2021b); a reliable administrative expenses assumption, that is, an annual cost per contract; the risk-free rate maturity structure for discounting contract cash outflows; a stochastic model to determine *gt*, that is, the segregated fund return rate for the period (*t* − 1, *t*) recognised at time *t*, which allows determining the variation of the annuity instalment *Rt*, using the Vasicek model.

The Vasicek model is largely used to evaluate the short-term evolution of a return rate, using a stochastic differential equation according to which shocks fluctuate around a long-term value as a function of volatility (for further details, see Vasicek (1977)):

$$dr\_t = (\alpha + \beta r\_t)dt + \sigma dZ\_t \tag{8}$$

where:

*rt* is the short-term interest rate at time *t*; *α* is the mean-reverting force of the shocks; *β* is the long-term interest rate mean; *σ* is the market volatility; *Zt* is a Wiener process.

### **3. Numerical Application**

The application is developed by analysing several different aspects of the definition of an immediate annuity contract. The first concerns the demographic scenario that evaluates the evolution over time of mortality considering the effects of the pandemic. Second, for the financial aspect, we observe the interest rate trend on which the variation of annuity instalment will be based. Third, the cash-flow analysis allows evaluating the differences in the premium in the function of the use of baseline or accelerated mortality tables. The last is the profitability analysis that allows quantifying the extra profit due to the adjustment of the mortality table.

### *3.1. Demographic Scenario*

The first step in evaluating profitability is to determine the demographic technical basis, that is, the individual death probabilities. In this sense, we use a stochastic model capable of projecting the probabilities of life over time. The model defines the probabilities of death with respect to two scenarios.

The baseline scenario assumes the absence of the COVID-19 pandemic, and the projections of survival probabilities are obtained through the Renshaw and Haberman (2003) estimation using the data on deaths collected in the Human Mortality Database,1 with reference to the entire Italian population, considering the historical series from 1950 to 2017 for all ages from 0 to 100.

The alternative scenario considers the COVID-19 pandemic as a mortality acceleration factor estimated through a multiplicative model, that is, the projections of the accelerated probability of death obtained from the product between the probabilities of the basic scenario model and the multiplicative factor that depends on age *x* and time *t*.

Cairns et al. (2020) define the multiplicative factor as a negative exponential function as follows:

$$
\pi(\mathbf{x},t) = \frac{a(\mathbf{x})}{\rho(\mathbf{x},t)} \exp\left(\frac{-t}{12\rho(\mathbf{x},t)}\right) \tag{9}
$$

where:

*α*(*x*) is the expected proportion of deaths by COVID-19 at age *x*; *ρ*(*x*, *t*) is the expected loss of years of life expectancy at age *x* and time *t*.

The *α* and *ρ* parameters in Formula (9) are computed from the COVID-19 deaths data and the all-causes mortality data of the Italian population for the year 2020. The data are collected weekly by the Italian Health Institute2 (ISS) and the Italian Statistical3 (ISTAT). *α*(*x*) is calculated as the ratio of the number of deaths due to COVID-19 infection and the total of deaths for the age *x*, while *ρ*(*x*, *t*) is calculated as the product of the life expectancy at age *x* and time *t* and the proportion of deaths due to COVID-19 at age x on the total mortality due to COVID-19. *π*(*x*, *t*) is calculated as a negative exponential function, aggregating the data in a monthly granularity, and it is used as a multiplicative coefficient to recalibrate the mortality projection obtained by the Renshaw–Haberman model. Figure 1 shows the death projections by age for 2021 per 100,000 population, distinguishing between deaths due to COVID-19 and all other causes. To make the data easier to read, we report them in logarithmic scale:

**Figure 1.** Non-COVID-19 and COVID-19 death projections by age for the year 2021.

As Figure 1 shows, the number of deaths from COVID-19 proportionally follows the trend in mortality for all causes of deaths, except for older ages where the proportion of deaths appears higher. This suggests a relationship between age and mortality from COVID-19, with a mortality shock currently present.

### *3.2. Financial Scenario*

We estimate the Vasicek model parameters *α*, *β*, and *σ* using EURO SWAP maturing at one year (1Y) and ten years (Y10) from 31 January 2005 to 31 December 2020.4 Figure 2 shows both the Y1 and Y10 EURO SWAP time series.

Figure 2 shows a generalised reduction in interest rates. The decreasing trend affects the values of the simulated interest rate structures using both the annuity instalments variation and the discounted cash flows best estimate. The Vasicek parameters are shown in Table 1.

As Table 1 shows, the parameter *α* is very close to zero, suggesting a strong persistence in the time series, as also observed in Figure 2, with no strong fluctuations with respect to the decreasing trend. The parameter *β* is estimated at around 1.20%, being affected by a period in which rates even exceeded 3% (up to 2009) and the most recent periods of negative rates (from 2016). The parameter *σ* is 0.48 suggesting quite high volatility. Therefore, according to the estimated model, a divergent trend from the mean is expected, consistent with the most recent time interval in which rates are downward and continue to decrease with non-negligible volatility.

**Figure 2.** Y1 and Y10 EURO SWAP time series from 31 January 2005 to 31 December 2020.

### *3.3. Cash Flow Analysis*

We perform the cash flow analysis considering the conditions shown in Table 2. Important to note is that the A62I unisex table with 50% male and 50% females is the most used by insurance companies for annuity contracts.

Using these data, the segregated fund return rate is simulated based on a zero-couponbond forward rate at one year for the period 2022 to 2142 for a total 1000 scenarios.

Tables 3 and Tables 5–7 show the effects of COVID-19 acceleration, comparing (for an immediate annuity contract for policyholders of age *x* = 20, 40, and 60), the pure premium (*PT*), the best estimate of liability (*BE*), the solvency capital requirement (*SCR*), the cost of capital (*CoC*), the expected value of the profit (*E*(*U*)), and the *RORAC*. Table 3 relates to an annuity contract signed in 2022.

**Table 3.** Effects of COVID-19 for an annuity contract signed in 2022.

As Table 3 shows, for policyholders aged 20, the contract is at a loss even without COVID-19 acceleration. This is due to the technical basis used to determine the expenses loaded premium, already inadequate to determine future longevity of the age considered. Furthermore, the *RORAC* is negative, and the mortality increase due to COVID-19 determines an increment of only 1.5%. For the year 2022, the pandemic acceleration causes an increase in profitability at most equal to 16.1% of *RORAC*. Furthermore, for ages 40 and 60, there is a huge profit for the insurance company even without considering the effects of the pandemic on mortality.

We further explore the profitability of annuity contracts in Table 4 showing the annual cash flows for the three ages considered for one hundred years forward, comparing the baseline mortality table, ignoring the pandemic effects, and the accelerated mortality table.

**Table 4.** Cash flow analysis for one hundred years forward.

As Table 4 for all ages, the expected cash flows for the baseline scenario and the accelerated scenario are similar, showing some differences only for very large values of *t* that do not affect the value of *BE*. Furthermore, no particular differences emerge when comparing the *BE* distributions by age and mortality basis. The results are shown in Figures 3–5. Therefore, both in terms of expected values and variability, cash flows and *BE*s are little affected by the mortality shock due to COVID-19.

**Figure 4.** *BE* distribution for baseline and accelerated mortality tables for age 40 (EUR/thousands).

**Figure 5.** *BE* distribution for baseline and accelerated mortality tables for age 60 (EUR/thousands).

Table 5 shows the results of an annuity contract signed on 1 January 2024.

**Table 5.** Effects of COVID-19 for an annuity contract signed in 2024.

As Table 5 shows, not all the contracts are in profit. For policyholders aged 20, the insurance company is at loss, with *RORAC* −87.8%, and the adjustment of the tables to the effects of COVID-19 allows partial recovery only at 1.5%. For policyholders aged 40 and 60, the contract always has positive expected profitability both with and without the COVID-19 acceleration adjustment. In summary, for an annuity contract signed in 2024, the COVID-19 acceleration allows increasing profitability by only 5.2% of *RORAC* for a policyholder age 60. Conversely, with very young policyholders, it does not allow full recovery of the loss due to the increase in longevity from 2022 to 2024.

Table 6 shows the effects of COVID-19 for an annuity contract signed on 1 January 2026.

**Table 6.** Effects of COVID-19 for an annuity contract signed in 2026.

As Table 6 shows, for a policyholder aged 20, the insurance company is at loss with huge negative *RORAC* −101.0%, which reduces only to −99.6% considering the COVID-19 effects. Contracts with policyholders aged at least 40 maintain reduced profitability compared to the previous two years but are still satisfactory, even more so considering an increase in *RORAC* with acceleration due to COVID-19 of at least 3.6%. In summary, considering the data relating to the year 2026, the acceleration of mortality due to COVID-19 entails a negligible increase in profitability compared to the increase in longevity from the year 2022 to the year 2026.

Table 7 shows the results for an annuity contract signed on 1 January 2032.

**Table 7.** Effects of COVID-19 for an annuity contract signed in 2032.

As Table 7 shows, for policyholders aged 20, the insurance company is heavily at loss with a reduction in *RORAC* compared to the previous decade (2022) equal to 65%. For all the ages considered in the year 2032, the acceleration of mortality due to Covd-19 entails a negligible increase in profitability compared to the increase in longevity from the year 2022 to the year 2032. Considering a balanced portfolio in terms of the age of policyholders, the empirical evidence suggests that in 2032, insurance companies need to update the currently used A62I mortality table.

### *3.4. Focus on Profitability*

Figures 6–8 show the *RORAC* indicator trend by year for the three ages considered. To note is that for all ages analysed, the impact of COVID-19 is very modest and decreases over the years, except for policyholders aged 60, for which in 2022 the impact of COVID-19 determines a consistent reduction in *RORAC*. In addition, as noted in Tables 3 and 5–7, for all three ages considered, the *RORAC* trend decreases with significant variations over the years. This trend confirms the significant weight of the longevity risk in the management of annuities.

Furthermore, looking more deeply at the longevity risk and how much it can affect profitability, the distributions of expected profit and *RORAC* for the year 2022 shown in Figures 9–11 indicate high variability of profit and consequently *RORAC* in all scenarios considered and for all ages:

**Figure 6.** *RORAC* by year for a policyholder aged 20.

**Figure 7.** *RORAC* by year for a policyholder aged 40.

**Figure 8.** *RORAC* by year for a policyholder aged 60.

**Figure 9.** Expected profit and *RORAC* distributions for a policyholder aged 20.

**Figure 11.** Expected profit and *RORAC* distributions for a policyholder aged 60.

Estimating the expected profit and relative distribution, we obtain the probability of a negative profit, which could prove very useful to understand that even in the presence of a positive profit value, the risk of obtaining a negative result could be high. For example, taking into account the 2022 contracts, the probability of a negative profit is equal to 0.596 for policyholders at age 20, 0.253 at age 40, and 0.012 at age 60. Therefore, while for policyholders aged 60 the probability of obtaining a negative result is negligible, for policyholders aged 40, despite a very positive value of profit and *RORAC*, this probability is to be taken into account and confirms the importance of a good pricing and longevity risk monitoring system.

### *3.5. Discussion*

The analysis of profitability is of great practical and policy value to study how the pandemic affects the insurance market. In particular, the study provides useful indications we can re-formulate as valuable recommendations. In order to increase the resilience of the life insurance business to the COVID-19 pandemic, proactive risk management is required. We suggest taking into account the profitability in the long run by implementing prompt premium adjustments. Due to the long-run dynamics of the mortality characterised by a decreasing effect of the COVID-19 acceleration, accurate safety loadings are necessary to guarantee stability for the insurance industry. In order to evaluate the effects of the acceleration of mortality in the life insurance business, we initially define a framework to operate. In this sense, we analyse the demographic scenario, which shows a greater acceleration of mortality due to COVID-19 for older ages, and the financial scenario, which shows that interest rates tend to decrease in the long term with a certain volatility. The two scenarios make it possible to define the contractual conditions of the immediate annuities for which the cash flow and profitability from 2022 to 2032 are analysed. With regard to cash flow, it is observed that the acceleration in mortality does not generate large differences, with the exception of the older ages and considering a rather long period of time. Regarding profitability, it is noted that age and time are the determining factors to be taken into consideration. In particular, if we consider a short time horizon, adapting the mortality tables to the acceleration from COVID-19 allows obtaining greater profitability for the older ages, while it does not allow to remedy the inadequacy of the tables themselves for the older ages, recording a loss. Similarly, the broader the time horizon, the lower the margin obtained from the use of tables that take acceleration into account while the improvement in longevity tapers the margin more and more, increasing the losses for the younger age and also recording losses for the middle ages. In accordance with our results, Harris et al. (2021) observe an adjustment in life insurance market profitability in presence of some particular health condition or old age, although to a limited extent.

### **4. Concluding Remarks**

The pandemic phenomenon has a non-material impact on the profitability of annuity contracts as it has an instant impact since in a pandemic event there is an increase in mortality only in the first years of the contract and therefore for medium and long-term contracts such as annuities, post-shock mortality quickly tends to mortality without considering the COVID-19 effect.

Therefore, for these contracts, with medium and long durations, this effect with an accidental and unsystematic nature leads to non-material increases in profitability with respect to the same contracts without COVID-19 effects with the same contractual conditions.

On the other hand, as regards the opposite phenomenon, that is the longevity risk, it is a systematic risk that has a material impact over the entire duration of the contract and for such contracts, with medium or long durations, this risk is significant. In fact, if we consider a contract with a very young insured, for example with age *x* = 20, we always have negative returns that increase in material measure passing from the marketing year 2022 to the year 2032 or to the year 2042. For example if we consider the RORAC for *x* = 20 passing from the year 2022 to 2024 we have a contract always with negative profitability and a loss of RORAC without the COVID-19 effect equal to 13.4% while with the COVID-19 effect this loss of RORAC is equal to 13.5%. If we consider the year 2026 the differences become even more significant, in fact without the COVID-19 effect there is a loss of RORAC equal to 26.6% while with the COVID-19 effect this loss is equal to 26.8%.

Therefore, we can conclude that in just two calendar years, the longevity risk and therefore the increase in the life expectancy of the insured lead to an increase in losses for the insurer of approximately 13% while considering four years this increase in loss exceeds 26%.

These conclusions are also fully consistent with the same ones in the paper by Carannante et al. (2021c) in which the authors analyse a pure mortality risk insurance product such as term insurance which cover the opposite risk compared to those of annuities.

Even for these contracts, the acceleration of mortality from COVID-19 would lead to price increases with the same profitability always lower than 0.5%.

Besides in the paper we show that COVID-19 would bring material increases in profitability for insurance companies only in the case of old insureds, in fact for instance if we consider in the commercial year 2022 an insured 60 years old we have, after COVID-19 mortality shock, a RORAC increase equal to 16%, but in our opinion, this case is not real because if we analyse a real new business Italian insurance portfolio the majority of the insured are younger than 50 years old. We conclude that from a theoretical point of view the COVID-19 phenomenon has brought benefits to insurers for non-young policyholders but from a real point of view given the real age of the policyholders of annuity insurance portfolios, the profit margins are not material.

Therefore, the study confirms that it is much more important how the estimated trend of post-COVID-19 mortality realigns to what was predicted in the ante-COVID-19 situation, rather than the shock level recorded in a very short period (1–2 years), i.e., during the pandemic.

The effects of COVID-19 are expected to continue hitting some property-casualty lines harder than others. Nevertheless, pension schemes and annuity portfolios are also exposed to the aftermath of the pandemic. According to Deloitte (2021), the growth and profitability in annuities and many non-term life insurance products will likely be impacted throughout 2021 and beyond with persistently low interest rates. The profitability of life offices also seems to be threatened by the temporary shock of mortality.

In light of these considerations, our paper explores how the COVID-19 pandemic mortality shock might affect the profitability of insurance companies considering immediate annuity contracts, as well as the financial and actuarial aspects. Unlike the commonly assumed post-pandemic effects, COVID-19 mortality acceleration did not and will not bring insurance companies a huge increase in annuity contract profitability, considering a risk portfolio with different ages.

On the other hand, the increasing longevity issue will remain the main problem over the years and will lead insurance companies to adjust their mortality tables with a frequency that never exceeds five years, particularly if the portfolio is composed of a rather low mean policyholder age (see Supplementary Materials).

**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/ 10.3390/risks10020040/s1.

**Author Contributions:** Conceptualization, V.D. and S.F.; methodology, S.F. and P.F.; software, G.M. and M.C.; validation, V.D. and M.C.; formal analysis, S.F., G.M.; data curation, M.C. and G.M.; writing—original draft preparation, S.F. and V.D.; writing—review and editing, M.C. All authors have read and agreed to the published version of the manuscript.

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

**Data Availability Statement:** Publicly available datasets were analysed in this study. Human Mortality Database data can be found here: https://www.mortality.org/, accessed on 28 September 2021. Euroswap data can be found here: https://www.eurex.com/ex-en/markets/int/eur-swap/eurswap-fut/, accessed on 25 September 2021.

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

### **Notes**

### **References**

Spiegelhalter, David. 2020. Use of "normal" risk to improve understanding of dangers of COVID-19. *BMJ* 370: 3259. [CrossRef]

### *Article* **Proposal to Extend Access to Loans for Serious Illnesses Using Open Data**

**Frédéric Planchet \*,†, Édouard Debonneuil \*,‡ and Marie Péju §**

ISFA—SAF Laboratory, Institut de Science Financière et d'Assurances (ISFA), Laboratoire SAF EA2429, University Lyon, Université Claude Bernard Lyon 1, F-69366 Lyon, France<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>(F.P<EMAIL_ADDRESS>(É.D.)

† Frédéric Planchet is Professor at ISFA and Partner Actuary at PRIM'ACT.

‡ Edouard Debonneuil is manager of ActuRx.

§ Marie Péju is an independent *data scientist*.

**Abstract:** In France, access to a loan requires one to obtain loan insurance and the presence of a pathology in the applicant may be a reason for refusal. Improving knowledge of health risks and pooling risks are two methods of broadening access to loans. We attempt to analyse these possibilities using open data and risk pooling scenarios. We find that the removal of medical selection can be ensured if the current framework is adjusted. We also demonstrate how to use open data to estimate loan insurance premiums for a variety of diseases.  Broadening access to borrowing would be beneficial for patients and for the development of the economy associated with these projects.

**Keywords:** borrower insurance; mortality; serious illnesses

**Citation:** Planchet, Frédéric, Édouard Debonneuil, and Marie Péju. 2022. Proposal to Extend Access to Loans for Serious Illnesses Using Open Data. *Risks* 10: 51. https://doi.org/ 10.3390/risks10030051

Academic Editors: Anna Rita Bacinello and Nadine Gatzert

Received: 12 November 2021 Accepted: 21 February 2022 Published: 28 February 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/).

### **1. Introduction**

Every year, borrowers are refused loans for the purchase of a property or the creation of a business because of their health status. The human impact is all the greater, as these patients did not choose to be ill. The impossibility of supplementing their financial resources with a loan to help them achieve a life goal can be perceived as discrimination. These refusals also have an economic impact beyond the loan itself, as these life projects would have contributed to the development of the economy.

Some of these refusals are due to insurers' insufficient knowledge of health risks. This refusal is the insurer's response to a risk of financial loss deemed too great due to the risks of death or temporary or permanent disability.

It may seem that insurers would have no issue in appreciating the risk. Insurers' actuaries generally know how to rate contracts on the basis of policyholder statistics. They also know how to deal with illnesses in various ways; in the underwriting process of protection insurance for products with death and disability benefits, premiums are increased notably when medical examinations reveal the presence of an illness. In some countries, such as in the United Kingdom, lower premiums are even applied for annuities in the presence of illness (Ridsdale 2012).

Still, for diseases that are currently refused, insurers do not have the corresponding statistics. It is not easy to find external information for reasons of health privacy. We have only partial access to data that enable us to construct statistics from such sources as health or life insurance data or national health data. Expert assessments of the risks associated with diseases are not unanimous; there are many diseases and associated risk factors, and scientific advances and changes in behaviour modify the risk levels. Faced with this lack of knowledge, insurers take a cautious approach (and they are obliged to do so to avoid bankruptcy and failure to honour their guarantees) and in practice have no choice but to replace excessively high rates with a refusal to accept them in order to avoid a significant image risk (and they are obliged not to exceed a usury rate).

Furthermore, pooling high risks with low risks comes up against population behaviour. Within the existing framework, an insurer who creates a loan contract without medical selection, therefore being maximally inclusive with regard to pathologies, risks seeing an influx of patients who have been refused a loan elsewhere without this influx being compensated for by a sufficient influx of low-risk individuals; the experiment would come to nothing.

Thus, the existing framework does not favour the detailed knowledge of risks associated with pathologies or the pooling of high health risks with lower risks. Here we try to use open data and our knowledge of insurance to find ways out of the current situation. Throughout the paper we take breast cancer and diabetes as application examples. After an overview of mortality risks and the insurability of removing medical selection, we describe a generic approach to estimating loan insurance pure premiums for patients and apply it to breast cancer and type 1 diabetes. Throughout, we distinguish between mortality and temporary or permanent disability risks and provide elements for inclusive approaches. We hope that these elements can contribute to widening access to borrowing.

### **2. Overall Picture Based on French Mortality Benchmarks**

For about fifty pathologies, the pathology sheets on the Ameli website (Ameli 2018a) provide information on the level of mortality in the general population. They give (at the top of their second page) mortality rates by age group. In Figure 1, we plot these mortality risks by age groups on a log scale for two pathologies, diabetes (Ameli 2018d) and breast cancer both the risky active breast cancers (Ameli 2018b) and the less risky under (SEER 2020) breast cancers (Ameli 2018c). We also plot the mortality risks of various French populations not defined by pathology in order to compare them with the mortality risks of patients.

**Figure 1.** Annual mortality rate in France between the ages of 25 and 65 for different populations (55% men and 45% women)1.

We can first observe the consistency of the risks of populations not defined by a pathology. The general French population had a similar risk (represented in black) in 2016 (noted general population in 2009–2013 (Blanpain 2016)) and 2018 (noted general population 2021–2016 (Blanpain 2018)) with a decreasing trend. When segmented in green by income twenties (Blanpain 2018) and in blue by diplomas (Blanpain 2016), the level of mortality falls as one's socio-professional level increases. In red, the average risk of insured individuals, from a table known as "IA 2013" (Tomas and Planchet 2014), is around the line of executives and higher intellectual professionals (which is the lowest blue curve). In yellow-dotted lines, an average of actuarial tables in loan insurance2, corresponding to the subscription to high incomes and an absence of severe pathology, demonstrates a risk lower than that in red of policyholders in general, which are, on average, less selected or even lower than the first twentieth in terms of income. The risk of death in case of disability, from tables known as "BCAC 2013" (Bagui 2013), is particularly high. Although it decreases in the early years (following an accident or illness, probably due to the death of those most at risk and a stabilization of health conditions for the others) before rising with age, it remains higher than for non-retired inactive individuals (disabled or not). We are used to the use of the latter curves and know that the order of magnitude is correct.

Secondly, the coherence observed between these reference curves allows us to reconstruct mentally and approximately what would be the mortality risks given by the Amelie pathology records if they were available in finer age brackets (the horizontal segments in Figure 1 would be inclined, similarly to the close reference curves, with approximately the same centres). We see that the risks of active breast cancer are of the order of those of disabled insured individuals in the years following the cause of disability and that the risks of the general population with breast cancer or diabetes are much higher than those of the borrower population, even in the case of breast cancer under surveillance. It is therefore necessary to study in greater detail the risks associated with different forms of disease and the transposition of risk to the socio-professional status of borrowers. This is conducted in Section 3.

Before moving to Section 3, however, it is interesting to note that Figure 1 helps us to appreciate what the risk of borrowers would be in the absence of medical selection. As the majority of loan applicants are executives, especially when weighted by amount, the risk should be close to the lowest blue curve. In the absence of medical selection, the risk should be similar to the red curve (tends to be higher due to this, tends to be lower due to the financial resources needed to borrow). Associating these professional categories with a twentieth, this would then correspond to the 5th twentieth of the French population, represented by the green-dotted line, i.e., an increase in risk of the order of 40% compared to current borrowers (yellow-dotted line)—or perhaps less, as patients with serious health conditions are likely to be less inclined to seek a loan and because fixed costs embedded into the insurance premiums absorb the risk (by definition they do not increase with the risk of borrowers). Given fixed insurance costs that do not increase with the risk of borrowers, commercial premiums that would keep current margins would be increased by less than 40% for all borrowers compared to current ones. However, given the almost 100-fold difference between these risks and those of people with active breast cancer, for example, this theoretical view's framework is insufficient and severe conditions are over-represented in one or a handful of insurers rather than being distributed evenly between insurers.

### **3. A Generic Method to Estimate Loan Insurance Premiums for Patients**

We observe that if the framework is to be adjusted, it is important to refine our knowledge of the risks according to factors related to the pathology to minimize patient exclusion. In this section, we describe a method of finding aggregate open data in a way that estimates loan insurance premiums for a variety of pathologies. We do not claim that this is optimal; on the contrary, it sets the scene for potential improvements by other authors. We focus on France, but the approach is not limited to France.

### *3.1. Mortality Risk of Patients in the General Population According to Various Risk Factors*

Finding mortality risks for specific diseases can typically be done with a Google or PubMed search on the disease and on keywords such as "survival", "age", "mortality rate", "death rate", "mortality", "death", "breast cancer", and "type 1 diabetes" (or their equivalents in French). As illustrated in Section 4 with breast cancer and diabetes, articles are found that model mortality from national datasets, clinical trials, or epidemiologic studies, along with various risk factors.

We performed this for a number of diseases. Trade-offs have to be made between choosing recent articles or articles with more data and more risk factors. Otherwise article results must be combined to get the most out of different articles. Sometimes no adequate study is found in France or the desired country and studies from other countries are transposed while adjusting the results to macro statistics in France.

This step is relatively specific to each pathology.

### *3.2. Transposition to Borrowers*

The mortality risks obtained so far correspond to patients in the general population, not to patients who have the typical wealth and lifestyle of borrowers. However, as observed in Figure 1, the mortality of current policyholders and insured borrowers is, on average, 2 to 2.5 times lower than that of the general population. Thus, do loan applicants with a medical condition have the normal risk of patients with that condition or about half the risk?

### 3.2.1. Theoretical Considerations and Definition of a Relative Risk Multiplier

It is logical that mortality in the event of illness is lower among loan applicants than the general population: they are, on average, more educated about health and therefore get screened earlier, have better living conditions and fewer co-morbidities and therefore recover more easily from health problems. Thus, the more the survival of a disease depends on these elements, the higher the survival of the insured persons compared to the general population.

A mathematical perspective provides an a priori idea of quantification for certain diseases. Insured populations are typically half as likely to die from all causes. This translates into a reference ratio for causes of death: the death rate for a given cause (death rate relative to the size of the population, not the diagnosed population) is 2 + x times less than that of the general population, where x is centred in some way around zero (sometimes positive, sometimes negative, depending on the pathology).

In particular, if a condition is very common, x is likely to be small in absolute terms (because if x were large, the other causes of death would have to have a large x on average in the other direction for all-cause mortality to be what it is). Mortality from this condition is itself decomposed as the product of incidence and mortality, if diagnosed. Here, depending on the disease considered, knowledge of the qualitative link between incidence and socioprofessional category gives an idea of post-diagnosis mortality. Thus, in the case of breast cancer, the death rate (relative to the population size) of insured women can be expected to be about half that of women in the general population. Considering that screening is much more frequent and early in the higher socio-professional status, the incidence in the diagnostic sense and not in relation to a stage of the pathology should tend to be greater in insured women: we can expect the mortality rate in the case of breast cancer to be less than half that in the general population.

Cancers are special in that their early detection radically changes the associated survival—hence the screening approaches for breast cancer, melanoma, or colon cancer. For other diseases, we expect a lower incidence in the insured population than in the general population due to better health on average. However, the insured population tends to see the doctor earlier, so there is a tendency not to be very far from the incidence in the general population. Thus, the factor two of all-cause mortality would be found more in terms of mortality in the case of pathology than in terms of incidence.

Thus, the consensus is that the relative risk between mortality in the presence or absence of a specific pathology—the "relative mortality risk" due to that pathology should be on average a little larger for insured individuals than for the general population, and smaller in the case of pathologies, such as cancers, for which insured individuals are screened particularly early. This is why we define the "relative risk multiplier" as the ratio

between these two relative risks, which should, therefore, on average, be a little higher than 1 but lower than 2, and probably lower than 1 for breast cancer.

### 3.2.2. Mathematical Definition of the Multiplier and How to Obtain it

Let us consider four populations: A , B , C , and D . The prime is, here, to indicate that they are chosen to compute a risk multiplier *ρ* that does not exactly correspond to any one insurance loan. Let us use the same letters to represent their annual mortality risk: *qx A* , *qx B* , *qx C* , and *qx D* . Table 1 shows how to choose the definition of these populations to obtain a relative mortality multiplier by combining their mortality risks.

**Table 1.** Risk multipliers.

(\*) Here, the word "healthy" is an abuse of language to indicate that the population has not been selected for a particular pathology.

The relative risk multiplier *ρ* is then defined by:

$$\rho' = \frac{q\_{\ge}^{A'} / q\_{\ge} B'}{q\_{\ge}^{C'} / q\_{\ge} D'}$$

It is very difficult to find *qx A* /*qx B* on the Internet as it requires articles that focus on a pathology in high socioeconomic populations, whereas articles that focus on a pathology tend to also focus on social inequalities and more particularly on low socioeconomic populations. However, the multiplier can be rewritten as follows:

$$\rho' = \frac{q\_{\ge} \,^{A'} / q\_{\ge} \,^{C'}}{q\_{\ge} \,^{B'} / q\_{\ge} \,^{D'}}$$

This might seem a pure change of notation, but it leads us to search for other types of articles in the scientific literature that are easier to find, as social inequalities are an important issue for social and health policies. It is what permits one to use open data to estimate the risks of borrowers with a disease without having to guess what *ρ* might be. *qx A* /*qx C* is found in articles that study the impact of socioeconomic status on survival outcome in the case of a disease. *qx B* /*qx D* is found in articles that study the impact of socioeconomic status on mortality in the general population; it is important to find the same definition of socioeconomic status for the two ratios to be comparable (diplomas, occupations, salaries, etc.).

The relative risk multiplier is therefore computed with the definitions of high socioeconomic status and the pathology found in such articles. Let us remove primes when defining the relative risk multiplier that would be based on borrowers with a specific definition of the pathology related to the insurance contract and underwriting process:

$$\rho = \frac{q\_{\mathbf{x}}{}^A/q\_{\mathbf{x}}{}^C}{q\_{\mathbf{x}}{}^B/q\_{\mathbf{x}}{}^D}$$

Of note, D and D represent the same general population. The mortality rate of loan applicants with a medical condition is then derived from the definition of *ρ*:

$$q\_{\mathbf{x}}{}^A = q\_{\mathbf{x}}{}^B \times \frac{q\_{\mathbf{x}}{}^C}{q\_{\mathbf{x}}{}^D} \times \rho$$

In this formula,

We make the assumption *ρ* ≈ *ρ* ; we assume that the multiplier established with populations A , B , C , and D' is sufficient to rate contracts if it is a mortality adjustment. To validate this hypothesis, for a given pathology it can be useful to establish *ρ* for various countries to ensure that it is robust with similar values despite different environments. Additionally, if *ρ* were greater than 150% or less than 75%, for example, this assumption would need to be adjusted, for example by considering different categories of the pathology and different categories of high socio-professional status (at the cost of difficult internet searches).

In the case of breast cancer, we obtain a multiplier *ρ* of 83% for French data. For diabetes, we obtain a multiplier of 115% for France, 108% for Korea, 111% for the United States, and 123% for Scotland. It should be noted that these multipliers are in line with the most likely expectations expressed at the beginning of the section: multipliers slightly below 1 when early diagnosis in the higher socio-professional status makes a difference to the risks and slightly above 1 for frequent pathologies.

At this stage, it is therefore possible for insurers to establish mortality tables for an inclusive or partially inclusive approach for various conditions, bearing in mind that mortality risk is the most important guarantee in loan contracts.

### 3.2.3. Temporary Disability

As for mortality, for each disease the first step is to look for sources of data or studies that can be used to model the transition to temporary disability, or at least to work stoppage, and the continuation of temporary disability. This search is not easy, as the available data mentioning a specific disease in connection with work stoppage are often very macroscopic indicators. Luckily, in terms of modelling, mortality is the most important guarantee of loan insurance, thus approximations are more acceptable than when modelling mortality risk.

A solution is to rely on the American open data database made available by the National Center for Health Statistics (NHIS 2016). As this is an American database, the risks of temporary disability are lower and the prevalence of certain pathologies, such as diabetes, is higher. It is then a question of comparing the macroscopic indicators to appreciate the differences, modelling the risk in the United States, and adjusting it to the French macroscopic indicators within the framework of the understanding obtained when comparing the indicators.

This American database was compiled from a questionnaire sent each year between 2010 and 2018 to different, randomly selected individuals. It consists of 284,809 individuals, 55% of whom are women. A total of 3.3% of the women were diagnosed with breast cancer and the mean and median ages of these women are 68 and 70 years, respectively. The mean age at diagnosis is 56 years. It is possible to test different combinations of variables and thus select the most relevant ones.

With this database, we can model the annual probability *ix* to get into work stoppage of more than d days, and the average duration of work stoppage that follows *dx*. The risk is then the product of both (*ixdx*). Similar to mortality, we can define populations A, B, C, and D and their associated risks *ix A*, *ix B*, ... , *dx D*, and we get the formula that defines a risk multiplier *ρ*:

$$i\_{\boldsymbol{x}}{}^{A}d\_{\boldsymbol{x}}{}^{A} = i\_{\boldsymbol{x}}{}^{B}d\_{\boldsymbol{x}}{}^{B} \times \frac{i\_{\boldsymbol{x}}{}^{C}d\_{\boldsymbol{x}}{}^{C}}{i\_{\boldsymbol{x}}{}^{D}d\_{\boldsymbol{x}}{}^{D}} \times \rho$$

Here, A and B still represent French borrowers, but C and D represent American workers, with a specific pathology in the case of A and C. *ix Cdx C* and *ix Ddx D* are modelled based on the American database, *ix Bdx B* comes from knowledge of French insurers and we make the assumption that the risk multiplier is the same as for mortality: *ρ* ≈ *ρ* (a higher mortality is associated with a more severe disease and greater risk of disability).

For applications for breast cancer and diabetes we decided to model incidence (*ix C* and *ix D*) and duration (*dx C* and *dx D*) separately to facilitate the modelling in a Markovian approach.

The use of regressions improves the understanding of the relationship between temporary disability and risk factors. It is important to look for the most relevant explanatory variables for the condition under study.

*ix B* is an average of placeholder borrower tables calculated by the laboratory using the same approach as for mortality. An insurer can use its own experience table. For *dx B* we used the BCAC 2013 tables (Bagui 2013).

The approach we described here is limited in terms of risk factors compared to the risk factors that are typically found in articles related to mortality risks. As cases that lead to higher mortality risk likely also lead to higher disability risks, we should include mortality-related risk factors with some proportionality rules and interpolations on the risk factors already modelled for disability risk.

### 3.2.4. Permanent Disability

As with temporary disability, it is particularly difficult to find permanent disability statistics related to diseases but we found an approach for modelling temporary disability with open data.

As shown in Table 2, a French study (Cuerq et al. 2008) provides the probabilities *P*3 and *P*10 of becoming disabled 3 years and 10 years after diagnosis for various pathologies. It is possible to interpolate and extrapolate these data to model permanent disability risk as a function of time.

**Table 2.** Percentage of patients in permanent disability 3 and 10 years after the start of long-term sickness in France.

Let us denote the annual probability of entering permanent disability as *px*. Here, also, let us consider four populations, A, B, C, and D, and let us reuse the mortality risk multiplier *ρ* (that transposes relative risk to policy-holders) so that the following can be written:

$$p\_x^A = p\_x^B \times \frac{p\_x^C}{p\_x^D} \times \rho$$

where A and B represent policy-holders, C and D the population of workers, and A and C are identified as having a specific pathology.

Using constant values of *px C* between 0 and 3 years following diagnosis, noting that *q x C* is the annual death rate of people from population C when in permanent disability, and considering the average values of *px C*, *qx C* and *q x C* over the first three years postdiagnostic, *P*3 is the probability of entering permanent disability in the first, second, or third year without dying:

$$P\_3 = p\_\mathbf{x}^\mathbb{C} \left( 1 - q\_\mathbf{x}^{\prime \ C} \right)^2 + \left( 1 - q\_\mathbf{x}^{\prime \ C} - p\_\mathbf{x}^{\prime \ C} \right) p\_\mathbf{x}^\mathbb{C} \left( 1 - q\_\mathbf{x}^{\prime \ C} \right) + \left( 1 - q\_\mathbf{x}^{\prime \ C} - p\_\mathbf{x}^{\prime \ C} \right)^2 p\_\mathbf{x}^{\prime \ C}$$

Considering that *px C*, *qx C*, and *q x C* are small and that the product of more than two of these numbers is negligible compared to 1, the probability of ending up with permanent disability after 3 years is:

$$P\_3 \approx p\_x \, ^\complement \left( 3 - 3q' \, \_x \text{C} - 3q \, \_x \text{C} - 3p \, \_x \text{C} \right) \, .$$

As a result, *px C* ≈ *P*3/3 in the first three years following diagnosis and this estimation is slightly prudent. A simpler approach is to neglect mortality, as we see that it is slightly prudent with respect to *px C*. Neglecting mortality, the probability of remaining out of permanent disability is:

$$1 - P\_{\mathbb{B}} \approx \left(1 - p\_x^{\mathbb{C}}\right)^{\mathbb{B}}$$

As a result, *px C* ≈ 1 − (1 − *P*3) 3 is a refined approximation that is still slightly prudent overall from diagnosis to 3 years later.

Similarly, if we neglect mortality up to 10 years following diagnosis, the probability of remaining out of permanent disability starting from the percentage 1 − *P*3 of people 3 years after diagnosis is:

$$\frac{1 - P\_{10}}{1 - P\_{\heartsuit}} \approx \left(1 - p\_x^{\heartsuit}\right)^{\heartsuit}$$

As a result, *px C* ≈ 1 − 1−*P*10 1−*P*3 1/7 is an estimation that is slightly prudent from year 3 to 10 after diagnosis. We use it for any year following the first three years post-diagnosis, and this might be prudent because, as diseases stabilise, the risk of entering long-term disability is lower.

For a given pathology, we have just defined *px C* as two numbers that represent weighted means over age, gender, and disease risk factors. This is why we define *px D* similarly. Concretely, we multiply the probability of entering disability during at least one month (Kusnick-Joinville et al. 2006) by the probability to remain in disability and enter permanent disability based on the BCAC 2013 tables as performed by (Bagui 2013), averaged over working ages. *px B* is an average of placeholder borrower tables calculated by the laboratory using the same approach as mortality.

We detailed how to compute the annual probability of entering permanent disability, *px A*. We also need to define the annual probability of dying when in permanent disability, *q x A*. In real life, it is greater than the death rate when not in permanent disability, but to evaluate loan insurance premiums it does not have great importance and *qx A* already represents the weighted average of the two death rates, thus *qx A* can be used as well.

### 3.2.5. Premium Calculation

A Markov model is a simple approximation for assembling mortality and temporary or permanent disability risks3. A four-state Markov model is shown in Figure 2.

**Figure 2.** Diagram of a Markov model with four states.

Each arrow corresponds to a probability; the probability of death from one of the three other states, probability of going into temporary disability, of going into permanent disability before the end of the three years in temporary disability (after three years the transition is automatic), and the probability of returning to the work state. These probabilities would typically be functions of age, pathology characteristics, and length of time in the state or length of time in the diagnosis in a semi-Markovian framework.

In practice, it is difficult to find data for each arrow. A three-state Markov model shall thus be used, as described in Figure 3, with steps of one year.

**Figure 3.** Diagram of a Markov model with three states.

At t = 0, the proportion of borrowers is 1 in the "Work or Disability" state and 0 in the other two states. It is therefore represented by a vector (1; 0; 0). At each subsequent time step, the proportion of borrowers in each state is the product of, on the one hand, the transition matrix made up of the annual transition probabilities and, on the other hand, the vector of the proportion of borrowers in the states of the previous time step.

Note that the matrix is as follows (in Table 3) and that its numerical value is different for each date (t = 0, 1, etc. over the duration of the contract) depending on the age and the initial characteristics of the pathology.

**Table 3.** Transition matrix.

The pure premium is the sum of the amounts the insurer must pay to the borrowers in the year for each state multiplied by the proportion of borrowers in the state. The simplest example is that of a loan covered by a borrower's insurance policy for death only, as it does not depend on parameters related to disability. Each year *t* ≥ 0 the proportion *St* of patients is alive:

$$\mathcal{S}\_0 = 1\\\mathcal{S}\_t = \prod\_{\mathbf{x} = \mathbf{x}\_0}^{\mathbf{x} = \mathbf{x}\_0 + (t - 1)} 1 - q\_\mathbf{x}^{A\_t}$$

who have borrowed an amount *C* at age *x*0 for a term *T* with an interest rate *r*. In the case of death, the insurer must take over the payment of the remaining balance, or capital remaining due (CRD):

$$CRD\_t = \left[\frac{\mathcal{C}}{T} \left(1 + r\right)^T\right] (T - t)^2$$

assuming in this loan that the same amount, in square brackets, is to be repaid every year. Then, the pure premium *PA* is:

$$P^A = \sum\_{t=0}^T S\_t \, q\_{x\_0 + (t-1)} \, ^A \, CRD\_t$$

Note that in a two-state model (alive, dead) *St* is obtained by reading the proportion in the first state; in the three-state model (work-or-disability, disability, and death) *St* is the sum of the proportions of the first two states.

The inclusion of charges (e.g., distribution, claims handling, and reinsurance costs) would result in a commercial premium that is higher than the pure premium.

This pure premium *PA* for a borrower with a medical condition can be compared to the pure premium *PB* for a borrower under usual conditions by replacing A with B in the transition matrix. The additional premium *PA* − *PB* due to the pathology can be defined as a percentage *p* of the usual premium or a permillage *m*:

$$p = \frac{P^A - P^B}{P^B} 100$$

$$m = \left(P^A - P^B\right) 1000$$

In practice, not all expenses are proportional to the risk: there are fixed costs. The fixed costs do not increase with risks; therefore, surcharges in real life should be lower than the one we computed here.

Examples are given below for breast cancer and diabetes.

### **4. Application to Two Diseases: Breast Cancer and Diabetes**

We apply the modelling to breast cancer and type 1 diabetes.

### *4.1. Breast Cancer*

4.1.1. Mortality by Risk Factor

Dabakuyo et al. (2008) modelled mortality in women with active invasive breast cancer. This study was based on data from 3831 French women with breast cancer. The data were collected between 1982 and 1997, with a median follow-up time of 9 years, on women aged 19 to 99 at diagnosis (mean and median age of 61 years). They model risk ratios (HR) based on 2615 women and the following risk factors:

The impact of these risk factors was calculated with a Cox model using the following reference population:

We do not provide the results of the model here; the goal is to illustrate the type of information one finds to model loan insurance risks.

Such a model of mortality risk must be transposed from the general population to the insured population using a multiplier defined as *ρ* = *qx A* /*qx C qxB* /*qxD* .

To obtain *qx A* /*qx C* , a ratio of mortality risks of women with breast cancer but different socio-professional groups, we relied on the study by (Gentil-Brevet et al. 2008). This study is based on a sample of 1150 French women with invasive breast cancer diagnosed between 1995 and 1997 who were followed up with to 2006. It is important to carefully note the distribution of socio-professional status in the study in order then to compute *qx B* /*qx D* accordingly:

The general population being approximately represented by as many SPS+ as SPS−, we computed *qx C* as the average of the mortality rate of the two populations: *qx C* = - 1 − 88.2%1 5 + 1 − 77.4%1 5 /2 over 5 years and *qx C* = - 1 − 83.0%1 5 + 1 − 69.4%1 5 /2. This leads to *qx A qxC* = 66.4% from diagnosis to 5 years later and *qx A qxC* = 68.1 % from

diagnosis to 7 years later. Thus, we have two possible values. *B D*

To obtain *qx* /*qx* , a ratio of mortality risks depending on socio-professional status, we used mortality tables by sex and standard of living quintile (Blanpain 2018). As in the article by (Gentil-Brevet et al. 2008), we separated the socio-professional status into two groups (SPS+ and SPS−). We used the same weighting as the article for the population affected by breast cancer: 25% of women are under 50 years of age, approximately 50% are between 50 and 70 years of age and 25% are between 71 and 85 years of age. We obtained the mortality rates of 686 and 860 per 100,000 persons, respectively, for the SPS+ and SPS− populations: *qx B qxD* = 686 860 = 80%.

Alternatively, *qx B* /*qx D* can be obtained with mortality tables by sex and the socioprofessional status provided by (Blanpain 2016). It needs to be combined with the distribution of socio-professional status among females, such as those provided by (Maruani and Meron 2012) to make weighted average mortality rates that have similar socio-professional statutes, as in (Gentil-Brevet et al. 2008). We do not detail the calculation, as it is long and does not bring much to this article—we obtained *qx B qxD* = 85%.

The multiplier is then one of four different values depending on the choice of the numerator and denominator: *ρ* = 66.4 % 80 % = 83% or 68.1 % 80 % = 85.1% or 66.4 % 85 % = 78.1% or 68.1 % 85 % = 81.1%. We chose the first value, as it is close to the average of the four values and slightly higher—thus closer to more prudent premiums (additionally, while building these values we felt it had the best match of assumptions between the numerator and denominator).

### 4.1.2. Short-Term Disability

Using the American database, incidence and duration were modelled among women as follows:

$$d\_{\mathbf{x}} = \frac{1}{\mathbf{1} + \mathbf{e}^{-(\alpha + \beta \mathbf{x} + \gamma \cdot 1\_{\text{transt}} \cdot \text{concr})}}$$

where 1*breast cancer* is 1 or 0 depending on whether the person was diagnosed with breast cancer or not. We then have *ix D* = 1 1+e−(*α*+*βx*) and *ix C* = 1 1+e−(*α*+*γ*+*βx*) . Our regression found that *β* = −0.019 (more frequent work stoppage at younger ages) and *γ* = 0.53 (effect of cancer). *α* was adjusted to match the average frequency of work stoppage in France (Kusnick-Joinville et al. 2006);

$$d\_{\mathbf{x}} = \mathbf{e}^{-(\mathbf{a} + \beta \mathbf{x} + \gamma \mathbf{ 1}\_{\text{brcast\\_concor}})},$$

We then have *dx D* = e−(*α*+*βx*) and *dx C* = e−(*α*+*γ*+*βx*). Here, *β* = 0.018 (longer work stoppage with age) and *γ* = 0.73 (effect of cancer). *α* was adjusted to match the average frequency of work stoppage in France (Kusnick-Joinville et al. 2006).

### 4.1.3. Long-Term Disability

In Table 2, we do not have data for breast cancer, only for cancers ("tumours"). We still considered the latter, which is prudent, as breast cancer has a similar prognostic value to most cancers.

### 4.1.4. Results

Table 4 illustrates additional premiums, in the last column, depending on various breast cancer cases described in the other columns.

The additional commercial premium may be smaller than the additional pure premium due to fixed costs (and commercial decisions).

It is possible to pool some cases, as shown in Table 5, to provide loan access to a maximum number of patients. The weighting used to calculate the average excess premiums is the distribution of such cancers found in SEER4, a free application that provides very precise statistics on cancers in the United States.

Figures 4 and 5 show the additional disability pure premium (this time expressed as a percentage *p*) as a function of the age of the borrower at the time of application. Here, all severities of breast cancers have been pooled.

Of note, it would not have made much sense to express additional mortality premiums as a percentage *p*. This is because it would have been particularly high for young women (who have a particularly low mortality risk in the absence of breast cancer)—according to the model we developed, we observe that Tables 3 and 4 are roughly valid for all ages. Additionally, it would not have made much sense to express additional disability premiums as a permillage *m*. This is because long-term disability is limited at higher ages.

**Table 4.** Detailed additional pure mortality premiums.

**Table 5.** Pooled additional pure mortality premiums.

**Figure 5.** Additional pure premium *p* for long-term disability as a function of age.

### *4.2. Diabetes*

There is much open data about diabetes in France but little about type 1 diabetes. Thus, to take a more complex example and appreciate the limits of what open data brings, we present the modelling of type 1 diabetes.

### 4.2.1. Mortality by Type of Diabetes

We failed to identify French articles that finely modelled type 1 diabetes mortality, thus we searched for articles from neighbouring countries. We selected a Danish study that investigated the evolution of relative and absolute mortality rates in 4821 type 1 diabetic patients followed between 2002 and 2011 (Jørgensen et al. 2013). Of these, 54% were male, the median age was 44 years, and the median follow-up time was 18 years.

Standard-mortality ratios (SMR) are calculated relative to the general population by sex, current age, age at diagnosis, and the presence or not of kidney disease, as diabetic nephropathy is a complication that can progress to renal failure. They are presented in Figure 6.

Unsurprisingly, mortality is higher for those diagnosed at a young age and for those with kidney disease. Given the high risk associated with kidney disease, in what follows we decided to limit the mortality risk to other cases of type 1 diabetes.

To model the mortality rates of diabetics in France, we apply the Danish SMR to the mortality rates of the general French population. The model must then be transposed from the general population to the insured population using a multiplier defined as *ρ* = *qx A* /*qx C qxB* /*qxD* .

We did not find articles to produce this multiplier specifically for type 1 diabetes in France, so we decided to simply estimate it for diabetes (essentially type 2).

**Figure 6.** SMR by age at diagnosis and current age5.

To obtain *qx A* /*qx C* , a ratio of mortality risks with diabetics from different socioprofessional groups, we relied on a French article (Piffaretti et al. 2016). They studied the mortality of 7218 individuals with type 2 diabetes who answered a questionnaire in 2001 and 2007 and were followed until 2013. Figure 7 shows the relative risks according to their socio-professional category.

**Figure 7.** Relative risks according to socio-professional category with executives as reference (source: Piffaretti et al. 2016).

Figure 8 shows the distribution of socio-professional status. Based on the two figures, we estimate *qx A qxC* = 1 *qxC* /*qx A* where A are executives with diabetes and C is decomposed into various categories:

$$\frac{q\_{\chi^{C}}^{A^{\prime}}}{q\_{\chi^{C}}^{C^{\prime}}} = \frac{1}{1 \ast 9.6\% + 1.36 \ast 9.0\% + 1.05 \ast 11.7\% + 1.20 \ast 18.4\% + 1.36 \ast 26.7\% + 1.36 \ast 6.3\% + 1.14 \ast 18.3\%} = 82\%$$