Optimal Reinsurance Under VaR and CTE Risk Measures When Ceded Loss Function is Concave

Abstract

Cai et al. (2008) explored the optimal reinsurance designs among the class of increasing convex reinsurance treaties under VaR and CTE risk measures. However, reinsurance contracts always involve a limit on the ceded loss function in practice, and thus it may not be enough to confine the analysis to the class of convex functions only. The object of this article is to present an optimal reinsurance policy under VaR and CTE optimization criteria when the ceded loss function is in a class of increasing concave functions and the reinsurance premium is determined by the expected value principle. The outcomes reveal that the optimal form and amount of reinsurance depend on the confidence level p for the risk measure and the safety loading θ for the reinsurance premium. It is shown that under the VaR optimization criterion, the quota-share reinsurance with a policy limit is optima, while the full reinsurance with a policy limit is optima under CTE optimization criterion. Some illustrative examples are provided.

Keywords:

• Optimal reinsurance
• Value-at-risk
• Conditional tail expectation
• Increasing concave function
• Quota-share reinsurance
• Full reinsurance

Mathematics Subject Classification:

• 60E15
• 62P05

Appendix

This Appendix collects some lemmas and the proofs of some results stated in the previous section.

Lemma A.1.

For any , there exists a sequence of functions such that for all x ⩾ 0.

Moreover, we have (A.1) and (A.2) as n → ∞.

Proof.

Since I(x) is increasing and concave, from Lemma A.2 in Cai et al. (2008), I(x) must be either strictly increasing on [0, ∞) or has the following representation: (A.3) for some strictly increasing and concave function h(x) and some constant x₀ ∈ (0, ∞). We will complete the proof of the first part by considering the following cases:

1. I(x) is strictly increasing on [0, ∞).

   Define and . It is well known that A_n → ∞ as n → ∞. By Schlömilch-Lemonnier inequality (see Alzer and Brenner, 1992), we have which leads to lim_{n → ∞}δ_n = 0 since .

   For any , define g_n(x) = ∑ⁿ_{k = 1}c_{n, k}(x − (x − d_{n, k})₊), n = 1, 2, …, with and d_{n, k} = kδ_n, k = 0, 1, …, n. We can easily verify that c_{n, k} > 0 and , which implies . After simple algebras, one can find that g_n(d_i) = I(d_i) for i = 1, 2, …, n. For any x ⩾ 0 and ε > 0, there exists N such that 0 ⩽ I(x) − g_n(x) ⩽ δ_n < ε when n > N, which implies that lim_{n → ∞}g_n(x) = I(x).

2. I(x) has the representation of (A.3(A.3) ).

   The assertion can be derived similarly to case (a) by letting , and

We now begin to show the last statement. It is easy to see that and R_I(x) are increasing continuous and . Then by Proposition A.1(b) in Cai et al. (2008) and dominated convergence theorem, we can obtain (A.4) which immediately yields (A.1(A.1) ).

Furthermore, from (2.8(2.8) ) and (2.11(2.11) ), we can find that . Then by (A.4(A.4) ) and dominated convergence theorem, we can obtain which proves A.2(A.2) .

The following lemma plays a key role in dealing with the problems (2.4(2.4) ) and (2.5(2.5) ). Similar to Lemma 3.1 and Lemma 4.1 in Cai et al. (2008), by applying Lemma A.1, its proof can be completed without difficulty and is omitted here.

Lemma A.2.

For any , then:

1. g(x) is an optimal solution to the problem (2.4(2.4) ) if it is an optimal one to the problem (2.6(2.6) ); and

2. g(x) is an optimal solution to the problem (2.5(2.5) ) if it is an optimal one to the problem (2.7(2.7) ).

Lemma A.3.

φ(t) = (1 + θ)∫^t₀S_X(x)dx − t is a continuous function with respect to t:

1. if θ* > p₀, then φ(t) is increasing on [0, VaR_X(θ*)) while decreasing on (VaR_X(θ*), ∞), that is, φ(t) reaches its maximum at VaR_X(θ*); and

2. if θ* ⩽ p₀, then φ(t) is deceasing on [0, ∞) and φ(t) < 0 on (0, ∞).

Proof.

1. One can find that . The proof follows from that if θ* > p₀, then φ′(t) > 0 for t ∈ [0, VaR_X(θ*)) and φ′(t) ⩽ 0 for t ∈ [VaR_X(θ*), ∞).

2. The proof follows from that φ′(t) ⩽ 0 when θ* ⩽ p₀ for any t ∈ [0, ∞) and φ(0) = 0, as well as φ′(t) < 0 when t ∈ (0, ∞).

Proof of Proposition 3.1.

Since ∫^t₀S_X(x)dx is increasing in t ⩾ 0, it follows from (3.1(3.1) ) that reaches its maximum at d* = (VaR_X(p), …, VaR_X(p)) when and (A.5)

(i) If φ(VaR_X(p)) > 0, then from Lemma A.3 and (3.2(3.2) ), one can find that φ(d_{n, k})(k = 1, …, i) ⩾ 0 when . Hence, we have (A.6) and the the minimum is attained at .

Similarly, the minimum of is attained at d* = (0, …, 0) when and (A.7) By comparing (A.5(A.5) ), (A.6(A.6) ), and (A.7(A.7) ), we prove (i).

(ii)If φ(VaR_X(p)) < 0, it follows from Lemma A.3 and (3.2(3.2) ) that the minimum of is attained at (VaR_X(p), …, VaR_X(p)) when . Then we have (A.8) When , it is easy to see that the minimum of is reached at d* = (VaR_X(p), …, VaR_X(p)) and (A.9) Combining the observations above, we complete the proof of (ii).

(iii) When φ(VaR_X(p)) = 0, it is easy to see that reaches its minimum at or (VaR_X(p), …, VaR_X(p)) when and (A.10) One can also find that the minimum of is attained at (VaR_X(p), …, VaR_X(p)) or (0, 0, …, 0) when and (A.11) which, together with A.5(A.5) and A.10(A.10) , completes the proof of case (iii).

Proof of Lemma 4.1.

From (2.1(2.1) ) and (2.3(2.3) ), we have (A.12)

1. When , the expressions of can be derived in the following two cases:

   1. When ∑ⁿ_{k = 1}c_{n, k} < 1, from (2.11(2.11) ) and Proposition A.1 in Cai et al. (2008), we have (A.13) and (A.14) From (2.8(2.8) ), one can find that (A.15) Substituting (A.13(A.13) ), (A.14(A.14) ), and (A.15(A.15) ) into (A.12(A.12) ), we can obtain

   2. When ∑ⁿ_{k = 1}c_{n, k} = 1, we have and

2. When , i = 1, 2, …, n − 1, from (2.9(2.9) ), (2.10(2.10) ), and (2.11(2.11) ), we can obtain (A.16) and (A.17) Moreover, it follows from (2.8(2.8) ) that (A.18) Substituting (A.16(A.16) ),(A.17(A.18) ), and (A.18(A.18) ) into (A.12(A.12) ), we get that

3. When VaR_X(p) ⩾ d_{n, n}, we have

Proof of Lemma 4.2

1. 1. By noticing that ∫^t₀S_X(x)dx is increasing in t ⩾ 0, one can easily find from (4.1) that if ∑ⁿ_{k = 1}c_{n, k} = 1, then the minimum CTE is attained at (VaR_X(p), …, VaR_X(p)) since d_{n, k} ⩾ VaR_X(p), k = 1, …, n, which leads to (4.1(4.1) ).

   1. By (i) of Lemma 4.1, when p > θ*, the minimum CTE is attained at (∞, …, ∞), and the minimum CTE is attained at (VaR_X(p), …, VaR_X(p)) when p > θ*, which implies (4.2(4.2) ).

2. By Lemma A.3 and (ii) of Lemma 4.1, we have

   (1) In case that p > θ* > p₀ and φ(VaR_X(p)) ⩾ 0, the the minimum CTE is attained at ;

   (2) In case that p > θ* > p₀ and φ(VaR_X(p)) < 0, or p > p₀ ⩾ θ*, the minimum CTE is attained at ;

   (3) In case that θ* ⩾ p > p₀, the minimum CTE is attained at .

   After simple calculations, we can obtain (4.3(4.3) ).

3. By Lemma A.3 and (iii) of Lemma 4.1, one can find the following.

   (1) In case that , or θ* ⩾ p > p₀, the minimum CTE is attained at (0, …, 0) and

   (2) In case that , or p > p₀ ⩾ θ*, the minimum CTE is attained at (VaR_X(p), …, VaR_X(p)) and

Proof of Proposition 4.1.

(a): From Lemma A.3, φ(VaR_X(p)) < 0 implies p > θ*, or equivalently δ < 0.

1. If , it follow from Lemma 4.2 that for i = 1, …, n − 1, and Thus, we have (A.19)

2. If , then we have for i = 1, …, n − 1, and which implies that (A.20)

Furthermore, we can obtain that (A.21) where the first inequality holds since . The result follows from (A.19(A.19) ), (A.20(A.20) ), and (A.21(A.21) ).

(b),(d) and (f):

Under the assumptions of (b), from Lemma 4.2, we can get that (A.22) (A.23) (A.24) for i = 1, …, n − 1, and (A.25) (A.26) Moreover, the following relation holds: (A.27) We now consider the following cases.

1. If , then from (A.23(A.23) ), we have (A.28) which, together with (A.22(A.22) )–(A.24(A.24) ), leads to (A.29) and (A.30)

   Similar to (A.21(A.21) ), one can get that (A.31)

2. If , similarly to (1), we can easily prove that (A.32) and (A.33) Furthermore, it is easy to see that (A.34) Putting case (1) and (2) together and using Lemma 4.2, we complete the proof of (b).

To prove (d), we first observe that φ(VaR_X(p)) > v(p) implies from (A.27(A.27) ). It follows from (A.22(A.22) )–(A.34(A.34) ) that (A.35) Now the desired result follows from (A.35(A.35) ) and Lemma 4.2.

(f) is an immediate consequence of the observations above.

(c), (e), and (g):

When θ* ⩾ p > p₀ and φ(VaR_X(p)) > 0, from Lemma 4.2, we have and for i = 1, …, n − 1, which imply that Moreover, we have If φ(VaR_X(p)) < v(p), then Since then we complete the proof of (c).

If φ(VaR_X(p)) > v(p), then we find which proves (e). (g) follows directly from the results above.