The Impact of Systemic Risk on the Diversification Benefits of a Risk Portfolio

Risk diversification is the basis of insurance and investment. It is thus crucial to study the effects that could limit it. One of them is the existence of systemic risk that affects all the policies at the same time. We introduce here a probabilistic approach to examine the consequences of its presence on the risk loading of the premium of a portfolio of insurance policies. This approach could be easily generalized for investment risk. We see that, even with a small probability of occurrence, systemic risk can reduce dramatically the diversification benefits. It is clearly revealed via a non-diversifiable term that appears in the analytical expression of the variance of our models. We propose two ways of introducing it and discuss their advantages and limitations. By using both VaR and TVaR to compute the loading, we see that only the latter captures the full effect of systemic risk when its probability to occur is low.


Introduction
Every financial crisis reveals the importance of systemic risk and, as a consequence, the lack of diversification.Diversification is a way to reduce the risk by detaining many different risks, with various probabilities of occurrence and a low probability of happening simultaneously.
Unfortunately, in times of crisis, most of the financial assets move together and become very correlated.The 2008/09 crisis is not an exception.It has highlighted the interconnectedness of financial markets when they all came to a stand still for more than a month waiting for the authorities to restore confidence in the system (see e.g. the Systemic Risk Survey of the bank of England available on line).For any financial institution, it is important to be aware of the limits to diversification, while, for researchers in this field, studying the mechanisms that hamper diversification is crucial for the understanding of the dynamics of the system (see e.g.[2], [3] and references therein).Both risk management and research on risk would enhance our capacity to survive the inevitable failures of diversification.A small fact, like turmoils in the sub-prime in the US real-estate market, a relatively small market compared to the whole US real-estate market, can trigger a major financial crisis that extends to all markets and all regions in the globe.Systemic risk manifests itself by a breakdown of the diversification benefits and the appearance of dependence structures that were not deemed important during normal times.
In this study, we introduce a simple stochastic modelling to understand and point out the limitations to diversification and the mechanism leading to the occurrence of systemic risk.The idea is to combine two generating stochastic processes that, through their mixture, produces in the resulting process a non-diversifiable component, which we identify to a systemic risk.
Depending on the way of mixing these processes, the diversification benefit appears with various strengths due to the emergence of the systemic component.The use of such a model, which is completely specified, allows us to obtain analytical expressions for the variance, and then to identify the non-diversifiable term.With the help of Monte Carlo simulations, we explore the various components of the model and check that we reproduce the analytical results.
The paper is organized as follows: in a first section, we introduce, to measure the effects on diversification, the standard insurance framework for pricing risk and define the risk measures that are used in this study.The second section is dedicated to the mathematical presentation of the model and its various approaches to systemic risk, as well as numerical applications.The obtained results are compared numerically and analytically in the third section, where we also discuss the influence of the choice of risk measure on the diversification benefits.We conclude the study and suggest new perspectives to extend it.

Insurance framework
Before moving to stochastic modelling, let us introduce the insurance framework in which we are going to compute the risk diversification.It is an example of application, but this study on systemic risk could easily be generalized to any financial institution.

The technical risk premium
In insurance, risk is priced based on the knowledge of the loss probability distribution.Let L be the random variable (rv) representing a loss defined on a probability space (Ω, A, IP).

• One policy case
For any policy incurring a loss L (1) , we can define, as in [4], the technical premium, P , that needs to be paid, as: η: the return expected by shareholders before tax K: the risk capital assigned to this policy e: the expenses incurred by the insurer to handle this case.
An insurance is a company in which we can invest.Therefore the shareholders that have invested a certain amount of capital in the company expect a return on investment.So the insurance firm has to make sure that the investors receive their dividends, which corresponds to the cost of capital the insurance company must charge on its premium.
This is what we have called η.
We will assume that the expenses are a small portion of the expected loss e = aIE[L (1) ] with 0 < a < < 1 , which transforms the premium as We can now generalize this premium principle (2) for N similar policies (or contracts).
• Case of a portfolio of N policies The premium for one policy in the portfolio, incurring now a total loss L (N ) = N × L (1) , can then be written as where K N is the capital assigned to the entire portfolio.

Cost of Capital and Risk Loading
First we have to point out that the role of capital for an insurance company is to ensure that the company can pay its liability even in the worst cases, up to some threshold.For this, we need to define the capital we have to put behind the risk.We are going to use a risk measure, say ρ, defined on the loss distribution.It allows to estimate the capital needed to ensure payment of the claim up to a certain confidence level.We then define the risk-adjusted-capital K as a function of the risk measure ρ associated to the risk1 L as since the risk is defined as the deviation from the expectation.
Note that we could have also defined K as K = ρ(L) − IE[L] − P since the premiums can serve to pay the losses.It would change the premium P defined in (1) into P defined by Such an alternative definition would reduce the capital but does not change fundamentally the results of the study.
Consider a portfolio of N similar policies, using the notation for the loss as in § 2.1.Let R denote the risk loading per policy, defined as the cost of the risk-adjusted-capital per policy.
Using (3), R can be expressed as a function of the risk measure ρ, namely

Risk Measures
We will consider for ρ two standard risk measures, the Value-at-Risk (VaR) and the tail Valueat-Risk (TVaR).Let us remind the definitions of these quantities (see e.g.[7]).
The Value-at-Risk with a confidence level α is defined for a risk L by where q is the level of loss that corresponds to a VaR α (simply the quantile of L of order α), and F L the cdf of L.
The tail Value-at-Risk at a confidence level α of L satisfies When considering a discrete rv L, it can be approximated by a sum, which may be seen as the average over all losses larger than VaR α : where q u i (L) = VaR u i (L) and ∆u i ≡ u i − u i−1 corresponds to the probability mass of the particular quantile q u i .

Stochastic modelling
Suppose an insurance company has underwritten N policies of a given risk.To price these policies, the company must know the underlying probability distribution of this risk, as seen in the previous section.In this study, we assume that each policy is exposed n times to this risk.
In a portfolio of N policies, the risk may occur n × N times.So, we introduce a sequence (X i , i = 1, . . ., N n) of rv's X i defined on (Ω, A, IP) to model the occurrence of the risk, with a given severity l.Note that we choose a deterministic severity, but it could be extended to a random one, with a specific distribution.
Hence the total loss amount, denoted by L, associated to this portfolio is given by

A first model, under the iid assumption
We start with a simple model, considering a sequence of independent and identically distributed (iid) rvs X i 's.Let X denote the parent rv and assume that it is simply Bernoulli distributed, i.e. the loss lX occurs with some probability p: Hence the total loss amount L = l S N n of the portfolio is modeled by a binomial distribution B(N n, p): with We are interested in knowing the risk premium the insurance will ask to a customer if he buys this insurance policy.So we compute the cost of capital given in (4) for an increasing number N of policies in the portfolio, which becomes for this model: since the notation L (N ) in ( 4) has been simplified to L in this section.
Note that the relative risk per policy defined by the ratio R/IE[L (1) ] is given by Numerical application.
We compute the quantities of interest by fixing the various parameters.We choose for instance the number of times one policy is exposed to the risk to be n = 6.Then we take as cost of capital η = 15%, which is a reasonable value given the fact that the shareholders will obtain a return on investment after taxes of approximately 10%, when considering a standard tax rate of 30%.For a discussion on the choice of the value of η, we refer to [4].The unit loss l will be fixed to l = 10.We choose in the rest of the study α = 99% for the threshold of the risk measure ρ.
We present in Table 1 an example of the distribution of the loss L = lS 1n for one policy (N = 1) when taking e.g.p = 1/n = 1/6 (same probability for each of the n exposures).This would be the typical distribution for the outcome of a particular value when throwing a dice (see [1]).
The expected total loss amount is given by IE[L] = lIE[S 1n ] = 10.We see that there is a 26.32% probability (corresponding to IP[S 1n > 10] = 1 − IP[S 1n ≤ 10]) that the company will turn out paying more than the expectation.Thus, we cannot simply ask the expected loss as premium.
This justifies the premium principle adopted in § 2.1.Now, we compute the cost of capital per policy given in (4) as a function of the number N of policies in the portfolio.The results are displayed in Table 2 for both risk measures VaR and TVaR, and when taking p = 1/6, 1/4 and 1/2, respectively.The expected total loss amount Note that, when considering a large number N of policies, the binomial distribution of L could be replaced by the normal distribution N (N np, N np(1 − p)) (for N n ≥ 30 and p not close to 0, nor 1; e.g.np > 5 and n(1 − p) > 5) using the Central Limit Theorem (CLT).The VaR of order α of L could then be deduced from the αth-quantile, q α , of the standard normal distribution N (0, 1), as: Thus the risk loading R would become, in the case of ρ being VaR: ever smaller as a function of N .
We can see in Table 2 that the risk loading drops practically by a factor 100 for a portfolio of 10'000 policies, compared with the one computed for one policy (N = 1) that represents 30% of the loss expectation (IE[L] = 10 in this case).We notice also numerically that, if R increases with p, the relative risk per policy R/IE[L (1) ] decreases when p increases.When considering the Gaussian approximation and the explicit VaR given in (10), the relative risk per policy when choosing ρ =VaR, is, as a function of p, of the order 1/ √ p, giving back the numerical result.
Finally, it is worth noting that the risk loading with TVaR is always slightly higher than with VaR for the same threshold, as TVaR goes beyond VaR in the tail of the distribution.
In this setting, a fair game is defined by having an equal probability of losing at each exposure: p = 1/n.The biased game will be when the probability differs from 1/n, generally bigger.We can thus define two states, one with a "normal" or equilibrium state (p = 1/n) and a "crisis" state with a probability q >> p.In the next section, we will introduce this distinction.

Introducing a structure of dependence to reveal a systemic risk
We propose two examples of models introducing a structure of dependence between the risks, in order to explore the occurrence of a systemic risk and, as a consequence, the limits to diversification.We still consider the sequence (X i , i = 1, . . ., N n) to model the occurrence of the risk, with a given severity l, for N policies, but do not assume anymore that the X i 's are iid.

A dependent model, but conditionally independent
We assume that the occurrence of the risks X i 's depends on another phenomenon, represented by a rv, say U .Depending on the intensity of the phenomenon, i.e. the values taken by U , a risk X i has more or less chances to occur.Suppose that the dependence between the risks is totally captured by U .Consider, w.l.o.g., that U can take two possible values denoted by 1 and 0; U can then be modeled by a Bernoulli B(p), 0 < p << 1.The rv U will be identified to the occurrence of a state of systemic risk.So if p could mathematically take any value between 0 and 1, but we choose it here to be very small since we want to explore rare events.We still model the occurrence of the risks with a Bernoulli, but with a parameter depending on U .Since U takes two possible values, the same holds for the parameters of the Bernoulli distribution of the conditionally independent rv's X i | U , namely We choose q >> p, so that whenever U occurs (i.e.U = 1), it has a big impact in the sense that there is a higher chance of loss.We include this effect in order to have a systemic risk (non-diversifiable) in our portfolio.
Looking at the total amount of losses S N n , its distribution can then be written, for k ∈ IN, as The conditional and independent variables, Sq := S N n | (U = 1) and Sp := S N n | (U = 0), are distributed as Binomials B(N n, q) and B(N n, p), with mass probability distributions denoted by f Sq and f Sp respectively.The mass probability distribution f S of S N n appears as a mixture of f Sq and f Sp (see e.g.[6]): Note that p = 0 gives back the normal state, developed in § 3.1.
The expected loss amount for the portfolio, denoting L = L (N ) , is given by whereas for each policy, it is from which we deduce the risk loading defined in (3) and ( 4).
Let us evaluate the variance var(S N n ) of S N n .Straightforward computations (see [1]) give which, combined with (12), provides from which we deduce the variance for the loss of one contract as 1 Notice that in the variance for one contract, the first term will decrease as the number of contracts increases, but not the second one.It does not depend on N and thus represents the non-diversifiable part of the risk.Numerical application.
For this application, we keep the same parameters n = 6 and p = 1/n as in § 3.1 , and we choose the loss probability during the crisis to be q = 1/2.We explore different probabilities p of occurrence of a crisis.The calculation consists in mixing the two Binomial distributions, according to (11), for an increasing number of policies N .Results for the two choices of risk

Conclusion
In this study, we have shown the effect of diversification on the pricing of insurance risk through a first simple modeling.Then, for understanding and analyzing possible limitations to diversification benefits, we propose two alternative stochastic models, introducing dependence between risks by assuming the existence of an underlying systemic risk.These models, defined with mixing distributions, allow for a straightforward analytical evaluation of the impact of the nondiversifiable part, which appears in the close form expression of the variance.We have adopted here purposely a probabilistic approach for modelling the dependence and the existence of systemic risk.It could be easily generalized to a time series interpretation by assigning a time-step to each exposure n.In the last model, the occurrence of the rv U = 1 could then be identified to the time of crisis.
In real life, insurers have to pay special attention to the effects that can weaken the diversification benefits.For instance, in the case of motor insurance, the appearance of a hail storm will introduce a "bias" in the usual risk of accident due to a cause independent of the car drivers, which will hit a big number of cars at the same time and thus cannot be diversified among the various policies.There are other examples in life insurance for instance with pandemic or mortality trend that would affect the entire portfolio and cannot be diversified away.Special care must be given to those risks as they will affect greatly the risk loading of the premium as can be seen in our examples.These examples might also find applications for real cases.This approach can be generalized to investments and banking; both are subject to systemic risk, although of different nature than in the above insurance examples.
The last model we suggested, introducing the occurrence of crisis, may find an interesting application for investment and the change in risk appetite of investors.It will be the subject of a following paper.Moreover, the models we introduce here allow to point out explicitly to the impact of dependence; they are simple enough to compute analytic expression and analyze the impact of the emergence of systemic risks.Yet, they are formulated in such a way that extensions to more sophisticated models, are easy and clear.In particular, it makes possible to obtain an extension to non identically distributed rv's, or when considering random severity.
Another interesting perspective would be to consider econometric models with multiple states.

Table 1 :
The loss distribution for one policy with n = 6 and p = 1/6.

Table 2 :
The Risk loading per policy as a function of the number N of policies in the portfolio (with n = 6).

Table 3 :
For Model (11), the Risk loading per policy as a function of the probability of occurrence of a systemic risk in the portfolio using VaR and TVaR measures with α = 99%.The probability of giving a loss in a state of systemic risk is chosen to be q = 50%.