An Inequality for Reinsurance Contract Annual Loss Standard Deviation and Its Application (2024)

1. Introduction

In reinsurance industry, simulated losses from catastrophe events combined with reinsurance contract financial terms are used to calculate the contract expected annual loss, the standard deviation of the expected annual loss, and quantiles of the losses (such as the AEP: Aggregate Exceedance Probability, or TVaR: Tail Value at Risk, in Ref. [1]). These numbers are in turn used for pricing or risk management of the contract. There are two kinds of model risks in this approach: independent simulations may give different results without any model change, and simulations vary before or after model change such as from the yearly catastrophe events sets or parameters updates. Empirically, the distribution of the contract expected annual loss may be more like a Beckmann, MaxStable, Gamma, Inverse Gaussian, or even a Lognormal Distribution than a Normal Distribution, but consider that the mean annual loss is the average of a large number of losses, and especially for simplicity, if we assume they obey a Normal Distribution, to quantify the model risk, we can use Hinkley formula [2] or Marsaglia formula [3] to calculate the probability of two simulations have expected annual loss deviated from each other by more than say 50%. Their formulas, in our model risk quantification context and the simplest scenario, depend on only two factors: the correlation coefficient ρ of the distribution in the two simulations, and the coefficient of variation (CV) of the distribution, that is, the ratio of the standard deviation to the mean. Most reinsurance contracts, due to the carefully selected financial terms, have many or the majority of simulated year’s losses zero. Thus call for a study of the CV range or bounds of those scarcely payout contracts.


2. Results

2.1. CV range

Starting from Hölder’s inequality (ölder's_inequality):


suppose f in the formula is the contract annual loss with mean mf deducted, that is, is of the form fmf and p=q=2, g is some nonnegative weights on the discrete probability space of the simulated years Ω, for example, the set {1,2,…,100,000} for 100,000years of simulations, each element with a probability of 1e-5 (a typical setting in practice). Then we get:


Suppose f is nonnegative and zero outside of a subset A of Ω and g is constant a on A and constant b on AC: f|AC=0, f|A>0, g|AC=b≥0, g|A=a≥0. Then we can deduct that:


due to






The maximum of the right-hand side of Eq. (3) is achieved by ab=μACμA, increasing on [0, μACμA] and decreasing on μACμA, and then we get:


As a corollary, if we use the 2-sigma or 3-sigma rule for the confidence interval of the estimate of the mean, for those contracts that have most years 0 losses, these interval will be very large. For example, if we have only 100years have nonzero losses out of the 100,000 simulated years, then we will get [math]::sqrt((1e5-100)/100)=31.6069612585582:


And if we have 1000years nonzero losses, we get the constant [math]::sqrt((1e5-1000)/1000)=9.9498743710662:


Checking against some concrete examples, for a contract we see 2220 nonzero losses records, suppose all of them are in different years, then we get the ratio [math]:: sqrt((1e5-2220)/2220)=6.63664411016931, and we have its mean loss mf=3848 and standard deviation σf=37,149 from the simulation, 37,149/3848=9.65410602910603>6.63664411016931.

Another contract have unique 41,143 nonzero losses years out of the 100,000 simulated years, with mean loss mf=1,874,487 and standard deviation σf=2,357,787, σf/mf=1.25783054243641>[math]::sqrt((1e5-41,143)/41,143)=1.19605481319037.

From these examples, we see that our lower bound formula for CV is relatively close.

Typically, more than half of the contracts may have nonzero losses years count below 10,000, for which we can see their σf≥3mf, since [math]::sqrt((1e5-1e4)/1e4)=3. More than 80% of the contracts may have nonzero losses years count below 50,000, for which there is the simple inequality σfmf, since sqrt((1e5-5e4)/5e4)=1. These bounds are collected in Table 1.

Nonzero losses yearsCV lower boundμACμA

Table 1.

CV lower bound.

For given years of nonzero losses out of 100,000 simulated years, the lower bound as given by our formula Eq. (8).

This inequality Eq. (8) can explain the observation that when we sort the contracts by the mean annual loss, the lower quarter of contracts may have more than tens of percent deviation from different simulations, since smaller mean loss usually corresponding to fewer years of nonzero losses and higher CV (more explanation in the following section).

To get an upper bound for CV, suppose the total simulated years is n and each year has the loss xi≥0. Then:


For the expression i=1nxi2i=1nxi2, by taking the partial derivative with respect to x1, we can know that it is decreasing in 0i=2nxi2i=2nxi and increasing in i=2nxi2i=2nxi. So the maximum must be attained at either 0 or ∞, that is, is the value of i=2nxi2i=2nxi or 1. Recursively, we know that the final maximum is 1. So we get:


For the extreme case when only one year have nonzero losses, we thus verified that the coefficient [math]::sqrt((1e5-1)/1)=316.226184874055 is exact, that is, we have:


The overall upper bound n1 for CV is approachable when we let all but one of the xi arbitrarily close to 0.

From the fact that the minimum of the expression in Eq. (11) is attained at i=2nxi2i=2nxi . we can see that:


More generally, if year is have possibly unequal probability pi of occurrence (such as when using importance sampling), then:


The minimum value of the right side is attained when x1=i=2nxi2pii=2nxipi. This can be used inductively to give an “elementary” proof of our lower bound results, and additionally can show that the lower bound is attained when all the nonzero losses are equally valued which using the Hölder’s inequality cannot arrive. Similarly we can show that the upper bound is max1p11p21pn1.

We summarize our deduction and discussion into the following:

Theorem 1. For reinsurance contract simulated annual losses f, the standard deviation σf with respect to the mean mf is bound below by:


where μ(AC) and μ(A) are the measure of the numbers of zero losses years and the numbers of non-zero losses years, respectively. The lower bound μACμA is attended:


if and only if all the non-zero losses are of the same value. The standard deviation σf with respect to the mean mf is bound above by:


where the pi is the probability of occurrence of year i. The upper bound is attained if and only if the smallest occurrence probability year is the only year of non-zero losses. And when only year i have nonzero losses:


For not necessarily nonnegative loss contracts (such as contracts with complex layers structure and hedging design), and for contracts that have significant concentration on the upper bound (due to limit and annual limit), replacing f by fm or Mf, where m and M are the minimum and the maximum annual loss, from the theorem we get the following lower bounds:

Corollary 1. For arbitrary reinsurance contract simulated annual losses f, the standard deviation σf with respect to the mean mf, minimum annual loss m, and maximum annual loss M, is bound below by:



where μ(LC) and μ(L) are the measure of the numbers of minimum losses years and the numbers of not-minimum losses years, μ(UC) and μ(U) are the measure of the numbers of maximum losses years and the numbers of not-maximum losses years, respectively. The equality hold if and only if f is a bivalued distribution.

From Theorem 1, we can get an upper bound for the average annual loss on an arbitrary subset of the years:

Corollary 2. 1For a nonnegative random variable f on a probability space Ω, an arbitrary subset BΩ, the averageBfdμμB is bound above by the standard deviation σf and the mean mf by:


Proof: Define two functions f1 and f2 from f such that they are the restrictions of f on the subset B and BC: f1|BC=0, f1|B=f|B, f2|BC=f|BC, f2|B=0. Then we have f=f1 + f2 and f1 f2=0. The standard deviation:





from Theorem 1 and the fact that the domain with zero value for f1 include the set BC and the domain with zero value for f2 include the set B. Hence:


The inequality Eq. (22) is arrived by the fact that mf=mf1+mf2 and


If we let the subset B be {x|f(x)>0}, then Eq. (22) become Eq. (16). If we let the subset B be {x|CDFf (x)≥q,0≤q≤1}, we get the so called AEP TVaR upper bound for the given quantile q or the return period r11q: TVaR(q) ≤ q1qσf+mf. For the usually used return period, the TVaR upper bound (now simply σfr1+mf) is in Table 2.

Return periodQuantileTVaR upper bound σfμBCμB+mf

Table 2.

TVaR upper bound.

For given year of return period, the upper bound as given by our formula Eq. (22).

Numerical example shows that our TVaR upper bound is relatively close in the quantile range [0.8,0.9], with the theoretical upper bound deviated from the simulated value by less than 20%, no matter what the distribution of the annual loss is.

Notice that the measure of the numbers of nonzero losses years is also called the probability of attaching in insurance, we can rearrange the terms in the formula Eq. (8) to get a lower bound for the probability of attaching:

Corollary 3. For a reinsurance contract simulated annual losses f, the probability of attaching, ProbAProb{f>0}, with respect to the CV is bound below by:


As an application of Corollary 3, we see that if CV≤3, then ProbA≥0.1, the 0.9 quantile of f is larger than zero. Equivalently, if the 0.9 quantile of f (the so called AEP in insurance) is zero, we know CV>3: then we will less prone to think that those zero quantiles is due to simulation inaccuracy. The CV bounds for commonly used AEP, related to probability of attaching by the formula 1ProbA1, is in Table 3.

CV upper boundProbA lower bound 1CV2+1Beginning quantile with nonzero lossReturn period 1ProbA

Table 3.

ProbA lower bound.

For given range of CV, the lower bound as given by our formula Eq. (29)

Similarly, from Corollary 2, we can easily rearrange terms to get an upper bound for the probability of exceeding a given loss, which is called the Cantelli’s inequality in the literature ( Chebyshev's_inequality):

Corollary 4. For a reinsurance contract simulated annual losses f, the probability of exceeding a given loss x, ProbEProb{fx}, with respect to the mean loss mf and the standard deviation σf, when xmf, is bound above by:


Specifically, if x=σf2mf+mf, then:


This bound gives a limitation on simulation with a given number N of simulated years where each year have equal probability of occurrence 1N. If 1CV2+1<1N, that is, CV>N1, then in theory no simulated loss can reach to σf2mf+mf, the lowest permissible exposure to allow the given mean mf and given standard deviation σf (to be shown in Lemma 1). In other words, if CV>N1, no such simulation can match both the given mean and the given standard deviation closely (see also inequality Eq. (12)).

Lemma 1. For a reinsurance contract simulated annual losses f that are bound up by M and with a given mean loss mf and a given standard deviation σf, we must have:


On the other hand, with the given max loss M and mean loss mf, the standard deviation σf must satisfy:


The maximum standard deviation given mf and M is attained only by a bivalued distribution of values either 0 or M, with probability q1mfM and pmfM, respectively, whose CV is then Mmf1=1p1. Similarly, the minimal exposure given mf and σf is attained only by a bivalued distribution of values either 0 or σf2mf+mf,with probabilityqσf2σf2+mf2 andpmf2σf2+mf2, respectively, whose CV is then σfmf=1p1.


Let g=fM, then g is a random variable with values in interval [0,1]. So:







This proves both of our inequalities. Without loss of generality, suppose any nonempty subset of Ω have nonzero measure, the equality hold in Eq. (34) and its subsequent inequalities if and only if g=0 or g=1.□

Because of the probability of mfM of taking value M, we cannot solve the limitation on CV by increasing M. The only solution is then by increasing N or using unequal probabilities (please refer to Eq. (18)), otherwise we may have to choose to only match the mean loss, and reduce the simulated standard deviation.

By examining the proof of Corollary 2 and Theorem 1Eq. (17), forcing the inequality in Eq. (25) to be an equality, we can prove that:

Corollary 5. For a nonnegative random variable f on a probability space Ω, an arbitrary subset B⊂Ω, assuming σf>0,μB>0,μBC>0,mf>0, the average BfdμμB attain its upper bound with respect to the standard deviation σf and the mean mf:


if and only if f is a nonzero constant function on the subset B and a constant function on the subset BC.

So the maximum TVaR distribution is bivalued, this corollary provides a guide for implementing relatively high CV distribution simulation: for CV close to N1 which do not simulate well, a conservative and simple selection is using bivalued distribution. Similar conclusion about bivalued distribution can be made for the maximum AEP distribution which are bound by TVaR’s bound and attain the same upper bound given in Eq. (39). Both conclusions give clue for a risk measure of the maximally likely or best compromise quantile by comparing simulated TVaR or AEP with the theoretical bound for the best match but that is the topic of a different research.

2.2. Simulation deviation

Typical correlation coefficient ρ from yearly model update range from 0.27 to 0.96, the CV range is 0.003–316 as we calculated in Table 1. With these parameters we used Mathematica Manipulate function to explore the probability of the ratio of two simulated annual-mean-loss be within the range of 0.5–1.5, assuming the annual-mean-loss obeys the Normal Distribution. We find that the probability is small when ρ is close to 0, and is decreasing when CV is increasing, but is stabilized after CV≥7. For an example ρ=0.822434, for almost half of the contracts, the simulated annual-mean-loss being within 50% to each other has probability of 0.459, that is, with probability of 0.551 we will see two simulation have simulated annual-mean-loss increased or decreased by more than 50% (Figures 1 and 2). These factors should be considered for model risk management or individual contract evaluation.

An Inequality for Reinsurance Contract Annual Loss Standard Deviation and Its Application (1)

An Inequality for Reinsurance Contract Annual Loss Standard Deviation and Its Application (2)

The Mathematica code for the plot in Figures 1 and 2 is in Appendix A. The two plots are identical even though their formulas are quite different and we do not know whether they can be analytically proved to be equivalent: our plots are numerical validation of both of their formulas.


3. Discussion

The lower bound for CV of reinsurance contract annual loss is established. The largest of those bound are also proved to be the upper bound for all CV. Applying this range information to ratio distribution, we can get theoretical value of the probability that different simulations will have simulated mean annual loss with deviation from each other less than a given percentage, under the Normal Distribution assumption of the mean annual loss. We think this assumption can be removed by using more suitable distributions, with numerical methods, but may still give the probability not too different. Typical example case numerical study confirmed this, and showed that the “Normal approximation” gives probability only a few percent (2–5%) less than using more suitable distributions that do not have explicit formula for the probability.

As the starting point and the application of the CV range, the ratio distribution and the model risk quantification results we get may be only rudimentarily correct due to other factors, such as the distribution modeling, the dependence modeling, and additional parameters dependence than just the CV and ρ, but our CV inequality itself is mathematically sound.

The less general upper bound n1 where all probabilities are equal is obtained by Katsnelson and Kotz in the literature [4, 5].

Using the same Hölder’s inequality and calculus technique which may not have a simple elementary inequality approach counterpart, we can prove a more complex formula:

Theorem 2. For reinsurance contract simulated annual losses f, the standard deviation σf with respect to the mean mf is bound below by:


when μ(m)+μ(M)<1, where σf is the standard deviation, mf is the mean, m is the minimum annual loss, M is the maximum annual loss, μ(m) denote the measure of the numbers of minimum losses years, μ(M) denote the measure of the numbers of maximum losses years.


In the inequality Eq. (2), we divide Ω into three subset and let the nonnegative function g be constant in each of the three sets:








due to fmffmf.

We get:



suppose b>0.

Using the negative form of the inequality fmfmff, we also get a dual form inequality:



suppose b>0.







The derivative


If B≤0, then Ft>0, F(t) take the maximum ACat ∞. If B>0, then F(t) increase on (0, ADBC) and decrease on (ADBC, ∞), attain the maximum A2C+B2D at ADBC.

Apply the same argument to


with (Mmf)μ(M)>0 since μ(m)+μ(M)<1, we have if (mfm)μ(m)−(Mmf)μ(M)>0, then BDattain the maximum Mmf2μM+mfmμmMmfμM21μmμM at Mmf1μmμMmfmμmMmfμM.

If (mfm)μ(m)−(Mmf)μ(M)=0, then BD is monotonically increasing with respect to cband attain the maximum MmfμM at ∞.

If (mfm)μ(m)−(Mmf)μ(M)<0, then use the inequality Eq. (50), we can follow the same steps to arrive at the same form of maximum formula. We thus proved the maximal


with the specific choice of cband ab.□

We can also prove by calculus that:

Theorem 3. In the terminology of Theorem 2, if m=0,





for t≥0.

Then F(t) is continuous at 0 and



The derivative of F(t) is


which is always nonnegative, so F(t)≥0 for any t≥0.□

Theorem 3 can be combined with the following form of the Hölder’s inequality:


to give an alternative proof of Theorem 2 and then Eq. (16) (or directly for Eq. (20) by using the set {f>m}).

So there is a complex but better lower bounds Eq. (40), and empirical study shows that when μ(m)>0.86, both bounds are close to the true σf to within 86% with the simple form Eq. (8) 3–4% lower than the complex form Eq. (40). Even though the complex form Eq. (40) is generally valid for any discrete random variable, it may not be as easily applicable as the simple form Eq. (8) when we need a fast first approximation, and hence of less practical interest.

With numerical simulation, we can get σf and CV directly, so these formulas seems not to be useful for the numerical results. But since each simulation may arrive at a different value, known a priori their approximate value will be a check for any possible simulation process problem. Our inequalities also reveal that the CV is intrinsically related to important value distribution characteristics of the annual loss random variable. This essentialness of CV is also confirmed by other studies, such as the correlation and cluster analysis of these random variables.


4. Conclusions

Lower bound for reinsurance contract annual loss standard deviation involving zero losses years counts are obtained, which imply a general upper bound for annual loss TVaR or AEP with no mention of zero losses years. Alternative forms of these bounds give inequalities for probability of attaching and exceeding. These bounds can explain the difficulties or instabilities observed in numerical simulations, show the major reason of the limitation of the simulation is high CV and give clue to alternative solutions.



This research is supported by Validus Research Inc.


Conflict of interest

The authors declare no conflict of interest.



The author thanks Nancy Wang for checking against a C++ application that validated the practical usefulness of our inequality.


hinkley[x_?NumericQ,c_,p_]:=1/Sqrt[2 Pi]/c (x+1) (1−p)/(xˆ2–2 p x+1)ˆ1.5 Exp[−1/2/cˆ2 (x−1)ˆ2/(xˆ2–2 p x+1)](CDF[NormalDistribution[0,1],1/c Sqrt[(1−p)/(1+p)] (x+1)/Sqrt[xˆ2–2 p x+1]]−CDF[NormalDistribution[0,1],−1/c Sqrt[(1−p)/(1+p)] (x+1)/Sqrt[xˆ2–2 p x+1]])+1/Pi Sqrt[1−pˆ2]/(xˆ2–2 p x+1) Exp[−1/cˆ2/(1+p)];


LogLinearPlot[Evaluate[H[1.5,x,0.822434]−H[0.5,x,0.822434]],{x,0.1,10},PlotRange>All, GridLines>Full,GridLinesStyle−>Directive[Gray,Dashed],Mesh>Automatic, ImageSize>Full,Frame>on];



q[t_,p_,c_]:=q[t,p,c]=With[{},1.0/c (1.0+(1.0−p) t/Sqrt[1.0−pˆ2])/Sqrt[1.0+tˆ2]]

marsaglia[t_,p_,c_]:=marsaglia[t,p,c]=With[{},Exp[−1.0/(1.0+p)/cˆ2]/Pi/(1.0+tˆ2)(1.0+q[t,p,c] Exp[q[t,p,c]ˆ2/2.0] Evaluate[Integrate[Exp[−yˆ2/2.0],{y,0.0,q[t,p,c]}]])]

M[v_,p_,c_]:=M[v,p,c]=With[{},CDF[NormalDistribution[0,1],1/c (1−p)/Sqrt[1−pˆ2]]+CDF[

NormalDistribution[0,1],1/c]−2 CDF[NormalDistribution[0,1],1/c (1−p)/Sqrt[1−pˆ2]] CDF[.



DistributeDefinitions [q,marsaglia,M,MA];

LogLinearPlot[Evaluate[MA[1.5,0.822434,x]−MA[0.5,0.822434,x]],{x,0.1,10},PlotRange>All, GridLines>Full,GridLinesStyle−>Directive[Gray,Dashed],Mesh>Automatic, ImageSize>Full,Frame>on]

I have a deep understanding of the content you provided, particularly in the field of reinsurance and risk management. The article delves into the calculation of standard deviation and quantiles for reinsurance contracts, considering factors like model risk and distribution types. The concepts of CV (coefficient of variation), probability of attaching, and simulation deviation are explored.

Now, let's break down the key concepts mentioned in the article:

  1. Reinsurance Industry and Model Risks:

    • Simulated losses from catastrophe events and reinsurance contract terms are used for risk assessment.
    • Model risks include variations in simulations and changes due to updates.
  2. Distribution Types:

    • The expected annual loss distribution may not be Normal but can be Beckmann, MaxStable, Gamma, Inverse Gaussian, or Lognormal.
    • The article simplifies by assuming a Normal Distribution for quantifying model risk.
  3. Coefficient of Variation (CV):

    • CV is crucial in model risk quantification.
    • Lower bounds for CV are established based on Hölder's inequality.
  4. Results and Bounds:

    • The article provides lower bounds for CV and explores Hölder's inequality to derive these bounds.
    • Upper bounds for CV are discussed, considering different scenarios.
  5. TVaR (Tail Value at Risk) Upper Bounds:

    • Upper bounds for TVaR are derived based on CV and other parameters.
  6. Probability of Attaching and Exceeding:

    • Lower bounds for the probability of attaching are discussed in terms of CV.
    • Upper bounds for the probability of exceeding a given loss are presented.
  7. Simulation Deviation:

    • Numerical simulations explore the probability of deviations in simulated annual-mean-loss under certain conditions.
  8. Theorems and Corollaries:

    • The article presents mathematical theorems and corollaries to support the derived bounds.
    • Theorems provide insights into the relationships between mean, minimum, and maximum losses.
  9. Empirical Studies:

    • Empirical studies are mentioned, confirming the closeness of bounds to true standard deviation.
  10. Conclusions:

    • The article concludes by summarizing the obtained results and their implications for risk management.

Overall, the article demonstrates a rigorous mathematical approach to understanding and quantifying risks in the reinsurance industry, particularly in the context of simulated losses and model uncertainties. If you have specific questions or need further clarification on any aspect, feel free to ask.

An Inequality for Reinsurance Contract Annual Loss Standard Deviation and Its Application (2024)


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