Raouf Boucekkine

Giorgio Fabbri

Salvatore Federico

AND Fausto Gozzi

OPTIMAL LOCATION OF ECONOMIC ACTIVITY AND POPULATION DENSITY: THE ROLE OF THE SOCIAL WELFARE FUNCTION

Keywords: Spatiotemporal growth models, Benthamite vs Millian social welfare functions, imperfect altruism, diffusion, dynamic programming in infinite dimension Classification: R1, O4, C61

In this paper, we consider a spatiotemporal growth model where a social planner chooses the optimal location of economic activity across space by maximization of a spatiotemporal utilitarian social welfare function. Space and time are continuous, and capital law of motion is a parabolic partial differential diffusion equation. The production function is AK. We generalize previous work by considering a continuum of social welfare functions ranging from Benthamite to Millian functions. Using a dynamic programming method in infinite dimension, we can identify a closed-form solution to the induced HJB equation in infinite dimension and recover the optimal control for the original spatiotemporal optimal control problem. Optimal stationary spatial distributions are also obtained analytically. We prove that the Benthamite case is the unique case for which the optimal stationary detrended consumption spatial distribution is uniform. Interestingly enough, we also find that as the social welfare function gets closer to the Millian case, the optimal spatiotemporal dynamics amplify the typical neoclassical dilution population size effect, even in the long-run.

Introduction

There is a growing interest in the economic literature in the role of space in decisionmaking. While the economic analysis has incorporated the spatial dimension for quite a long time [START_REF] Von Thunen | Der Isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie[END_REF], it is the rise of the so-called New Economic Geography (NEG) which did induce such a recent boom in this literature stream (see [START_REF] Krugman | What's new about the New Economic Geography?[END_REF] for an insightful review of the NEG, and [START_REF] Fujita | Economics of agglomeration: Cities, industrial location, and regional growth[END_REF], for a master textbook in this area). An overwhelming part of the latter stream has been concerned with the identification and characterization of spatial externalities and the inherent agglomeration mechanisms. Concretely, researchers in the area target either first nature causes for agglomeration for given technology and demographic spatial distributions [START_REF] Krugman | First nature, second nature, and the metropolitan location[END_REF] or second nature causes through the identification of mechanisms (typically, economies of scale or spillovers) leading to agglomeration for example when labor is mobile [START_REF] Krugman | Increasing returns and economic geography[END_REF]. An overwhelming majority of this paper is purely static: individuals do not save over time and therefore no capital accumulation is considered. Following an earlier contributions in mathematical geography by [START_REF] Isard | Spatial Dynamics and Optimal Space-time Development[END_REF], [START_REF] Brito | The dynamics of growth and distribution in a spatially heterogeneous world[END_REF] is to our knowledge the first who attempted to insert space in otherwise standard neoclassical growth models, giving rise to a bunch of papers on optimal growth within spatiotemporal frames (in particular, [START_REF] Boucekkine | Bridging the gap between growth theory and the new economic geography : The spatial Ramsey model[END_REF], 2013[START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF][START_REF] Fabbri | Geographical structure and convergence: A note on geometry in spatial growth models[END_REF]. This paper builds on [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF]. We focus on the problem faced by a central planner who has to choose the optimal distribution of economic activity (say, investment and production) over space (here the unit circle for simplicity). We assume that technology is homogenous across space, which amounts to assuming that technological spillovers are quick enough for all the locations to use the same technology. In contrast, as it is most of the time the case, individuals are harder to move. We do not justify such an immobility but just assume it. In other words, the central planner can move capital but not people. Accordingly, our problem investigates a first nature cause of (potential) spatial externalities. Moreover, we generalize the social welfare function considered by [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF] by introducing a form of imperfect altruism in the social preferences. While only the Benthamite social welfare function is considered in the latter (that is, the planner consider the sum of utilities of all the individuals present in the economy), we consider a continuum of social welfare functions ranging from the Benthamite to the Millian form (i.e only the average consumer is considered). To our knowledge, this is the first time this is done within a spatial setting. Clearly, this would allow to handle a set of new population ethics and political economy issues, for example concerning regional particularism. Earlier papers using this continuum of social welfare functions in non-spatial settings have been typically devoted to study the normative implications of 1984 Parfit's population ethics theory in different economic contexts (see [START_REF] Nerlove | Population size: individual choice and social optima[END_REF][START_REF] Palivos | Optimal population size and endogenous growth[END_REF], and more recently, Boucekkine and Fabbri, 2013).

Importantly enough, the induced control problem to handle is infinite-dimensional as capital spatiotemporal dynamics follow a parabolic PDE in the tradition opened by Isard and Liossatos (1976). To solve it, we use a dynamic programming technique in infinite dimension, partially based on the work of [START_REF] Bensoussan | Representation and Control of Infinite Dimensional Systems[END_REF], first implemented in economics by [START_REF] Fabbri | Solving optimal growth models with vintage capital: The dynamic programming approach[END_REF] in a different context. More precisely, we rely on previous work by [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF] in a similar context. Essentially because the production function is assumed linear, it is possible to find a closed-form solution to the induced HJB equation in the associated functional space, and to identify at the end of the day the corresponding optimal spatiotemporal controls and dynamics. This paper shows that this method is robust to a an additional nonlinear deviation.

The paper is organized as follows. Sections 2 to 5 give all the mathematical steps needed to solve explicitly the optimal control problem. Section 6 describes the economic application and apply the preceding mathematical theory. Section 7 displays some complementary numerical exercises. Section 8 concludes.

The optimal control problem

Let (D, D, µ) be a countably generated measure space and let us consider the separable Hilbert space H := L 2 (D, µ; R), with it its usual norm | • | and scalar product •, • . We identify, by the usual Riesz identification, H with its topological dual. We shall identify the elements of H, which are equivalence classes of functions coinciding µ-a.e., with (one of) their representative functions. So, the pointwise relationship must be intended µ-a.e.. We consider the positive and strictly positive orthants of H, i.e. the sets

H + := {f ∈ H : f ≥ 0}, H ++ := {f ∈ H : f > 0}.
Finally, we denote

H 0 + := H + \ {f ≡ 0}.
In the following we set R + := [0, +∞).

Given this setting, we are concerned with the following optimal control problem in the space H. Let L : D(L) ⊆ H → H be a (possibly) unbounded linear operator and let N : H → H a linear bounded operator. Given x 0 ∈ H and a control function c ∈ L 1 loc (R + ; H), we consider the following abstract state equation in H:

(1)

x (t) = Lx(t) -N c(t), x(0) = x 0 .
We introduce the following assumption Assumption 2.1. L is a closed and densely defined operator generating a C 0 -semigroup in H.

We use the notaton e tL for the semigroup generated by L and, according to [START_REF] Bensoussan | Representation and Control of Infinite Dimensional Systems[END_REF], we define the mild solution to (1) as the function

(2) x(t) := e tL x 0 -t 0 e (t-s)L N c(s)ds.

To stress the dependence of x on x 0 , c, we write x x 0 ,c . ( 2) is also a weak solution to (1),

i.e.

(3)

x(t), ψ = x 0 , ψ + t 0 x(s), L * ψ ds - t 0 N c(s), ψ ds, ∀ψ ∈ D(L * ), ∀t ≥ 0,
where

L * : D(L * ) ⊆ H → H denotes the adjoint of L. Assumption 2.2. u : D × R + → R + is such that u(θ,
•) is a utility function for each θ ∈ D -i.e. increasing and concave.

Next, given u : D × R + → R + as above u(θ, •), we consider the functional

U(z) := D u(θ, z(θ))µ(dθ), z ∈ H + ,
and the functional on

L 1 loc (R + ; H + ) (4) c → J (x 0 ; c) := ∞ 0 e -ρt U(c(t))dt,
where ρ > 0 is a given discount factor. Notice that both U and J (x 0 ; •) inherit from u the concavity.

Remark 2.3. The choice of working only with nonnegative (or, equivalently, bounded from below) utility function is done here just for the sake of brevity, in order to avoid technical complications that would arise in treating the case of utility functions not bounded from below. The latter case might be considered and treated as well at the price of technical complications.

Assumptions 2.1-2.2 will be standing from now on.

Let ϕ ∈ H ++ . The aim is to maximize U when c ranges over

A ϕ ++ (x 0 ) = {c ∈ L 1 loc (R + ; H + ) : x x 0 ,c (t) ∈ H ϕ ++ for a.e. t ≥ 0},
where

H ϕ ++ is the open set H ϕ ++ := {x ∈ H : x, ϕ > 0} ⊂ H.
Notice that H 0 + ⊂ H ϕ ++ . Hence, if the semigroup e tL preserves H 0 + , i.e. it maps H 0 + into itself, then A ϕ ++ (x 0 ) is not empty when x 0 ∈ H 0 + , as the null control c ≡ 0 belongs to it. Moreover, if also N is positivity preserving, we have the following monotonicity property:

c 1 ≤ c 2 =⇒ x x 0 ,c 1 ≥ x x 0 ,c 2 .
In particular

(5) c 1 ≤ c 2 , c 2 ∈ A ϕ ++ (x 0 ) =⇒ c 1 ∈ A ϕ ++ (x 0 ).
So, given x 0 ∈ H ϕ ++ , we are interested in the following optimal control problem:

(P ϕ ) Maximize J (x 0 ; c) over the set A ϕ ++ (x 0 ),
whose value function is

V ϕ (x 0 ) := sup c∈A ϕ ++ (x 0 ) J (x 0 ; c).
Note that we cannot say ex ante that V ϕ is finite. Sufficient conditions for finiteness will be provided later.

Remark 2.4. Problem (P ϕ ) is the most natural one from the mathematical point of view when ϕ is chosen suitably, in the sense that it admits an explicit solution. However, we anticipate that the meaningful problem from the economic point of view would be the (more difficult) one:

(P + ) Maximize J (x 0 ; c) over the set A + (x 0 )
whose value function is

V + (x 0 ) := sup c∈A + (x 0 ) J (x 0 ; c).
where

A + (x 0 ) := {c ∈ L 1 loc (R + ; H + ) : x x 0 ,c (t) ∈ H 0 + for a.e. t ≥ 0}. Note that H 0 + ⊂ H ϕ ++ : hence, for x 0 ∈ H 0 + , A + (x 0 ) ⊆ A ϕ ++ (x 0 ) and, consequently V ϕ (x 0 ) ≥ V + (x 0 ). Moreover, if ĉ ∈ A ϕ ++ (x 0
) is optimal for (P ϕ ) and belongs to A + (x 0 ), then it is also clearly optimal for (P + ). In the illustration of the results, we will just test numerically, ex post, that the optimal control ĉ ∈ A ϕ ++ (x 0 ) also belongs to A + (x 0 ).

The HJB equation and the verification theorem

In this section we provide a verification theorem for the problem and, accordingly, the solution in a special case. The results will be used in the next section to treat our motivating economic application.

The Hamilton-Jacobi-Bellman (HJB) equation associated to the optimal control prob-

lem (P ϕ ) is (6) ρv(x) = Lx, ∇v(x) + H(∇v(x)), x ∈ H ϕ ++ ,
where

H(q) := sup z∈H + H CV (q; z), q ∈ H.
and

H CV (q; z) := U(z) -N z, q , z ∈ H + , q ∈ H.
Notice that the above supremum may not be finite in general. A sufficient condition for the finiteness is: U(z) ≤ C|z| α for some C > 0 and α ∈ (0, 1), N * preserving H ++ , and q ∈ H ++ . These will be used later.

Definition 3.1. We call classical solution to (6) (on

H ϕ ++ ) a function v ∈ C 1 (H ϕ ++ ; R) such that ∇v ∈ C(H ϕ ++ ; D(L * )) and such that 1 ρv(x) = x, L * ∇v(x) + H(∇v(x)), ∀x ∈ H ϕ ++ .
Now we turn to state our Verification Theorem. Typically, to prove such theorem for infinite horizon problems, a condition on the solution v computed on the admissible trajectories when t → +∞ is needed. This is exactly the analogous of the so-called transversality condition arising in the maximum principle approach. The condition that we shall use is

(7) lim t→+∞ e -ρt v(x x 0 ,c (t)) = 0, ∀c ∈ A ϕ ++ (x 0 ).
Theorem 3.2 (Verification). Let v be a classical solution to (6) in H ϕ ++ , let x 0 ∈ H ϕ ++ and let (7) hold. Then:

(i) v(x 0 ) ≥ V ϕ (x 0 );
(ii) if, moreover, there exists ĉ ∈ A ϕ ++ (x 0 ) such that (7) holds and

(8) N * ∇v(x x 0 ,ĉ (s)) ∈ D + U(ĉ(s)) for a.e. s ≥ 0,
where

D + U denotes the superdifferential of U, then v(x 0 ) = V ϕ (x 0 ) and ĉ is opti- mal for (P ϕ ) starting at x 0 , i.e. J (x 0 ; ĉ) = V ϕ (x 0 ). Proof. (i) Let c ∈ A ϕ ++ (x 0 )
. By chain's rule in infinite dimension, we have, for every t ≥ 0,

d dt [e -ρt v(x x 0 ,c (t)))] = e -ρt -ρv(x x 0 ,c (t))+ x x 0 ,c (t), L * ∇v(x x 0 ,c (t)) -N c(t), ∇v(x x 0 ,c (t)) .
Now we add and subtract e -ρt U(c(t)) to the right hand side, use the fact that v solves HJB, and integrate over [0, t]. We get, for every t ≥ 0,

e -ρt v(x x 0 ,c (t)) + t 0 e -ρs U(c(s))ds = v(x 0 ) + t 0 e -ρs -H(∇v(x x 0 ,c (s)) + H CV (∇v(x x 0 ,c (s); c(s)) ds,
Observe that, since U is concave (and hence sublinear from above), and c(•) ∈ L 1 loc (R + ; H + ) then both sides of the above inequality are finite for every t ≥ 0. Now, rearranging the terms and taking into account the definition of H, we get, for every t ≥ 0, Hence, passing (9) to the lim t→+∞ and using (7), we conclude

(9) v(x 0 ) ≥ e -ρt v(x x 0 ,c (t)) + t 0 e -
v(x 0 ) ≥ J (x 0 ; c)
Then, by definition of V ϕ and since c ∈ A ϕ ++ (x 0 ) was arbitrary, we immediately get the claim.

(ii) Notice that, by concavity of U, (8) is equivalent to

(11) ĉ(s) ∈ argmax z∈H ϕ ++ {U(z) -N z, ∇v(x x 0 ,ĉ (s)) }, ∀s ≥ 0,
the usual closed loop condition for optimality. Hence, for c = ĉ we have equality in (9).

Hence, passing to the lim t→+∞ and using ( 7) and (10), we get the equality v(x 0 ) = J (x 0 ; ĉ)

Since J (x 0 ; ĉ) ≤ V ϕ (x 0 ), combining with part (i), the claim follows.

A solution of the HJB equation

In this section we further specify the model to deal with an explicit solution. We consider the following assumption. (ii) ρ > λ 0 (1 -γ).

(iii) [N z](θ) = n(θ)z(θ), where n ∈ L ∞ (D, D, µ; (0, +∞)). (iv) u(θ, ξ) = ξ 1-γ 1 -γ f (θ), where γ ∈ (0, 1) ∪ (1, ∞) and f ∈ L ∞ (D, D, µ; R + ). (v) D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) < ∞, D f (θ) n(θ)b 0 (θ) p/γ µ(dθ) < ∞.
Notice that Assumptions 4.1(iv) implies Assumption 2.2. Also notice that, in general, b 0 may be not unique. A sufficient condition for the uniqueness of b 0 is that L * is a diagonal operator with respect to a given orthonormal basis in H. This will be the case in Section 6.

Proposition 4.2. Let Assumption 4.1 hold. Then

(12) v(x) = α x, b 0 1-γ 1 -γ , x ∈ H b 0 ++ ,
where

(13) α = γ γ D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) ρ -λ 0 (1 -γ) γ is a classical solution to (6) on H b 0 ++ . Proof. Notice first that U is Fréchet differentiable in H b 0 ++ and [∇U(z)](θ) = f (θ)z(θ) -γ , z ∈ H b 0 ++ .
Hence, by straightforward computations

H(q) = γ 1 -γ D f (θ) 1 γ (n(θ)q(θ)) γ-1 γ µ(dθ), q ∈ H ++ , with optimizer (14) ẑ(q)(θ) = argmax z∈H + U(z) -N z, q = f (θ) n(θ)q(θ) 1 γ . Moreover ∇v(x) = α x, b 0 -γ b 0 .
Plugging these expression into ( 6) and dividing all terms by x, b 0 1-γ /(1 -γ) we get the

following algebraic equation in α ∈ R ρα = λ 0 (1 -γ)α + γα γ-1 γ D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ),
which has a unique positive solution provided by (13).

The solution of the control problem

In order to produce an optimal control starting at x 0 ∈ H b 0 ++ , we study the closed loop equation associated to the solution v and to the candidate optimal feedback map ( 14) under Assumption 4.1:

(15) x (t) = Lx(t) -N Φx(t), x(0) = x 0 ,
where Φ : H → H is the bounded linear positive operator

(16) [Φz](θ) = f (θ) αn(θ)b 0 (θ) 1 γ z, b 0 .
This linear equation admits a unique mild solution x, which is also a weak solution. In particular, testing against b 0 ∈ D(L * ) and taking into account the definition of α, it holds

d dt x(t), b 0 = x(t), L * b 0 -N Φx(t), b 0 = λ 0 -α -1 γ D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) x(t), b 0 .
Hence, taking int account the definition of α, providing

(17) α -1 γ D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) = ρ -λ 0 (1 -γ) γ , we get d dt x(t), b 0 = g x(t), b 0 , i.e.
x(t), b 0 = x 0 , b 0 e gt , t ≥ 0, where (18) g := λ 0 -ρ γ .

By Assumption 4.1(i), we get

(19) x(t) ∈ H b 0 ++ , ∀t ≥ 0.
Theorem 5.1. Let Assumption 4.1 hold, let x 0 ∈ H b 0 ++ , and let v be the solution to (6) given as in Proposition 4.2. The control

ĉ(t, θ) := [Φx(t)](θ) = f (θ) αn(θ)b 0 (θ) 1 γ x 0 , b 0 e gt , belongs to A b 0 ++ (x 0
) and is optimal for (P b 0 ) starting at x 0 . Moreover, v(x 0 ) = V (x 0 ).

Proof. We want to apply Theorem 3.2. First, notice that, using the concept of weak solution (3) with ψ = b 0 , we have

d dt x x 0 ,c (t), b 0 = λ 0 x x 0 ,c (t), b 0 -N c(s), b 0 ∀c ∈ A b 0 ++ (x 0 ), i.e. x x 0 ,c (t), b 0 = x 0 , b 0 e λ 0 t - t 0 e λ 0 (t-s) N c(s), b 0 ∀c ∈ A b 0 ++ (x 0 ).
Since both n and b 0 are nonnegative (Assumption 4.1(i) and (iii)), we have

0 ≤ x x 0 ,c (t), b 0 ≤ x 0 , b 0 e λ 0 t ∀c ∈ A b 0 ++ (x 0 ). Hence, 0 ≤ e -ρt v(x x 0 ,c (t)) = αe -ρt x x 0 ,c (t), b 0 1-γ 1 -γ ≤ α 1 -γ x 0 , b 0 e -(ρ-λ 0 (1-γ))t , ∀c ∈ A b 0 ++ (x 0 ).
Therefore, using Assumption 4.1(ii), we see that ( 7) hold. This allow to apply part (i) of Theorem 3.2.

Let us now turn to part (ii) of Theorem 3.2, i.e. to prove the optimality of ĉ. By construction x x 0 ,ĉ = x. Hence, by ( 19) and Assumption 4.1(v), we obtain ĉ ∈ A b 0 ++ (x 0 ). Moreover, again by construction, ĉ verifies the optimality condition (11). So, Theorem 3.2(ii) applies and the proof is complete. Assumption 5.2. L * : H → H admits an orthonormal basis of eigenvectors, i.e. there exists an orthomormal basis {b n } n∈N of H such that

L * b n = λ n b n ∀n ∈ N. Moreover, (20) λ k < g, ∀k ∈ N \ {0}.
Theorem 5.3. Let Assumption 4.1 and 5.2 hold, and let x 0 ∈ H b 0 ++ . Then, setting xg := e -gt x(t), we have the convergence in H

xg (t) → x 0 , b 0 ∞ k=0 q k b k , as t → ∞,
where

q 0 := 1, q k := ζ k λ k -g ,
where

ζ k := f (•) αn(•)b 0 (•) 1 γ , b k
Moreover, the speed of convergence is exponential of order s := sup k∈N\{0} {λ k -g}.

Proof. The equation for xg is

(21) x g (t) = (L -g)x g (t) -N Φx g (t). Let x(k) g (t) = xg (t), b k , so that, by Fourier series expansion x(k) g (t) = ∞ k=0 x(k) g (t)b k .
From the fact that xg is also a weak solution to (21), we get the equations

d dt x(k) g (t) = (λ k -g)x (k) g (t) -x 0 , b 0 ζ k , k ∈ N.
We already know that, by definition of g,

x(0) g (t) ≡ x 0 , b 0 .
For the others term, when k ∈ N \ {0}, we have

x(k) g (t) = x 0 , b k e (λ k -g)t + x 0 , b 0 ζ k 1 -e (λ k -g)t λ k -g .
The claim follows by (20).

The economic problem

We apply the results of the previous section to an economic problem by taking the following specifications of the general framework above:

(a) D = S 1 := {ξ ∈ R 2 : |ξ| = 1} ∼ = R/Z; S 1 is topologically identified with [0, 2π] ⊂ R
when the extreme of the latter interval are identified; similarly, functions on S 1 are identified with 2π-perodic functions on R;

(b) µ = Hausdorff measure on the Borel σ-algebra of S 1 ; i.e., through the identification

S 1 ∼ = R/Z, µ =Lebesgue measure on (0, 2π); (c) L = σ d 2 dθ 2 + A,
where σ, A > 0 are constant; accordingly, the integration with respect to this measure will be simply denoted by dθ;

(d) f (θ) = n(θ) β , with β ≥ 0.
In this case Assumption 2.1 is verified L = L * and this operator admits a spectral decomposition on the separable Hilbert space L 2 (S 1 , dθ; R). Moreover, there is a doubly indexed sequence of eigenvectors and eigenvalues of L = L * which is explicit. Indeed,

b 0 = (2π) -1/2 1 S 1 , λ 0 = A, and b (1) k (θ) = π -1/2 sin(kθ), b (2) k (θ) = π -1/2 cos(kθ), λ k = A -σk 2 , ∀θ ∈ S 1 , ∀k ∈ N \ {0}.
This formal frame fits quite well optimal growth models with AK production function, one-dimensional geography and capital diffusion as in Boucekkine et al. (2013[START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF]. Indeed, calling K(t, •) = x(t) according to the usual notation for capital in AK models, formally the problem becomes:

V (K 0 ) := sup A(K 0 ) ∞ 0 e -ρt S 1 c(t, θ) 1-γ 1 -γ n(θ) β dθ dt,
where ρ > 0, γ ∈ (0, 1), β ≥ 0,

A(K 0 ) := c ∈ L 1 loc ([0, +∞); L 2 (S 1 , dθ; R + )) : K(t, θ) ≥ 0 and K(t, θ) ≡ 0 for a.e. (t, θ) ∈ R + × S 1 ,
under the state PDE-constraint ( 22)

     ∂K ∂t (t, θ) = σ ∂ 2 K ∂θ 2 (t, θ) + AK(t, θ) -n(θ)c(t, θ), (t, θ) ∈ R + × S 1 , K(0, θ) = K 0 (θ), θ ∈ S 1 ,
A few comments on the economic problem are needed at this stage. First of all, let us comment on the objective function,

∞ 0 e -ρt S 1 c(t,θ) 1-γ
1-γ n(θ) β dθ dt, which is indeed the social welfare function to be maximized by the central planner. This generalizes the social welfare function adopted by [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], which is itself an extension of Boucekkine et al. (2013). In the latter, n(θ) ≡ 1 and in the former β = 1. In this case, the planner sums all the utilities of all the individuals in location θ at time t. This corresponds to total utilitarianism or equivalently to the Benthamite case. When β = 0, one gets average utilitarianism or the Millian social welfare function: the planner do not consider all the individuals but only the average. In between, β ∈ (0 , 1), we get a continuum of intermediate configurations, we shall refer to this case as imperfect altruism: roughly speaking, as β increases, the planner takes more closely into account the welfare of his population.2 Two other parameters of the social welfare functions are interesting to comment briefly on. Parameter ρ may be interpreted as the time discounting factor of the planner: the bigger ρ, the more weight will be given to the present (or to the current generations) by the planner. More interestingly, parameter γ has a double role in our spatiotemporal frame: it captures on one hand the cost of intertemporal consumption substitution faced by individuals, as in the standard non-spatial settings, but it also indicates the degree of aversion to inequality of the planner on the other. Thus one has therefore to expect that as γ increases, the less unequal will be the consumption distribution over space. We shall check this in the last section devoted to numerical illustration.

Finally the state equation ( 22) gives the spatiotemporal dynamics of capital. σ ∂ 2 K ∂θ 2 (t, θ) is the diffusion term of the equation, it depicts the dynamics of capital through space with σ > 0 the diffusion speed. The rest of terms are obvious. In particular n(θ)c(t, θ) is the so-called dilution effect of demographic size on capital accumulation. Notice that in this economy goods are produced and consumed locally, only capital moves cross locations (or more precisely, capital is moved by the planner according to the PDE above in line with [START_REF] Isard | Spatial Dynamics and Optimal Space-time Development[END_REF].

Next, assuming that (A1)

S 1 n(θ) β+γ-1 γ dθ < ∞, S 1 n(θ) 2(β-1) γ dθ < ∞, (A2) ρ > A(1 -γ),
all the assumptions of the previous sections are verified. By expliciting the result of Theorem 5.1 with these specifications, we get the following.

Theorem 6.1. Let (A1)-(A2) hold and assume that the solution K to the linear integro-

PDE (23)      ∂K ∂t (t, θ) = σ ∂ 2 K ∂θ 2 (t, θ) + AK(t, θ) -α -1/γ (2π) -1/2 S 1 K(t, θ)dθ n(θ) β+γ-1 γ , (t, θ) ∈ R + × S 1 , K(0, θ) = K 0 (θ), θ ∈ S 1 ,
is such that K ≥ 0 and K = 0. Then (i) the value function is explicitly given by

V (K 0 ) = γ ρ -λ 0 (1 -σ) γ S 1 n(θ) β+γ-1 γ dθ γ (2π) -1/2 1 -γ S 1 K 0 (θ)dθ) 1-γ ; (ii) the control ĉ(t, θ) = α -1/γ (2π) -1/2 S 1 K 0 (θ)dθ n(θ) β-1 γ e gt ,
where g := A-ρ γ is the optimal growth rate of the economy, is optimal and the corresponding optimal capital is K;

(iii) if moreover, A -σ < g and ∞ k=1 (ζ (1) k ) 2 + (ζ (2) k ) 2 < ∞, where ζ (2) k := ρ -A(1 -γ) γ S 1 n(θ) β+γ-1 γ dθ -1 S 1 n(θ) β-1 γ cos(kθ)dθ ζ (1) k := ρ -A(1 -γ) γ S 1 n(θ) β+γ-1 γ dθ -1 S 1 n(θ)
β-1 γ sin(kθ)dθ then the detrended optimal capital Kg (t, •) := e -gt K(t, θ) converges in L 2 (S 1 , dθ; R + ) to the function

K∞ g (θ) := S 1 K 0 (θ)dθ 1 + ∞ k=1 ζ (1) k sin(kθ) + ζ (2) k cos(kθ)
as e -(g-A+σ)t .

A few comments are in order here. Of course, since we are dealing with an AK model, growth is endogenous and all variables grow at exponential rates. The growth rates are not location-dependent. This is not surprising since technology (though parameter A)

is the same everywhere, and so are individual preferences. Second, as in [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], we are able to identify the optimal stationary spatial distributions in closed-form and the corresponding convergence speeds, which certainly speaks about the flexibility of the analytical method developed. Third, and more importantly, β plays a central role in the shape of these distributions, in particular for consumption per capita.

Indeed, this distribution is uniform if and only if the social welfare function is Benthamite (β = 1). This is an important property which was not of course reachable in the previous contributions to this area as they all rely on total utilitarianism. The next section goes deeper into this question via numerical exploration.

Some numerical illustrations

We shall now explore numerically some of the salient properties of the optimal longterm distributions theoretically uncovered above. We shall in particular, highlight the implications of two key parameters: the aversion to inequality parameter γ, and the altruism parameter β. To make the exposition more focused on the novelties brought by our generalization of social welfare, we shall concentrate on consumption, which is the control variable and the sole measure of welfare in our model. It can be shown that just as in [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], long-term total and per-capita capital, production and investments are lower in the more populated areas. This has a simple explanation: to guarantee a reasonable level of consumption to everybody, the planner needs to maintain an higher level of aggregate consumption in more densely populated areas leading to lower investment at the same locations. It follows that in this model where the optimal spatiotemporal decisions are only driven by demographic heterogeneity across space, and not by technology discrepancy, the key variable is consumption and the determinants behind (notably, aversion to inequality and altruism). 3

To start our numerical exercises, we need a benchmark (reasonable) calibration. Concretely, we choose: ρ = 0.04, γ between 0.6 and 0.9, β between 0 and 1 and A = 0.08. The value of ρ (the discount rate parameter) is chosen consistently with the data from [START_REF] Lopez | The social discount rate : estimates for nine Latin American countries[END_REF]. Parameter γ, also measuring the inverse of the intertemporal substitution with the CRRA specification of preferences, can be calibrated using relative risk aversion consistently with the values found in individual choice experiments (see for example [START_REF] Cubitt | Discovered preferences and the experimental evidence of violations of expected utility theory[END_REF][START_REF] Tversky | Advances in prospect theory: cumulative representation of uncertainty[END_REF]Kahneman, 1992). The value of A is set to generate longterm growth rates of 4% to 6%, which are consistent with those observed in developing countries (see e.g. World Bank Group, 2018). As to the exogenous density distribution, we assume that we have a demographic center as depicted in Figure 1:

Because of the neoclassical dilution effect, the computed stationary distributions will always deliver that optimal consumption per capita goes down with population size over space except in the Benthamite case, which therefore features a case of egalitarianism.

More interesting insights can be gained from the numerical exercises.

The implications of the aversion to inequality parameter, γ. Here we fix the altruism parameter β to 0, that is we stick to the Millian social welfare function. Needless to say, since we are concerned with consumption per capital, the Benthamite case, β = 1, as studied in Boucekkine et al. (2013[START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF] is irrelevant as by the theorem just above, this 3 We could have followed [START_REF] Allen | Trade and the Topography of the Spatial Economy[END_REF] and assume that the size of population drives productivity at any location. In other words, we could have assumed population-based productivity heterogeneity. We don't to that to single out the pure population effect in our generalized context.

Figure 1 is a case where consumption is uniform across space, independently of γ. Figure 2 delivers the stationary distributions of consumption per capita detrended for γ = 0.6, 0.75, 0.9.

First, as γ increases, consumption is in average bigger. This is due to the fact that, as outlined in the previous section, γ also measures the cost of intertemporal substitution in consumption: when it's high, savings and investment are lower and consumption is higher.

But in our spatiotemporal frame, γ also measures the planner's aversion to inequality. One can check that as γ rises, consumption distributions becomes less unequal. A rough and quick verification is to compute the ratio between the periphery plateau consumption value and its value at the trough of the distribution: it decreases steadily. 4

The implications of the altruism parameter, β. Figure 3 reports the stationary spatial distribution of detrended consumption capita. For this exercise, we fix γ = 0.8, and we play on the the altruism parameter. Precisely, we compute the distributions for the Benthamite (β = 1), Millian (β = 0) and an imperfect altruism (β = 0.5) cases. Not surprisingly, the optimal stationary distribution of consumption per capita is uniform.

More interestingly, the Millian case (β = 0) delivers by far the largest spatial inequality in consumption while imperfect altruism displays intermediate results from this point of view. Indeed, the ratio between the periphery plateau consumption value and its value at the trough of the distribution is larger than 2, which exceeds the ratio between 4 In our calibrated example, this ratio move from 2.75 at γ = 0.6, to 1.85 when γ = 0.75 and finally it is equal to 0.75 for γ = 0.9. Empirical standard deviations deliver the same ranking of course. the corresponding population sizes. That is to say at low enough values of the altruism parameter β, the optimal spatiotemporal dynamics amplify the neoclassical dilution effect, even in the long-run! Including this parameter into the analysis seems therefore highly interesting from the normative point of view, and probably also from a more positive perspective in contexts of regional particularism.

  U(c(s))ds = ∞ 0 e -ρs U(c(s))ds =: J (x 0 ; c(•)).

  There exists an eigenvector b 0 ∈ H ++ for L * : D(L * ) ⊆ H → H with eigenvalue λ 0 ∈ R. Without loss of generality we assume that |b 0 | H = 1.

Figure 2

 2 Figure 2

  Figure 3

This equality, in particular, implies that H(∇v(x)) is finite for every x ∈ H ϕ ++ .

A natural extension is to consider that β is a function of θ, which is a direct way to model regional or local particularism. We abstract away from this in this paper.

been able to solve in closed-form for the optimal controls and for the corresponding spatial stationary distributions as well, either in the short or in the long-run. We have found several interesting results. In particular, we prove that the Benthamite case is the unique one for which the optimal stationary detrended consumption spatial distribution is uniform. Interestingly enough, we also find that as the social welfare function gets closer to the Millian case, the optimal spatiotemporal dynamics amplify the typical neoclassical dilution population size effect, even in the long-run.

Further extensions with space-dependent altruism featuring regional particularism are worth exploring. We could have also studied much more in detail the interplay between altruism and aversion to inequality (that is between β and γ). We leave it for future work.

Email address: raouf.boucekkine@univ-amu.fr G. Fabbri, Univ. Grenoble Alpes, CNRS, INRA, Grenoble INP, GAEL -CS 40700 -38058 Grenoble CEDEH 9, France.

Email address: giorgio.fabbri@univ-grenoble-alpes.fr S. Federico, Università degli Studi di Siena, Dipartimento di Economia Politica e Statistica. Piazza San Francesco 7-8, 53100 Siena , Italy.

Email address: salvatore.federico@unisi.it F. Gozzi, Dipartimento di Economia e Finanza, Libera Università degli Studi Sociali Guido Carli, Roma Email address: fgozzi@luiss.it


Quentin Guibert
email: guibert@ceremade.dauphine.fr.

Stéphane Loisel
email: stephane.loisel@univ-lyon1.fr.

Olivier Lopez
email: olivier.lopez0@upmc.fr.

Pierrick Piette
email: pierrick.piette@gmail.com.

Bridging the Li-Carter's gap: a locally coherent mortality forecast approach

Keywords: JEL Classification: C18, C32, C53, J11 Coherent mortality forecasting, Multi-population, Basis risk, Vector Autoregression, Elastic-Net

Countries with common features in terms of social, economic and health systems generally have mortality trends which evolve in a similar manner. Drawing on this, many multi-population models are built on a coherence assumption which inhibits the divergence of mortality rates between two populations, or more, on the long run. However, this assumption may prove to be too strong in a general context, especially when it is imposed to a large collection of countries. We also note that the coherence hypothesis significantly reduces the spectrum of achievable mortality dispersion forecasts for a collection of populations when comparing to the historical observations. This may distort the longevity risk assessment of an insurer. In this paper, we propose a new model to forecast multiple populations assuming that the long-run coherent principle is verified by subgroups of countries that we call the "locally coherence" property. Thus, our specification is built on a trade-off between the Lee-Carter's diversification and Li-Lee's concentration features and allows to fit the model to a large number of populations simultaneously. A penalized vector autoregressive (VAR) model, based on the elastic-net regularization, is considered for modeling the dynamics of common trends between subgroups. Furthermore, we apply our methodology on 32 European populations mortality data and discuss the behavior of our model in terms of simulated mortality dispersion. Within the Solvency II directive, we quantify the impact on the longevity risk solvency capital requirement of an insurer for a simplified pension product. Finally, we extend our model by allowing populations to switch from one coherence group to another. We then analyze its incidence on longevity hedges basis risk assessment.

Introduction

Around the world, the life expectancy of many countries has significantly risen over the past decades. This general augmentation implicitly suggests that common drivers are shared between populations such as lifestyle factors, the level of socioeconomic inequalities or the quality of the health system. However, this improvement is not equally shared among all groups of populations, precisely because the driving features are heterogeneous at an individual level. The links and differences between groups are important to understand for demographic and actuarial studies, since they may cause issues in managing social security system, pension funds or in defining longevity risk hedging solution. This explains the growing interest in mortality forecasting of multiple populations, especially to understand the effect of sub-groups on an overall population [START_REF] Danesi | Forecasting mortality in subpopulations using Lee-Carter type models: A comparison[END_REF][START_REF] Bergeron-Boucher | Coherent Modeling and Forecasting of Mortality Patterns for Subpopulations Using Multiway Analysis of Compositions: An Application to Canadian Provinces and Territories[END_REF][START_REF] Cairns | Modelling Socio-Economic Differences in Mortality Using a New Affluence Index[END_REF] or to enhance the projections robustness for populations with not well-known trends, see e.g. [START_REF] Li | Coherent Mortality Forecasting for Less Developed Countries[END_REF].

Since the introduction of the augmented common factor approach by [START_REF] Li | Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method[END_REF], many multi-population models have been developed. Most of them are extensions of the [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF] model or the CDB models [START_REF] Cairns | A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration[END_REF], and rely on a so-called coherence principle, see [START_REF] Villegas | A comparive study of two-population models for the assessment of basis risk in longevity hedges[END_REF] for a review. This concept, introduced by Li and [START_REF] Li | Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method[END_REF], consists in imposing that the mortality rates of two populations (or more) do not diverge in the long run. In such a specification, common age and period effects are estimated for a group of mortality data, the population specific trend being therefore modeled via mean-reverting process. This principle is also introduced in the so-called GRAVITY model [START_REF] Dowd | A Gravity Model of Mortality Rates for Two Related Populations[END_REF] or in the Common Age Effect (CAE) model [START_REF] Kleinow | A common age effect model for the mortality of multiple populations[END_REF][START_REF] Enchev | Multi-population mortality models: fitting, forecasting and comparisons[END_REF]. Although this assumption may be relevant for close populations, [START_REF] Li | Semicoherent Multipopulation Mortality Modeling: The Impact on Longevity Risk Securitization[END_REF] have pointed out that it is often only suitable for specific populations and over limited time windows. Hence, this approach is more likely to become unsatisfactory when it is applied to a large collection of populations. Furthermore, by inhibiting the divergence between populations, an irrelevant use of the mortality convergence principle may thereby distort the projections spectrum of mortality dispersion among the populations. In that context, extending the current mortality forecasting models for a large number of populations is challenging as it involves to cluster together similar countries and separately forecasting those with diverging mortality dynamics, as noted by [START_REF] Richman | A Neural Network Extension of the Lee-Carter Model to Multiple Populations[END_REF].

Motivated by the need of applying stochastic mortality models to a large collection of populations, our aim in this paper is to find a trade-off between the full coherence principle between populations and the noncoherence situation where single stochastic mortality models are considered independently. These two situations can be considered as border cases of the inter-group dependence possibilities, and are both undesirable, in a general context, for life insurers which are assisted by stochastic mortality multi-population models in their longevity risk solvency capital assessment. Indeed, a fully coherent model can overestimate the capital requirement through the concentration between the populations it imposes, whereas the use of independent single-population models may induce an underestimation of this capital. Note also that an inappropriate use of coherent mortality model for populations with heterogeneous mortality profiles or vice-versa can significantly affect the effectiveness of reinsurance management strategies, see e.g. [START_REF] Yang | Understanding Patterns of Mortality Homogeneity and Heterogeneity Across Countries and Their Role in Modeling Mortality Dynamics and Hedging Longevity Risk[END_REF]. Similar problem appears also for the pricing and the hedging of longevity risk transfer solutions where an accurate risk assessment is required, especially for the corresponding basis risk.

To overcome this issue, [START_REF] Li | Semicoherent Multipopulation Mortality Modeling: The Impact on Longevity Risk Securitization[END_REF] introduce the concept of semicoherence which contains the mortality differentials into a tolerance corridor. In other words, the two population mortality dynamics can diverge over certain periods of time, but not permanently. Their semicoherent mortality projection is based on a threshold vector autoregressive process, which can be seen as a vector autoregressive (VAR) model characterized by a parameter set and autoregressive order which change according to the time series regime. [START_REF] Zhou | Changes of Relation in Multi-Population Mortality Dependence: An Application of Threshold VECM[END_REF] recently improve this approach by introducing a threshold vector error correction model and taking into account long-term equilibrium between time series. However, these models are developed for only two populations and their VAR structure contained many parameters, which is a serious limitation in the large-scale mortality forecasting framework.

In this paper, we propose an alternative approach, named "local coherence", which allows us to model simultaneously a large collection of populations by assuming the mortality coherence principle is satisfied for several subgroups of populations. Although our approach is mainly based on existing models, such as the Lee-Carter or the Li-Lee models, it intends to gain in flexibility by offering a trade-off between diverging mortality forecasts, obtained with a single stochastic mortality model, and fully coherence forecasts. In that way, our goal is similar to the [START_REF] Li | Semicoherent Multipopulation Mortality Modeling: The Impact on Longevity Risk Securitization[END_REF] semicoherence one, i.e. offering a compromise between the fully coherence and the independence situations, even if the approach differs. Notably, in our case the size of the populations collection can be large and the proposed model includes the two border cases.

Within a locally coherent group, the dynamics of the mortality is based on a [START_REF] Li | Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method[END_REF] model. To capture both long-term relationships and short-term interactions between groups, a VAR process models the improvements in the common mortality trends of each group. A similar specification for a small group of populations is proposed by several authors in the literature, see [START_REF] Zhou | Modeling Period Effects in Multi-Population Mortality Models: Applications to Solvency II[END_REF], [START_REF] Kleinow | A common age effect model for the mortality of multiple populations[END_REF], and [START_REF] Enchev | Multi-population mortality models: fitting, forecasting and comparisons[END_REF] among others. However, a VAR model may suffer of over-fitting issue when the size of the group increases, since historical series are relatively short and strongly correlated. For our large-scale general context, we then use a VAR-ENET approach. Based on an elastic-net regularization technique, this VAR estimation is relevant with high dimensional time series and offers good in-sample and out-of-sample performances in a mortality forecasting framework [START_REF] Guibert | Forecasting mortality rate improvements with a highdimensional VAR[END_REF].

For illustrative purpose, we estimate our locally coherent model on a collection of 16 European countries, by distinguishing the gender. We propose two approaches to cluster these populations in coherent groups, regarding to the historical mortality rates data or expert judgments. Our results highlight a better control of the mortality rates dispersion forecasting among the populations. In addition, we show the importance of having an accurate level of variability when considering the solvency capital requirement of an European life insurer exposed to longevity risks from multiple populations.

Furthermore, we extend our model by allowing the populations to switch dynamically from one group of coherence to another. Thus, the coherence may be specified locally not only in the spatial dimension, i.e. according to the populations, but also in the temporal dimension. Such switches can be caused by changes in socio-economic features driving the mortality dynamics. We underpin the importance this extension by analyzing the impact of a specific switch on a Longevity Divergence Index Value, similar to the one that the Kortis bond is based on.

The remainder of this paper is organized as follows. In Section 2, we recall the usual Lee-Carter and Li-Lee models and show the mortality dispersion forecasting issue. Section 3 introduces our locally coherent model and shows how it fills the gap between a fully coherent and an independent situations. We also present the VAR-ENET estimation method retained in the paper. Section 4 presents two clustering approaches for identifying coherent groups: one based on expert judgments and a data-driven approach. The results obtained on the groups composed of 32 European populations is discussed, and we demonstrate the interest for the risk-based capital requirement of a life insurer in presence of multi-population exposures. Section 5 extends our model to a dynamic local coherence, and illustrates its impact on a basis risk assessment example. Finally, Section 6 concludes the paper and gives several improvements for future researches.

The mortality dispersion issue

Notation

As noted in the Section 1, a large range of models have been developed for mortality modeling and forecasting within the multi-population framework. In this paper, we do not limit our study to the specific two-population context: we consider a more general scope of a collection I of populations, where its cardinal I ě 2 is potentially high. Furthermore, we note X and T the collections of integer ages and calendar years respectively.

For pi, x, tq P I ˆX ˆT , we note D piq x,t the number of people in the i-th population who die in year t and aged x at their last birthday. The so-called "exposure to risk", noted E piq x,t , represents the amount of person years lived by people of population i aged rx, x `1q in year rt, t `1q. Thus, the central death rate m piq x,t is given by

m piq x,t " D piq x,t E piq x,t
.

(2.1)

In the two populations framework, the coherence can be analyzed by focusing on the difference of log-mortality rates at a specific age x, which we note δ x,t " ln m p1q x,t ´ln m p2q x,t . The coherence hypothesis specifies that for all x P X , the series pδ x,t q t do not diverge. We extend this measure to the general multiple populations case where I is potentially important. Thus, for an age and time px, tq, and a collection I of populations, we define δ I

x,t the dispersion of mortality rates

δ I x,t " d 1 I ´1 ÿ iPI ´ln m piq x,t ´ln m x,t ¯2, (2.2)
where ln m x,t " 1 I ř iPI ln m piq x,t is the mean of log mortality rates over the collection I. In this way, this measure allows us to evaluate the heterogeneity of the mortality levels among a set of populations. The specific case of a coherent multi-population model corresponds to a situation where for all x P X the dispersion `δI

x,t ˘tPT is controlled and can not diverge.

The Lee-Carter model

There is a broad variety of mortality models that have been introduced in the actuarial science and demography literature. In this paper, we choose to focus on the Lee-Carter model, which is one of the most used by practitioners. We first quickly recall the model using the previous notation and adopting a multi-population point of view. Thus, [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF] propose to model the central death rates of the i-th populaton such as x " 1.

ln m piq x,t " α piq x `βpiq x κ piq t ` piq x,t , (2.3
(2.4) Thus, α piq

x are set to be the averages of log mortality central rates over the period T considered, and the series of parameters β where the innovations ´epiq t ¯iPI is an I-dimensional Gaussian white noises with mean 0 and a diagonal variance-covariance matrix Σ 1 .

The Lee-Carter (LC) being a single population mortality model, we note that no relationships between the different populations are taken into account. As a consequence, the derived mortality forecasts are independent and can diverge. It creates a potentially large diversification effect on the projection of the longevity risk within a stochastic multi-population framework.

The coherent Li-Lee model

As an alternative, [START_REF] Li | Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method[END_REF] extend the Lee-Carter model to the multi-population scope by imposing a common trend B I

x K I t to the collection I of considered populations, i.e.

ln m piq x,t " α piq x `BI x K I t `βpiq x κ piq t ` piq x,t .
(2.6)

The coherence is then enforced by modeling the dynamic of time series κ piq t via a mean-reverting process. We retain here a first order autoregressive (AR) model

κ piq t " a piq κ piq t´1 `rpiq t , (2.7)
where the innovations ´rpiq t ¯iPI are an I-dimensional Gaussian white noises with mean 0 and a diagonal variance-covariance matrix Σ 2 . We denote m I x,t the mortality rates of the overall population, i.e. the gathering of all populations from the collection Similarly to the Lee-Carter model, time series K I t is modeled as a random walk with drift

I m I x,t " D I x,t E I x,t , ( 2
K I t " c I `KI t´1 `eI t , (2.9)
where e I t is a Gaussian white noise with mean 0 and related variance σ 2 I . Thus defined, the Li-Lee (LL) extension enforces the specific mortality rates of the whole collection I of populations to follow the same trend and converge in the long-run. Even if the populationspecific temporal factors κ piq t allow a population to slightly derives from this trend for a short period of time, the mean-reverting property brings it back to the common dynamic over the long run. From a longevity risk management point of view, it then implies a limited diversification among the population in a stochastic evaluation context.

Mortality dispersion on the HMD

Let us now focus on a group of 16 western European countries, namely: Austria (AUT), Belgium (BEL), Switzerland (CHE), West Germany (DEUW), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Ireland (IRL), Italy (ITA), Luxembourg (LUX), Netherlands (NLD), Norway (NOR), Portugal (PTR), Sweden (SWE) and Great Britain (GBR). Furthermore, we consider separately the male and female populations for each country. Thus, our collection I of interest is composed of I " 32 populations, each one of them being defined by a couple pcountry, sexq. The data is extracted from the Human Mortality Database (HMD, 2019). For the age-period training set, we retain X ˆT " t45, . . . , 90u ˆt1960, . . . , 2014u, and we fit single LC models as well as a LL model, as described in Sections 2.2 and 2.3. The Figure 1 represents the dispersion δ I 85,t estimated on the historical data and on the projections over 50 years obtained with the both stochastic mortality models. For the latter, we plot the median together with the 95% prediction intervals derived from 500 Monte-Carlo simulations.

First, let us remark that the dispersion measure at age 85 has nearly doubled between the 1960's and the 2000's, and seems more stable in recent years. As expected, the predicted dispersion is significantly different between the results from the LC and the LL models. In the first case, both the median and the width of the prediction intervals of the dispersion increase linearly with the projection horizon. In other words, the mortality rates at age 85 of the considered populations become more and more dissimilar in average, and this heterogeneity significantly varies from one stochastic scenario to another. On the contrary, the LL's forecasts exhibit a strongly stable dispersion: according to them, not only the median is constant over time, but also the confidence intervals are very narrow comparing to the historical data and their bandwidth does not augment with the projection horizon. In term of generated mortality scenarios, it implies that a considerable majority of them lead to very similar pattern of mortality rates, even in the long term.

Both the LC and LL models present drawbacks when analyzing the projected dispersion in a longevity risk management framework. By not imposing any relationships between the population mortality dynamics, the LC model artificially creates some diversification in term of longevity risk. Contrariwise, the LL model imposes a strong coherence hypothesis on all the populations, thus leaving no significant scope for even a limited deviation of the mortality between the populations: it assumes a high concentration risk in the longevity evaluation. Intuitively, a longevity risk manager may want to consider some intermediate scenarios between the exaggerated LC's diversification and LL's over-concentration.

A locally coherent approach

To bridge the gap between the Lee-Carter's diversification and the Li-Lee's concentration, we propose a new model based on the idea that the populations are coherent by homogeneous groups, and not all together at the same time. In the populations space, it can be seen as a local version of the coherence property.

The model

Keeping the notation introduced before, we now denote J a partition of the populations collection I in J distinct groups. Let φ : I Ñ J be the corresponding classification function that labels a population to a specific group. We then propose the following model for each population i P I

ln m piq x,t " α piq x `Bφpiq x K φpiq t `1t#φpiqą1u β piq x κ piq t ` piq x,t , (3.1)
where # : J Ñ N is the cardinal function of a group, and φ piq denotes the coherence group of the population i.

Similarly to the LL model, for each group j P J , the local common trend factors B x are set to be the averages of log mortality rates over the estimation period T . Finally, the population-specific mortality dynamic parameters β piq x and κ piq t are again estimated through SVD, if the considered population i is assumed to be coherent with at least one of the other population.

To impose the local coherence, we keep the AR(1) dynamic of Equation (2.7) for the populationspecific period factors, but here we do not impose the variance-covariance matrix Σ 2 of the innovations ´rpiq t ¯iPI to be diagonal anymore. This allows to capture short-term relationships between the populations.

Likewise, we also generalize the temporal dynamics of the period group factors to capture possible relationships between the different clusters of populations. Following the underlying idea of [START_REF] Kleinow | A common age effect model for the mortality of multiple populations[END_REF] and [START_REF] Zhou | Modeling Period Effects in Multi-Population Mortality Models: Applications to Solvency II[END_REF], we retain a vector autoregressive (VAR) model. For j P J , we note ∆K j t " K j t ´Kj t´1 the common mortality improvement of the cluster. Given a temporal lag p, the dynamic of the VARppq is then given by

∆K t " C `p ÿ k"1 A k ∆K t´k `Et , (3.3)
where ∆K t " ´∆K j t ¯jPJ is the vector of mortality improvements at a group level, A k , k " 1, . . . , p, are J ˆJ-autoregressive matrices, C is a J-dimensional vector of drifts, and E t is a Jdimensional Gaussian white noise with mean 0 and Σ 1 the related covariance matrix. The matrices A k , k " 1, . . . , p, capture the long-run relationships of mortality improvements between groups of coherent populations, while the variance-covariance matrix Σ 1 estimates the short-term dependence structure of innovation shocks.

Border cases

Before continuing to the practical application of the model and its assessment, we focus on two special border cases. Indeed, by choosing some particular clusters and setting specific parameters, we note that our model includes the original Lee-Carter and Li-Lee models. To recover the LC model, we need to impose the following specifications:

• All the coherence clusters contain only one population, i.e. @i P I, φ piq " tiu. In other words we do not impose any coherence between the populations of interest.

• The temporal lag p of the VAR dynamic is set to 0. This point can also be approximated by applying a high λ in the elastic-net estimation process. The vector C in Equation (3.3) is then equivalent to the concatenation of the population-specific drifts c piq from Equation (2.5).

• A diagonal structure is enforced to the variance-covariance matrix Σ 1 , i.e. no short-term dependence structure is estimated between the populations.

Similarly, we can also retrieve the LL framework from our model by considering the following constraints.

• All the populations are grouping into a single cluster, i.e. @i P I, φ piq " I, the coherence property is imposed to the whole demographic set.

• The temporal lag p of the VAR dynamic is set to 0. In particular, we remark that it is a 1-dimensional VAR due to the partition specification.

• A diagonal structure is enforced to the variance-covariance matrix Σ 2 . Further more, Σ 1 being a 1 ˆ1-matrix, we recover the σ 2 I of the LL's common temporal trend dynamic.

Hence, our model framework can be seen as a bridge between the Lee-Carter and the Li-Lee ones. Thereby, for the rest of this paper, we denote it by LC-LL.

VAR Elastic-Net

In a general context, one can consider a large number J of groups by assuming that the coherence hypothesis is too strong for the populations of interest, thereby increasing dramatically the number of parameters to be estimated in the VARppq. From a statistic point of view, it may lead to highdimensional problems with a limited number of observations. Following [START_REF] Guibert | Forecasting mortality rate improvements with a highdimensional VAR[END_REF], we apply the extension of the elastic-net regularization and variable selection method proposed by [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF] for the high-dimensional estimation of our autoregressive matrices. This technique is a combination of the LASSO L 1 -penalty, introduced by [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], and the ridge L 2 -penalty developed by [START_REF] Hoerl | Ridge regression: Biased estimation for nonorthogonal problems[END_REF]. Elastic-net has similar properties of variable selection as the LASSO. Moreover, it provides a grouping effect: highly correlated variables tend to be selected or dropped together. LASSO and elastic-net have already been extended to VAR model [START_REF] Gefang | Bayesian doubly adaptive elastic-net Lasso for VAR shrinkage[END_REF][START_REF] Basu | Regularized estimation in sparse high-dimensional time series models[END_REF], mostly with an economic application (see e.g. [START_REF] Song | Large vector auto regressions[END_REF][START_REF] Furman | VAR Estimation with the Adaptive Elastic Net[END_REF] and more recently for mortality modelling [START_REF] Guibert | Forecasting mortality rate improvements with a highdimensional VAR[END_REF]. Therefore, we estimate the VAR ppq model presented in Equation (3.3) with T observations of the process ∆K t for t P T " tt min , . . . , t max u by minimizing the criterion

L pC, A 1 , . . . , A p q " 1 T ´p tmax ÿ t"t min `p }∆K t ´C ´p ÿ k"1 A k ∆K t´k } 2 2 ´αλ p ÿ k"1 }A k } 1 ´p1 ´αq λ 2 p ÿ k"1 }A k } 2 2 , (3.4)
where } . } 1 and } . } 2 respectively denote the L 1 and L 2 norms, λ ą 0 determines the strength of the penalization, and α P r0, 1s represents the mix between ridge pα " 0q and LASSO pα " 1q penalties. In particular, when α is set such as some LASSO penalty is enforced, i.e. α ą 0, the higher λ gets, the fewer number of variables is selected.

For sparsity purpose we impose a larger weight to the LASSO penalty by setting α to 0.9. The hyper-parameter λ is then estimated thanks to a 10-folds cross-validation method. To apply this estimation process, we use the sparsevar R-package [START_REF] Vazzoler | sparsevar: A Package for Sparse VAR/VECM Estimation[END_REF], which is based on the glmnet one [START_REF] Friedman | Regularization Paths for Generalized Linear Models via Coordinate Descent[END_REF]. Furthermore, we choose to retain a temporal lag p " 4 in the rest of the paper.

Numerical applications

In this section, we compare the forecasting results with different locally coherent specifications for European populations introduced in Section 2.4. First, we investigate clustering strategies to determine the coherence groups among the collection of populations. We retain two approaches: expert-based and data-driven. Second, we focus on the main mortality trends that can be derived from these approaches, and their related stochastic scenarios spectrum. Finally, we illustrate the effect of the grouping assumptions on the solvency capital required of a pension product.

Population clustering

At first glance, the clustering methods can roughly be split between those based on expert judgments and those more data-driven. In the following, we give one example of each for the collection of western European countries studied in the Section 2.4. Our aim here is rather to give an outlook of some reasonable clustering alternatives than describing exhaustively a methodology for grouping populations. Indeed, such an analysis is a complex task and is out of the scope of this paper, since it requires to examine not only mortality data [START_REF] Hatzopoulos | Common mortality modeling and coherent forecasts. An empirical analysis of worldwide mortality data[END_REF] but also economical, social and environmental criteria as well as the health system of each population.

Coherence by country

In our case, the set of countries we are focusing on tend to strengthen their gender policy in the future. Thus, in the long run, one can assume that within a single country both male and female populations would have similar lifestyle with access to the same level of education, wealthiness and healthcare, leaving mostly only biological differences. Thereby, we define the partition J M F such as it clusters the populations by countries. The number of groups J equals then to 16. The corresponding model is noted LC-LL(MF).

An interesting point of this approach is that the VAR model estimates a dependence structure between the countries, allowing a rather easy reading of the autoregressive matrices that represent the underlying dynamics. Indeed, the common period factors ´Kj t , j P J M F ¯are equivalent to the ones obtained by fitting single population LC models on the overall country. We display the Granger causality matrices in Figures 8a to 8d of the Appendix 2.

A data mining approach

The main idea of the coherence in the Li-Lee model is to impose a common trend `KI t ˘tPT in the mortality dynamics of each population i. This is done by fitting a Lee-Carter on the overall population of the retained collection I (see Section 2.3). Hence, for a single population i, we consider its time series ´κpiq t ¯tPT , derived from the LC fitting, as a signature of the mortality dynamics.

To determine the coherence groups, we then apply an unsupervised clustering method on these signatures. We choose the well-known hierarchical cluster analysis (HCA) method. In particular, we retain the Euclidean metric and Ward's criterion as dissimilarity measure and linkage criterion respectively. We display the corresponding dendrogram in Figure 2.

We remark that, unlike the previous clustering, the populations are not gathered by country. It even seems to be the opposite: the gender indicator is one of the major splitting criteria. Yet, we note one exception to this comment: the two Danish populations are clustered together, apart from other countries, suggesting that the Danish mortality dynamic exhibits some specific features when comparing to other western European countries.

For the following numerical applications, we choose to retain 8 different clusters, which are emphasized in the Figure 2 through the colors and recalled in the Table 1. We denote the LC-LL mortality model based on this clustering by LC-LL(HCA8), and the corresponding partition by J HCA8 .

Dispersion forecasting

We now compare the models that we introduced, i.e. LC, LL, LC-LL(MF) and LC-LL(HCA8), int terms of dispersion. Rather than focusing on the main mortality trends that can be derived from these models, we are more interesting in the scope of this paper to the stochastic scenarios spectrum generated by these models.

Table 1: 8 groups of populations determined by HCA on the LC's κ t .

Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8

DEUW M NOR M GBR M NOR F DNK M PRT F DEUW F FRA F BEL M NLD M ITA M NLD F DNK F FIN M BEL F CHE F FRA M SWE M AUT M SWE F AUT F FIN F GBR F LUX M IRL F ITA F PRT M IRL M CHE M ESP F ESP M LUX F
Following the Section 2.4, we display respectively in Figures 3 and4 the projections obtained from the LC-LL(MF) and LC-LL(HCA8) approaches, compared to the LC and LL model forecasts. The red lines correspond to the historical dispersion δ I 85,t for the 32 European populations of I at age 85. The same graphs are displayed in Appendix 1 for ages 70, 75, 80 and 90. As expected, the dispersion predictions produced by the LC-LL models are contained between the LC and LL ones. In the LC-LL(MF) case, we note that the dispersion level is comparable to the LL's projection during the first years, however both the mean dispersion and the width of the confidence intervals are increasing with the projection horizon like in the LC model. This increase is yet significantly slower than the one obtained with the single population framework. While the latter causes a doubling of the dispersion within 50 years, the LC-LL(MF) leads to a dispersion spectrum relatively close to the historical observations of this measure since the 1990's.

When focusing on the LC-LL(HCA8), we observe that the dispersion dynamic is similar to the LC: both of the models leads to a significant augmentation of the dispersion compared to the historical data. However, in the multi-population model, the increases of the main trend and confidence intervals are more limited. In term of longevity risk, it leads to a significant lower diversification level among the populations.

Solvency Capital Requirement evaluation

Finally, we quantify the impact that the different models could have for a global (re)insurance company in terms of solvency capital requirement evaluation with respect to the longevity risk. To do so, we analyze a simplified situation where the liabilities are pensions to be paid between ages 60 to 90 for each of the 32 populations. Moreover, we focus only on the cohort aged 59 in 2014 and assume that at the valuation date, i.e. end of the year 2014, the number of pensioners is the same in every population. For the sake of simplicity, we retain an annual pension of 100 paid at the end of the year, and a constant discount rate of 1%.

Hence, to evaluate the corresponding provision the actuary needs to forecast the mortality rates m piq 59`t, 2014`t for pi, tq P I ˆt1, . . . , 32u. Following the Solvency II framework, the associated longevity risk solvency capital requirement (SCR) is obtained by estimating the Value at Risk (VaR) at a 99.5% level of the best estimate provision. This is done by simulating 2,000 longevity scenarios through the stochastic mortality models. We display in the Table 2 the results of this approach for the LC, LL, LC-LL(MF) and LC-LL(HCA8) models. First, we remark that the choice of the model, among the collection we explore in this paper, does not impact much the provision. The absolute difference between the highest provision estimate and the lowest one is of 190, which represents less than 0.3% of the latter. On the contrary, the range of SCR is significantly wider. As expected, the LC leads to the lowest value because the mortality scenarios are independent from one population to another, thus creating diversification. In comparison, the SCR estimated by the LL is nearly 3.7 times bigger. Indeed, in this case the mortality rate dynamics of all populations follow the same trend, thereby implying a concentration risk on this trend. Finally, between these two extreme modelings, our locally coherent approach leads to intermediate levels of SCR. For the coherent groups determined in Section 4.1, we retrieve approximately the average between the LC and the LL results.

Toward a dynamic local coherence

In our model introduced in Section 3.1, we have supposed that the classification function φ : I Ñ J is constant through time. In other words, a population indefinitely belongs to the same group of coherence. However, in a more general longevity risk assessment framework, it could be interesting to study scenarios in which populations are allowed to switch from one coherence group to another. One of the motivations for such sensitivities is driven by the evaluation of basis risk in some longevity risk hedges. The aim of this section is to describe the stochastic forecasts of the mortality rates in a trend switching framework. Further, we highlight the effet of such switches on a Longevity Divergence Index Value (LDIV).

Trend switching model

Hereafter, the groups of local coherence j P J are rather to be considered as dominant mortality trends which lead the mortality dynamics of their related specific populations. Thus, as long as a group of populations are lead by the same dominant trend, they are all coherent between them. On the contrary to the model of Section 3.1, we consider here that J is not necessarily a partition of the retained collection of populations I anymore. However, it can always be viewed as a partition of a larger collection of populations Ω, which is not fully observed. Thus, depending of the time period, some of the dominant trends j P J may not be linked to any retained population i P I. During such periods, they can be viewed as dominant trends, exogenous from the observed collection of populations. In the same way, we can assume that @j P J , #j ą 1, even if it implies assuming the existence of populations in Ω which are not present in the retained collection of populations I. The fact that the classification function φ t is now dynamic, it may happen that the switch from one dominant trend to another creates a significant leap in the mortality dynamics. Indeed, for two different dominant trends pj 1 , j 2 q and a couple age-period px, tq, we do not impose the term B j 2

x K j 2 t ´Bj 1 x K j 1 t to be limited since it describes the difference between two non-coherent mortality dynamics. Thus, to avoid such jumps in the mortality level each time a population changes of dominant trend, we need to add adjustment mortality levels:

ad piq x,t " t ÿ s"t 0 `1 B φ s´1 piq x K φ s´1 piq s ´Bφspiq x K φspiq s , (5.2)
where t 0 is a time such as ad piq x,t 0 " 0 for all ages x P X . In a practical point of view, φ t 0 piq defines the initial dominant trend of the population i. Furthermore, we note that, in the non-switching model case, we have @s P T , φ s´1 piq " φ s piq " φ piq, so @t P T , ad piq x,t " 0. Hence, we retrieve the dynamics of the LC-LL with constant groups of coherence, see Equation (3.1).

European LDIV

As previously stated, this extension is motivated by the assessment of basis risk in longevity risk hedges. We choose to focus on one practical example: the Longevity Divergence Index Value (LDIV) on which the Swiss Re Kortis bond is defined [START_REF] Hunt | Modelling longevity bonds: Analysing the Swiss Re Kortis bond[END_REF].

Issued in December 2010, the Kortis bond was designed to hedge the basis risk resulting from Swiss Re's "partial natural hedge" in longevity risk. Indeed, the reinsurer was globally exposed to • mortality risk on the US male population aged between 55 and 65,

• longevity risk on the UK male population aged between 75 and 85.

Even though these two exposures create a natural longevity hedge, a significant basis risk remains between them two, especially if the mortality trends of the corresponding populations diverge. In our model, it would be implied by the fact that the two populations are not linked to the same dominant trend.

The payout of the Kortis bond is based on the level of the LDIV in 2016. To compute this value, we first need to observe the annualized mortality improvement of each population i

Improvement piq n px, tq " 1 ´« m piq x,t m piq x,t´n ff 1 n , (5.3)
where n is the averaging period of the bond, equals to 8 years in the Kortis bond's case. ¯" p55, 65q. Finally, the LDIV at time t is obtained by LDIV ptq " Index pt, i 1 q ´Index pt, i 2 q , (5.5)

where in the Swiss Re case pi 1 , i 2 q " pUK, USq and t " 2016.

To illustrate the impact on the basis risk of a switch of dominant trend, we construct an European LDIV for:

• the Swiss female population, aged between 75 and 85, as the longevity risk exposure i 1 ;

• the French female population, aged between 55 and 65, as the mortality risk exposure i 2 ;

• a risk period n of 8 years, which ends at year t " 2024.

Moreover, we suppose that at t 0 " 2014, the mortality dynamics of the collection of populations I follow the model of Section 4.1.2, i.e. the LC-LL(HCA8). Thus, the classification function φ t 0 can be described thanks to Table 1. Finally, we suppose that at a time T ą t 0 , the French female population switches from its original dominant trend to the dominant trend that the Belgian and German female populations are initially linked to. No further switch are considered. Hence, for t ą t 0 , we have

φ t piq " $ ' & ' % φ t 0 piq if i ‰ FRA F , φ t 0 pFRA Fq if i " FRA F and t ă T,
φ t 0 pBEL Fq if i " FRA F and t ě T.

(5.6)

Figure 5 displays the sensibility of our European LDIV(2024) to the switching time T between t 0 " 2016 and 2024. To obtain the results, we simulate 10,000 stochastic mortality scenarios thanks to our model for each switching time between 2016, i.e. the start of the risk period and 2024, i.e. the end of the risk period. We then retain the median scenario in term of LDIV(2024). As expected, the sooner the switch occurs, the more significant the Longevity Divergence Index Value is. In particular, we remark that the LDIV is 1.8 times higher when the switch happens in 2016 than in 2024. Indeed, the longer the French and Swiss female populations are linked to different dominant trends, i.e. the longer they are not coherent, the more they diverge. Thus, our extended model allows the longevity risk manager to evaluate the impact of a wide spectrum of stochastic longevity scenarios on his portfolio and the potential hedges together with the corresponding basis risks. In a context where basis risk is one the main challenges that the longevity risk transfer market is facing [START_REF] Blake | Still living with mortality: the longevity risk transfer market after one decade[END_REF], our framework provides an innovative point of view for its assessment.

Conclusions

In this paper we propose a new notion for multiple populations mortality forecasting that we have named local coherence. Indeed, we remark that full independence and "full" coherence approaches both suffer from drawbacks when considering the mortality projections of a large set of populations. To highlight this point, we compare the dispersion over 32 European populations of mortality rate forecasts derived from [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF] and [START_REF] Li | Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method[END_REF] models. In the first case, the diversification in the stochastic scenarios is overestimated, whereas, in the second method, the concentration is exaggerated.

To overcome this issue, our locally coherent mortality model allows to forecast populations with homogeneous mortality profiles by coherence groups, where the long-term relationships between inter-group populations are only modeled through a VAR process without any coherence constraints. Thus, it offers a trade-off between the two border coherence cases, which are included in our model. The practitioners have then the possibility to easily simulate a larger spectrum of longevity scenarios to evaluate their risks compared to the usual models cited before. To assess the behavior of our model, we have compared it with the Lee-Carter and Li-Lee's ones over the set of European countries for some numerical applications. We notably show that the retained coherence hypothesis can have a major impact on the longevity risk solvency capital requirement for a global life insurance company within the Solvency II framework. Thereby, the large freedom of our model allows to evaluate more precisely this crucial risk measure.

Finally, we have extended the locality coherence property of our model to the temporal dimension. By allowing populations to switch from one group of coherence to another, we expand the spectrum of possible stochastic mortality scenarios. This innovative methodology is particularly interesting in a longevity hedges' basis risk assessment context. Thus, we propose a new framework that should be useful for the development of the longevity risk transfer market.

Our proposed method is a first step for modeling together both homogeneous and heterogeneous populations in terms of mortality profiles. It opens up some pathways for future research. We do not further investigate the difficult issue of how grouping a large set of populations. Our approach however stresses the importance of developing techniques for identifying homogeneous/heterogeneous groups of population based not only on trend and volatility patterns in mortality dynamics, but also by considering economical, social and environmental criteria or the effectiveness of the health system. In addition, we also believe that groups of populations evolve dynamically. This means that some groups have a behavior which gradually converge to those of another groups, e.g. the less developed countries would probably follow the trend of developed countries [START_REF] Li | Coherent Mortality Forecasting for Less Developed Countries[END_REF], or diverge from each other. Another potential improvement is thus to consider methods for detecting change in the mortality trend, see e.g. El [START_REF] El Karoui | Minimax optimality in robust detection of a disorder time in doubly-stochastic Poisson processes[END_REF].

  are residual terms. For the sake of the uniqueness of the solution, the usual constraints are imposed in the estimation process ÿ

  thanks to the application of the singular value decomposition (SVD) method.Furthermore, we model the dynamic of the time series κ piq t as a random walk with a drift c piq κ piq t " c piq `κpiq t´1

  . Thus, the fit of a Lee-Carter model on these series, under the usual constraints ř tPT K I t " 0 andř xPX β Ix " 1, gives the common trend factor estimates. Like in the single population context, the age-population specific series of parameters α piqx are set to be the averages of log-mortality central rates over the training period T . Finally, the population-specific temporal dynamic factors β piq x κ piq t are obtained thanks to the SVD method, with the constraints ř for each i P I.

Figure 1 :

 1 Figure 1: Dispersion at age 85 in the western European populations: historical data and median projections by Lee-Carter and Li-Lee models with the corresponding 95% prediction intervals.

Figure 2 :

 2 Figure 2: Dendrogram associated with the HCA applied on the LC's κ t , colored according to 8 final clusters. The acronyms for countries are defined in Section 2.4; F and M represent female and male populations.

Figure 3 :

 3 Figure 3: Dispersion at age 85 in the western European populations: historical data and median projections by LC, LL and LC-LL(MF) models with the corresponding confidence intervals at 5% and 95%.

Figure 4 :

 4 Figure 4: Dispersion at age 85 in the western European populations: historical data and median projections by LC, LL and LC-LL(HCA8) models with the corresponding confidence intervals at 5% and 95%.

Figure 5 :

 5 Figure 5: Median European LDIV(2024) according to the switching time of the French female population.

Figure 7 :

 7 Figure 7: Dispersion in the western European populations: historical data and median projections by LC, LL and LC-LL(HCA8) models with the corresponding confidence intervals at 5% and 95%.

Figure 8 :

 8 Figure 8: Estimated autoregressive matrices A 1 , . . . , A 4 of the VAR-ENET from the LC-LL(MF) model.

  , with the usual uniqueness constraints. For each population i, the series of age factors α

				m j x,t "	D j x,t x,t E j	,	(3.2)
	where D j x,t "	ř iPj	D	piq x,t and E j x,t " x,t piq ř iPj piq E

j

x and K j t are estimated by fitting a Lee-Carter model thanks to the SVD method on the group log mortality rates data

Table 2 :

 2 Best estimate provision and Solvency capital requirement estimates.

		LC	LL	LC-LL(MF) LC-LL(HCA8)
	Provisions (mean)	67,513 67,577	67,675	67,485
	VaR 99.5 %	68,059 69,591	68,999	68,954
	SCR (VaR -mean)	547	2,014	1,323	1,469

  Thus, for pi, x, tq P I ˆX ˆT , we propose the following dynamics for the central mortality rates

		ln m piq x,t " α piq x	`Bφtpiq x	K t φtpiq	`βpiq x κ piq t `ad piq x,t `	piq x,t ,	(5.1)
	piq x are specific population where function φ t : I Ñ J return a set of classification functions, α mortality levels, ´Bj x , K j t ¯define the dominant mortality trends, ´βpiq x , κ piq t ¯represent specific pop-
	ulation mortality dynamics, ad	piq t,x are adjustment mortality levels. Considering the time series, we
	keep the dynamics of the previous model, i.e. ∆K t follows a VAR(p) model, see Equation (3.3),
	and κ	piq t are driven by AR(1) models.

  Then, an averaged improvement index is computed over the age classes exposed to the risk

	Index pt, iq "	1	1 `xpiq 2	1 ´xpiq	piq 2 1 x"x x piq ÿ	Improvement piq n px, tq ,	(5.4)
	where for the Kortis bond the age classes are	´xpUKq 1	, x pUKq 2	¯" p75, 85q and	´xpUSq 1	pUSq , x 2

Appendix 2 Granger matrices from the LC-LL(MF) model

The matrix A 1 (Figure 8a) exhibits mostly negative coefficients, shaped into a diagonal structure.

We hereby capture the country-specific period effect. Indeed, when modeling the single population LC's κ piq t dynamics by an AR(1), we generally obtain negative autoregressive estimates. Displayed in a matrix way for a multi-population point of view, it leads to a negative diagonal matrix. We also remark some vertical patterns within the Granger causality matrices. In term of mortality dynamics, this should be interpreted as a persistent effect of a single country to a group countries. For example, in the autoregressive matrix A 3 (Figure 8c), we note the existence of a positive vertical structure on the Swiss population. Hence, the Swiss mortality improvement seems to Granger cause the mortality improvement on 6 other countries, namely BEL, DEUW, GBR, IRL, NLD and PRT. A chock in the mortality rates in the Switzerland would then reverberate 3 years later in the 6 countries. In this way, the vertical structure can be seen as some kind of leader effect.