Value-at-Risk substitute for non-ruin capital is fallacious and redundant
aa r X i v : . [ q -f i n . R M ] M a y VALUE-AT-RISK SUBSTITUTE FOR NON-RUINCAPITAL IS FALLACIOUS AND REDUNDANT
Vsevolod K. Malinovskii
Abstract.
This seemed impossible to use a theoretically adequate but too sophis-ticated risk measure called non-ruin capital, whence its widespread (including regu-latory documents) replacement with an inadequate, but simple risk measure calledValue-at-Risk. Conflicting with the idea by Albert Einstein that “everything shouldbe made as simple as possible, but not simpler”, this led to fallacious, and even de-ceitful (but generally accepted) standards and recommendations. Arguing from thestandpoint of mathematical theory of risk, we aim to break this impasse.
1. Introduction
The basis of Solvency II system (see Directives [ ], [ ]) is the Value-at-Risk set as themain measure of risk. Various criticisms (see, e.g., [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ],[ ], [ ]) are directed against this basis. Floreani (see [ ]) expressed it categorically: the Solvency II regime uses an inadequate risk measure to compute the Sol-vency Capital Requirement . . . The metric used by regulators, which is basedon a total risk measure such as the Value-at-Risk, is not a balanced solutionbetween effectiveness and simplicity, but is simply wrong and could lead tosignificant adverse side effects, ultimately resulting in a generalized Europeaninsurance industry crisis in the case of a hard market shortfall.
Surprisingly, the text of Directives [ ], [ ] contains clear evidence of controversyregarding risk measures: the phrase from Directive [ ] that the Solvency Capital Re-quirement (SCR) “shall correspond to the Value-at-Risk of the basic own funds of aninsurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period” dramatically differs from the phrase in the same Directive [ ], that theSCR determines the economic capital which an insurance company must hold in orderto guarantee a one-year ruin probability of at most 0.5%.Every risk theory expert knows that, dealing with solvency, it is appropriate toaddress the occurrence of ruin within a year (and its probability), rather than the capital Key words and phrases.
Insurance solvency, risk measures, Value-at-Risk, non-ruin capital.This work was supported by RFBR (grant No. 19-01-00045). See Directive [ ], Article 101: Calculation of the Solvency Capital Requirement. In the preambular paragraph (64) for Directive [ ], it is said as follows: “the Solvency CapitalRequirement should be determined as the economic capital to be held by insurance and reinsuranceundertakings in order to ensure that ruin occurs no more often than once in every 200 cases”. deficit at the end of this year (and its probability). Stepping back one step more, theaggregate claim amount distribution and the probability of ruin within finite time, whichare the basis for determining the Value-at-risk and non-ruin capital respectively, werealways clearly distinguished in the risk theory.In this paper, relying on inverse Gaussian approximations in the problem of levelcrossing by a compound renewal processes and on associated results for the level which acompound renewal process crosses with a given probability, obtained in [ ] and [ ], weshow that the non-ruin capital, being a theoretically sound risk measure, is not inferiorto the Value-at-risk in simplicity even in a fairly general risk model.The rest of this paper is arranged as follows. In Section 2, we introduce the riskmodel, focussing on the difference between the Value-at-Risk and non-ruin capital. InSection 3, using a series of well-known results, we show that in the exponential risk modelthe analysis of non-ruin capital is not much harder than the analysis of Value-at-Risk. InSection 4, we show that in the general risk model most results so much discussed in theliterature are fit for the analysis of Value-at-Risk, but not of non-ruin capital. We turnto recent advances in the direct and inverse level crossing problems (see [ ], [ ]) whichare suitable for a deep insight into the structure of non-ruin capital. In Section 5, wepresent numerical results obtained by both analytical technique and direct simulation,in order to illustrate the non-ruin capital’s structure. The final conclusion of this paper,given in Section 6, is that the analytical structure of non-ruin capital is simple enough,and this measure of risk can be used per se, without resorting to any substitute.
2. Model and main definitions
In the risk theory, the quantitative analysis is based on the annual model thatformalizes the concept of collective risk. Given that time is operational and t is thelength of the year, for 0 s t the claim arrival process is N s = max n n > n X i =1 T i s o , (2.1)or 0, if T > s , the cumulative claim payout process is V s = N s X i =1 Y i , (2.2)or 0, if T > s , and the balance of income and outcome is modeled by the risk reserveprocess R s = u + cs − V s , (2.3)which starts at time zero at the point u >
0, called initial capital. Here c > T i d = T , i = 1 , , . . . , are i.i.d.intervals between claims, and Y i d = Y , i = 1 , , . . . , are i.i.d. claim sizes. It is generallyassumed that these sequences are independent of each other.In what follows, α is a reasonably small positive real number, e.g., α = 0 . Definition . The
Value-at-Risk u [ VaR ] α,t ( c ), c >
0, is a positive solution to theequation P { R t < } = α ; (2.4) This model, traditionally called Lundberg’s collective risk model, is most useful (see [ ]) as abuilding block for multi-year models. From the angle of Directive [ ], this modeling is very close tobuilding an internal model. ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 3
Figure 1.
Graphs ( X -axis is c ) of u [ VaR ] α,t ( c ) (red) and u α,t ( c ) (blue), drawnfor T and Y exponentially distributed with parameters δ = 1, ρ = 1, and α = 0 . t = 200. Horizontal grid line: u α,t ( δ/ρ ) = 40 . δ/ρ = 1. for those c , for which this solution is negative, we set u [ VaR ] α,t ( c ) equal to zero. The non-ruincapital u α,t ( c ), c >
0, is a positive solution to the equation P (cid:8) inf s t R s < (cid:9) = α ; (2.5)for those c , for which this solution is negative, we set u α,t ( c ) equal to zero.Note that the left-hand side of (2.5) is the probability of ruin within time t , i.e., ψ t ( u, c ) = P (cid:8) inf s t R s < (cid:9) = P { Υ u,c t } , where Υ u,c = inf { s > V s − cs > u } , or + ∞ , if V s − cs u for all s >
0, is the timeof the first ruin. In these terms, equation (2.5) rewrites as P { Υ u,c t } = α. (2.6)Obviously, to investigate solvency in the usual sense of non-ruin , we must focus on u α,t ( c ), rather than on u [ VaR ] α,t ( c ). Since inf s t R s is always less than or equal to R t , wehave u [ VaR ] α,t ( c ) u α,t ( c ) , c > , (2.7)and u [ VaR ] α,t ( c ) always underestimates u α,t ( c ). The underestimating of u α,t ( c ) by u [ VaR ] α,t ( c )can be significant (see Fig. 1). To deal with this problem quantitatively, rather thanqualitatively, we must calculate both u α,t ( c ) and u [ VaR ] α,t ( c ) in the risk model (2.1)–(2.3),striving for the most general assumptions about T and Y . Traditionally, it has beenconsidered possible to achieve success in this endeavor for u [ VaR ] α,t ( c ), but not for u α,t ( c );our aim is to break this impasse. Remark . Let us write c ∗ = E Y / E T and introduce the ultimate ruin probability ψ ∞ ( u, c ) = P (cid:8) inf s > R s < (cid:9) , which is equivalently written as P { Υ u,c < ∞} . It hasbeen studied in detail (see, e.g., [ ]). Plainly, P { Υ u,c t } P { Υ u,c < ∞} . In risk theory, the event of ruin is traditionally synonymous with bankruptcy, and solvency isusually measured by the probability of ruin.
VSEVOLOD K. MALINOVSKII
Figure 2.
Graphs ( X -axis is c ) of u α,t ( c ) (blue) and u α ( c ) (red), drawnfor T and Y exponentially distributed with parameters δ = 1, ρ = 1, and α = 0 . t = 200. Horizontal grid line: u α,t ( c ∗ ) = 40 .
08. Vertical grid line: c ∗ = 1. In [ ], Chapter 6, Section 2, the insurer’s risk is measured by the ultimate ruinprobability P { Υ u,c < ∞} and the “minimal admissible initial capital” is introduced as asolution to the equation P { Υ u,c < ∞} = α. (2.8)In our notation, this is u α ( c ), c > c ∗ . Plainly (see Fig. 2), u α,t ( c ) u α ( c ), c > c ∗ , and u α ( c ) is tending to infinity, as c → c ∗ .Assuming that the “insurer wants to attract as many clients as possible keeping therelative safety loading at the lowest possible level” ([ ], pp. 172–173), in [ ] focussed is u α ( c ), as c → c ∗ . Thus, the problem to explore “the initial capital securing a prescribedrisk level when the relative safety loading tends to zero” ([ ], p. 27) is put forth.In our opinion, the focus on u α ( c ), as c → c ∗ , does not help the insurer “to attract asmany clients as possible keeping the relative safety loading at the lowest possible level”([ ], pp. 172–173), given that “the insurer accepts at most α as an acceptable risklevel” ([ ], p. 172). And even worse, this is hard to accept that it “can help the insurerto determine whether the initial capital suffices to start the business” ([ ], p. 175)because the theory developed in [ ] claims that when the insurer’s price c decreases tothe equilibrium price c ∗ , what often happens in some years of the real insurance businessand what is far from tragic, “the initial capital securing a prescribed risk level” is tendingto infinity.
3. Value-at-Risk and non-ruin capital in exponential case
The additional assumption that T and Y in the model (2.1)–(2.3) are exponentiallydistributed with parameters δ and ρ , yields many items of our interest in the analyticalform, in terms of elementary or special functions, such as modified Bessel functions I k ( x ), x >
0, of the first kind of order k .In what follows, we denote the cumulative distribution function (c.d.f.) of a standardGaussian distribution by Φ (0 , ( x ), x ∈ R . The corresponding probability density function(p.d.f.) is denoted by ϕ (0 , ( x ), x ∈ R . The (1 − α )-quantile of this distribution is denotedby κ α = Φ − , (1 − α ). Plainly, 0 < κ α < κ α/ for 0 < α < / In the expo-nential case, for the aggregate claim amount V t we have the following widely known ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 5 closed-form results: E ( V t ) = ( δ/ρ ) t, D ( V t ) = 2 ( δ/ρ ) t, (3.1)and P { V t x } = e − tδ + e − tδ ∞ X n =1 ( tδ ) n n ! ρ n Γ( n ) Z x e − ρz z n − dz = e − tδ + e − tδ (cid:0) δρ t (cid:1) / Z x z − / I (cid:0) p δρ tz (cid:1) e − ρz dz (3.2)for x >
0, and zero otherwise.An important observation is that equation (2.4) rewrites as P { V t > u + ct } = α, (3.3)and its solution u [ VaR ] α,t ( c ) is (see, e.g., [ ], Section 14.3.2) a percentile (or quantile) ofthe distribution of the aggregate claim amount distribution at the year-end time point t .Using equality (3.2), we express equation (3.3) in a closed form, whence u [ VaR ] α,t ( c ) isan implicit function defined by the equation e − tδ + e − tδ (cid:0) δρ t (cid:1) / Z u + ct z − / I (cid:0) p δρ tz (cid:1) e − ρz dz = 1 − α. (3.4)This implicit function can not be found in a closed form, but it can be calculated numer-ically. The graph of u [ VaR ] α,t ( c ), c >
0, was drawn in Fig. 1 in this way.Since V t is (see (2.2)) the sum of N t i.i.d. random variables, where E ( N t ) = δt , itseems natural to address the asymptotic analysis of u [ VaR ] α,t ( c ), as t → ∞ . The assumptionthat t is large, which allows us to turn to the central limit theory, is sensible in termsof applications for the following reasons: time in the model (2.1)–(2.3) is operational(see, e.g., [ ] p. 219), rather than calendar. This time, measured in monetary units, isproportional to the ball-park figure of the annual financial transactions of the company.Consequently, the assumption that t → ∞ means that this ball-park figure is large, i.e.,the insurer’s portfolio size is large.Since V t is asymptotically normal with mean and variance given in (3.1), equation(3.4) is closely related to the equation Φ (0 , u + ct − ( δ/ρ ) t p δ/ρ ) t ! = 1 − α, (3.5)whose solution ( δ/ρ − c ) t + (cid:0) √ δ/ρ (cid:1) κ α √ t is straightforward. Applying simple argumentsbased on the proximity of two implicit functions, we conclude that for all c > u [ VaR ] α,t ( c ) = max n , (cid:0) δ/ρ − c (cid:1) t + √ δρ κ α √ t (1 + o (1)) o , t → ∞ . (3.6) In the exponential case, we havethe following widely known (see, e.g., [ ], Remark 2) closed-form result: P { Υ u,c t } = P { Υ u,c < ∞} − π Z π f ( x ) dx, (3.7)where P { Υ u,c < ∞} = , δ/ ( cρ ) > ,δcρ exp {− u ( cρ − δ ) /c } , δ/ ( cρ ) < , VSEVOLOD K. MALINOVSKII - - Figure 3.
Graph ( X -axis is x ) of z α,t ( x ), drawn for T and Y exponentiallydistributed with parameters ρ = 1, δ = 1, and α = 0 . t = 200. Horizontalgrid lines: κ α = 1 .
645 and κ α/ = 1 . and f ( x ) = ( δ/ ( cρ )) (cid:0) δ/ ( cρ ) − p δ/ ( cρ ) cos x (cid:1) − × exp n uρ (cid:0)p δ/ ( cρ ) cos x − (cid:1) − tδ ( cρ/δ ) × (cid:0) δ/ ( cρ ) − p δ/ ( cρ ) cos x (cid:1)o × (cid:0) cos (cid:0) uρ p δ/ ( cρ ) sin x (cid:1) − cos (cid:0) uρ p δ/ ( cρ ) sin x + 2 x (cid:1)(cid:1) . Using equality (3.7), we express the left-hand side of equation (2.6) in a closed form,whence u α,t ( c ) is an implicit function defined by this equation. The same as u [ VaR ] α,t ( c ), thisimplicit function can not be found in a closed form, but can be calculated numerically.The graph of u α,t ( c ), c >
0, was drawn in Fig. 1 in this way.The function u α,t ( c ), c >
0, can be analyzed asymptotically (see, e.g., [ ], The-orems 3.1 and 3.2; this analysis was based on the properties of Bessel functions). Inparticular, we have (see [ ], Theorem 3.2) u α,t ( c ) = ( δ/ρ − c ) t + √ δρ z α,t (cid:18) ρ (cid:0) δ/ρ − c (cid:1) √ δ √ t (cid:19) √ t, c c ∗ , √ δρ z α,t (cid:18) ρ (cid:0) δ/ρ − c (cid:1) √ δ √ t (cid:19) √ t, c > c ∗ , (3.8)where c ∗ = δ/ρ and the function z α,t ( x ), x ∈ R , is (see Fig. 3) continuous, monotoneincreasing, as x increases from −∞ to 0, monotone decreasing, as x increases from 0 to ∞ ,and such that lim x →−∞ z α,t ( x ) = 0, lim x →∞ z α,t ( x ) = κ α , and z α,t (0) = κ α/ (1 + o (1)),as t → ∞ . In particular, we have u α,t (0) = ( δ/ρ ) t + √ δρ κ α √ t (1 + o (1)) , t → ∞ ,u α,t ( c ∗ ) = √ δρ κ α/ √ t (1 + o (1)) , t → ∞ . (3.9)First equality in (3.9) is straightforward; see, e.g., (2.4) in [ ]. Second equality in (3.9)is Theorem 3.1 in [ ]. ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 7
4. Value-at-Risk and non-ruin capital in general case
It is widely believed that in the general case, the situation described in Section 3deteriorates dramatically, and the non-ruin capital becomes intractable. First, we clarifythe reasons for this belief. Second, we show that the situation in the general case is notso bad due to several innovative approaches.
In the generalcase, there is no hope to get explicit equalities like (3.1) or (3.2) for all t . But since theasymptotic analysis, as t → ∞ , is based on the fairly general central limit theory, it iseasy to obtain analogues for (3.5) and (3.6). To be specific, P { V t x } is approximatedby Φ ( M V t,D V t ) ( x ), as t → ∞ , where M N = 1 / E T, M V = E Y / E T,D N = D T / ( E T ) , D V = E ( T E Y − Y E T ) / ( E T ) . This approximation, being a version of the central limit theorem, is valid under well-known mild technical condition on T and Y and can be applied to (3.3).Therefore, though in the general case equation (3.3) cannot be written in terms ofelementary or special functions, as it was done (see (3.4)) in the exponential case, for t sufficiently large (3.3) is close to the equation (cf. (3.5)) Φ (0 , (cid:18) u + ct − M V tD V √ t (cid:19) = 1 − α, (4.1)whose closed-form solution ( M V − c ) t + κ α D V √ t is straightforward. Applying simplearguments based on the proximity of two implicit functions, we conclude that (cf. (3.6))for all c > u [ VaR ] α,t ( c ) = max n , ( M V − c ) t + κ α D V √ t (1 + o (1)) o , t → ∞ . (4.2)It is easy to see that the analysis in the general case differs a little, regarding both appliedtechnique and results, from the analysis in the exponential case. In the general case (except for somevery special subcases), there is no hope to express P { Υ u,c t } in terms of elementaryor special functions for all t . There is even less hope of finding in a closed form for all t the implicit function u α,t ( c ), c >
0, defined by the corresponding equation (2.6), even ifits left-hand side could be represented in such terms.Moreover, the results of asymptotic analysis, as u → ∞ , so much discussed in theliterature, are unsatisfactory from the angle of their further application to asymptotical,as t → ∞ , analysis of the non-ruin capital u α,t ( c ), c >
0. We will show this by referringto the normal (or Cram´er’s) and diffusion approximations that are best known. We startwith the former and point out its deficiencies. It is noteworthy that E ( N t ) = M N t + D T − ( E T ) E T ) + o (1), D ( N t ) = D N t + o ( t ), E ( V t ) = M V t + E Y ( D T − ( E T ) )2 ( E T ) + o (1), D ( V t ) = D V t + o ( t ), t → ∞ . What is said below about this approximation is folklore of the risk theory and can be found inmany standard textbooks, e.g., in [ ]. VSEVOLOD K. MALINOVSKII
Normal approximation.
The primary assumption is that there exists a positivesolution κ , called adjustment coefficient, to the equation (w.r.t. r ) M X ( r ) = 1 , (4.3)called Lundberg’s equation. Here M X ( r ) = E ( e rX ) is the moment generating functionof X d = Y − c T ; plainly, M X (0) = 1. This assumption is a significant limitation of themodel. It implies that M X ( r ) has to exist in a neighborhood of 0 or, in other words,that the right tail of c.d.f. F X is exponentially bounded above. The latter follows fromMarkov’s inequality 1 − F X ( x ) e − κ x E ( e κ X ) = e − κ x , x > . (4.4)Starting with c.d.f. F XT ( x, t ) = P { X x, T t } and having κ > , whose c.d.f. F ¯ X ¯ T ( x, t ) = P { ¯ X x, ¯ T t } is defined by the equality F ¯ X ¯ T ( x, t ) = Z x − ct Z t e κ z F XT ( dz, dw ) . Plainly, this is a proper probability distribution.Recall that c ∗ = E Y / E T . The normal (or Cram´er’s) approximation is formulatedseparately for 0 c < c ∗ and for c > c ∗ , with the case c = c ∗ excluded. For 0 c < c ∗ ,i.e., for E X = E Y − c E T >
0, we write m ▽ = E T / E X, D ▽ = E ( X E T − T E X ) / ( E X ) . Plainly, we have m ▽ > D ▽ > Proposition c < c ∗ ) . Assume that p.d.f. of the random vector ( T, Y ) is bounded above by a finite constant and < D ▽ < ∞ . Then d u = sup t> (cid:12)(cid:12) P { Υ u,c t } − Φ ( m ▽ u,D ▽ u ) ( t ) (cid:12)(cid:12) = o (1) , u → ∞ . If, in addition, E ( Y ) < ∞ , E ( T ) < ∞ , then d u = O ( u − / ) , as u → ∞ . For c > c ∗ , i.e., for E X = E Y − c E T <
0, we write m △ = E ¯ T / E ¯ X, D △ = E ( ¯ X E ¯ T − ¯ T E ¯ X ) / ( E ¯ X ) , C = 1 κ E ¯ X exp n − ∞ X n =1 n P { S n > } − ∞ X n =1 n P { ¯ S n } o , where ¯ X i d = ¯ X , i = 1 , , . . . , and ¯ T i d = ¯ T , i = 1 , , . . . , are associated random variables,and ¯ S n = P ni =1 ¯ X i , and ¯ Z n = P ni =1 ¯ T i , n = 1 , , . . . , are associated random walks. Proposition c > c ∗ ) . Assume that a solution κ > to equation (4.3) exists, p.d.f. of the random vector ( T, Y ) is bounded above by a finite constant, and < D △ < ∞ . Then d u = sup t> (cid:12)(cid:12) e κ u P { Υ u,c t } − C Φ ( m △ u,D △ u ) ( t ) (cid:12)(cid:12) = o (1) , u → ∞ . If, in addition, E ( T ) < ∞ , then d u = O ( u − / ) , as u → ∞ . See, e.g., Example (b) in [ ], Chapter XII, Section 4. Commonly used shorthand notation for it is F ¯ X ¯ T ( dx, dt ) = e κ z F XT ( dx, dt ). ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 9
Figure 4.
Graphs ( X -axis is c ) of P { Υ u,c t } (blue), of the approximationsof Proposition 4.3 (red), and of simulated values (∆ c = 0 . N = 1000) of P { Υ u,c t } , drawn for T and Y exponentially distributed with parameters ρ = 1, δ = 1, and t = 1000, u = 50. Horizontal grid line: P { Υ u,c ∗ t } = 0 . Figure 5.
Graphs ( X -axis is c ) of P { Υ u,c t } (blue), of M u,c ( t ) (red), andof simulated values (∆ c = 0 . N = 1000) of P { Υ u,c t } , drawn for T and Y exponentially distributed with parameters ρ = 1, δ = 1, and t = 1000, u = 50.Horizontal grid line: P { Υ u,c ∗ t } = 0 . In the exponential case, when T and Y are exponentially distributed with parameters δ > ρ >
0, straightforward calculations (see [ ], Proposition 2.3) yield c ∗ = δ/ρ , C = δ/ ( cρ ) , κ = ρ (1 − δ/ ( cρ )) ,m ▽ = − c (1 − δ/ ( cρ )) , D ▽ = − δ/ ( cρ )) c ρ (1 − δ/ ( cρ )) ,m △ = δ/ ( cρ ) c (1 − δ/ ( cρ )) , D △ = 2 ( δ/ ( cρ )) c ρ (1 − δ/ ( cρ )) , (4.5)and Propositions 4.1 and 4.2 are fused together, as follows. Proposition . In the renewal model with T and Y exponentially distributed withparameters δ > and ρ > , we have for c < c ∗ sup t> (cid:12)(cid:12)(cid:12) P { Υ u,c t } − Φ ( m ▽ u,D ▽ u ) ( t ) (cid:12)(cid:12)(cid:12) = o (1) , u → ∞ , where m ▽ > , D ▽ > are defined in (4.5) , and for c > c ∗ sup t> (cid:12)(cid:12)(cid:12) e κ u P { Υ u,c t } − C Φ ( m △ u,D △ u ) ( t ) (cid:12)(cid:12)(cid:12) = o (1) , u → ∞ , where κ > , and < C < , m △ > , D △ > are defined in (4.5) . These results, if used for an asymptotic analysis of non-ruin capital, u α,t ( c ), c > Deficiency . Proposition 4.2 requires restrictive technicalcondition: Y must be (see (4.4)) light-tailed. Deficiency c = c ∗ ) . Besides the fact that the case c = c ∗ is formallyexcluded, the normal (or Cram´er’s) approximation fails for c in a neighborhood of c ∗ .This is illustrated in Fig. 4. Deficiency . The structure of the approximation in Propo-sitions 4.1 and 4.2 is significantly different from the structure of CLT–type approximationused to get (4.1) and (4.2). This is particularly evident when c = 0, i.e., when the left-hand sides of equations (2.4) and (2.6) are the same and can be written as P { V t > u } .It is clear that, being solutions to the same equation, u α,t (0) and u [ VaR ] α,t (0) coincide witheach other. The CLT–type approximation is P { V t > u } ≈ Φ (0 , u − M V (cid:12)(cid:12) c =0 tD V (cid:12)(cid:12) c =0 √ t ! , t → ∞ , (4.6)whereas the approximation in Proposition 4.1 is P { V t > u } ≈ Φ (0 , t − m ▽ (cid:12)(cid:12) c =0 uD ▽ (cid:12)(cid:12) c =0 √ u ! , u → ∞ . (4.7)When T and Y are exponentially distributed with parameters δ and ρ , m ▽ (cid:12)(cid:12) c =0 = ρ/δ , D ▽ (cid:12)(cid:12) c =0 = 2 ρ/δ in (4.6), and M V (cid:12)(cid:12) c =0 = δ/ρ , D V (cid:12)(cid:12) c =0 = 2 δ/ρ in (4.7).It is worth noting that if u α,t (0) would not tend to infinity, as t → ∞ , the approxi-mation (4.7) (unlike (4.6)) would be useless to get information about u α,t (0), as t → ∞ .Fortunately, u α,t (0) ∼ M V | c =0 t , which tends to infinity, as t → ∞ . Indeed, u α,t (0) isequal to u [ VaR ] α,t (0) and (see (4.2)) u [ VaR ] α,t (0) = M V (cid:12)(cid:12) c =0 t + κ α D V (cid:12)(cid:12) c =0 √ t (1 + o (1)) , t → ∞ . For c sufficiently larger than c ∗ , i.e., for c > Kc ∗ with K > u α,t ( c )is finite regardless of t . Therefore, the structural difference between the approximationsof Propositions 4.1 and 4.2 matters.4.2.2. Diffusion approximation.
Because of the space limitations, we describe thedeficiencies of simple and corrected diffusion approximations for P { Υ u,c t } verbally.The idea of a simple diffusion approximation is that the original risk reserve process (2.3)has some similarities (regarding the properties of distributions, rather than trajectories)with the diffusion process, although its trajectories are continuous. Matching the originaland the auxiliary diffusion processes, one finds that the distribution of the first levelcrossing time for the former process is approximated by the distribution of the first level What is said below about this approximation is folklore of the risk theory and can be found inmany standard textbooks, e.g., in [ ]. Using (see, e.g., [ ]) Donsker’s theorem, called also Donsker–Prohorov’s invariance principle. ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 11
Figure 6.
Graph ( X -axis is c ) of M u,c ( t ) (red) and of simulated values(∆ c = 0 . N = 1000) of P { Υ u,c t } (blue), drawn for T which is 2-mixturewith parameters δ = 1, δ = 2, p = 2 / Y which is Pareto with parameters a Y = 4 . b Y = 0 .
35, and t = 1000, u = 40. Figure 7.
Graph ( X -axis is c ) of M u,c ( t ) (red) and of simulated values(∆ c = 0 . N = 1000) of P { Υ u,c t } (blue), drawn for T which is Erlangwith parameters δ = 6 . k = 4, Y which is Pareto with parameters a Y = 4 . b Y = 0 .
4, and t = 1000, u = 40. crossing time for the latter process. This observation is productive because the firstpassage probabilities in the diffusion model are found in a closed form.The diffusion process is skip-free; the idea behind the simple diffusion approximationignores the presence of overshoot in the original process with discontinuous trajectories.The corrected diffusion approximation takes into account this and other similar featuresof the initial process.Congenital deficiency of both simple and adjusted diffusion approximations is De-ficiency 1: these results often require that the distribution of Y has a light tail. Inaddition, these results are valid under the assumption that c → c ∗ + 0. Such regime iscertainly a structural drawback; this impedes the analysis of u α,t ( c ), c > For M = E T / E Y, D = (( E T ) D Y + ( E Y ) D T ) / ( E Y ) , (4.8) Often formulated as “the safety loading is small and positive”; it is often added that this is justthe same as the heavy traffic in the queuing theory.
Figure 8.
Graph ( X -axis is c ) of M u,c ( t ) (red) and of simulated values(∆ c = 0 . N = 1000) of P { Υ u,c t } (blue), drawn for T which is Paretowith parameters a T = 4 . b T = 0 . Y which is Pareto with parameters a Y = 4 . b Y = 0 .
4, and t = 1000, u = 40. we write M u,c ( t ) = Z ctu x + 1 ϕ (cid:16) cM ( x +1) , c D u ( x +1) (cid:17) ( x ) dx. (4.9)Bearing in mind that c ∗ = E Y / E T equals to 1 /M and denoting p.d.f. of inverseGaussian distribution by F ( x ; µ, λ ) = Φ (0 , (cid:18)r λx (cid:18) xµ − (cid:19)(cid:19) + exp (cid:26) λµ (cid:27) Φ (0 , (cid:18) − r λx (cid:18) xµ + 1 (cid:19)(cid:19) , x > , we can show by elementary calculations that M u,c ( t ) = (cid:16) F (cid:16) ctu + 1; µ, λ (cid:17) − F (cid:0) µ, λ (cid:1)(cid:17) (cid:12)(cid:12)(cid:12) µ = − cM ,λ = uc D , < c c ∗ , exp n − λ ˆ µ o (cid:16) F (cid:16) ctu + 1; ˆ µ, λ (cid:17) − F (cid:0)
1; ˆ µ, λ (cid:1)(cid:17) (cid:12)(cid:12)(cid:12) ˆ µ = cM − ,λ = uc D , c > c ∗ . The following theorem, as well as its refinements like Edgeworth expansions (see [ ]and [ ], [ ], [ ]), is called the inverse Gaussian approximation for P { Υ u,c t } . Theorem . Assume that p.d.f. f T ( x ) and f Y ( x ) are bounded above by a finiteconstant, D > , E ( T ) < ∞ , E ( Y ) < ∞ . Then for any c > t> | P { Υ u,c t } − M u,c ( t ) | = o (1) , t, u → ∞ . In a nutshell, the proof of Theorem 4.1 is based on Kendall’s identity which rep-resents the first level crossing time’s distribution in terms of the convolution powers ofp.d.f. f T ( x ) and f Y ( x ); then the well-developed central limit theory is applied to theseconvolution powers. Remark . The inverse Gaussian distribution F ( x ; µ, λ ) is concentrated on thepositive half-line; its mean is µ , variance is µ /λ , and the third central moment is3 µ /λ . The appearance of this skewed distribution in Theorem 4.1 sheds light onnumerous claims by many practitioners (see, e.g., [ ], [ ], [ ]) that the “world of Theorem 4.1 shows that it is central in the following approximation, called (see [ ]) inverseGaussian. ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 13
Table 1.
Models in Figs. 5–8.
T Y M D Fig. 5: exponentially distributed; δ = 1 exponentially distributed; ρ = 1 1 2Fig. 6: 2-mixture; Pareto; 0 . . δ = 1, δ = 2, p = 2 / a Y = 4 . b Y = 0 . . . δ = 6 . k = 4 a Y = 4 . b Y = 0 . . a T = 4 . b T = 0 . a Y = 4 . b Y = 0 . normal or, more generally, elliptically contoured risk distributions” is chosen wronglywhen the solvency problems are considered, whereas the “world of skewed distributions”is adequate in this framework.Conditions of Theorem 4.1 are very general for both T and Y , and the accuracy ofapproximation of P { Υ u,c t } by M u,c ( t ) is high for all c for which these values are notnegligibly small; this is a satisfactory approximation for the left-hand side of equation(2.6) that defines u α,t ( c ), c >
0, as an implicit function.To demonstrate this in a spectacular way, we compare Figs. 4 and 5. This shows thedifference in the accuracy of the normal (or Cram´er’s) and inverse Gaussian approxima-tions. The advantages of the latter are especially noticeable in that domain (includingthe point c ∗ and its neighborhood) where P { Υ u,c t } assumes not too small values.To emphasize that the inverse Gaussian approximation works well for heavy-tailed Y , we address Figs. 6–8 (see Table 1), where t = 1000, u = 40. To get simulated values of P { Υ u,c t } according to the algorithm described in [ ], we take ∆ c = 0 .
05 (and evenless in the vicinity of c ∗ , where this function’s flexure is considerable), and N = 1000.In Fig. 6, we draw the graph of M u,c ( t ) calculated by means of numerical integrationin (4.9) for T δ = 1, δ = 2, p = 2 / Y Pareto with parameters a Y = 4 . b Y = 0 .
35, whence c ∗ = 1 . M = 0 . D = 2 . T Erlang with parameters δ = 6 . k = 4 and Y Pareto with parameters a Y = 4 . b Y = 0 .
4, whence c ∗ = 1 . M = 0 .
8, and D = 1 . T Pareto with parameters a T = 4 . b T = 0 . Y Pareto withparameters a Y = 4 . b Y = 0 .
4, whence c ∗ = 1, M = 1, and D = 1 . The analytical technique which yields Theorem 4.1 (see[ ] and [ ], [ ], [ ]) is suitable for asymptotic analysis of non-ruin capital in thegeneral risk model. The following theorem (see [ ], Theorem 1) gives an asymptoticrepresentation for u α,t ( c ) at the points c = 0 and c = c ∗ , where (see (4.8)) c ∗ = 1 /M ;this generalizes asymptotic equalities (3.9). Theorem . Assume that p.d.f. f T ( x ) and f Y ( x ) are bounded above by a finiteconstant, D > , E ( T ) < ∞ , E ( Y ) < ∞ . Then u α,t (0) = tM + DM / κ α √ t (1 + o (1)) , t → ∞ ,u α,t ( c ∗ ) = DM / κ α/ √ t (1 + o (1)) , t → ∞ . The following theorem (see [ ], Theorem 2) gives an asymptotic representation for u α,t ( c ), c >
0, which generalizes asymptotic equality (3.8).
Theorem . Assume that p.d.f. f T ( x ) and f Y ( x ) are bounded above by a finiteconstant, D > , E ( T ) < ∞ , E ( Y ) < ∞ . Then u α,t ( c ) = ( c ∗ − c ) t + DM / z α,t (cid:18) M / ( c ∗ − c ) D √ t (cid:19) √ t, c c ∗ ,DM / z α,t (cid:18) M / ( c ∗ − c ) D √ t (cid:19) √ t, c > c ∗ , where for t sufficiently large the function z α,t ( x ) , x ∈ R , is continuous, monotone in-creasing, as x increases from −∞ to , monotone decreasing, as x increases from to ∞ , and such that lim x →−∞ z α,t ( x ) = 0 , lim x →∞ z α,t ( x ) = κ α and z α,t (0) = κ α/ (1 + o (1)) , t → ∞ . Let us construct simple bounds for u α,t ( c ), c >
0. First, Theorem 4.3 yields thefollowing bilateral asymptotic bounds: for 0 c c ∗ , we have( c ∗ − c ) t + DM / κ α √ t (1 + o (1)) u α,t ( c ) ( c ∗ − c ) t + DM / κ α/ √ t (1 + o (1)) , t → ∞ . (4.10)Second, looking for upper bounds for u α,t ( c ), c > c ∗ , we note that for c sufficiently largerthan c ∗ , i.e., for c > Kc ∗ with K > u α,t ( c ) is finite regardless of t .Thus, we focus (see Remark 1) on u α ( c ), c > c ∗ , which is a natural upper bound for u α,t ( c ), or on any sensible upper bound for u α ( c ).Bearing in mind the widely known theory (see, e.g., [ ]) built for the ultimate ruinprobability P { Υ u,c < ∞} , let us focus on the following cases. M ( i ) : exponential case. When T and Y are exponentially distributed with param-eters δ and ρ , we have c ∗ = δ/ρ , κ = ρ − δ/c . For c > δ/ρ , we have (see, e.g., [ ]) P { Υ u,c < ∞} = (1 − κ /ρ ) e − κ u for all u >
0. This rewrites as P { Υ u,c < ∞} = ( δ/ ( cρ )) exp {− ( ρ − δ/c ) u } , c > δ/ρ, and by simple calculations we have u α,t ( c ) max (cid:26) , − ln ( αcρ/δ ) ρ − δ/c (cid:27) , c > δ/ρ. (4.11) M ( ii ) : Poisson claims arrival and Y light-tailed. When T is exponentially distributedwith parameter δ and the distribution of Y is light-tailed, but non-exponential, specialcases of which are, e.g.,( a ) T exponentially distributed and Y b ) T exponentially distributed and Y Erlang,we have c ∗ = δ E Y , equation (4.3) rewrites as E exp { κ Y } = 1 + c κ /δ , and κ is itspositive solution. For c > c ∗ , we have (see, e.g., [ ]) P { Υ u,c < ∞} e − κ u for all u > u α,t ( c ) − ln α/ κ , c > δ E Y, and the problem comes down to finding κ in a closed form. In order to have more freedom of action, especially for finding compact formulas.
ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 15 M ( iii ) : Poisson claims arrival and Y heavy-tailed. Special cases are, e.g.,( a ) T exponentially distributed and Y Pareto,( b ) T exponentially distributed and Y Kummer .Any upper bounds for u α,t ( c ), c > c ∗ , which assumes small, rather than large values, istightly related to particulars of the probability P { Υ u,c < ∞} for small, rather than large values of u , which is a problem beyond the scope of this article. M ( iv ) : renewal claims arrival and Y exponentially distributed. When Y is exponen-tially distributed with parameter ρ and the distribution of T is arbitrary, special casesof which are, e.g.,( a ) T Y exponentially distributed,( b ) T Erlang and Y exponentially distributed,( c ) T Pareto and Y exponentially distributed,( d ) T Kummer and Y exponentially distributed,we have c ∗ = 1 / ( ρ E T ), equation (4.3) rewrites as E exp {− κ c T } = 1 − κ /ρ , and κ isits positive solution. For c > c ∗ , we have P { Υ u,c < ∞} = (1 − κ /ρ ) e − κ u for all u > − κ /ρ
1, we have (see [ ], Corollary 6.5.2) u α,t ( c ) − ln α/ κ , c > / ( ρ E T ) , (4.12)and the problem comes down to finding κ in a closed form. M ( v ) : renewal claims arrival and Y light-tailed. When Y is light-tailed, but non-exponential, and the distribution of T is arbitrary, special cases of which are, e.g.,( a ) T Erlang and Y Erlang,( b ) T Erlang and Y X d = Y − c T , whose c.d.f. is F X , denote by F X ( x ) = 1 − F X ( x ) is tailfunction, and write x = sup { x : F X ( x ) < } . For c > c ∗ = E Y / E T , we have (see [ ],Theorem 6.5.4) b ⊖ e − κ u P (cid:8) Υ u,c < ∞ (cid:9) b ⊕ e − κ u for all u > , (4.13)where κ is a positive solution to (4.3) and b ⊕ = inf x ∈ [0 ,x ] e κ x F X ( x ) R ∞ x e κ y dF X ( y ) , b ⊖ = sup x ∈ [0 ,x ] e κ x F X ( x ) R ∞ x e κ y dF X ( y ) . Alternatively (see [ ], Theorem 6.5.5), the inequalities (4.13) hold with b ∗⊕ = inf x ∈ [0 ,x ∗ ] e κ x F Y ( x ) R ∞ x e κ y dF Y ( y ) , b ∗ ⊖ = sup x ∈ [0 ,x ∗ ] e κ x F Y ( x ) R ∞ x e κ y dF Y ( y ) , where x ∗ = sup { x : F Y ( x ) < } ; the inequalities 0 b ∗ ⊖ b ⊖ b ⊕ b ∗⊕ u α ( c ), c > c ∗ , which is a solution to equation (2.8),and therefore upper bounds for u α,t ( c ), c > c ∗ , is easy to get from (4.13), and we leavethis to the reader. M ( vi ) : renewal claims arrival and Y heavy-tailed. Special cases are, e.g.,( a ) T is 2-mixture and Y is Pareto,( b ) T is Erlang and Y is Pareto,( c ) T is Pareto and Y is Pareto. Definition of the Kummer distribution see, e.g., in [ ]. Figure 9.
Model M ( i ): upper bound ( X -axis is c ) on u α,t ( c ) and simulatedvalues of u α,t ( c ), drawn for T and Y exponentially distributed with parameters δ = 3 / ρ = 4 /
5, and α = 0 . t = 200. Vertical grid line: c ∗ = 4 / u α,t ( c ∗ ) = 59 . Any upper bounds for u α,t ( c ), c > c ∗ , which assumes small, rather than large values, istightly related to particulars of the probability P { Υ u,c < ∞} for small, rather than large values of u , which is a problem beyond the scope of this article.
5. Numerical illustrations of non-ruin capital’s structure
Let us compare numerically the results formulated in Section 4.4 with the simulationresults taken as exact values; the algorithm of simulation is the same as in [ ], or in[ ]. For completeness, we return to Section 3.2 and start with T and Y exponentiallydistributed with parameters δ and ρ , whose p.d.f. are f T ( x ) = δ e − δx , f Y ( x ) = ρ e − ρx , x > . M ( i ): exponential case. Elementary calculations yield E ( T k ) = k ! /δ k , E ( Y k ) = k ! /ρ k , k = 1 , , . . . , whence E T = 1 /δ, D T = 1 /δ , E Y = 1 /ρ, D Y = 1 /ρ , and E e − κ c T = δ Z ∞ e − ( κ c + δ ) x dx = δ/ ( δ + c κ ) , E e κ Y = ρ Z ∞ e ( κ − ρ ) x dx = ρ/ ( ρ − κ ) . Plainly, c ∗ = E Y / E T is equal to δ/ρ , the constants defined in (4.8) are M = E T / E Y = ρ/δ,D = (cid:0) ( E T ) D Y + ( E Y ) D T (cid:1) / ( E Y ) = 2 ρ/δ , and for c > δ/ρ the positive solution κ to the Lundberg equation (4.3), which rewritesas the quadratic equation (cid:0) ρ − κ (cid:1) (cid:0) δ + c κ (cid:1) − δρ = 0, is κ = ρ − δ/c .In Fig. 9, the upper bounds (4.10) in the case 0 c δ/ρ , and (4.11) in the case c > δ/ρ , are drawn for t = 200, α = 0 . δ = 4 / ρ = 3 /
5, whence c ∗ = 1 . M = 0 .
75, and D = 1 . u α,t ( c ). ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 17
Figure 10.
Model M ( iv ): graphs ( X -axis is c ) of two-sided bounds (4.10),when 0 c c ∗ , of the upper bound, when c > c ∗ , and of simulated valuesof u α,t ( c ), drawn for T which is Erlang with parameters δ = 8 / k = 2 and Y exponentially distributed with parameter ρ = 3 /
5, and α = 0 . t = 200.Vertical grid line: c ∗ = 4 /
3. Horizontal grid line: u α,t ( c ∗ ) = 48. M ( iv ): Erlang T and exponentially distributed Y . This modelis a particular case of Model M ( v ), where T is Erlang with parameters k integer and δ > Y is Erlang with parameters m integer and ρ >
0, whose p.d.f. are f T ( x ) = δ k x k − Γ( k ) e − δx , f Y ( x ) = ρ m x m − Γ( m ) e − ρx , x > . Elementary calculations yield E T = k/δ, D T = k/δ , E Y = m/ρ, D Y = m/ρ , and E e − κ c T = δ k Γ( k ) Z ∞ e − ( κ c + δ ) x x k − dx = δ k ( δ + c κ ) k , E e κ Y = ρ m Γ( m ) Z ∞ e ( κ − ρ ) x x m − dx = ρ m ( ρ − κ ) m . Plainly, c ∗ = E Y / E T is equal to ( mδ ) / ( kρ ), the constants defined in (4.8) are M = E T / E Y = kρ/ ( mδ ) ,D = (( E T ) D Y + ( E Y ) D T ) / ( E Y ) = k ( k + m ) ρ/ ( m δ ) , and for c > ( mδ ) / ( kρ ) the positive solution κ to the Lundberg equation (4.3), whichrewrites as ( ρ − κ ) m ( δ + c κ ) k − δ k ρ m = 0, is easy to find numerically; this κ is notexplicit, except for m = 1.In Fig. 10, the upper and lower bounds (4.10) in the case 0 c δ/ρ , and the upperbound (4.12) in the case c > δ/ρ , are drawn for t = 200, α = 0 . δ = 8 / k = 2, ρ = 3 /
5, whence c ∗ = E Y / E T = 1 . M = 0 .
75, and D = 1 . u α,t ( c ). ++++++++++++++++++++++++++++++++++++++++ Figure 11.
Model M ( iii ): graphs ( X -axis is c ) of two-sided bounds (4.10)for 0 c c ∗ and of simulated values of u α,t ( c ), drawn for T exponentiallydistributed with parameter δ = 4 / Y which is Pareto with parameters a Y = 10, b Y = 0 .
05 (dots), a Y = 3, b Y = 0 . α = 0 . t = 200.Vertical grid lines: c ∗ = 1 . c ∗ = 1 . u α,t ( c ∗ ) = 80 (the same for dots and crosses). M ( iii ): exponentially distributed T and Pareto Y . For T ex-ponentially distributed with parameter δ > Y whose distribution is Pareto withparameters a Y > b Y >
0, p.d.f. are f T ( x ) = δ e − δ x , f Y ( x ) = a Y b Y ( x b Y + 1) a Y +1 , x > , elementary calculations yield E T = 1 /δ, D T = 1 /δ , E Y = 1 / (( a Y − b Y ) , D Y = a Y / (( a Y − ( a Y − b Y ) . Plainly, c ∗ = E Y / E T is equal to δ/ (( a Y − b Y ), the constants defined in (4.8) are M = E T / E Y = ( a Y − b Y δ ,D = (( E T ) D Y + ( E Y ) D T ) / ( E Y ) = 2 ( a Y − b Y δ ( a Y − , and the adjustment coefficient does not exist.In Fig. 11, the upper and lower bounds (4.10) in the case 0 c c ∗ are drawn.Bounds for c > c ∗ are beyond the scope of this article and are not considered, althoughthe essence of the complexity in their construction is clear. By dots, drawn are simulatedvalues of u α,t ( c ), c >
0. We note that for a Y = 3, b Y = 0 . E ( Y ) is not finite, and Fig. 11 suggests that the moment conditions in Theorems 4.2and 4.3 may be somewhat relaxed. However, this will significantly complicate the proof,which lies beyond the scope of this article. M ( iii ): exponentially distributed T and Kummer Y . For T exponentially distributed with parameter δ > Y whose distribution is Kummer ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 19 ++++++++++++++++++++++++++++++++++++++++
Figure 12.
Model M ( iii ): graph ( X -axis is c ) of two-sided bounds (4.10)for 0 c c ∗ and of simulated values of u α,t ( c ), drawn for T exponentiallydistributed with parameter δ = 4 / Y which is Kummer with parameters k Y = 5, l Y = 5 (dots), k Y = 200, l Y = 200 (crosses), α = 0 .
05, and t = 200.Vertical grid line: c ∗ = 1 . c ∗ = 0 . u α,t ( c ∗ ) = 102 (dots) and u α,t ( c ∗ ) = 36 (crosses). with parameters k Y > l Y >
0, p.d.f. are f T ( x ) = δ e − δ x , f Y ( x ) = k Y (cid:0) k Y + l Y (cid:1) Γ (cid:0) k Y (cid:1) U (cid:18) l Y , − k Y , k Y l Y x (cid:19) , x > . Elementary calculations yield E T k = k ! /δ k , E Y k = Γ (cid:0) k Y + k (cid:1) Γ (cid:0) l Y − k (cid:1) Γ (cid:0) k Y (cid:1) Γ (cid:0) l Y (cid:1) l kY k − kY , k < l Y , k = 1 , , . . . . In particular, E T = 1 /δ, D T = 1 /δ , E Y = l Y l Y − , D Y = l Y ( 4( l Y −
2) + k Y l Y ) k Y ( l Y − ( l Y − . Plainly, c ∗ = E Y / E T is equal to δ l Y / ( l Y − M = E T / E Y = ( l Y − δ l Y ,D = (cid:0) ( E T ) D Y + ( E Y ) D T (cid:1) / ( E Y ) = 2 (2 + k Y )( l Y − δ k Y ( l Y − l Y , and the adjustment coefficient does not exist.In Fig. 12, the upper and lower bounds (4.10) in the case 0 c c ∗ are drawn.Bounds for c > c ∗ are beyond the scope of this article and are not considered, althoughthe essence of the complexity in their construction is clear. By dots, drawn are thesimulated values of u α,t ( c ), c >
6. Conclusion
This paper provides a mathematical investigation of an observation (see, e.g., [ ])made empirically, that the Value-at-Risk is not a good solution to the problem of risk For other equivalent formulas for f Y ( x ) see [ ]. measure’s choice balanced for efficiency and simplicity. Regarding efficiency, the Value-at-Risk is not a good substitute for non-ruin capital, which can be seen even from definitions.Regarding simplicity, it turns out in fairly general risk models that the structure of non-ruin capital is as simple as the structure of Value-at-Risk. References [1] Artzner, P., Delbaen, F., Eber, J.M., and Heath, D. (1999) Coherent measures of risk.
MathematicalFinance , Vol. 9 (3), 203–228.[2] Billingsley, P. (1999) Convergence of Probability Measures. 2-nd ed., John Wiley & Sons, New York.[3] Culp, C.L., Miller, M.H., and Neves, A.M.P. (1997) Value-at-Risk: uses and abuses,
Journal ofApplied Corporate Finance , Vol. 10, 26–38.[4] Cummins, D., Harrington, S. and Niehaus, G. (1994) An economic overview of risk-based capitalrequirements for the property-liability industry,
Journal of Insurance Regulation , Vol. 11, 427–447.[5] Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 onthe taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II), Brussels, 25November 2009.[6] Directive 2014/51/EU of the European Parliament and of the Council of 16 April 2014 amendingDirectives 2003/71/EC and 2009/138/EC and Regulations (EC) No 1060/2009, (EU) No 1094/2010and (EU) No 1095/2010 in respect of the powers of the European Supervisory Authority (Euro-pean Insurance and Occupational Pensions Authority) and the European Supervisory Authority(European Securities and Markets Authority), Brussels, 16 April 2016.[7] Doff, R.R. (2008) A critical analysis of the Solvency II proposals,
The Geneva Papers on Risk andInsurance – Issues and Practice , Vol. 33, Issue 2, 193–206.[8] Eling, M., and Schmeiser, H. (2010) Insurance and the credit crisis: impact and ten consequencesfor risk management and supervision,
The Geneva Papers on Risk and Insurance – Issues andPractice , Vol. 35, Issue 1, 9–34.[9] Eling, M., Schmeiser, H., and Schmit, J. (2007) The Solvency II process: overview and criticalanalysis,
Risk Management and Insurance Review , Vol. 10, 69–85.[10] Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. II. 2-nd ed.,John Wiley & Sons, New York, etc.[11] Floreani, A. (2013) Risk measures and capital requirements: a critique of the Solvency II approach,
The Geneva Papers on Risk and Insurance – Issues and Practice , Vol. 38, 189–212.[12] Heep-Altiner, M., Mullins, M., and Rohlfs, T., Eds. (2018) Solvency II in the Insurance IndustryApplication of a Non-Life Data Model. Springer.[13] Kalashnikov, V. (1997) Geometric Sums: Bounds for Rare Events with Applications. Kluwer Aca-demic Publishers, Dordrecht.[14] Malinovskii, V.K. (1998) Non-Poissonian claims arrivals and calculation of the probability of ruin.
Insurance: Mathematics and Economics , Vol. 22, 123–138.[15] Malinovskii, V.K. (2012) Equitable solvent controls in a multi-period game model of risk,
Insurance:Mathematics and Economics , Vol. 51, 599–616.[16] Malinovskii, V.K. (2017) On the time of first level crossing and inverse Gaussian distribution;https://arxiv.org/pdf/1708.08665.pdf[17] Malinovskii, V.K. (2017) Generalized inverse Gaussian distributions and the time of first level cross-ing; https://arxiv.org/pdf/1708.08671.pdf[18] Malinovskii, V.K. (2018) Approximations in the problem of level crossing by a compound renewalprocess,
Doklady Akademii Nauk , Vol. 483, No. 5, 622–625.[19] Malinovskii, V.K. (2020) The level which a compound renewal process crosses with a given proba-bility,
Doklady Akademii Nauk , To appear.[20] Malinovskii, V.K. (2020) Insurance Planning Models. Price Competition and Regulation of Finan-cial Stability. World Scientific. To appear.[21] Malinovskii, V.K., and Kosova, K.O. (2014) Simulation analysis of ruin capital in Sparre Andersen’smodel of risk,
Insurance: Mathematics and Economics , Vol. 59, 184–193.[22] Malinovskii, V.K., Malinovskii, K.V. (2017) On approximations for the distribution of first levelcrossing time; https://arxiv.org/pdf/1708.08678.pdf.[23] Marano, P., and Siri, M., Eds. (2017) Insurance Regulation in the European Union Solvency II andBeyond. Palgrave Macmillan.
ALUE-AT-RISK SUBSTITUTE FOR NON-RUIN CAPITAL 21 [24] Mittnik, S. (2011) Solvency II calibrations: where curiosity meets spuriosity. Munich: Center forQuantitative Risk Analysis (CEQURA), Department of Statistics, University of Munich.[25] Pfeifer, D., and Straßburger, D. (2008) Solvency II: stability problems with the SCR aggregationformula,
Scandinavian Actuarial Journal , Vol. 1, 61–77.[26] Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999) Stochastic Processes for Insurance andFinance. John Wiley & Sons, Chichester, etc.[27] Sandstr¨om, A. (2011) Handbook of Solvency for Actuaries and Risk Managers: Theory and Practice.Chapman & Hall / CRC, Taylor & Francis Group. Boca Raton, etc.[28] Sparre-Andersen, E. (1957) On the collective theory of risk in case of contagion between the claims.In book: Transactions of the XV-th International Congress of Actuaries, Vol. 2, 219–229.
Central Economics and Mathematics Institute (CEMI) of Russian Academy of Science,117418, Nakhimovskiy prosp., 47, Moscow, Russia
E-mail address : [email protected], [email protected]
URL ::