Cost-of-Captal Margn for a General Insurance Lablty Runoff Robert Salzmann and Maro V Wüthrch Abstract Under new solvency regulatons, general nsurance companes need to calculate a rsk margn to cover possble shortfalls n ther lablty runoff A popular approach for the calculaton of the rsk margn s the so-called cost-of-captal approach The cost-of-captal approach nvolves the consderaton of multperod rsk measures Because multperod rsk measures are complex mathematcal obects, varous proxes are used to calculate ths rsk margn Of course, the use of proxes and the study of ther qualty rases many questons, see IAA poston paper 7 In the present paper we derve the frst mathematcally rgorous multperod cost-of-captal approach for a general nsurance lablty runoff wthn a chan ladder framework We derve analytc formulas for the rsk margn whch allow to compare the dfferent proxes used n practse Moreover, a case study nvestgates and answers questons rased n 7 Key words Solvency, rsk margn, cost-of-captal approach, multperod rsk measure, rsk bearng captal, general nsurance runoff, clams reservng, outstandng loss labltes, clams development result, chan ladder model 1 Introducton The runoff of general nsurance labltes outstandng loss labltes usually takes several years Therefore, general nsurance companes need to buld approprate reserves provsons for the runoff of the outstandng loss labltes These reserves need to be TH Zurch, Department of Mathematcs, 8092 Zurch 1
ncessantly adusted accordng to the latest nformaton avalable Under new solvency regulatons, general nsurance companes have to protect aganst possble shortfalls n these reserves adustments wth rsk bearng captal In ths sprt, ths work provdes a frst comprehensve dscourse on multperod solvency consderatons for a general nsurance lablty runoff and answers questons rased n the IAA poston paper 7 Ths dscourse nvolves the descrpton of the cost-of-captal approach n a multperod rsk measure settng In a cost-of-captal approach the nsurance company needs to prove that t holds suffcent reserves frstly to pay for the nsurance labltes clams reserves and secondly to pay the costs of rsk bearng captal cost-of-captal margn or rsk margn Hence, at tme 0, the nsurer needs to hold rsk-adusted clams reserves that comprse best-estmate reserves for the outstandng loss labltes and an addtonal margn for the coverage of the cashflow generated by the cost-of-captal loadngs Such rsk-adusted clams reserves are often called a market-consstent prce for the runoff labltes n a marked-to-model approach, see eg Wüthrch et al 14 Because the multperod cost-of-captal approach s rather nvolved, state-of-the-art solvency models consder a one-perod measure together wth a proxy for all later perods In ths paper we consder four dfferent approaches denoted by R 1,, R4 : R 1 : The Regulatory Solvency Approach currently used n practse Ths approach s rsk-based wth respect to the next accountng year k 1, but t s not rsk-based for all successve accountng years t uses a proxy for later accountng years k 2 R 2 : The Splt of the Total Uncertanty Approach Ths approach presents a rskbased adapton of the frst approach to all remanng accountng years Partcularly, the rsk measures quantfy the rsk n each accountng year k 1 wth respect to the nformaton avalable at the begnnng, e at tme 0 R 3 : The xpected Stand-Alone Rsk Measure Approach Ths approach ncorporates rsk measures for each accountng year whch are rsk-based, e measurable wth respect to the prevous accountng year Moreover, the rsk-adusted clams reserves are self-fnancng n the average but they lack protecton aganst possble shortfalls n the cost-of-captal cashflow 2
R 4 : The Multperod Rsk Measure Approach Ths approach gves a complete, methodologcally consstent vew va multperod rsk measures, but as a consequence, t s much more techncal and complex compared to R 1, R2, R3 Note that throughout ths paper we only consder nomnal values A frst approach to dscounted clams reserves for solvency consderatons s provded n Wüthrch-Bühlmann 15 They present a model for the one-year runoff wth stochastc dscountng whch s smlar to Approach 1 Organsaton of the paper In Secton 2 we ntroduce the clams development result whch descrbes the adustments The underlyng model for the predcton of the outstandng loss labltes s dscussed n Secton 3 In Secton 4 we dscuss the four dfferent approaches n order to determne the cost-of-captal margns Fnally, Secton 5 presents a case study for a general nsurance lablty runoff portfolo 2 The Clams Development Result Let C, > 0 denote the cumulatve payments of accdent year {0,, I} after development year {0,, J} wth J I The ultmate clam of accdent year s then gven by C,J and for the nformaton avalable at tme k for k 0,, J we wrte D I+k {C, ; + I + k, 0 I, 0 J} Snce loss reservng s bascally a predcton problem, we are manly nterested n the predctons Ĉk,J of the ultmate clam C,J, gven nformaton D I+k The outstandng loss labltes for accdent year at tme k are defned by C,J C,I +k assume that I + k + J At tme k these are predcted by the clams reserves R k Ĉk,J C,I +k Note that for every further accountng year more data become avalable and we have to adapt the predctons accordng to the latest nformaton avalable Therefore, we consder the successve best-estmate predctons of the ultmate clams C,J, e Ĉ 0,J, Ĉ1,J,, ĈJ+ I 1,J 3, Ĉ J+ I,J C,J 21
Ther ncrements determne the so-called clams development result CDR; see Bühlmann et al 2 and Ohlsson-Lauzenngks 10 In new solvency regulatons, the CDR s the central obect of nterest for the reserve rsks and has to be thoroughly studed Defnton 21 For accountng year k and accdent year the CDR s defned by CDR k Ĉk 1,J Ĉk,J 22 It refers to the change n the balance sheet n accountng year k so that we always have best-estmate predctons, e the outstandng loss labltes are covered by clams reserves accordng to the latest nformaton avalable New solvency approaches see eg Swss Solvency Test 13 and AISAM-ACM 1 requre protecton aganst possble shortfalls n ths one-year CDR by rsk bearng captal Ths means that nsurance companes gve a yearly guarantee that the best-estmate predctons are always covered by a suffcent amount of clams reserves modulo the chosen rsk measure Therefore, n a busness year or solvency vew, we need to study the sequence of CDR s For a fxed accdent year we consder whch n terms of the clams reserves satsfy CDR k Ĉk 1,J CDR 1,, CDR J + I, Ĉk,J k 1 R X,I +k + k R, where X,I +k C,I +k C,I +k 1 are the ncremental payments for accdent year n accountng year k 3 Outstandng Loss Lablty Model In the followng the clams reservng s descrbed n a Bayesan chan ladder model framework Ths Bayesan framework provdes a unfed approach for a successve nformaton update n each accountng year, e recent nformaton s mmedately absorbed by the Bayesan model see also Bühlmann et al 2 We defne the ndvdual clams development factors F, C, /C, 1 for 1,, J Then the cumulatve payments C, are gven by C, C,0 F,m 4 m1
The frst payment C,0 plays the role of the ntal value of the process C, 0,,J and F, are the multplcatve changes In a Bayesan chan ladder framework, we assume that the unknown underlyng parameters are descrbed by the random varables Θ 1 1,, Θ 1 J Gven these, we further assume that C, satsfes a chan ladder model Model Assumptons 31 Gamma-Gamma Bayes Chan Ladder Model Condtonally, gven Θ Θ 1,, Θ J, the cumulatve payments C, for dfferent accdent years are ndependent C,0, F,1,, F,J are ndependent wth F, Θ Γ σ 2, Θ σ 2, for 1,, J, where the σ s are gven postve pror constants C,0 and Θ are ndependent Θ 1,, Θ J are ndependent wth Θ Γγ, f γ 1 wth pror parameters f > 0 and γ > 2 Remark The gamma-gamma Bayes chan ladder model defnes a model that belongs to the exponental dsperson famly wth assocate conugate prors see eg Bühlmann- Gsler 3 Ths famly of dstrbutons delvers an exact credblty case because the Bayesan estmators concde wth the lnear credblty estmators Hence, t s possble to explctly calculate the posteror dstrbuton of Θ, gven the observatons F, We restrct ourselves to the gamma-gamma case because t allows for an explct calculaton of the predcton uncertantes whch s essental for multperod rsk measure decompostons For most other models, only numercal solutons are avalable or one needs approxmatons smlar to Theorem 65 n Gsler-Wüthrch 6 Condtonally, gven Θ, we obtan a chan ladder model wth frst two moments gven by C, Θ, C,0,, C, 1 C, 1 F, Θ C, 1 Θ 1, Var C, Θ, C,0,, C, 1 C 2, 1 Var F, Θ C 2, 1 σ 2 Θ 2 5
Henceforth, Θ 1 plays the role of the chan ladder factor see Mack 9 Note that the varance s proportonal to C 2, 1, ths assumpton s crucal for the calculaton of feasble standard devaton loadngs n the multperod cost-of-captal approach but s dfferent from Mack s 9 classcal dstrbuton-free chan ladder model That s, we need to modfy the classcal chan ladder varance assumpton n order to have a tractable model otherwse only numercal solutons are applcable In the case study below, the numercal dfferences between these dfferent models are analyzed see Secton 5 below Moreover, for the pror moments we have Θ 1 f, Θ 2 f 2 γ 1 γ 2 and Var Θ 1 f 2 1 γ 2 Ths shows that we have a pror mean for the chan ladder factor of f In order that the pror second moments exst we need to assume that γ > 2 31 The Parameter Update Procedure At tme k 0, we have nformaton D I+k and we need to predct the outstandng loss labltes that correspond to the random varables C,J C,I +k Ths means that we have to update our model accordng to the nformaton generated by the runoff portfolo for successve accountng years The followng proposton descrbes the parameter update procedure for the posteror dstrbutons of Θ Proposton 32 Under Model Assumptons 31 we have that the condtonal posteror dstrbutons of Θ, gven D I+k, are ndependent gamma dstrbutons wth parameters γ k I + k I + 1 γ + σ 2 and c k f γ 1 + σ 2 I+k I 0 F, Proof of Proposton 32 We denote the dstrbuton of C,0 by π Then the ont densty of Θ, D I+k s gven by π Θ, D I+k + I+k Θ σ 2 σ 2 Γ σ 2 F σ 2, J f γ 1 γ 1 Γ γ 1 exp { Θ σ 2 F, } Θ γ 1 exp { f γ 1 Θ } I π C,0 0 6
Ths mples that the posteror densty of Θ, gven D I+k, satsfes the followng proportonalty property π Θ D I+k J Θ 1 γ + I+k I+1 σ 2 1 exp f γ 1 + σ 2 These are ndependent gamma denstes whch proves the proposton The above result mples the followng corollary Corollary 33 Under the assumptons of Proposton 32 we have f k Θ 1 DI+k def where we have defned γ k c k 1 αk 2 c k Θ 2 DI+k γ k 1 γ k 2 I+k I 0 F, F k + 1 α k f, 2 k γ k 1 f γ k 2, Θ F k α k 1 I + k I + 1 I+k I 0 F,, I + k I + 1 I + k I + 1 + σ 2 γ 1 Remark Note that the F k s dffer from the chan ladder estmates resultng from volume weghtng n Mack s 9 model It s well-known that the parameter updatng procedure can also be done recursvely see Gerber-Jones 5, Kremer 8, Sundt 12 and Bühlmann-Gsler 3, Theorem 96 In our case ths leads to the helpful representaton: Corollary 34 For k 1 and k we have f k a k F I+k, + 1 a k k 1 f, where the credblty weght a k s gven by a k I + k + 1 + σ 2 γ 1 1 Proof The proof follows from Corollary 33 7
32 Ultmate Clam Predcton The followng proposton determnes the best-estmate predcton of the ultmate C,J n our Bayesan chan ladder framework Proposton 35 Suppose the assumptons of Proposton 32 to hold The predctor for the ultmate C,J that has mnmal condtonal varance, gven D I+k, s gven by Ĉk,J C,J D I+k For I + k < + J we obtan Ĉ k,j C,J D I+k C,I +k J I +k+1 f k Proof The proof easly follows from the fact that condtonal expectatons mnmse L 2 -dstances and from the posteror ndependence of the Θ s see Proposton 32 Remark Note that ths s a so-called Bayes chan ladder model see Bühlmann et al 2 where the chan ladder factors f k are credblty weghted averages between the pror estmates f and the observatons F k For unnformatve prors, e γ 1 and therefore α k 1, we obtan a frequentst s chan ladder factor estmator that s only based on the observatons Note that for the second posteror moment to exst, we need to have γ k > 2 Ths may exclude the consderaton of the asymptotc unnformatve case In the remander of ths paper we are gong to characterse the uncertantes n the CDR Defnton 21 Our frst Corollary states that best-estmate predctons 21 form a martngale Corollary 36 Under Model Assumptons 31 we have CDR k D I+k 1 0 Moreover, the CDR s are uncorrelated, e for k 1, l J and m < maxk, l t holds that CDR kcdr l D I+m 0 8
Proof Note that ths best-estmate predctons form a martngale ths follows from the tower property of condtonal expectatons Hence, t easly follows that the expected CDR s zero The second clam then easly follows because martngales have uncorrelated ncrements Remark At ths stage t turns out to be crucal that we have an exact credblty model Otherwse one obtans a bas term n the CDR as eg n Proposton 31 n Bühlmann et al 2 that s dffcult to control 4 Cost-of-Captal Margn For a fxed accdent year, CDR k consttutes the amount by whch we need to adust the clams reserves to have best-estmate clams reserves n each accountng year k From a regulatory or polcyholder pont of vew the nsurance company needs to protect aganst possble shortfalls n CDR k by an approprate rsk margn n each accountng year otherwse the company wll not be able to balance ts proft & loss statement n accountng year k Further dscusson on the use of rsk margns s provded n Secton 6 of the IAA poston paper 7 We choose a rsk measure ρ k for the protecton aganst shortfalls n accountng year k, e ths rsk measure corresponds to the rsk bearng captal requred by the regulator that protects aganst adverse developments n the best-estmate predctons In a cost-of-captal approach the nsurance company does not need to hold the captal ρ k tself but rather the prce for ths captal cost-of-captal margn, rensurance premum We assume that the cost-of-captal rate s gven by a constant c > 0, hence the regulatory prce for protecton aganst adverse development n accountng year k s defned by cρ k Note that ths exactly descrbes the prce of rsk but t does not tell us anythng about the organsaton of the rsk bearng, e n addton to the cost-of-captal margn cρ k the regulator also needs to make sure that ths captal s used for organsng the rsk bearng Below we defne dfferent rsk measures ρ k whch wll lead to dfferent cost-of-captal loadngs cρ k Such a cost-of-captal loadng can be vewed as a lablty towards the party 9
that provdes the rsk bearng captal ρ k Consequently, n addton to the clams reserves R k, the nsurance company needs to buld reserves for the cost-of-captal cashflow Assume that ĈoCk c ρ k+1,, c ρ J+ I are the reserves cost-of-captal margn that cover the aggregated cost-of-captal cashflow c ρ k+1,, c ρ J+ I Hence, the rsk-adusted clams reserves at tme k based on the nformaton D I+k are gven by R k R k + ĈoCk The k R s can be nterpreted as a market-consstent prce for the runoff labltes n an ncomplete market settng, e the captal equpment for the outstandng loss labltes and a prce for the rsk at whch the outstandng loss labltes can be transferred to a thrd party at tme k n a marked-to-model vew As descrbed n Secton 1, we develop 4 dfferent models for the choce of the rsk measure k ρ k that wll lead to 4 dfferent rsk-adusted clams reserves approaches,, R 4 k for R k R 1 41 Regulatory Solvency Proxy Approach For I + k J + and, k 1 we defne the constant β,k σi+k 2 + 1 γk 1 I+k 1 J 2 σ γ k 1 I+k 2 a k 2 + 1 γk 1 1 γ k 1 2 1 + 1 I+k +1 Note that we have β,k > 1 and snce the a k s and γ k s do not depend on the observatons see Proposton 32 and Corollary 34, β,k s also unaffected Ths s a crucal property of Model Assumptons 31 that we are gong to use n the dervatons below Proposton 41 Under Model Assumptons 31 we have, for I + 1 J +, Var CDR 1 D I Var Ĉ1,J D I Proof The proposton easly follows from Theorem 42 below 10 Ĉ0,J 2 β,1 1
Many regulators use an approach that s smlar to the followng proxy for the estmaton of the cost-of-captal charge see eg Swss Solvency Test 13, Sandström 11, Secton 68 or Appendx C3 n the IAA poston paper 7 The expected outstandng loss labltes at tme k vewed from tme 0 are gven by Hence, r 0 r k Rk D I,, r J+ I 1, r J+ I,J C,I +k D I Ĉk Ĉ0,J Ĉ0,I +k 0 descrbes the expected runoff of the outstandng loss labltes vewed from tme 0 In the frst cost-of-captal approach the rsk measure n accountng year k s chosen to be the upper ndex n the rsk measure notaton labels the approaches ρ 1,k rk 1 r 0 φ Var CDR 1 D I 1/2 Ĉ 0,J Ĉ0,I +k 1 Ĉ 0,J C,I φ Ĉ0,J β,1 1 1/2, where φ s a fxed postve constant determnng the securty level In ths setup, the approprate rsk measure for accountng year 1 s gven by ρ 1,1 φ Var CDR 1 D I 1/2 φ Ĉ0,J β,1 1 1/2 That s, we choose an approprate rsk measure ρ 1,1 determned by a standard devaton loadng The rsk measures ρ 1,k for the frst accountng year whch s years k 2 are then obtaned by the D I -measurable volume scalng r k for later accountng descrbng the expected runoff of the outstandng loss labltes The underlyng assumpton s that ths volume measure s a good proxy for the runoff of the CDR uncertanty The rsk-adusted clams reserves at tme 0 are then gven by R 1 0 R 0 J+ I + c φ Ĉ0,J β,1 1 1/2 Ĉ 0,J Ĉ0,I +k 1 Ĉ 0,J C,I 41 We refer to 41 as the regulatory solvency proxy approach We see that the calculaton of the rsk-adusted clams reserves s very smple It only requres the study of the CDR for the frst accountng year and all the remanng uncertantes are proportonal to the uncertanty n the frst accountng year Hence, ths approach meets the smplcty requrements often wanted n practse However, snce ths approach s not rsk-based for later accountng years, the rsk-adaptaton for later accountng years s rather questonable 11
42 Splt of the Total Uncertanty Approach Corollary 36 mples that the total uncertanty vewed from tme 0 can be splt nto the sngle one-year uncertantes for dfferent accountng years as follows J+ I J+ I Var C,J D I Var CDR k D I Var CDR k D I J+ I CDR 2 k J+ I DI Var Ĉk,J D I+k 1 DI For the second equalty to hold, we have used the uncorrelatedness of the CDR s Theorem 42 Under Model Assumptons 31 we have, for I + k J +, Var Ĉk,J D I+k 1 Var CDR k D I An empty product s set equal to 1 Ĉk 1,J Ĉ0,J 2 β,k 1, 2 k 1 β, β,k 1 The proof s provded n the appendx Theorem 42 mmedately mples: Corollary 43 Aggregated One-Year Rsks Under Model Assumptons 31 we have J+ I Var C,J D I Var CDR k D I Ĉ0 J+ I 2 Ĉ0 2 J σ 2,J β,k 1,J + 1 γ0 1 1 γ 0 2 The proof s provded n the appendx 1 I +1 Remark Corollary 43 gves the predcton uncertanty for the total runoff of the outstandng loss labltes Ths s smlar to the famous Mack formula Mack 9 n the classcal chan ladder model and to the formula n the Bayesan chan ladder model consdered n Gsler-Wüthrch 6 Theorem 42 then states how ths total uncertanty factorses across the sngle accountng years We defne the rsk measures ρ 2,k approach by n ths second ρ 2,k φ Var CDR k D I 1/2 φ Ĉ0,J k 1 1 β 1/2, β,k 1 1/2 12
Ths means that we analyse the CDR uncertanty for each accountng year vewed from tme 0 Snce all the underlyng terms are D I -measurable, we can defne the rsk-adusted clams reserves at tme 0 for the rsk measure ρ 2,k by R 2 0 R 0 + c φ Ĉ0,J J+ I k 1 We refer to 42 as the splt of the total uncertanty approach 1 β 1/2, β,k 1 1/2 42 43 xpected Stand-Alone Rsk Measure Approach Note that the rsk measures ρ 1,k and ρ2,k are both D I-measurable However, one would expect that the rsk measure ρ k should be D I+k 1 -measurable whch reflects the clams development up to accountng year k Note that due to Theorem 42 we have Ĉk 2 Var CDR k D I+k 1 Var D I+k 1 β,k 1 Hence, we defne the thrd rsk measure by,j Ĉk 1,J ρ 3,k φ Var CDR k D I+k 1 1/2 φ Ĉk 1,J β,k 1 1/2 Ths corresponds exactly to the regulatory rsk bearng captal usng a standard devaton approach that the nsurance company needs to hold n order to run ts busness n accountng year k and havng clams experence D I+k 1 Note that the cost-of-captal margn c ρ 3,k s a D I+k 1-measurable cashflow for whch we need to put reserves asde at tme 0 Therefore, we defne the rsk-adusted clams reserves for ρ 3,k by R 3 0 R 0 + c J+ I ρ 3,k D I R 0 + c φ Ĉ0,J J+ I β,k 1 1/2 43 We refer to 43 as the expected stand-alone rsk measure approach Note that these rsk-adusted clams reserves are self-fnancng n the average whch means that we have exactly reserved for the expected value of the cashflow X,I +1 + c ρ 3 1,, X,J + c ρ 3 J+ I Because β, > 1 or due to Jensen s nequalty we can easly see that the followng corollary holds true: Corollary 44 We have R 3 0 13 0 2 R
44 Multperod Rsk Measure Approach Reserves that are self-fnancng n the average as above see formula 43 do not account for the rsk nherent n the cost-of-captal cashflow tself In a multperod dynamc rsk measure approach see, eg, Föllmer-Penner 4 we addtonally quantfy the uncertanty n the cost-of-captal cashflow c ρ k Ths then needs a recursve calculaton of the necessary rsk measures whch s explaned n ths subsecton We start wth a schematc llustraton based on backward nducton Fx accdent year > I J, then the last accountng year for ths accdent year s gven by J + I cost-of-captal cashflow by ĈoCJ+ I clams reserves are defned by where ρ 4 R 4 k,k+1 φ Var φ Var R k + Rk+1 ĈoC k+1 Hence, we ntalse the reserves for the 0 For k 0,, J + I 1 the rsk-adusted D I+k + c ρ 4,k+1 + X,I +k+1 + ĈoCk+1 CDR k + 1 + Note that the rsk measure ρ 4,k+1 ĈoC k+1 def 1/2 D I+k D I+k R k ĈoCk+1 + ĈoCk, DI+k 1/2 quantfes the CDR uncertanty n the clams cashflow X, and n the cost-of-captal cashflow c ρ 4,, k +1 as well The total reserves at tme 0 for the cost-of-captal cashflow are gven by ĈoC0 We defne for k 1,, J + I b,k 1 + c φ β,k 1 1/2 Proposton 45 The cost-of-captal reserves at tme k 0,, J + I 1 are gven by ĈoC k Ĉk,J J+ I mk+1 b,m 1 The proof goes by nducton and s provded n the appendx Remark As a consequence of our model assumptons, the cost-of-captal cashflow turns out to be lnear n Ĉk,J Ths fact allows for an analytc calculaton n the multperod rsk measure approach for sngle accdent years 14
Henceforth, n the multperod rsk measure approach we have rsk-adusted clams reserves at tme 0 gven by R 4 0 R 0 1 + ĈoC D I + c ρ 4,1 R 0 R 0 + ĈoC0 J+ I + Ĉ0,J 44 b,k 1, referred to as the multperod rsk measure approach Corollary 46 We have R 3 0 0 4 R The deeper reason for Corollary 46 to hold s that n addton to the expected cost-ofcaptal cashflow the rsk-adusted clams reserves n the multperod rsk measure approach ncorporate a margn aganst possble shortfalls n ths cashflow The orderng of R 4 0 and R 2 the choce of the securty level φ We need to compare { J+ I } J+ I b,k 1 1 + cφ β,k 1 1/2 1 wth 0 Lemma 47 For a 1,, a R, we have depends on the choce of the cost-of-captal rate c and { cφ J+ I k 1 1 β 1/2, β,k 1 1/2 } 1 + a k 1 k 1 1 + a a k 1 The lemma s proved n the appendx Therefore, the queston smplfes to comparng { J+ I } { k 1 1 + cφ β, 1 1/2 J+ I } k 1 β,k 1 1/2 wth β 1/2, β,k 1 1/2 1 Corollary 48 Assume for I J + 2,, I c φ 1 1/2 1/2 max β 1/2, 1 / β 1/2, + 1 1,,J+ I 1 45 Then we have R 2 0 15 0 4 R
The proof s provded n the appendx Remarks 1/2 Note that for c φ beng smaller than mn β 1/2, 1 / β 1/2, + 1 1/2 we obtan that the splt of total uncertanty approach 42 gves hgher rsk-adusted clams reserves than the multperod rsk measure approach 44 However, n every other case we cannot say whch rsk-adusted clams reserves are more conservatve In the practcal examples we have consdered, the assumptons of Corollary 48 were always fulflled Ths means that the multperod rsk measure approach turned out to result n the most conservatve rsk-adusted clams reserves Further, we have notced that β, 1 whch mples that β, 1 1/2 1 Moreover, the securty level and the cost-of-captal margn typcally are such that c φ 03 Ths mmedately mples that the two terms n 45 are almost equal and hence, very often n practcal stuatons we observe that R 2 0 R 4 0 45 Aggregaton of Accdent Years In the prevous subsectons we have studed the cost-of-captal margn for one sngle accdent year only Fnally, of course, we would lke to measure the uncertanty over all accdent years {I J +1,, I} Hence, the total CDR n accountng year k 1,, J s defned by CDRk I I+k J CDR k Note that the statement of Corollary 36 also holds true for CDRk Therefore, we prove a smlar theorem lke Theorem 42 for the aggregated CDR For, m I + k J we have the followng covarance decompostons Ĉk Var,J + Ĉk m,j D I+k 1 Ĉk Var D I+k 1 + Var and,j Var CDR k + CDR m k D I Ĉk m,j D I+k 1 + 2 Cov Ĉk,J, Ĉk m,j D I+k 1, Var CDR k D I + Var CDR m k D I + 2 Cov CDR k, CDR m k D I 16
Thus, t remans to study the covarance terms; the other terms are already consdered n Theorem 42 For I + k J + we defne σ δ,k β,k a k I+k + 1 a k 2 I+k I+k + 1 1 γ k 1 I+k 2 γ k 1 I+k 1 > 1 Theorem 49 Under Model Assumptons 31 we have, for m > I + k J, Ĉk Cov,J, Ĉk m,j D I+k 1 Ĉk 1,J Cov CDR k, CDR m k D I Ĉ0,J The proof s provded n the appendx Ĉ k 1 m,j δ,k 1, Ĉ0 m,j k 1 δ, δ,k 1 1 451 Regulatory Solvency Proxy Approach 41 We defne the rsk measure for the frst accountng year by ρ 1 1 φ Var CDR1 D I 1/2 φ Var φ φ I I+1 J I I+1 J Var CDR 1 D I + 2 Ĉ0,J 2 β,1 1 + 2 I I+1 J I+1 J <m I I+1 J <m I CDR 1 D I 1/2 Cov CDR 1, CDR m 1 D I Ĉ 0,J Ĉ0 m,j δ,1 1 1/2 The aggregated rsk-adusted clams reserves n the regulatory solvency proxy approach 41 at tme 0 are gven by 1/2 R 10 I I+1 J R 0 + c ρ 1 1 J I I+k J Ĉ0,J Ĉ0,I +k 1 I I+1 J Ĉ0,J C,I 46 The last term descrbes the expected runoff pattern of the outstandng loss labltes over all accdent years 17
452 Splt of Total Uncertanty Approach 42 We defne the rsk measure for accountng year k 1 measurable at tme I + t, t < k, by I 1/2 ρ D I+t k φ Var CDRk D I+t 1/2 φ Var CDR k D I+t I+k J 1/2 I φ Cov CDR k, CDR m k D I+t Var CDR k D I+t + 2 I+k J <m I Ĉt k 1 2 φ,j β, β,k 1 + 2 Ĉ t,j t+1 <m I+k J For t 0 we defne ρ 2 k Ĉt m,j k 1 t+1 δ, δ,k 1 1/2 ρ D I k Then the aggregated rsk-adusted clams reserves at tme 0 n the splt of total uncertanty approach 42 are gven by I J R 20 + c I+1 J R 0 453 xpected Stand-Alone Rsk Measure Approach 43 We defne the rsk measure for accountng year k by ρ 3 k ρ 2 k 47 ρ D I+k 1 k Then the aggregated rsk-adusted clams reserves n the expected stand-alone rsk measure approach 43 are gven by R 30 I I+1 J R 0 + c J ρ 3 k D I 48 The term on the rght-hand sde of 48 cannot be calculated n closed form If we use Jensen s nequalty as follows X X 2 1/2 we obtan that R 30 R 20 An mportant remark s that the approaches 47 and 48 allow for dversfcaton between accdent years: Corollary 410 For n 2, 3 and k 0 we have I ρ n k I+k J ρ n,k Ths means that we have subaddtve rsk measures Proof The proof easly follows from the fact that for any random varables X 1,, X m wth fnte second moment we have Var m 1 X 1/2 m 1 Var X 1/2 18
454 Multperod Rsk Measure Approach 44 The aggregaton n the multperod rsk measure approach s more nvolved and unfortunately does not allow for an analytc soluton Moreover, smulaton results are often too tme-consumng because the dmensonalty of the problem s rather large exponental growth Formally, the multperod rsk measure approach s gven recursvely The reserves of the cost-of-captal cashflow are ntally gven by ĈoCJ k 0,, J 1 where ρ 4 ĈoC k k+1 φ Var CDRk + 1 + I φ Var I+k+1 J def k+1 ĈoC DI+k + c ρ 4 k+1, Ĉ k+1,j 0 and for k+1 ĈoC DI+k ĈoCk+1 1/2 DI+k 1/2 + ĈoCk+1 D I+k Thus, the aggregated rsk-adusted clams reserves n the multperod rsk measure approach 44 are gven by R 40 I I+1 J R 0 + ĈoC0 49 Let us analyse ths expresson If we start the backward nducton at accountng year J we see that only accdent year I s stll actve n ths accountng year Therefore, ρ 4 J ρ 4 I,J and for the reserves of the cost-of-captal cashflow n accountng year J 1 t follows that ĈoC J 1 ĈoCJ 1 I One perod before, we then obtan ĈoC J 2 J 1 ĈoC DI+J 2 c φ ĈJ 1 I,J β I,J 1 1/2 + c ρ 4 J 1 c φ ĈJ 2 I,J β I,J 1 1/2 + ρ 4 J 1, where usng Theorems 42 and 49 ρ 4 J 1 φ Var CDR I J 1 1 + cφ β I,J 1 1/2 1/2 + CDR I 1 J 1 D I+J 2 φ ĈJ 2 I,J 2 1 + cφ β I,J 1 1/2 2 ĈJ 2 2 βi,j 1 1 + I 1,J βi 1,J 1 1 + 2 ĈJ 2 I,J Ĉ J 2 I 1,J 1 + cφ β I,J 1 1/2 δ I 1,J 1 1 1/2 19
At ths stage we lose the lnearty property n the volume measures ĈJ 2 I,J and ĈJ 2 I 1,J For ths reason we cannot further expand the calculaton analytcally, that s, we can nether calculate the condtonally expected value of ĈoC J 2, gven D I+J 3, nor s t possble to calculate the rsk measure ρ 4 J 2 Therefore, we cannot calculate R 40 n closed form Smlar dffcultes occurred n the expected stand-alone rsk measure approach 48 By neglectng dversfcaton effects between accdent years, we easly fnd an upper bound for the aggregated rsk-adusted clams reserves as follows: R 40 I R 4 I+1 J 0 410 Another more sophstcated upper bound that consders dversfcaton between accdent years wthn accountng years s gven by: Proposton 411 For c φ < 1 we have ĈoC 0 0 def ĈoC J 1 + k 1 2 2 1 c φ c ρ k The proof s provded n the appendx Ths proposton motvates the followng rskadusted clams reserves R 4 0 Note that for c φ < 1 we have the order R 4 0 I I+1 J R 0 { max R 20, R3 0, 0} R4 0 + ĈoC 411 For the remander we wll refer to 41-44 as Approaches 1-4, respectvely 5 Case Study We present a case study for the dfferent cost-of-captal approaches on a real dataset from practse The clams data are gven by a loss trangle see Table 1 representng the observed hstorcal cumulatve clams payments C, Further, the data also nclude pror values for the development factors f and γ and the correspondng standard devaton σ as well as the observed chan ladder factors F 0 The standard devaton σ s obtaned 20
from smlar busness In a full Bayesan approach ths parameter should also be modelled wth the help of a pror dstrbuton But then the model s no longer analytcally tractable Therefore, we use an emprcal Bayesan vewpont usng a plug-n estmate from smlar busness Further, note that we work wth vague prors for Θ, e γ s close to 2 whch results n hgh credblty weghts α 0 see Table 1 cumulatve clams payments C, \ 0 1 2 3 4 5 6 7 8 9 0 122 058 183 153 201 673 214 337 227 477 237 968 261 275 276 592 286 337 298 238 1 132 099 193 304 213 733 230 413 243 926 258 877 269 139 284 618 295 745 2 132 130 186 839 207 919 222 818 237 617 253 623 267 766 284 800 3 127 767 187 494 207 759 222 644 237 671 256 521 271 515 4 127 648 179 633 196 260 213 636 229 660 245 968 5 125 739 181 082 203 281 219 793 237 129 6 117 470 172 967 190 535 204 086 7 117 926 172 606 191 108 8 118 274 171 248 9 119 932 F 0 14530 11065 10750 10680 10650 10629 10599 10372 10416 f 14500 11100 10750 10700 10650 10630 10600 10500 10400 γ 21 30 41 43 47 48 51 64 88 σ 00202 00080 00078 00073 00117 00233 00031 00026 00022 α 0 10000% 10000% 10000% 10000% 9999% 9995% 10000% 10000% 10000% Table 1: Observed hstorcal cumulatve clams payments C,, averages over the ndvdual clams development factors F 0, pror development factors f, pror parameters γ, standard devaton parameters σ, credblty weghts α 0 from Corollary 33 Accordng to the prevous secton, we compute the cost-of-captal margns for ths runoff portfolo for all the dfferent approaches Table 2 presents an overvew of the numercal results The cost-of-captal rate and the securty level are chosen to be c 8% and φ 3 Ths choce s reasonable accordng to the IAA poston paper 7, page 79 Fgure 1 summarses the results for each sngle accdent year Dscusson of the results As expected, we observe that the regulatory solvency approach Approach 1 essentally dffers from the other approaches Ths s because t s not rsk-based for later accountng years In partcular, the regulatory solvency approach may not 21
reserves ultmate clam cost-of-captal margns CoC d 0 % of reserves br 0 predctor C d 0,J A1 A2 A3 A4 A1 A2 A3 A4 1 12 292 308 037 231 231 231 231 19% 19% 19% 19% 2 22 861 307 661 403 461 461 462 18% 20% 20% 20% 3 39 369 310 884 569 723 723 724 14% 18% 18% 18% 4 53 394 299 362 4 412 2 529 2 529 2 533 83% 47% 47% 47% 5 70 239 307 368 2 917 3 562 3 562 3 575 42% 51% 51% 51% 6 78 429 282 515 2 233 3 867 3 867 3 886 28% 49% 49% 50% 7 93 284 284 392 2 686 4 496 4 495 4 522 29% 48% 48% 48% 8 110 718 281 966 2 976 5 055 5 054 5 091 27% 46% 46% 46% 9 166 991 286 923 5 853 6 551 6 549 6 611 35% 39% 39% 40% Total 647 577 22 280 27 475 27 470 27 634 34% 42% 42% 43% aggregated case 15 590 18 196 18 194 22 688 24% 28% 28% 35% dversfcaton effect 30% 34% 34% 18% Table 2: Summary of numercal results of the cost-of-captal margns for the Approaches 1-4 denoted by A1-A4 for sngle accdent years and for the aggregated case The brackets n column A3 ndcate that ths value s generated by numercal smulaton and for A4 t means that we calculated the upper bound gven n Proposton 411 All the other values can be calculated exactly 7'000 6'000 5'000 CoC margn 4'000 3'000 2'000 1'000 0 1 2 3 4 5 6 7 8 9 accdent year CoC Approach 1 CoC Approach 2 CoC Approach 3 CoC Approach 4 Fgure 1: Cost-of-captal margn for sngle accdent years provde suffcent protecton n our example accdent years 5-9 or on the contrary superfluous protecton aganst shortfalls accdent year 4 22
Note that Approaches 2 and 3 serve as good approxmatons to Approach 4 From a mathematcal pont of vew, Approach 4 provdes a methodologcal comprehensve model n order to quantfy the uncertanty for ths multperod rsk consderaton On the other hand, n the practcal examples that we have looked at, the dfferences between Approaches 2 and 3 and Approach 4 turned out to be margnal Hence, due to ts smplcty, Approach 2 s probably preferable from a practtoners pont of vew We see that Approach 4 s more conservatve than Approach 3 Ths confrms the result of Corollary 46 and corresponds to our ntuton snce Approach 4 addtonally quantfes the rsk n the cost-of-captal cashflow Due to the choce of the cost-of-captal rate c and the securty level φ, our computaton shows that the assumpton of Corollary 48 s fulflled Therefore, we fnd that n ths example Approach 2 s slghtly less conservatve than Approach 4 Intutvely, the further an accdent year s developed the less uncertanty there s n the predcton For Approaches 2-4 the cost-of-captal margn s growng for consecutve accdent years On the other hand, the regulatory solvency approach stll shows a trend but wth more fluctuaton whch mght lead to counterntutve stuatons see Fgure 1, observe the decrease n the cost-of-captal margn gong from accdent year 4 to accdent year 5 Important observaton for premum calculaton: for the last accdent year, we compute that the cost-of-captal expenses wth respect to Approaches 2-4 account for approxmately 23% of the ultmate clams predcton Ĉ0,J Ths can be read from Table 2, eg for accdent year 9, one dvdes the cost-of-captal margn 6 611 of Approach 4 by the predcton for the ultmate clam 286 923 Ths mples that the cost-of-captal loadngs for the runoff labltes result n a substantal premum calculaton element that s of about 2% of the total premum! In partcular, ths means that f ths element s neglected n premum calculatons, the P&L gan s substantally reduced by regulatory captal costs Below we extend the above results from sngle accdent years to the study of the cost- 23
of-captal charge for all accdent years smultaneously Fgure 2 presents the results n percentage of the clams reserves Note that n the aggregated case see Fgure 2, only Approach 1 and 2 are analytcally tractable and allow for drect computaton Snce Approach 3 lacks a closed form calculaton, evaluaton has been done by Monte Carlo smulaton Snce numercal computaton s too tme consumng, we have no vable algorthm for Approach 4 Therefore, we only computed the bound gven n Proposton 411 45% 40% CoC margn / clams reserves 35% 35% 30% 25% 20% 15% 10% 10% 05% 00% Total: aggregated case Total: sngle accdent years CoC Approach 1 CoC Approach 2 CoC Approach 3 CoC Approach 4 Fgure 2: Cost-of-captal margns n percentage of the clams reserves for the aggregated case left panel wth dversfcaton and the total over sngle accdent years rght panel wthout dversfcaton accordng to, eg the rght-hand sde of formula 410 for the cost-of-captal margns only As before, the regulatory solvency approach for aggregated accdent years turns out to be the less conservatve one Ths means that also the cost-of-captal charge for the uncertanty over all accdent years may not be suffcent compared to Approaches 2-4 If we sum the cost-of-captal margns over sngle accdent years, we mmedately get upper bounds for Approaches 2-4 see Fgure 2 Dversfcaton effects between accdent years account for substantal releases of over 34% for Approach 2 and 3 and 18% for the upper bound of Approach 4 24
We observe that for ths example, the upper bound of Approach 4 turns out to be an upper bound for all the other Approaches For Approach 3, ths s mmedately clear and, snce cφ < 1, the results holds true for Approach 2 Note that n general, Approach 4 s not necessarly more conservatve than Approach 1 The example further confrms that Approach 2 s more conservatve than Approach 3 but ust by a small margn The fact that Approach 2 s analytcally tractable for aggregated accdent years makes t a preferable approxmaton to the multperod rsk measure approach The computed upper bound for Approach 4 n the aggregated case only allows for a rough statement about the precson of ths approxmaton If we compare the results for sngle accdent years, we observe that the uncertanty n the cost-of-captal cashflow accounts ust for a margnal proporton Fgure 2 ndcates that the upper bound for Approach 4 s conservatve Fnally, we would lke to compare the gamma-gamma Bayes chan ladder model used n ths dscusson wth the classcal dstrbuton-free chan ladder model presented n Mack 9 A man devaton les n the fact that the chan ladder factors are calculated dfferently, we use an average over the observed ndvdual clams development factors F,, whereas the classcal chan ladder model takes a volume weghted average thereof We denote by msep C,J D Ĉ,J 0 I Var C,J D I, 0 the mean square error of predcton MSP of Ĉ,J and by msep CDR 1 D I 0 Var CDR1 D I, the MSP of the clams development result of the frst accountng year for the gammagamma Bayes chan ladder model see Corollary 43 and Subsecton 451 For the classcal chan ladder model, the estmators of the MSP are denoted by msep Mack C,J D Ĉ,J CL I and msep BDGMV CDR CL 1 D I 0 The frst s estmated wth the classcal Mack formula Mack 9, the latter s calculated accordng to Remark 411 n Bühlmann et al 2 We only observe margnal devatons, that s, our model choce does not sgnfcantly change rsk assessment compared to a classcal chan ladder model 25
gamma-gamma Bayes chan ladder «0 1/2 reserves msep C,J D Ĉ I,J % of reserves msep CDR 1 D 0 1/2 % of reserves I 1 12 292 961 8% 961 8% 2 22 861 1 372 6% 1 091 5% 3 39 369 1 770 4% 1 247 3% 4 53 394 7 981 15% 7 822 15% 5 70 239 9 087 13% 4 288 6% 6 78 429 8 642 11% 2 791 4% 7 93 284 9 014 10% 2 929 3% 8 110 718 9 251 8% 2 958 3% 9 166 991 11 226 7% 6 371 4% Total 647 577 31 317 5% 19 402 3% Table 3: Gamma-gamma Bayes chan ladder, the last row provdes the aggregated case Mack model 9 reserves msep Mack CL C,J D Ĉ,J 1/2 % of reserves msep BDGMV I CDR CL 0 1/2 1 D I % of reserves 1 12 292 965 8% 965 8% 2 22 869 1 380 6% 1 102 5% 3 39 379 1 770 4% 1 248 3% 4 53 212 7 946 15% 7 783 15% 5 70 083 8 957 13% 4 232 6% 6 78 263 8 822 11% 2 840 4% 7 93 112 9 177 10% 2 946 3% 8 110 561 9 454 9% 2 993 3% 9 166 722 11 406 7% 6 482 4% Total 646 494 31 345 5% 19 300 3% Table 4: Classcal chan ladder model presented n Mack 9, the MSP for the oneyear CDR s calculated accordng to Bühlmann et al 2 and the last row provdes the aggregated case Concluson and Outlook We have studed the clams development result for a multperod general nsurance lablty runoff portfolo Our paper drectly addresses some open questons dscussed n the IAA poston paper 7 concernng the calculaton of an approprate cost-of-captal margn For the four dfferent approaches dscussed n ths paper, the numercal example ndcated that the Splt of the Total Uncertanty Approach Approach 2 provdes a good approxmaton to the mathematcally comprehensve Multperod Rsk Measure Approach Approach 4 The example further confrms that the cost-of-captal margns have a substantal mplcaton on premums whch should be accounted for n premum calculatons 26
We performed our calculaton n a Bayesan model framework where recent nformaton s mmedately absorbed by the model The underlyng dstrbutons and model parameters are chosen n such a way that Approaches 1-4 for sngle accdent years as well as Approach 2 n the aggregated case have analytc solutons The study of other stochastc models and clams reservng methods goes beyond the scope of the present paper but s an mportant topc for further research So far, our dscusson only consders nomnal values Undscounted values then ncorporate hdden reserves whch contrbute substantally to the fnancal strength of general nsurance companes A next step s to extent the results on dscounted clams reserves presented n Wüthrch-Bühlmann 15 to a multperod general nsurance lablty runoff portfolo, smlar to the dscusson n ths paper Ths then provdes a cost-of-captal margn for dscounted clams reserves Further research should also nvestgate the role of other rsk measures as well as stochastc cost-of-captal rates c, dependency modellng along accountng years, and the ncorporaton of other nformaton avalable A Proofs Proof of Theorem 42 For I + k J + we obtan J Var D I+k 1 Var Ĉk,J C,I+k I+k +1 f k D I+k 1 Usng Corollary 34 we rewrte the chan ladder factor estmators as follows f k α k F k + 1 α k From ths we see that condtonally, gven D I+k 1, f a k F I+k, + f k 1 a k f k 1 s only random n F I+k, Usng the posteror ndependence of Θ 1,, Θ J, gven D I+k 1, and that fact that all random varables nvolved only depend on dfferent accdent years and development years, we see 27
k k that C,I+k, f I+k +1,, f J are ndependent, gven D I+k 1 Ths mples that J Var C,I+k f k D I+k 1 I+k +1 C,I+k 2 DI+k 1 C,I+k D I+k 1 2 C 2,I+k 1 Further, F 2,I+k F 2,I+k J I+k +1 J I+k +1 DI+k 1 2 k f D I+k 1 f k J I+k +1 2 D I+k 1 2 k f D I+k 1 DI+k 1 F 2,I+k Θ, DI+k 1 DI+k 1 J I+k Var F,I+k Θ + F,I+k Θ 2 DI+k 1 σ 2 I+k + 1 Θ 2 I+k σ 2 I+k + 1 f k 1 I+k D I+k 1 2 γ k 1 I+k 1 γ k 1 I+k 2, 2 k 1 f and 2 k k f D I+k 1 Var f a k k 1 D I+k 1 + f 2 k 1 Var FI+k, D I+k 1 + f 2 2 2 k 1 σ f a k 2 + 1 γk 1 1 γ k 1 2 1 2 + 1 Ths proves the frst clam of the theorem Moreover, we have Var CDR k D I CDR 2 k DI Var Ths mples wth the frst statement of the theorem that Var Ĉk,J D I+k 1 DI β,k 1 β,k 1 Var Ĉk 1,J Iteratng ths procedure completes the proof 28 Ĉk 1,J D I+k 2 + Ĉk,J 2 D I Ĉk 2,J D I+k 1 DI 2 D I
Proof of Corollary 43 From Theorem 42 we obtan J +I Ĉ0 J +I 2 Var CDR k D I,J Calculatng the last sum gves the frst clam Because of J +I Var CDR k D I Var C,J D I, a straghtforward calculaton gves Var C,J D I C,J 2 DI C,J D I 2 C,J 2 Ĉ0 Θ, DI+J 1 DI,J 2 k 1 β, β,k 1 Var C,J Θ, D I+J 1 + C,J Θ, D I+J 1 2 Ĉ0 DI,J σj 2 + 1 Θ 2 J C,J 1 2 Ĉ0 2 DI,J Iteratng ths procedure and usng posteror ndependence of Θ, gven D I, we obtan Var C,J D I C 2,I Ths proves the corollary Ĉ0,J J I +1 2 J I +1 1 σ 2 + 1 Θ 2 Ĉ0 DI,J σ 2 + 1 γ0 1 1 γ 0 2 2 2 Proof of Proposton 45 We calculate the cost-of-captal reserves nductvely For k J + I 1 we obtan see Theorem 42 ĈoC J+ I 1 0 + c ρ 4,J+ I c φ Var CDR J + I D J+ 1 1/2 c φ ĈJ+ I 1,J β,j+ I 1 1/2 Ths proves the clam for k J + I 1 Inducton step: Assume the clam holds true for k + 1 We then have that J+ I ĈoC k k+1 ĈoC D I+k + c ρ 4,k+1 Ĉk,J b,m 1 + c ρ 4,k+1, 29 mk+2
where,k+1 φ Var CDR k + 1 + ρ 4 φ Var φ J+ I mk+2 Ĉ k+1,j b,m Var + Ĉk+1,J Ĉk+1,J ĈoC k+1 J+ I mk+2 D I+k b,m 1 ĈoCk+1 1/2 D I+k 1/2 D I+k Ĉk,J J+ I mk+2 DI+k 1/2 b,m φ β,k+1 1 1/2, where we have used Theorem 42 n the last step Ths completes the proof Proof of Lemma 47 The proof goes by nducton It s obvous that the result holds for 1, 2 Hence we do the nducton step + 1 Usng the nducton assumpton we get +1 1 + a k 1 1 + a +1 1 + a k 1 k 1 1 + a a k + 1 + a +1 1 + a k 1 Ths proves the result 1 k 1 1 + a a k + a +1 1 + a k 1 Proof of Corollary 48 In vew of 45 t suffces to prove that for all, 1,, J + I 1, or equvalently 1 + c φ β, 1 1/2 β 1/2,, c φ β, 1 1/2 β 1/2, 1 1/2 If we rewrte the left-hand sde β, 1 1/2 β 1/2, 1 ths s equvalent to the assumpton β 1/2, + 1 1/2 we obtan that whch completes the proof 1/2 1/2 β, 1 c φ, β 1/2, + 1 30
Proof of Theorem 49 For m > I + k J we obtan Ĉk Cov,J, Ĉk m,j Cov D I+k 1 C,I+k J I+k +1 f k, C m,i+k m I+k 1 I+k m+1 f k J I+k f k D I+k 1 Usng Corollary 34 we decouple the problem nto ndependent problems smlar to Theorem 42 Ths mples Ĉk Cov,J, Ĉk m,j D I+k 1 C m,i+k m D I+k 1 J I+k +1 f k I+k 1 I+k m+1 2 D I+k 1 k k f D I+k 1 C,I+k f Ĉk 1,J Ĉ k 1 m,j The only dfference to Theorem 42 s that the calculaton of F 2 replaced by F,I+k k f I+k F,I+k a k a k I+k F 2 D I+k 1 1 a k I+k I+k F,I+k + D I+k 1 + 1 a k,i+k I+k,I+k I+k k 1 f DI+k 1 I+k f k 1 I+k 2 k 1 f I+k a k I+k σ 2 I+k + 1 γk 1 I+k 1 γ k 1 I+k 2 + 1 ak I+k Ths proves the frst clam of the theorem Moreover, we have Ĉk Cov CDR k, CDR m k D I Cov,J, Ĉk m,j D I+k 1 DI Ths mples wth the frst statement of ths theorem that Ĉk Cov,J, Ĉk 1 Ĉk m,j D I+k 1 DI δ,k 1,J δ,k 1 Cov D I+k 2 Ĉk 1,J, Ĉk 1 m,j Iteratng ths procedure completes the proof 31 Ĉ k 1 m,j + Ĉk 2,J 2 D I D I+k 1 DI+k 1 s now Ĉ k 2 m,j D I
In order to prove Proposton 411 we need the followng lemma Lemma A1 Choose y x 0 then we have, for p 0, 1, p x + 1 p y 2 x 2 1/2 1 p y 2 + 2p 1 x 2 1/2 Proof of Lemma A1 We defne the dscrete random varable Y by P Y 2x p and P Y y 1 p for p 0, 1 Hence we have Y 2 x 2 1/2 p x + 1 p y 2 x 2 1/2, and on the other hand usng Jensen s nequalty Y 2 x 2 1/2 Y 2 x 2 1/2 p x 2 + 1 p y 2 x 2 1/2 Ths proves the lemma 1 p y 2 + 2p 1 x 2 1/2 Proof of Proposton 411 For k {0, 1,, J 2} fxed, we have that ĈoC k k+1 ĈoC DI+k + c φ Var CDRk + 1 ĈoCk+1 1/2 DI+k c ρ D I+k k+1 + ĈoC k+1 DI+k + c φ Var ĈoC k+1 DI+k 1/2 We rewrte the above expresson n such a way that t s more sutable for teraton For ths, let p 0, 1 and k+1 ĈoC k+1 DI+k + c φ Var ĈoC 1/2 DI+k p 1 c φ ĈoC k+1 DI+k 1 p + c φ p ĈoC k+1 DI+k + 1 p Var ĈoC k+1 1/2 DI+k 1 p Now we apply Lemma A1 to the last term whch provdes the followng upper bound k+1 ĈoC DI+k 1 c φ p 1 p k+1 + c φ Var ĈoC 1/2 DI+k k+1 ĈoC DI+k + c φ 1 p 1 p k+1 ĈoC 2 k+1 D I+k + 2p 1 ĈoC 2 1/2 DI+k 32
Note that the term 2p 1 s postve for p 1/2 and wth Jensen s nequalty appled to the last term for p 1/2, 1 we obtan k+1 ĈoC DI+k 1 c φ p 1 p k+1 + c φ Var ĈoC 1/2 DI+k k+1 ĈoC DI+k + c φ p1/2 1 p ĈoC k+1 2 D I+k 1/2 Note that c φ < 1 mples c φ + 1 1 > 1/2 Hence, for p 1/2, c φ + 1 1 we see that c φ p/1 p < 1 and consequently wth Jensen s nequalty appled to the frst term on the rght-hand sde of the above nequalty we obtan k+1 ĈoC k+1 DI+k + c φ Var ĈoC 1/2 DI+k 1 + c φ p1/2 p ĈoC k+1 2 1/2 D I+k 1 p For p 1/2, the rght-hand sde s mnmal n p Defne κ 1 + 2 1 c φ, hence k+1 ĈoC k+1 DI+k + c φ Var ĈoC 1/2 k+1 DI+k κ ĈoC 2 1/2 D I+k By teraton we fnd that ĈoC k c ρ D I+k k+1 + κ ĈoC k+1 2 D I+k 1/2 c ρ D I+k k+1 + κ By Mnkowsk s nequalty we obtan c ρ D I+k+1 k+2 + κ k+2 ĈoC 2 1/2 2 D I+k+1 D I+k 1/2 ĈoC k 2 c ρ D I+k k+1 + κ c ρ D I+k+1 1/2 k+2 D I+k + κ 2 By teratng ths procedure we obtan ĈoC k+2 2 D I+k 1/2 ĈoC k J k 1 0 κ c ρ D 2 I+k+ 1/2 k++1 D I+k Note that we have ρ D 2 I+ 1/2 +1 D I ρ D I +1 ρ2 +1 Ths proves the result 33
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