EFFICIENT STOCHASTIC MODELING FOR LARGE AND CONSOLIDATED INSURANCE BUSINESS: INTEREST RATE SAMPLING ALGORITHMS

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1 EFFICIENT STOCHASTIC MODELING FOR LARGE AND CONSOLIDATED INSURANCE BUSINESS: INTEREST RATE SAMPLING ALGORITHMS Yvonne C. M. Chueh* Despie he availabiliy of faser, more powerful compuers, he complexiy of he models required o replicae complicaed asses and liabiliies has kep pace, leaving he need for more creaive soluions (for sochasic cash-flow-esing). Alasair G. Longley-Cook ABSTRACT One of he challenges of sochasic asse/liabiliy modeling for large insurance businesses is he run ime. Using a complee sochasic asse/liabiliy model o analyze a large block of business is ofen oo ime consuming o be pracical. In pracice, he compromises made are reducing he number of runs or grouping asses ino asse caegories. This paper focuses on he sraegies ha enable efficien sochasic modeling for large and consolidaed insurance business blocks. Efficien sochasic modeling can be achieved by applying effecive ineres rae sampling algorihms ha are presened in his paper. The algorihms were esed on a simplified asse/liabiliy model ASEM (Chueh 999) as well as a commercial asse/liabiliy model using asses and liabiliies of he Aena Insurance Company of America (AICA), a subsidiary of Aena Financial Services. Anoher mehodology using he New York 7 scenarios is proposed and could become an enhancemen o he Model Regulaion on cash flow esing, hus requiring all companies o do sochasic cash flow esing in a uniform, nononerous manner. * Yvonne C. M. Chueh is an Assisan Professor in he Mahemaics Deparmen a Cenral Washingon Universiy, 400 Eas 8h Avenue, Ellensburg, Washingon , [email protected]. I. INTRODUCTION Valuaion acuaries play a crucial role in proecing he solvency of insurance companies by exercising heir professional judgmen in evaluaing he adequacy of reserves. Under he Sandard Valuaion Law (SVL) and he Acuarial Opinion and Memorandum Regulaion (AOMR), every company doing business in a sae mus submi he opinion of a qualified acuary concerning reserves for all lines of business. To mee his annual regulaory requiremen, acuaries design, consruc, and validae cash-flow-esing models o help evaluae a company s financial srengh by analyzing relevan model oupus. These cashflow-esing models (mainly sochasic asse/liabiliy models) ofen prove o be valuable ools ha can provide acuaries wih more han yearend requiremens. For example, in addiion o C-3 risk applicaions, he models can be used o perform sensiiviy analysis on various assumpions, especially policyholder and agen behavior. The models can help gain insighs ino oher risks encounered by all lines of business. As anoher applicaion, pricing acuaries may need sochasic asse/liabiliy models o es he profiabiliy of new producs being priced. By using a sochasic asse/liabiliy model, pricing acuaries as well as valuaion acuaries can calculae a disribuion of economic values ha helps analyze and guide decisions for various lines of business. The well-known challenge of running a sochasic asse/liabiliy model is he run-ime issue. Using a complex sochasic asse/liabiliy model o analyze a large block of business in a full sochasic fashion is ofen oo ime consuming. In prac- 88

2 INTEREST RATE SAMPLING ALGORITHMS 89 ice, modelers reduce he number of runs or group asses ino asse caegories o reduce he run ime for each scenario. This paper focuses on he sraegies ha allow efficien sochasic modeling o be done for large and consolidaed insurance business blocks by developing powerful ineres rae sampling algorihms. The algorihms are independen of an ineres rae generaor, so any mulifacor (or he less popular one-facor) generaor can be considered. However, o keep he sampling algorihms simple ye effecive, only shor-erm (one-year) ineres raes are needed o perform sampling. The algorihms were esed on a simplified asse/liabiliy model ASEM (Chueh 999) as well as a commercial asse/liabiliy model using asses and liabiliies of he Aena Insurance Company of America (AICA), a subsidiary of Aena Financial Services. Anoher modeling sraegy using he New York 7 scenarios and disance algorihm is proposed in his paper. liabiliy cash flows, aking ino accoun he asse/ liabiliy wo-way ineracions. By comparing he sampling disribuion of key saisics (e.g., presen value of ending surplus, economic value,..., ec.) beween he full run and he sample run, he effeciveness of sampling algorihms can be evaluaed. Saisical goodnessof-fi ess such as he Kolmogorov-Smirnov es can be applied o compare he various sampling mehods. The advanage of using he ASEM model o es hese sampling algorihms is is flexibiliy and speed in processing. Using exising commercial asse and liabiliy sysems o es he sampling algorihms would be complicaed, ime consuming, and inflexible. The ASEM model, alhough buil on a simplified asse porfolio, is flexible and complee enough o represen he key drivers impacing he performance of he accumulaion line of business. II. WHY INTEREST RATE SAMPLING? In he discussion of he paper Applicaion of Risk Theory o Inerpreaion of Sochasic Cash- Flow-Tesing Resuls (Robbins, Cox, and Phillips 997), Alasair G. Longley-Cook poined ou ha Despie he availabiliy of faser, more powerful compuers, he complexiy of he models required o replicae complicaed asses and liabiliies has kep pace, leaving he need for more creaive soluions. His commen has moivaed his research. A pracical soluion is o use a represenaive sample of sochasic scenarios. However, since he main use of sochasic models is o examine performance a he ails of he disribuion, a represenaive sample mus provide comparable resuls o he full scenario run in hese regions. The objecive of his paper is o develop powerful ineres rae sampling mehods ha produce excellen resuls a he ails. The ASEM model (Chueh 999) consising of boh an asse model and a liabiliy model wih asse/liabiliy ineracions can be used o es he fi of he sampling mehod. The liabiliies include a block of deferred annuiies (in accumulaion sage), while asses are a number of callable bonds wih a variey of paymen schedules, yields, and mauriies. The ASEM model is a mulifacor model ha projecs asse cash flows and III. EXISTING INTEREST RATE SAMPLING ALGORITHMS A grea deal of financial lieraure deals wih he erm srucure of ineres raes, and many models have been designed o fi various heories. However, an ineres rae sampling mehod capable of capuring he ail disribuions of imporan saisics (e.g., presen value of he accumulaed surplus) has no ye been found when i comes o complicaed sochasic models such as asse/liabiliy models across differen lines of business. When building economic valuaion models o help evaluae fuure uncerainies encounered by business blocks, companies favor sochasic ineres rae scenarios. In 996, by using leassquares bes-fi echniques, Longley-Cook (996) proposed mapping a universe of ineres raes pahs o he required New York 7 scenarios. He assigned a probabiliy o each of he New York 7 scenarios depending on he number of sochasic ineres raes pahs mapped ono i. His paper suggesed ha we search ou addiional scenarios (no jus he required New York 7 scenarios) o improve he map. In 998 Chrisiansen proposed selecion of a sample of equally likely scenarios ha reflec he probabiliy disribuion of he full scenario run (i.e., approximaely he same mean, median, range, and variance).

3 90 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 6, NUMBER 3 This paper seeks a sampling sraegy ha no only can effecively selec represenaive ineres rae scenarios for all ypes of business lines (i.e., no jus annuiies) bu is easy o implemen in any complex asse/liabiliy model. Three ineres rae sampling algorihms are developed and described beginning in Secion IV. The process begins by defining a disance measure beween wo ineres rae scenarios, and hen describes he procedures for choosing a represenaive subse of ineres rae scenarios. Finally, he probabiliy associaed wih each of he represenaive scenarios is calculaed. IV. THE REASONING BEHIND THE PROPOSED SAMPLING ALGORITHMS As menioned in Secion II, he main use of sochasic asse/liabiliy models is o examine performance a he ails of he disribuion. To achieve his, a represenaive sample mus provide comparable resuls o he full scenario run in he ail regions. Keeping his in mind, when developing sampling algorihms, he reasonable algorihms mus be able o selec exreme scenarios as well as moderae ones so ha he resuling model oupus disribuion of he sample run will resemble or reproduce ha of he full run and, in paricular, in he low and high perceniles. To judge wheher a scenario is exreme or no, his paper incorporaes he concep of disance in mahemaical field. Two mehods are inroduced: () Relaive Disance By randomly selecing a scenario as he firs pivo (represenaive), he scenario farhes away from his chosen one is considered exreme and hus should be seleced as he nex pivo (represenaive). (2) Absolue Disance Each ineres rae scenario is given a measure of is poenial ime value, called he significance value, which measures is exremeness. The firs and second algorihms o be inroduced in Secions V and VI use he relaive disance concep, whereas he hird algorihm in Secion VII uses he absolue disance concep. I is imporan o noe ha he disance formulas should be ighly conneced wih he desirable asse/liabiliy models oupus, commonly ending surplus, and economic value of a block of business. Inuiively and ideally, he good disance formulas can capure or predic hese models oupus so ha he exreme scenario will mos likely produce exreme ending surplus or economic value. Also, o be mahemaically correc, he disance formulas canno conradic he fundamenal definiion of disance such as he riangle inequaliy rule. An iniial aemp was made by viewing ineres rae scenarios as vecors and hen calculaing heir disances using he adjused Euclidean Disance formula (see Secion V). This disance formula refleced he imporance of early projecion years by inroducing a weigh facor V. The drawback of he adjused Euclidean Disance formula was is small scale of measuremen ha made he calculaion less accurae due o numerical rounding errors. Is sensiiviy o he value of he weigh facor V also made i less desirable as a disance measuremen since he sampling effeciveness may be affeced easily. I is never a simple ask o ie he disance formula wih he asse/liabiliy models oupus. Neverheless, if he model oupus are ending surplus and economic value, hen i makes sense o ake ino accoun he presen value or fuure value of a sream of unknown ne cash inflows. Since i is impossible o know for sure wha he fuure ne cash inflows are like in he nex years before running he model, o keep he formula simple, his paper uses he presen value of $ o be received in years,,2,3,..., (cash flow esing is done for -year projecion period) o calculae relaive and absolue disances. When creaing some form of a meric o measure similariy or dissimilariy beween differen ineres rae scenario pahs, coninuiy is desirable. The coninuiy means if wo pahs are close in he domain of a funcion, he corresponding funcion oupus (sochasic model oupus) will be close. The funcion here is he sochasic asse/ liabiliy model funcion ha generaes oupus, he payoff value (economic value or ending surplus) for a scenario. The condiion (*) ha guaranees he coninuiy under he defined disance is given as follows: Condiion: An ineres rae pah consiss of shor-erm (oneyear) ineres raes for years. Given wo iner-

4 INTEREST RATE SAMPLING ALGORITHMS 9 es rae pahs for years x (i, i 2,...,i ) and s (i, i 2,...,i ), define k D k k i i. The condiion is he inequaliy ij D i D j 0. * ij Claim: If we assume bounded cash flows for every projecing year under any ineres rae scenario, and he above condiion (*) is saisfied, hen he coninuiy of he payoff value for he scenario can be guaraneed. Proof: To show his, he presen value of he sum of hese discouned cash flows will be used as an example. In mahemaical noaion, his presen value PV can be wrien as PV CF k i k, where cash flows are projeced for a number of years, in paricular years, and are bounded by he amoun M. I is clear ha PV is a funcion of he ineres rae scenario: PV f(i, i 2,...,i ), where PV f : X 3 Y, X R, Y R, and he disance beween wo poins x and s in he domain X, d X (x, s), is mea- sured by k i k k i k2 (according o he disance defined in he second sampling algorihm of Secion VI). The PV funcion f is a real funcion defined on a -dimensional domain X and is said o be coninuous on X if f is coninuous a every poin (scenario) of X. Now we wan o show ha if he condiion (*) is me, given an arbirary ineres rae scenario s, for every ε 0 here exiss a ε/2m 0 such ha d Y ( f(x), f(s)) ε for all poins x X for which d X (x, s) : d 2 Y f x, fs CF k 2 2M CF k CF k k i k 2 i CF i k k i k i k k 2 i k 2M 2 k if he condiion (*) is saisfied, k i k 2M 2 d X 2 x, s ε 2. k 2, i k2 Thus, d Y 2M d X ε. This complees he proof. e Unforunaely, he condiion (*) has o be saisfied by every pair of ineres rae pahs in he universe in order o guaranee he uniform coninuiy of he model oupu. I is difficul o verify or saisfy his condiion because of is mahemaical complexiy and ime consrain (even he fases compuers are easily overwhelmed by he ask). In addiion, i is difficul o creae a realisic ineres rae model ha always saisfies his condiion (*). The firs and second sampling algorihms using disance formulas o group similar scenarios employ an idea similar o cluser analysis (Johnson and Wichern 992) in saisics lieraure. However, since his paper is aimed a developing efficien sampling algorihms o be used in sochasic modeling, he focus is on he echnique and reasoning of selecing pivo (or represenaive) scenarios and calculaing heir probabiliies raher han on he analysis of mulivariae relaionships of daa in cluser analysis. To see how effecive he sampling algorihms are, a number of ess and resuls are presened laer in he paper. Ideally, by selecing a predeermined number of pivo (represenaive) ineres rae scenarios, he sample size is reduced o save modeling run ime while he probabiliy disribuions of he sochasic model oupus are capured.

5 92 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 6, NUMBER 3 V. FIRST SAMPLING ALGORITHM: MODIFIED EUCLIDEAN DISTANCE METHOD This secion firs defines a disance measure ha is similar o he Euclidean Disance and hen demonsrae he seps o selec represenaive (pivo) scenarios. A probabiliy is hen assigned o each represenaive scenario. Definiion of Disance The disance beween a given -year ineres rae pah and a pivo ineres rae pah is defined as D i i P 2 V, where i,,2,...,isanineres rae pah consising of shor-erm (one-year) ineres raes for years i P,,2,...,isapivo (i.e., represenaive) ineres rae pah consising of shor-erm (oneyear) ineres raes for years and V is a weigh facor wih a value beween 0 and ha disinguishes he relaive imporance of he ineres rae of each projecing year. I is fel ha he value is around he one-year discoun rae, and he value used in his paper is /.06. To illusrae how o perform he sampling algorihm, selec 00 represenaive ineres rae pahs ou of a large number, say, N, of ineres rae pahs by aking he following seps: Sep. Choose an arbirary ineres rae pah ou of he N simulaed ones and call i Pivo #. Sep 2. Calculae he disances from Pivo # o he remaining N ineres rae pahs. Sep 3. Name he ineres rae pah wih he larges disance o Pivo # as Pivo #2. Randomly decide among ies. Sep 4. Calculae he disances of he N 2 nonpivo ineres rae pahs o Pivo # and Pivo #2. Sep 5. Assign each of he N 2 ineres rae In pracice, running 00 sochasic scenarios is manageable and someimes he maximum. I is no unusual o ake more han one hour o projec cash flows for one scenario. pahs o he closes of Pivo # or Pivo #2, hus forming wo disjoin ses of ineres rae pahs. Flip a coin if he disances are equal. Each of he N 2 ineres rae pahs now has a unique disance o is pivo scenario. Sep 6. Rank hese N 2 disances in descending order. The ineres rae pah producing he op disance is called Pivo #3. (Break ies randomly.) Sep 7. Follow he above procedure o selec he addiional 96 pivo scenarios, Pivo #4, Pivo #5,...,Pivo #00. Sep 8. If he number of ineres rae pahs associaed o a pivo scenario is N k, hen assign a probabiliy of N k /,000 o his pivo scenario. VI. SECOND SAMPLING ALGORITHM: RELATIVE PRESENT VALUE DISTANCE METHOD A disance measure beween an ineres rae pah and a pivo scenario is given as shown below. Noe ha his disance measure does no require a weighing facor. Insead, he ineres rae for a given year is discouned by he ineres raes in he earlier years: D 2 k i k k P2, i k where i,,2,...,isanineres rae pah consising of shor-erm (one-year) ineres raes for years, and i P,,2,...,isapivo (i.e., represenaive) ineres rae pah consising of shor-erm (one-year) ineres raes for years. Repea he sampling algorihm in he previous secion o selec a sample of represenaive ineres rae pahs, and assign a probabiliy o each of he represenaive ineres rae pahs as before. VII. THIRD SAMPLING ALGORITHM: SIGNIFICANCE METHOD In his secion a much quicker bu sill effecive way o sample ineres scenarios is inroduced by firs giving a new disance definiion and hen aking a sample in a uniform way. This mehod assigns an equal probabiliy o scenarios. The significance of an ineres rae pah is defined as

6 INTEREST RATE SAMPLING ALGORITHMS 93 S k, i k2 where i k, k,2,...,isanineres rae pah consising of shor-erm (one-year) ineres raes for years. Selec 00 represenaive ineres rae pahs ou of a large number, say, N, of ineres rae pahs by aking he following seps: Sep. Calculae he significance measure of all he N ineres rae pahs based on he above formula. Sep 2. Sor he ineres rae pahs by significance measure in ascending order. Denoe he sored pahs by I [], I [2],...,I [N]. Sep 3. Choose he 00 represenaive ineres rae pahs as follows: For k,2,...,00, he k-h ineres rae pah is I [0k5]. The probabiliies of he seleced represenaive ineres rae pahs are se equal (/00 in his case). The sampling seps proposed above can be generalized o selec n pahs insead of jus 00 pahs. These sampling mehods (algorihms) discussed above direcly address he run-ime issue in doing asse/liabiliy modeling because run ime will be significanly reduced if one of hese sampling mehods is applied. They also have a broad use in any sochasic modeling ha involves sochasic ineres raes. Anoher use of hese algorihms would be in sochasic models ha have sochasic scenarios in he form of ime series (e.g., exchange rae, inflaion rae, equiy reurn,..., ec.). Insead of running a large number of sochasic scenarios, acuaries can use he above sampling mehods o reduce he number of scenarios while preserving he behavior of ail disribuions. VIII. VALIDATION AND TESTING FOR THE SAMPLING ALGORITHMS This secion uses he ASEM model (Chueh 999) as he basis o examine and validae our sampling mehods. The ineres rae generaor used is called he Sochasic Variance Model, and is formulas are provided in he Appendix. This generaor was developed by he SOA C-3 risk ask force. The resuls provide srong evidence o suppor he effeciveness and efficiency of he sampling algorihms. Figures 4 show empirical probabiliy disribuions derived from four sochasic runs. One is a,500-scenario full run, and he oher hree are 50-scenario sample runs in he ASEM model calculaing he Economic Value (EV) using baseline assumpions. Figure illusraes he EV disribuion based on,500 sochasic scenarios; Figures 2 4 illusrae he disribuion of EVs based on he hree sampling algorihms. Table compares perceniles of he EV disribuions, under he firs, second, and hird sampling algorihms, respecively, agains he full scenario run. Table 2 shows he Kolmogorov-Smirnov (K-S) es resuls. Figure 2 clearly shows ha he sample poins deermined by he 50 seleced ineres rae scenarios using he firs sampling algorihm cluser in he cener of he EV disribuion and is a weaker fi a he ails. The main reason for his is in he arbirariness of he weighing facor V ha direcly affecs he scenarios being seleced. This facor V is relaed o he fuure lifeime of he conracs being modeled as well as he duraion of he underlying asses and liabiliies. I requires some experimens o deermine he value for V. In esing he algorihms, V /.06 is used. The second disance formula (Relaive Presen Value Disance formula) avoids his subjecive selecion of a discoun facor and demonsraes a significan improvemen on he sample EV disribuion as refleced in Figure 3. Figure 3 shows a more accurae EV disribuion. The sample poins are evenly spread hrough he inerval (22,000, 32,000), and he Figure Cumulaive Disribuion Funcion for Economic Value:,500 Scenarios

7 94 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 6, NUMBER 3 Figure 2 Cumulaive Disribuion Funcion for Economic Value: 50 Represenaive Scenarios, Firs Sampling Algorihm Figure 4 Cumulaive Disribuion Funcion for Economic Value: 50 Represenaive Scenarios, Third Sampling Algorihm disribuion makes a smooh curve ha is commonly seen on cumulaive disribuion funcions. When compared wih Figure, he graph based on,500 runs, we observe an excellen fi onhe overall disribuion and ails. Since he Relaive Presen Value Disance formula makes such a large improvemen, we waned o see if he disance formula and he sampling procedures could be simplified while mainaining he improved accuracy. This moivaed he hird sampling algorihm: he significance mehod. Figure 4 shows he cumulaive disribuion funcion of economic value based on he 50 seleced represenaive scenarios using he hird sampling algorihm. We observe ha he 50 sample poins from he hird sampling algorihm form a smooh, solid Figure 3 Cumulaive Disribuion Funcion for Economic Value: 50 Represenaive Scenarios, Second Sampling Algorihm disribuion curve. However, some exreme sample poins are missing so ha he EV observed values fall in a shorer inerval (23,000, 33,000). This is anicipaed since we seleced 50 represenaives by dividing he,000 scenarios ino 50 groups and hen aking he midpoin of each group as a represenaive scenario. When he sampled perceniles a he lef ail are less han hose for he full run, oversampling a he lef ail occurs. However, when he sampled perceniles a he lef ail are greaer han hose for he full run, undersampling a he lef ail occurs. Similarly, when he sampled perceniles a he righ ail are greaer han hose for he full run, oversampling a he righ ail occurs. However, when he sampled perceniles a he righ ail are less han hose for he full run, undersampling a he righ ail occurs. When comparing Figures 2 4 wih Figure, i is imporan o know wheher or no he hree sampling algorihms oversample he exreme scenarios. Perhaps he hird sampling algorihm undersamples he exreme scenarios and hus misrepresens he ail-end disribuion. To answer he quesion of oversampling or undersampling? we urn o quaniaive evidence: percenage difference of perceniles of wo economic value disribuions, one based on represenaive scenarios, he oher based on a,500- scenario full sochasic run. The numerical resuls based on each sampling algorihm are lised in Table. The firs wo columns afer he sub column give he EV perceniles from wo disribuions: one from,500 sochasic runs and he

8 INTEREST RATE SAMPLING ALGORITHMS 95 Table Comparison of Perceniles of Two EV Disribuions:,500 Runs versus 50 Represenaive Runs EV Percenile,500 (Full Run) 50 Runs s Algorihm 50 Runs 2nd Algorihm 50 Runs 3rd Algorihm Percenage Difference s Algorihm Percenage Difference 2nd Algorihm Percenage Difference 3rd Algorihm s 23,442 9,404 22,647 23, h 24,897 9,404 24,663 25, h 26,03 23,587 25,743 25, h 26,558 26,705 26,503 26, h 26,886 27,83 26,567 27, h 27,060 27,24 26,706 27, h 27,82 27,244 27,243 27, h 27,552 27,365 27,452 27, h 27,979 28,065 27,969 28, h 28,256 28,280 28,50 28, h 28,56 28,32 28,83 28, h 29,088 29,47 29,65 29, h 29,67 29,739 29,644 29, h,552,548,495, h 32,087 32,24 32,36 3, Average of Absolue Value oher from 50 represenaive runs using he firs sampling algorihm. Table and Figure 2 indicae ha he low perceniles (s, 5h, and 0h perceniles) based on he firs sampling algorihm are significanly oversampled. The high perceniles, he 90h and 99h percenile, is also slighly oversampled. As a resul, he firs sampling oversaes he low perceniles by 7 25% and oversaes he high percenile (he 99h percenile only) by 0.48%. Table indicaes ha he second sampling mehod slighly oversamples boh ails of he disribuion. The s percenile is oversaed by 3.39% of rue value ha is assumed o be he s percenile based on he,500 sochasic runs. Considering he low percenage of all he differences (no more han 3.39%) and he small sample size seleced (50 ou of,500), he overall disribuion is well sampled by he second sampling algorihm. Table indicaes sligh undersampling a boh ails for he hird sampling algorihm. The larges percenage difference occurs a he 5h percenile, a 3.3%. The percenage differences in general are lower and comparable wih he second sampling algorihm. Thus he hird sampling algorihm also well samples he disribuion. For each sampling algorihm, boh he resuling overall EV disribuion and he wo-ail EV disribuion were examined by he modified Kolmog- Table 2 K-S Tes Resuls, Three Sampling Algorihms K-S Tes Values of N, N 2 D n y Pr(Y > y) Firs Sampling Algorihm Overall N 50, N 2, Overall N 00, N 2, Two-ail N 37, N Second Sampling Algorihm Overall N 50, N 2, Overall N 00, N 2, Two-ail N 22, N Third Sampling Algorihm Overall N 50, N 2, Overall N 00, N 2, Two-ail N 4, N Noe: Two-ail refers o he boom 20% and op 20% of he disribuion.

9 96 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 6, NUMBER 3 orov-smirnov es (Eland and Johnson 980). The modified K-S es involved comparing he cumulaive probabiliies of EV resuls from he sampled N (50 and 00, respecively) scenarios wih hose from he universe of N 2 (here,500) scenarios. When comparing wo empirical disribuions wih a K-S es, ND N is replaced wih N N 2 D N N N,N 2. We can use he following limiing 2 probabiliy o show he probabiliy of maching he wo empirical disribuions when N and N 2 50 (see London 988): lim N3 PrND N y 2 j exp2j 2 y 2. j For he overall disribuion, he K-S es indicaed he probabiliy ha he overall disribuion from he full run equals he disribuion from he firs sampling algorihm was low. The probabiliy was only for he 50-scenario sample run and for he 00-scenario sample run. The second and hird sampling algorihms had a higher probabiliy, , of maching. For he disribuion a wo ails (i.e., es of fi for he boom 20% and op 20% of he disribuion), he probabiliy of maching for he firs sampling algorihm was increased dramaically o Thus, we found no significan evidence agains he hypohesis of maching, alhough our resul was suspec since we had only N 37. All hree algorihms displayed a very high probabiliy of maching he wo ails of he disribuion, alhough he resuls were suspec due o small N.I is ineresing o see ha alhough he hird sampling algorihm is much simpler and quicker han he firs wo algorihms, he resuls for boh overall and wo-ail disribuions seem promising. Anoher es was conduced for he 00 scenarios chosen from he randomly generaed 0,000 ineres rae scenarios using he simple random sampling as well as he hree sampling algorihms proposed in his paper. The resuling EV sampling disribuions were compared agains ha of he 0,000-scenario full run. The descripive saisics and modified K-S es resuls are shown in Table 3. Based on he resuls in Table 3, i appears ha he simple random sampling also gives good resuls (comparable wih hird algorihm) in erms of he sampled perceniles no oo far from hose of he full Table 3 Descripive Saisics of EV and K-S Tes Resuls: 0,000-Scenario Full Run versus 00-Scenario Sample Run EV Saisics EV Percenage Difference EV 0,000 Full Run Simple Random s Algorihm 2nd Algorihm 3rd Algorihm Simple Random s Algorihm 2nd Algorihm 3rd Algorihm Minimum 9,69 2,296 9,69 9,69 22, s Percenile 22,624 22,437 2,807 22,024 22, h Percenile 23,484 23,206 22,368 23,62 23, h Percenile 24,207 24,372 23,950 24,26 23, h Percenile 25,08 25,206 24,463 24,944 24, Q 26,485 26,69 26,392 26,502 26, Q 2 27,259 27,88 27,49 27,64 27, Q 3 27,626 27,488 27,624 27,556 27, h Percenile 28,258 27,735 28,42 28,78 27, h Percenile 29,86 28,62 29,225 28,970 28, h Percenile 29,96 29,383,337 29,567 29, h Percenile,827,40,86,653, Maximum 35,24,898 35,24 35,24 3, Average of Absolue Value 26,96 26,472 26,73 26,837 26, Simple Random s Algorihm 2nd Algorihm 3rd Algorihm K-S Tes (Overall) (N, N 2 ) (00, 0,000) (00, 0,000) (00, 0,000) (00, 0,000) Probabiliy F(00) F(0,000) K-S Tes (Two-ail) (N, N 2 ) (32, 400) (7, 400) (66, 400) (36, 400) Probabiliy F(00) F(0,000) Noe: Two-ail refers o he boom 20% and op 20% of he disribuion.

10 INTEREST RATE SAMPLING ALGORITHMS 97 run. The firs and second sampling algorihms have superior resuls in perceniles. This does no surprise us since he algorihms aemp o selec he low and high perceniles. We hen expec ha when he model oupus form a heavy-ail disribuion, he firs and second algorihms will do a good job. As he firs and second sampling algorihms allow us o selec he mos exreme scenarios, hey are likely o selec he maximum and minimum economic values equal o hose of he full run. In his es, hey have. We can see ha he percenage differences (error) in he maximum and minimum economic values beween he sample run and he full run were zero. All he hree sampling algorihms passed he modified K-S es a 0% significance level. The second sampling algorihm had he lowes average (absolue) difference (error) in perceniles. The firs and hird sampling algorihms, however, had he higher probabiliy of maching he full run based on he K-S es. Overall, he second algorihm performs bes in he ails based on he es. For he firs and second sampling algorihms, he resuling represenaive (pivo) pahs depend on he pahs iniially seleced. In fac, he firs pivo pah deermines he subsequen pivo selecions. To assess he sensiiviy of he sampling effeciveness o he iniial pah selecion, five samples using a differen iniial (firs) pivo scenario were examined under he firs and second sampling algorihms (he hird algorihm does no use a pivo), respecively. Table 4 shows he percenage difference in EV saisics beween full run and sample run under he firs and second sampling algorihms, respecively. Table 5 shows he modified K-S es resuls for each iniial pivo scenario under he firs and second sampling algorihms, respecively. EV Perceniles under boh sampling algorihms do no appear o be very sensiive o he iniial pivo according o he resuls in Table 4. We observe ha he EV Perceniles sampled under boh sampling algorihms are quie sable regard- Table 4 Sensiiviy Tes of Iniial Selecion of Pivo Scenario: EV Saisics (Percenage Difference beween Full Run and Sample Run) Iniial Pivo Selecion EV Firs Algorihm Minimum s Percenile h Percenile h Percenile h Percenile Q Q Q h Percenile h Percenile h Percenile h Percenile Maximum Average of Absolue Value Second Algorihm Minimum s Percenile h Percenile h Percenile h Percenile Q Q Q h Percenile h Percenile h Percenile h Percenile Maximum Average of Absolue Value

11 98 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 6, NUMBER 3 Table 5 Sensiiviy Tes of Iniial Selecion of Pivo Scenario: Modified K-S Tes Probabiliy F(00) F(0,000) Iniial Pivo Selecion Firs Algorihm (N ) Second Algorihm (N 2 ) K-S Tes (Overall) N 00, N 2 0, Average Sandard Deviaion K-S Tes (Two-ail) N 2 4, (69) (68) (67) (68) (65) (64) (68) (62) (69) (66) Average Sandard Deviaion Noe: Two-ail refers o he boom 20% and op 20% of he disribuion. less of he iniial pivo. The second algorihm consisenly shows a beer fi o he full run han he firs algorihm in erms of Perceniles. Table 5 indicaes ha alhough he overall fi of EV disribuions under boh sampling algorihms seems sensiive o he iniial pivo (he firs algorihm is more sensiive han he second algorihm), he fis of wo-ail EV disribuions are no sensiive o he iniial pivo. For boh overall and wo-ail EV disribuions, he second algorihm appears o be a more sable and effecive sampling approach han he firs. IX. STOCHASTIC MODELING USING THE REQUIRED NEW YORK 7 SCENARIOS The final simplified sochasic algorihm ha also addresses he run-ime problem of asse/liabiliy modeling is he New York 7 asse approach. In conras o he previously discussed sampling algorihms, his approach is direcly linked o he curren deerminisic approach required in cash flow esing and uses he same seven asse cash flow projecions. The auhor believes ha his mehod shows he greaes promise for a sandardized sochasic cash flow esing procedure o be required for all companies. I requires he leas change from wha companies currenly have o do and ye produces an excellen sochasic disribuion of scenarios. The New York 7 asse approach is implemened as follows:. Firs run a commercial asse modeling sofware sysem o projec asse cash flows of a company s curren asses for only he required New York 7 scenarios. 2. Nex, generae a universe of sochasic ineres rae scenarios by an appropriae sochasic ineres rae model and map each scenario o one of he New York 7 scenarios, using he relaive presen value disance formula. 3. For each sochasic ineres rae scenario, use he iniial asse cash flows deermined by is associaed New York 7 ineres rae scenario. However, coninue o use he specific ineres rae scenario for he reinvesmen of liabiliy cash flows, principal repaymens, coupons, and any borrowing. This would be done in he asse/liabiliy model (TAS or PTS) where he run-ime problem is no an issue. Since i is he asse projecion process ha akes mos of he run ime, he use of only seven asse projecion scenarios involves he same run ime as is currenly incurred in deerminisic cash flow esing. This approach preserves he exreme asse projecions, since he New York 7 scenarios were originally designed o es exreme condiions. This approach is no only ime efficien bu also capable of generaing a disribuion Figure 5 Cumulaive Disribuion Funcion for Economic Value:,500 Scenarios Using New York 7 Approach

12 INTEREST RATE SAMPLING ALGORITHMS 99 Table 6 Comparison of Perceniles of EV Disribuions for Full Run and Efficien New York 7 Asse Approach Percenile s 5h 0h 5h 20h 25h h 50h 70h 75h 80h 85h 90h 95h 99h,500/,500 (Full Run) 23,442 24,897 26,03 26,558 26,886 27,060 27,82 27,552 27,979 28,256 28,56 29,088 29,67,552 32,087,500/New York 7 23,460 25,7 26,02 26,679 26,984 27, 27,237 27,567 27,999 28,258 28,585 29,07 29,666,558 32,054 Percenage Difference wih ails comparable wih a full sochasic run (as shown in Figures and 5 and Table 6). By assigning each scenario o one of he New York 7 we do no overemphasize he exreme ineres rae scenarios since each of he New York 7 is auomaically given a probabiliy proporional o he number of sochasic scenarios assigned o i. Wih he approximaed asse cash flows assigned o every sochasic scenario, each scenario can hen properly inerac wih is associaed liabiliies and allow he asse/liabiliy model o projec financial oucomes in he fuure years. In order o demonsrae he effeciveness of he New York 7 asse approach, we firs perform a full sochasic run of economic values in our ASEM model by inpuing he baseline assumpions o he ASEM model and hen running an asse/liabiliy projecion based on,500 sochasic ineres rae scenarios (Figure ). Then we repea he process, limiing asse cash flow projecions o only he required New York 7 scenarios. As described above, liabiliy cash flows are projeced sill based on he full,500 sochasic scenarios. I can be observed ha he probabiliy disribuion shown in Figure 5 closely resembles Figure. In order o ake a closer look a he above wo disribuions, firs compare he Perceniles of he probabiliy disribuions of he EVs. Table 6 shows ha he percenage difference beween any Percenile in he able does no exceed.%. These daa srongly suppor ha he wo disribuions derived from wo differen sochasic runs, one full run and he oher by using our efficien New York 7 asse modeling approach, have very similar paerns a he ails. Table 7 gives he K-S es resuls using he proposed New York 7 asse modeling approach. The resuls indicae a high probabiliy of maching. To help see he EV disribuions for he full and sample runs, hisograms are provided in Figures 6 4. X. VALIDATION AND TESTING OF SAMPLING ALGORITHMS USING COMMERCIAL ASSET/LIABILITY MODELS Independen validaion and esing of he proposed sampling algorihms have been conduced agains he asses and liabiliies of he Aena Insurance Company of America (AICA) using commercial asse/liabiliy models: BondEdge and TAS. Three algorihms (uniform mehod he hird sampling algorihm; pivo mehod he second sampling algorihm; and New York 7 New York 7 asse approach) were esed agains 500 ineres rae scenarios on he AICA accumulaion business. One hundred ineres rae scenarios were sampled based on respecive sampling algorihms. The es saisic measured is he presen value of he ending surplus. The AICA porfolio conains a relaively diverse se of asses and Table 7 K-S Tes Resuls, New York 7 Asse Approach K-S Tes New York 7 Asse D n y Pr(Y > y) Overall N,500, N 2, Two-ail N 409, N

13 00 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 6, NUMBER 3 Figure 6 Hisogram of Economic Value:,500 Scenarios, Full Run Figure 8 Hisogram of Economic Value: 50 Represenaive Scenarios Seleced from,500, Second Sampling Algorihm hence is proper o be used in esing. The porfolio composiion is as follows: Treasuries: 2.4% Pass-hroughs: 5.6% CMOs: 7.6% ABS: 7.8% Indusrials: 3.6% Elecric and gas: 3.2% Telephone: 2.6% Finance: 28.3% Inernaional: 4.4% Cash: 0.8% Oher: 5.7% A summary of asse daa is as follows: Number of asses: 02 Average qualiy: A Marke-weighed average coupon: 6.556% Marke value: $27,872,000 Modified duraion: 5.79 Effecive duraion: 5.33 Convexiy: 0.9. Table 8 summarizes he es resuls. The resuls based on AICA asses and liabiliies are very promising. The large P values for all he hree sampling algorihms suppor he goodness of he Figure 7 Hisogram of Economic Value: 50 Represenaive Scenarios Seleced from,500, Firs Sampling Algorihm Figure 9 Hisogram of Economic Value: 50 Represenaive Scenarios Seleced from,500, Third Sampling Algorihm

14 INTEREST RATE SAMPLING ALGORITHMS 0 Figure 0 Hisogram of Economic Value: New York 7 Approach Based on,500 Scenarios Figure 2 Hisogram of Economic Value: 00 Represenaive Scenarios Seleced from 0,000, Firs Sampling Algorihm fi o he full sochasic disribuion of he presen value of he ending surplus. XI. CONCLUSIONS AND FUTURE RESEARCH The sampling algorihms developed and analyzed in his paper are applicable o all business lines of he insurance indusry. This paper provides a pracical and saisically sound mehodology for even small companies o perform sochasic analysis of heir business. The New York 7 mehodology in paricular is somehing companies could easily adop since hey are required by law o use he New York 7 scenarios in analyzing heir business. I is easy o develop a sofware program o generae a large, predeermined number of ineres rae scenarios and assign hem o he corresponding New York 7 scenarios using he relaive disance formula. This mehodology could easily hen become an enhancemen o he Model Regulaion on cash flow esing, requiring all companies o do sochasic cash flow esing in a uniform, nononerous manner. The sampling algorihms were esed using he ASEM model wih a simplified porfolio of asses. More research needs o be done on analyzing how well hese algorihms hold up wih a more com- Figure Hisogram of Economic Value: 0,000 Scenarios, Full Run Figure 3 Hisogram of Economic Value: 00 Represenaive Scenarios Seleced from 0,000, Second Sampling Algorihm

15 02 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 6, NUMBER 3 plicaed porfolio of asses ha include morgagebacked securiies, CMOs, and oher derivaive securiies. The es resuls based on he Aena Insurance Company of America s asses and liabiliies are very promising. All hree sampling algorihms (uniform, pivo, and New York 7) show a good fi o he full sochasic disribuion. These algorihms should also be esed on heir robusness if changes are made on asse assumpions such as prepaymen raes, defaul raes, and changes in reinvesmen assumpions. ACKNOWLEDGMENTS I am graeful o Dr. Charles Vinsonhaler and Dr. Jeyaraj Vadiveloo for heir guidance and suppor during his research. I wan o hank he hree anonymous reviewers for heir valuable commens and suggesions ha grealy helped he clariy and exposiion of he paper. Special hanks are due o Alasair Longley-Cook for his inspiring his paper. Mos of he work was compleed during my sudy a he Mahemaics Deparmen of he Universiy of Connecicu and my inernship a Aena Reiremen Services (now ING-Aena). The discussions wih Terry Boucher, David Diamond, and Jagdish Shah of ING-Aena Financial Services on cash flow esing and insurance modeling were very helpful o his research. Table 8 Tesing Resuls on he AICA Porfolio Sampling Algorihm Uniform Pivo New York 7 K-S Tes: Toal P Value 98% 85% 75% K-S Tes: Tails P Value 98% 95% 88% K-S es: oal Kolmogorov-Smirnov es on he full sample. K-S es: ails K-S es on jus he boom 20% and op 20% sample values. REFERENCES CHRISTIANSEN, SARAH L. M A Pracical Guide o Ineres Rae Generaors for C-3 Risk Analysis. Transacions of he Sociey of Acuaries 992, Vol. 44. CHRISTIANSEN, SARAH L. M Represenaive Ineres Rae Scenarios. Norh American Acuarial Journal 2: No. 3, CHUEH, YVONNE C. M Sochasic Economic Modeling for he Deferred Annuiy (Accumulaion) Line of Business: Efficien Modeling Approaches for Large and Consolidaed Business Blocks. Ph.D. hesis, Universiy of Connecicu. ELANDT, R., AND N. L. JOHNSON Survival Models and Daa Analysis. New York: John Wiley & Sons. LONDON,DICK Survival Models and Their Esimaion. 2nd ed. Wespor, CT: ACTEX Publicaions. LONGLEY-COOK, ALASTAIR G Probabiliies of Required 7 Scenarios (and a Few More). Financial Reporer 29: 6. ROBBINS, EDWARD L., SAMUEL H. COX, AND RICHARD D. PHILLIPS Applicaion of Risk Theory o Inerpreaion of Sochasic Cash-Flow-Tesing Resuls. Norh American Acuarial Journal : No. 2, RUBNICH, RONALD Ineres Rae Generaor for C3 Projec. C-3 Task Force of he Sociey of Acuaries. Figure 4 Hisogram of Economic Value: 00 Represenaive Scenarios Seleced from 0,000, Third Sampling Algorihm APPENDIX In he following are lised he exac formulas, parameers, and assumpions for he Sochasic Variance model chosen as a ineres rae generaor for he ASEM model. The Sochasic Variance model generaor, based on he documen prepared by Edward Robbins (998), conains hree variables ha vary over ime and one consan parameer: The naural log of he long erm ineres rae (20-year rae) a ime The excess of he shor-erm rae (one-year rae) over he long-erm rae a ime ( is generally negaive) The naural log of he variance of ; /2 The naural log of he variance of ;

16 INTEREST RATE SAMPLING ALGORITHMS 03 /2 ( is modeled as a consan, based on he hisorical daa) Each of he variables will be assumed o follow a mean reversion random process. 2 The variables and will be modeled wih monhly ime seps, while will be modeled wih an annual ime sep. The parameer is assumed o be consan. Specifically, 3 ln(( ) 2 ) (0.3809% he average hisorical monhly volailiy 4 from 95 o 995). The following equaion governs he evoluion of : Z. 5 Here Z is a random variable wih a sandard normal disribuion. The following formulas for and assume ha and for beween inegers n and n are equal o heir respecive values a n; ha is, 2 When raes developed by a random process ge ou of bounds, merely applying bounds causes sickiness (a sequence of consan raes). A mean reversionary process is one in which he raes being generaed are consanly being pulled oward a prese goal (he expeced mean rae) by he use of a correcion facor (Chrisiansen 992). 3 Hisorical daa show ha he absolue difference essenially follows a normal disribuion ha does no depend on. Thus, i makes sense o ake as a consan when modeling his erm. Based on he hisorical daa, he sandard deviaion of he change in he difference beween he shor-erm and long-erm rae is 0.38%. 4 Monhly volailiy is defined as he sandard deviaion of he monhly change in he excess of he shor-erm rae over he longerm rae. 5 is a sochasic variance. The ieraion formula for is given here. and say unchanged for he enire year and change values only a he beginning of every year: / ln(0.0655) e 0.5 Z ; 6 / ln e Z Z 2. 7 Here Z and Z 2 are independen sandard normal random variables. Afer he one-year rae (shor erm) and 20- year rae (long erm) have been generaed, he remainder of he Treasury yield curve is derived from inerpolaion formulas. Firs, he hreemonh rae is calculaed as a linear funcion of boh he one-year and 20-year raes. The hreemonh and 20-year raes are used o calculae he oher raes (six-monh, one-year, wo-year, hreeyear, five-year, seven-year, 0-year, and -year). The formulas used come from a linear regression performed on he monhly Treasury raes covering he period from 977 hrough 997. Discussions on his paper can be submied unil January, The auhor reserves he righ o reply o any discussion. Please see he Submission Guidelines for Auhors on he inside back cover for insrucions on he submission of discussions ( ln (0.0655)) and 0.20( 0.005) are wo correcion facors for mean reversion purposes ( 0.005) and ( ln (0.0655)) are wo correcion facors for mean reversion purposes.