Optimal holding capital for insurance companies

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companies

Irene Doelman

Thesis for the

Bachelor Actuarial Science University of Amsterdam

Faculty Economics and Business Amsterdam School of Economics

Author: Irene Doelman Student i.d.: 10428380

Email: irene.doelman@student.uva.nl Date: June 24, 2015

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Abstract

The Solvency II regulation obliges insurance companies to hold capital to make sure that the probability of insolvency is less than 0.5%. This thesis is focused on the questions whether there is an ideal amount of capital for the policy holder and, if so, what the ideal amount of capital is, given various company sizes and claim distributions. It is proven that there is such an ideal amount of capital if the claims are indepentently distributed. This ideal amount is higher than the required amount of capital. For a claim distribution with a common shock effect, the required amount of capital is very high, which makes the risk uninsurable.

Keywords Insurance, Capital Requirement, Optimal holding capital, Solvency, Lim-ited Liability, Expected utility

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Although I started this project unsure whether I was going to like writing a thesis, I quite enjoyed it. It allowed me to use my creativity on the topic and gave me an insight on academic research in practice. I would like to thank my supervisor dr. Tim Boonen for helping me during this proces. His ideas and feedback were indispensable and he motivated me to improve my thesis. I would also like to thank drs. Nancy Bruin for the academic writing seminar.

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Chapter 1 Introduction

Consumers face monetary losses for which insurance is possible. When they do, they are either happy to be insured or sad to have to face the loss. A real setback it would be when they are insured, but the insurance company went bankrupt and can therefore not reimburse their loss. They then face a double setback; they both face the realized loss and they lose the paid premium (Biffis and Millossovich, 2012). During the financial crisis, this topic has become more and more relevant, since people have realized this is a risk which should not be underestimated.

Policy makers have anticipated the risk of bankruptcy of insurers. Since 2009, the European Union obliges insurance companies through the Solvency II directive to reduce the probability of insolvency to less than a benchmark of 0.5%. Reducing the insolvency probability is done by requiring the company to have an obligatory amount of capital, the height of which is calculated by a standard formula in the directive.

In 2014, the European Insurance and Occupational Pensions Authority (EIOPA) wrote a report on the underlying assumptions in the standard formula. However, it remains unclear why the benchmark of 99.5% chance of solvency was chosen. Given the large differences between insurance products and companies, this raises the question whether this benchmark is ideal for policy holders. Therefore, this thesis is focused on the questions whether there is an ideal amount of capital and, if so, what the ideal amount of capital is, given various company sizes and claim distributions.

In order to answer these questions, the second chapter of this thesis focuses on finding realistic assuptions for the model, which is specified in Chapter 3. In Chapter 4 the model is analysed analytically and numerically. Finally, conclusions are drawn in Chapter 5.

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Assumptions

In this chapter, various models from earlier studies are discussed in order to anal-yse what assumptions can be useful to come to a realistic model.

Fillipovi´c, Kremslehner and Muermann (2014) study the conflict of interest between the insurer and policy holder. Even though this is not the object of this thesis, some underlying assumptions of their model are very useful. They look at the non-life insurance market, and assume a one period economy with two agents; the insurer and the policy holder. The latter faces a random loss X, for which he can buy insurance. Furthermore, they assume limited liability. All these assumptions are also made in this thesis.

The model used by Fillipovi´c et al. (2014), is designed such that the insurance company can maximize its profits given the preferences of the policy holder. This is in contrast to Ibragimov, Jaffee and Walden (2010), who design a model for pricing insurance products in a competitive, complete market, where the insurance company is not able to maximize its profits.

2.1 The insurer

In the model of Biffis and Millossovich (2012) there is also a one period economy with two agents. The insurer has starting capital A > 0 ∈ R and receives premium P > 0 at the start of the period. At the end of the period, the insurer will have A + P − I(X), where I is the indemnity function and I(X) the claim height. Assumed is 0 ≤ I(X) ≤ X. Because of limited liability, the wealth of the insurer can not be negative, so the insurer will have max{A + P − I(X), 0} at the end of the period. This is noted as (A + P − I(X))+.

This model only has one policy holder. In this thesis a model with more than one policy holder is used, so some adjustments have to be made. With n the number of policy holders and Xi the random loss of the ith policy holder, the

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Optimal holding capital — Irene Doelman 3

wealth of the insurer will be

(A + nP −

n

X

j=1

I(Xj))+. (2.1)

It is assumed that the Xi have an identical distribution and that Xi ≥ 0. It is

also assumed that I(Xi) = Xi. The function I will therefore be left out of the

equations.

All beforementioned models use the assumption that the discount rate is zero for simplicity. In this thesis, the same assumption is made. However, like Ibrag-imov et al. (2010), the costs of obtaining capital A are considered. This will be noted as a portion δ of A. It is assumed that δ ≥ 0. Because of the market assupmtion, this means that the expected wealth of the insurer at the end of the period will be equal to the starting capital plus the costs of obtaining that capital:

E(A + nP −

n

X

j=1

Xj)+ = (1 + δ)A. (2.2)

This can also be seen in another way. Bernard and Ludkovski (2012) have a similar value in their model which they explain as the minimal expected profit the insurer wants to achieve. It could also be seen as both capital and minimal expected profit: that δ is the minimum amount of expected profit the stakeholders demand in exchange for their capital contribution.

The beforementioned models differ on the concequences of insolvency. In the models of Fillipovi´c et al. (2014) and Cummins and Mahul (2003), no payment is made to the policy holder. We make the same assumption in this thesis.

2.2 The policy holder

In order to find the optimum for the policy holder, assumptions about his utility function need to be made. Fillipovi´c et al. (2014), Biffis and Millosovich (2012) and Cummins and Mahul (2003) all assume that the utility functions of the policy holders are identical, and can be written as u : R → R. This function u is twice continuously differentiable, with u0 > 0 and u00 < 0. This means all policy holders are risk-averse. Specifically, the utility function

u(x) = −e−λx, (2.3)

is used in this thesis. This is called the CARA (Constant Absolute Risk Aversion) utility function and satisfies all beforementioned requirements.

With these assumptions, the model can be specified, which is done in the next chapter.

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The model

In the previous chapter, models from earlier studies were discussed and the un-derlying assumptions were analysed. Taking those as a starting point, the model is further specified in this chapter.

3.1 The standard model

The wealth of the insurer at the end of the period was previously found to be

E(A + nP −

n

X

j=1

Xj)+ = (1 + δ)A. (3.1)

For the insolvency event, an indicator function 1S is used, which is

1S =    1, if A + nP −Pn j=1Xj ≥ 0, 0, otherwise. (3.2)

This means Equation 3.1 can also be written as

E((A + nP −

n

X

j=1

Xj) · 1S) = (1 + δ)A. (3.3)

From this we can derive that

A · P(1S = 1) + nP · P(1S = 1) − E( n

X

j=1

Xj · 1S) = (1 + δ)A. (3.4)

From Equation 3.4 we can derive the premium P > 0 for the insurance. This leads to P = (1 + δ)A + E( Pn j=1Xj · 1S) − A · P(1S = 1) n · P(1S = 1) (3.5) = (1 + δ)A + E( Pn j=1Xj) · E(1S) + cov( Pn j=1Xj, 1S) − A · P(1S = 1) n · P(1S = 1) = (1 + δ)A + E( Pn j=1Xj) · P(1S = 1) + cov( Pn j=1Xj, 1S) − A · P(1S = 1) n · P(1S = 1) . 4

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For simplicity, we assume that this covariance is 0, which means the claim height of the ith policy holder is independent of the solvency event. The error made with this assumption is negligible for large n. This leads to

P = E(Xi) + 1 + δ − P(1 S = 1)

n · P(1S = 1)

· A. (3.6)

For simplicity, we use the 99.5 percent benchmark as P(1S = 1) instead of the

actual value. This is also an error, since varying A means varying the probability of insolvency, but this error is small and it makes the premium P much easier to calculate. Furthermore, this error is not unrealistic. Because of a lack of transparancy, the policy holder’s perception of the default risk is likely to differ from the actual risk (Cummins and Mahul, 2003). The actual equation used to calculate P is therefore

P = E(Xi) +

1 + δ − 99.5%

n · 99.5% · A, (3.7)

where we will verify later that P(1S = 1) ≥ 99.5%.

Using these assumptions, the corresponding expected utility for the policy holder can be specified as

E(u(w − Xi− P + Xi· 1S)), (3.8)

with w > 0 the initial wealth of the policy holder. In this thesis, this expected utility is maximized over A for several values of n:

max

A E(u(w − P − Xi+ 1S· Xi)). (3.9)

First, we will prove that a maximum can be found. Then, the model will be solved both numerically. The effects of three different loss distributions are simulated.

First the loss Xi will be randomly drawn from a lognormal distribution, and

the assumption is made that the Xi are identically and independently distributed.

For reference, the benchmark A is calculated using the Central Limit Theorem. Secondly, a binomial distribution is used to calculate the losses. Assumed is that there is a certain probability of facing a fixed loss. For both distributions, the function is maximized over A for fixed n.

Typically, the losses are not completely independent. Therefore, the third loss distribution takes a common shock effect into account. Xi is modelled as Z + Yi,

with Yi lognormal and i.i.d. The function is maximized over A, for several values

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3.2 Adjustments to the model

Instead of a fixed percentage as costs of capital, the assumption is made that δ is increasing in A. The intuition behind this is that for a very large number of n, funding is increasingly costly because the insurers need the capital market. The model is therefore adjusted and simulated again for the lognormal distribution. The increasing costs of capital are modelled such that for the regulatory minimal amount and lower, the costs are equal to the costs in the previous situation. For values of A that are larger, the costs are equal to δ A

Amin

2

, with Amin the

regulatory minimum. This function is convex in A, but takes into account that it is relatively more costly for small firms to attract capital than for big firms. The function is maximized over A for fixed n.

The results of the analytical and numerical analysis can be found in the next chapter.

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Chapter 4 Results and analysis

The model described in the previous chapter is analysed analytically and numer-ically in this chapter. In the first section, we prove that there is a maximum. In Sections 2 and 3, the model is solved numerically for a lognormal and a Bernoulli distribution. A model with claim distributions which are not independent is anal-ysed in Section 4. Finally, we look at the model for a lognormal distribution with increasing costs of capital.

4.1 The analytical maximum expected utility

We define the function

V (A) = E(u(w − P (A) − Xi+ 1S(A) · Xi)). (4.1)

The object of this thesis is to find the maximum of this function over A.

Theorem 4.1

The function V has a maximum on the domain [0, ∞).

In order to prove Theorem 4.1, four lemmas are needed.

Lemma 1

If a function h is concave and a function g is non-decreasing and concave, and f (x) = g(h(x)), f is concave.

This lemma is proven in Appendix A.

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Lemma 2

The function P(1S = 1) is concave on the domain [0, ∞).

This lemma is proven in Appendix A. Assumed is that the cumulative density function (CDF) of thePn

j=1Xj is twice differentiable, and the probability density

function (PDF) of the Pn

j=1Xj is non-increasing on the interval [0, ∞). These

assumptions hold ifPn

j=1Xj ∼ N (E(P Xj), σ), which is suggested by the Central

Limit Theorem. Assumed is also that δ is independent of A.

Lemma 3

The function V is concave on the domain [0, ∞).

In order to prove this, we assume that Xi and

PN

j=1Xj are independent. The

error made by this assumption is negligible for large n.

Proof Lemma 3

A function is concave if and only if

f ((1 − t)a1+ ta2) ≥ (1 − t)f (a1) + tf (a2), (4.2)

for all t ∈ [0, 1]. The function V is concave if

EXi,P(Xj)(u(w − 1 + δ − 99.5% n · 99.5% · ((1 − t)a1+ ta2) − EXi(Xi) − Xi+ 1S((1 − t)a1+ ta2) · Xi)) (4.3) ≥ (1 − t)EXi,P(Xj)(u(w − 1 + δ − 99.5% n · 99.5% · a1− EXi(Xi) − Xi+ 1S(a1) · Xi)) +tEXi,P(Xj)(u(w − 1 + δ − 99.5% n · 99.5% · a2− EXi(Xi) − Xi+ 1S(a2) · Xi)).

Because of independence, it holds that

EXi,P(Xj)(u(w − 1 + δ − 99.5% n · 99.5% · A − EXi(Xi) − Xi+ 1S(A) · Xi)) = P(1S(A) = 0) · EXi(u(w − 1 + δ − 99.5% n · 99.5% · A − EXi(Xi) − Xi)) + P(1S(A) = 1) · EXi(u(w − 1 + δ − 99.5% n · 99.5% · A − EXi(Xi)). (4.4)

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From Equation 4.3 and 4.4 we derive that V is concave if:

P(1S((1 − t)a1+ ta2) = 0)· EXi(u(w − 1 + δ − 99.5% n · 99.5% · ((1 − t)a1+ ta2) − EXi(Xi) − Xi)) +P(1S((1 − t)a1+ ta2) = 1)· EXi(u(w − 1 + δ − 99.5% n · 99.5% · ((1 − t)a1+ ta2) − EXi(Xi)) ≥ (4.5) (1 − t)P(1S(a1) = 0) · EXi(u(w − 1 + δ − 99.5% n · 99.5% · a1− EXi(Xi) − Xi)) +(1 − t)P(1S(a1) = 1) · EXi(u(w − 1 + δ − 99.5% n · 99.5% · a1− EXi(Xi))) +tP(1S(a2) = 0) · EXi(u(w − 1 + δ − 99.5% n · 99.5% · a2− EXi(Xi) − Xi)) +tP(1S(a2) = 1) · EXi(u(w − 1 + δ − 99.5% n · 99.5% · a2− EXi(Xi))).

From Equation 4.5 and Lemma 1 we derive that V is concave if

P(1S((1 − t)a1 + ta2) = 0)· EXi(w − 1 + δ − 99.5% n · 99.5% · ((1 − t)a1+ ta2) − EXi(Xi) − Xi) +P(1S((1 − t)a1 + ta2) = 1)· EXi(w − 1 + δ − 99.5% n · 99.5% · ((1 − t)a1+ ta2) − EXi(Xi)) ≥ (4.6) (1 − t)P(1S(a1) = 0) · EXi(w − 1 + δ − 99.5% n · 99.5% · a1− EXi(Xi) − Xi) +(1 − t)P(1S(a1) = 1) · EXi(w − 1 + δ − 99.5% n · 99.5% · a1− EXi(Xi)) +tP(1S(a2) = 0) · EXi(w − 1 + δ − 99.5% n · 99.5% · a2− EXi(Xi) − Xi) +tP(1S(a2) = 1) · EXi(w − 1 + δ − 99.5% n · 99.5% · a2− EXi(Xi)). Equation 4.6 holds if w −1 + δ − 99.5% n · 99.5% · ((1 − t)a1+ ta2) − EXi(Xi) − P(1S((1 − t)a1+ ta2) = 0) · EXi(Xi) ≥ (4.7) (1 − t)(w − 1 + δ − 99.5% n · 99.5% · a1− EXi(Xi)) − (1 − t)P(1S(a1) = 0) · EXi(Xi) t(w − 1 + δ − 99.5% n · 99.5% · a2− EXi(Xi)) − tP(1S(a2) = 0) · EXi(Xi). This holds if −P(1S((1 − t)a1+ ta2) = 0) · EXi(Xi) ≥ −(1 − t)P(1S(a1) = 0) · EXi(Xi) (4.8) −tP(1S(a2) = 0) · EXi(Xi).

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Equation 4.8 is true if 1 − P(1S((1 − t)a1+ ta2) = 0) = P(1S((1 − t)a1+ ta2) = 1) ≥ (4.9) 1 − (1 − t)P(1S(a1) = 0) − tP(1S(a2) = 0) = (1 − t)(1 − P(1S(a1) = 0)) + t(1 − P(1S(a2) = 0)) = (1 − t)P(1S(a1) = 1) + tP(1S(a2) = 1).

Equation 4.9 follows from Lemma 2, so this concludes the proof of Lemma 3.

Lemma 4

It holds that

lim

A→∞V (A) = −∞.

Proof Lemma 4

Bearing in mind that

lim

A→∞P = limA→∞E(Xi) +

1 + δ − 99.5%

n · 99.5% · A = ∞, and that

lim

A→∞1S = 1,

and given a utility function with u0 > 0,

lim

A→∞V (A) = limA→∞E(u(w − P − Xi+ 1SXi)) = E(u(w − ∞)) = −∞.

Proof Theorem 4.1

From Lemma 3 and 4 we can derive that the function V is concave on [0, ∞) and that the maximum is not at A = ∞. This means there is a maximum of V on the interval [0, ∞). This concludes the proof of Theorem 4.1. What this maximum is exactly, is not solved analytically in this thesis.

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4.2 Simulation with a lognormal claim

distribu-tion

Figure 4.1: The average utility for a variable amount of holding capital A and a various number of policy holders, using a lognormal claim distribution. The vertical line indicates the required amount of capital.

In order to answer the question which amount of capital is ideal for the policy holder in case of a lognormal claim distribution, the model is simulated for various values of n and A. The scripts used can be found in Appendix B. Assumed is that the losses have a lognormal distribution with mean 1 and variance 1. The cost of capital δ is set at 0.15. This value is realistic for insurance companies, according to historical estimates (Klumpes and Morgan, 2008). The initial wealth of the policy holder is set at 4, but this has no impact on the results, because we use the CARA utility function. The coefficient of risk aversion λ is set at 0.2. It is verified in Appendix C that with this coefficient the amount a policy holder is willing to pay for a certain insurance is realistic.

The results of 10,000 simulations are displayed in Figure 4.1 and 4.2. In both figures, the line at A = 1 represents the Solvency II regulatory benchmark, calculated using the normal approximation. As expected, an optimal amount of

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Figure 4.2: The optimal amount of capital held for a various number of policy holders, using a lognormal claim distribution. The horizontal line indicates the required amount of capital.

capital can be found. However, this seems to increase in n and is higher than the benchmark for every n in our simulation. For values of A greater than or equal to the regulatory minimum, the average utility is also higher for higher n.

4.3 Simulation with a Bernoulli claim

distribu-tion

In order to answer the question which amount of capital is ideal for the policy holder in case of a Bernoulli claim distribution, the model is simulated for various values of n and A. Assumed is that the losses have a Bernoulli distribution with a claim probability of 0.3 and a claim height of 3. The mean value of the claims is therefore 0.9 and the variance 1.89. The values of δ and λ are set the same as in the previous section.

The results of 100,000 simulations are displayed in Figure 4.3 and 4.4. These results are very simular to the results using the lognormal distribution. There is an optimal amount of capital which is higher than the benchmark, and the average utility is higher for the higher values of n.

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Figure 4.3: The average utility for a variable amount of A and a various number of policy holders, using a Bernoulli claim distribution. The vertical line indicates the required amount of capital.

Figure 4.4: The optimal amount of capital held for a various number of policy holders, using a Bernoulli claim distribution. The horizontal line indicates the required amount of capital.

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4.4 Simulation with a common shock effect

Figure 4.5: Fraction of solvent insurers for a variable amount of A and a various number of policy holders. The values of A used are relative to the minimal amount of capital needed in Section 4.2.

For the common shock effect, Xi is modelled as Z + Yi. Z has a lognormal

distribution with mean of 0.2 and a variance of 0.3. Y also has a lognormal distribution, but with mean 0.8 and variance 0.7. This means the mean and variance of X are 1, which is the same as in Section 4.2. In order to be able to compare the utilities, the values of δ, w and λ are also the same.

In Section 4.1 was proven that the expected utility function is concave. How-ever, the assumption that Xi and

Pn

j=1Xj are independent was used. This

as-sumption does not hold with a common shock effect, which means the function is not necessarily concave.

It is not possible to calculate the minimal amount of capital using the nor-mal approximation, because the Xi are not independent. Much more capital is

needed to ensure that the probability that the insurer is solvent is 99.5%. This is illustrated in Figure 4.5. For instance, for n=10,000, the capital needed is over a 100 times the capital that was needed in Section 4.2. This is very costly, and makes the premium so high that the expected utility of not being insured is higher

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Figure 4.6: The average utility of the policy holder when the required amount of capital is held for a various number of policy holders. The horizontal line indicates the average utility without insurance.

than the expected utility with insurance, as shown in Figure 4.6. This practically means that if the insurance company is unable to diversify this risk in another way, the risk is uninsurable.

Figure 4.7 shows the optimal amount of capital held. These are lower than the required amount, and do not meet the criteria of solvency. Table 4.8 shows the fractions of solvent insurers when the ideal amount of capital is held.

This raises the question if the benchmark is too high. If we set the benchmark at 95%, much less capital is needed, as we can see in Figure4.5. The average utility of the policy holder is now higher, as shown in Figure4.9. However, it is still lower than the average utility without insurance, so this risk is still uninsurable, even with a lower benchmark.

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Figure 4.7: The average utility for a variable amount of A and a various number of policy holders with the common shock effect.

Number of policy holders Fraction of insurers solvent Average utility in optimum

50 0.9762 -0.5774 100 0.9670 -0.5760 500 0.9620 -0.5782 1000 0.9792 -0.5764 2000 0.9462 -0.5750 3000 0.9472 -0.5755 4000 0.9712 -0.5742 5000 0.9682 -0.5756 6000 0.9504 -0.5747 7000 0.9554 -0.5750 8000 0.9720 -0.5750 9000 0.9772 -0.5760 10000 0.9788 -0.5758

Figure 4.8: Fraction of simulations where the insurer was solvent for the optimal amount of capital in the simulations.

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Figure 4.9: The average utility of the policy holder when the required amount of capital is held for a various number of policy holders, if the benchmark is set at 95% or 99.5%. The horizontal line indicates the average utility without insurance.

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4.5 Simulation with a lognormal claim

distribu-tion and increasing costs of capital

Figure 4.10: The average utility for a variable amount of A and a various number of policy holders, using a lognormal claim distribution and increasing costs of capital.

The optimal amount of capital might be different if the costs are increasing. In order to be able to compare this to the model where the costs are constant, the same values are used as in Section 4.2.

The results 100,000 simulations are displayed in Figure 4.10 and 4.11. The optimum still seems to increase in n. We also still see that for values of A greater than or equal to the regulatory minimum, the average utility is higher for higher n. However, the optimum does lie closer to the regulatory minimum.

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Figure 4.11: The optimal amount of capital held for a various number of policy holders, using a lognormal claim distribution and increasing costs of capital. The horizontal line indicates the required amount of capital.

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Conclusion

In order to answer the central question whether there is an optimal amount of holding capital for the ensurer, we proved that, for independent claim distribu-tions, it is possible to maximize the expected utility of the policy holder over the amount of capital held. This is possible because this is a concave function which goes to −∞ as A goes to ∞.

However, it is not easy to find an answer to the question what that optimum is. The benchmark set by the Solvency regulation is not always optimal. For a lognormal and Bernoulli claim distribution, the optimal amount of capital is relatively increasing in n, even if the costs of attracting this capital are also rising. For larger n, the optimal amount of capital was found to be larger than the benchmark. It was also found that for the same relative amount of capital, the average utility was higher for larger n.

When a model with a common shock effect is used, the situation is more complicated. Because calculating the benchmark correctly is problematic, the benchmark was found through simulation. Unfortunately, the capital needed to comply with the regulation was so high that the policy holder was better off without insurance. This is still the case if the benchmark is set at 95% solvency. This means that the risk is uninsurable if there is no other way of diversification possible.

5.1 Suggestions for further research

It would be interesting to research whether the losses in a common shock model are still uninsurable if an indemnity function I would be used. There would be many ways to model this, which might lead to an insurance complient with the regulation and also favorable for the policy holder.

Another possibility for further research would be to see how robust the opti-mum is for costs of capital. If these costs are very low the benchmark might be

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much lower than ideal for the policy holder, whereas the insurability of certain losses might be at stake if the costs are very high.

Furthermore, it would be interesting to see what would happen if the assump-tion of identically distributed losses was forfeited. This would be realistic for car insurance, among others. It is likely that this would also lead to differences in premium.

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Proof lemma 1

We need to prove that

f ((1 − t)a + tb) ≥ (1 − t)f (a) + tf (b),

for all a,b with a≤ b and t ∈ [0, 1]. We get that

f ((1 − t)a + tb) = g(h((1 − t)a + b)). Since h is concave, h((1 − t)a + tb) ≥ (1 − th(a) + th(b). Since g is non-decreasing, g(h((1 − t)a + tb)) ≥ g((1 − t)h(a) + th(b)). Since g is concave,

g((1 − t)h(a) + th(b) ≥ (1 − t)g(h(a)) + tg(h(b)) = (1 − t)f (a) + tf (b).

So f ((1 − t)a + tb) ≥ (1 − t)f (a) + tf (b). This concludes the proof.

Proof lemma 2

P(1S = 1) = P(A + nP − n X j=1 Xj ≥ 0) = P(1 + δ − 99.5% 99.5% · A + E( n X j=1 Xj) + A − n X j=1 Xj ≥ 0) = P( n X j=1 Xj ≤ E( n X j=1 Xj) + 1 + δ 99.5% · A) = FPn j=1Xj E( n X j=1 Xj) + 1 + δ 99.5%· A, 22

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with FPn

j=1Xj twice differentiable and f non-increasing.

∂ ∂AF Pn j=1Xj E( n X j=1 Xj)+ 1 + δ 99.5%·A = f Pn j=1Xj(E( n X j=1 Xj)+ 1 + δ 99.5%·A)· 1 + δ 99.5% > 0.

These probabilities are greater than zero by definition.

∂2 ∂2_AFPnj=1Xj(E( n X j=1 Xj)+ 1 + δ 99.5%·A) = ∂ ∂Af Pn j=1Xj((E( n X j=1 Xj)+ 1 + δ 99.5%·A)· 1 + δ 99.5% = fP0 n j=1Xj(E( n X j=1 Xj) + 1 + δ 99.5% · A) · ( 1 + δ 99.5%) 2_.

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Matlab script 1: Lognormal distribution

function [ U1, S ] = Logn( EX, EVX, n, p, A )

%Given a lognormal distribution, an expected value of x and %expected variance, this function simulates n xs and gives the %utility of a random policy holder (the first), and answers the %question whether the insurer has defaulted or not.

% set parameter values w=4; lambda=0.2; % generate x’s mu = log((EX^2)/sqrt(EVX+EX^2)); sigma = sqrt(log(EVX/(EX^2)+1)); X = lognrnd(mu,sigma,1,n);

% calculating whether the insurer is solvent SX = sum(X); if A + n*p - SX < 0; S=0; else S=1; end

% find utility of first policy holder U1 = -exp(-lambda*(w-p-X(1)+S*X(1)));

end

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Matlab scipt 2: Simulation with a lognormal claim

distribution

% set variables n = [50 100 500 1000 2500 5000 7500 10000]; optA=zeros(1,length(n)); reloptA=zeros(1,length(n)); optU=zeros(1,length(n))-10; optP=zeros(1,length(n)); optPerc=zeros(1,length(n)); Avar = 0.5:0.05:3; nsim = 10000; EX = 1; EVX = 1; delta=0.15; qmin=0.005; percS=zeros(1,length(Avar)); UAv=zeros(length(n),length(Avar)); p=zeros(length(n),length(Avar)); %Simulation for ii=1:length(n)

%find values of A and corresponding Premium

Amin = norminv(1-qmin)*sqrt(n(ii))*sqrt(EVX)/(1+delta); A=Amin*Avar; for i= 1:length(A); p(ii,i)=((1+delta-(1-qmin))*A(i))/(n(ii)*(1-qmin))+EX; Usum=0; SumS=0; for j=1:nsim; O=cell(1,2);

[O{:}] = Logn(EX, EVX, n(ii), p(ii,i), A(i)); Usum=Usum+O{1};

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end UAv(ii,i)=Usum/nsim; if (UAv(ii,i) > optU(ii)) optU(ii)=UAv(ii,i); optA(ii)=A(i); reloptA(ii)=Avar(i); optP(ii)=p(ii,i); optPerc(ii)=SumS/nsim; end end end

%Plotting the average utility against the relative amount of %capital figure plot(Avar,UAv(1,:),’:o’) hold on plot(Avar,UAv(2,:),’:o’) plot(Avar,UAv(3,:),’:o’) plot(Avar,UAv(4,:),’:o’) plot(Avar,UAv(8,:),’:o’) plot([1 1], [-0.6 -0.54]) axis([0.3 3.1 -0.6 -0.54]) legend(’n=50’,’n=100’, ’n=500’, ’n=1000’, ’n=10000’, ’Location’, ’southeast’) xlabel(’Relative amount of capital held’)

ylabel(’Average utility in simulations’) hold off

%Plotting the optimal amount of capital against the number of %policy holders figure plot(n,reloptA,’:o’) hold on plot([0 10000], [1 1]) axis([0 10000 0.6 2])

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ylabel(’Relative amount of capital held’) hold off

Matlab script 3: Bernoulli distribution

function [ U1, S ] = Bin(pclaim,claim, n, p, A)

%Given a binomial distribution and an expected value of x %this function simulates n xs and gives the utility of the

%first policy holder, and answers the question whether the insurer %has defaulted or not.

% set parameter values w=4;

lambda=0.2;

% generate x’s

X = claim*binornd(1,pclaim,1,n);

% find utility of first policy holder U1 = -exp(-lambda*(w-p-X(1)+S*X(1))); end

Matlab script 4: Simulation with a Bernoulli

dis-tribution

% set variables

n = [50 100 500 1000 2500 5000 7500 10000]; optA=zeros(1,length(n));

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reloptA=zeros(1,length(n)); optU=zeros(1,length(n))-20; optP=zeros(1,length(n)); optPerc=zeros(1,length(n)); Avar = 0.5:0.05:3; nsim = 100000; pclaim=0.3; claim=3; EX = pclaim*claim; EVX = pclaim*(1-pclaim)*claim^2; delta=0.15; qmin=0.005; percS=zeros(1,length(Avar)); UAv=zeros(length(n),length(Avar)); p=zeros(length(n),length(Avar)); %Simulation for ii=1:length(n)

%find values of A and corresponding Premium

Amin = norminv(1-qmin)*sqrt(n(ii))*sqrt(EVX)/(1+delta); A=Amin*Avar; for i= 1:length(A); p(ii,i)=((1+delta-(1-qmin))*A(i))/(n(ii)*(1-qmin))+EX; Usum=0; SumS=0; for j=1:nsim; O=cell(1,2);

[O{:}] = Bin(pclaim ,claim , n(ii), p(ii,i), A(i)); Usum=Usum+O{1}; SumS=SumS+O{2}; end UAv(ii,i)=Usum/nsim; if (UAv(ii,i) > optU(ii)) optU(ii)=UAv(ii,i); optA(ii)=A(i);

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Optimal holding capital — Irene Doelman 29 reloptA(ii)=optA(ii)/Amin; optP(ii)=p(ii,i); optPerc(ii)=SumS/nsim; end end end

%Plotting the optimal amount of capital against the number of %policy holders figure plot(n,reloptA,’:o’) hold on plot([0 10000], [1 1]) axis([0 10000 0.9 1.4])

xlabel(’Number of policy holders’)

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Matlab script 5: Common shock

function [ U1, S ] = CS( EY, EVY, EZ, EVZ, n, p, A )

% set parameter values w=4; lambda=0.2; % generate z muz = log((EZ^2)/sqrt(EVZ+EZ^2)); sigmaz = sqrt(log(EVZ/(EZ^2)+1)); Z = lognrnd(muz,sigmaz,1,1); % generate Y’s muy = log((EY^2)/sqrt(EVY+EY^2)); sigmay = sqrt(log(EVY/(EY^2)+1)); Y = lognrnd(muy,sigmay,1,n); % calculating Xi X=Z+Y;

% find utility of first policy holder U1 = -exp(-lambda*(w-p-X(1)+S*X(1)));

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Matlab script 6: Simulation with the common

shock effect

% set variables n=[50 100 500 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000]; optA=zeros(1,length(n)); reloptA=zeros(1,length(n)); optU=zeros(1,length(n))-10; optP=zeros(1,length(n)); optPerc=zeros(1,length(n)); Avar = 1:1:120; nsim = 100000; EZ = 0.2; EVZ = 0.3; EY = 0.8; EVY = 0.7; delta=0.15; qmin=0.005; % or 0.05 percS=zeros(length(n),length(Avar)); UAv=zeros(length(n),length(Avar)); p=zeros(length(n),length(Avar)); H=zeros(1,length(n)); Abench=zeros(1,length(n)); Ubench=zeros(1,length(n)); %Simulation for ii=1:length(n)

%calculate premium and find average utility

Amin = norminv(1-qmin)*sqrt(n(ii))*sqrt(EVY+EVZ)/(1+delta); A=Amin*Avar; for i= 1:length(A); p(ii,i)=((1+delta-(1-qmin))*A(i))/(n(ii)*(1-qmin))+EY+EZ; Usum=0; SumS=0;

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for j=1:nsim; O=cell(1,2);

[O{:}]=CS(EY, EVY, EZ, EVZ, n(ii), p(ii,i), A(i)); Usum=Usum+O{1};

SumS=SumS+O{2}; end

UAv(ii,i)=Usum/nsim; percS(ii,i)=SumS/nsim;

if (H(ii) == 0) && (percS(ii,i)>(1-qmin)) H(ii)=1; Abench(ii)=Avar(i); Ubench(ii)=UAv(ii,i); end if (UAv(ii,i) > optU(ii)) optU(ii)=UAv(ii,i); optA(ii)=A(i); reloptA(ii)=Avar(i); optP(ii)=p(ii,i); optPerc(ii)=SumS/nsim; end end end

%Simulating utility without insurance UsumNI=0;

for k=1:nsim;

UsumNI=UsumNI+CS(EY,EVY,EZ,EVZ,1,0,0); end

UNI=UsumNI/nsim;

%Plotting fraction of solvent insurers against the capital held %for a few values of n

figure plot(Avar,percS(1,:),’:o’) hold on plot(Avar,percS(3,:),’:o’) plot(Avar,percS(8,:),’:o’) plot(Avar,percS(13,:),’:o’)

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axis([0 120 0.90 1])

legend(’n=50’,’n=500’,’n=5000’, ’n=10000’, ’Location’,’southeast’) xlabel(’Relative amount of capital held’)

ylabel(’Fraction of solvent insurers’) hold off

%Plotting the average utility against number of policy holders figure

plot(n,Ubench,’o:’) hold on

plot([0 10000], [UNI UNI]) axis([0 10000 -0.75 -0.5]) hold off

ylabel(’Average utility in simulations’)

%Plotting the optimal amount of capital against number of policy %holders

figure

plot(n,reloptA,’*:’,n,Abench,’o:’) xlabel(’Number of policy holders’)

ylabel(’Relative amount of capital held’)

legend(’Optimal’,’Required’,’Location’,’northwest’)

%Plotting optimal utility against amount of policy holders figure

plot(n,optU,’o:’)

ylabel(’Average utility for optimal capital’)

Matlab script 7: Simulation with a lognormal

dis-tribution and increasing costs of capital

% set variables

n = [50 100 500 1000 2500 5000 7500 10000]; optA=zeros(1,length(n));

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reloptA=zeros(1,length(n)); optU=zeros(1,length(n))-10; optP=zeros(1,length(n)); optPerc=zeros(1,length(n)); Avar = 0.5:0.05:3; nsim = 100000; EX = 1; EVX = 1; qmin=0.005; percS=zeros(1,length(Avar)); UAv=zeros(length(n),length(Avar)); p=zeros(length(n),length(Avar)); %Simulation for ii=1:length(n)

%find values of A and corresponding Premium deltaAmin=0.15; Amin = norminv(1-qmin)*sqrt(n(ii))*sqrt(EVX)/(1+deltaAmin); A=Amin*Avar; delta=deltaAmin*(max(Avar,1)).^2; for i= 1:length(A); p(ii,i)=((1+delta(i)-(1-qmin))*A(i))/(n(ii)*(1-qmin))+EX; Usum=0; SumS=0; for j=1:nsim; O=cell(1,2);

[O{:}] = Logn(EX, EVX, n(ii), p(ii,i), A(i)); Usum=Usum+O{1}; SumS=SumS+O{2}; end UAv(ii,i)=Usum/nsim; if (UAv(ii,i) > optU(ii)) optU(ii)=UAv(ii,i); optA(ii)=A(i); reloptA(ii)=optA(ii)/Amin;

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Optimal holding capital — Irene Doelman 35 optP(ii)=p(ii,i); optPerc(ii)=SumS/nsim; end end end

%Plotting the average utility against the premium figure

plot(n,reloptA,’:o’) hold on

plot([0 10000], [1 1]) axis([0 10000 0.8 1.5])

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In order to find a realistic value of λ, 100,000 numbers are drawn from a log-normal distribution with mean 1 and variance 1. Next, the average utility over the simulation was calculated as if these were the losses, for various values of λ. Then the price was calculated for which the utility was equal to this average (for a 100% solvent insurer). The different prices are displayed in Table 5.1. From these prices, we find that λ = 0.2 is realistic.

λ p 0.01 0.9985 0.05 1.0193 0.1 1.0489 0.2 1.1269 0.4 1.5382 0.5 2.2098 1 9.1085

Figure 5.1: Prices policy holders are willing to pay with various coefficents of risk aversion.

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References

Bernard, C. & Ludkovski, M. (2012). Impact of counterparty risk on the reinsur-ance market. North American Actuarial Journal, 16 (1), 87-111.

Biffis, E. & Millossovich, P. (2012). Optimal insurance with counterparty default risk. Available at SSRN 1634883.

Cummins, J. & Mahul, O. (2003). Optimal insurance with divergent beliefs about insurer total default risk. Journal of Risk and Uncertainty, 27 (2), 121-138. European Commissuon (EC) (2009). Directive of the European Parliament and

the Council on the taking-up and pursuit of the business of Insurance and Rein-surance (Solvency II)(2009/138/EC). Official Journal of the European Union, L 335/1.

European Insurance and Occupational Pensions Authority. (2014). The underlying assumptions in the standard formula for the Solvency Cap-ital Requirement calculation. Frankfurt, Germany. Retrieved from https://eiopa.europa.eu/Publications/Standards/EIOPA-14-322

Underlying Assumptions.pdf

Filipovi´c, D., Kremslehner, R., & Muermann, A. (2014). Optimal investment and premium policies under risk shifting and solvency regulation. Journal of Risk and Insurance. Forthcoming.

Ibragimov, R., Jaffee, D., & Walden, J. (2010). Pricing and capital allocation for multiline insurance firms. Journal of Risk and Insurance, 77 (3), 551-578. Klumpes, P. J., & Morgan, K. (2008). Solvency II versus IFRS: Cost of Capital

Implications for Insurance Firms. Proceedings of the 2008 ASTIN Colloquium.

Optimal holding capital for insurance companies

companies

Irene Doelman

Abstract

Contents

Chapter 1

Introduction

Assumptions

2.1

The insurer

2.2

The policy holder

The model

3.1

The standard model

3.2

Adjustments to the model

Chapter 4

Results and analysis

4.1

The analytical maximum expected utility

4.2

Simulation with a lognormal claim

distribu-tion

4.3

Simulation with a Bernoulli claim

distribu-tion

4.4

Simulation with a common shock effect

4.5

Simulation with a lognormal claim

distribu-tion and increasing costs of capital

Conclusion

5.1

Suggestions for further research

Proof lemma 1

Proof lemma 2

Matlab script 1: Lognormal distribution

Matlab scipt 2: Simulation with a lognormal claim

distribution

Matlab script 3: Bernoulli distribution

Matlab script 4: Simulation with a Bernoulli

dis-tribution

Matlab script 5: Common shock

Matlab script 6: Simulation with the common

shock effect

Matlab script 7: Simulation with a lognormal

dis-tribution and increasing costs of capital

References