STRESS TESTING OF PROBABILITY OF DEFAULT OF INDIVIDUALS

Transcription

1 STRESS TESTING OF PROBABILITY OF DEFAULT OF INDIVIDUALS Per Kadeøábek, Aleš Slabý, Josef Vodièka Absrac: This paper inroduces a model for sress esing of probabiliy of defaul of individuals. The model ress on assumpion ha he individual defauls if his savings fall below zero. The probabiliy of defaul is hen described as a funcion of several macroeconomic indicaors, such as wages, unemploymen and ineres raes. Sress esing is carried ou by applying exogenous sress scenarios for developmen of hese indicaors. The model implies ha sensiiviy of probabiliy of defaul o he sress is mainly driven by insallmen o income raio and for morgages also by loan mauriy. Hence insallmen o income raio is suggesed as he appropriae ool o manage credi risk of reail porfolios. Keywords: banking; credi risk; sress esing; probabiliy of defaul JEL Classificaion: G21, E32, E21 1. Inroducion Complex credi risk managemen in financial insiuions addresses hree levels of credi risk: expeced losses, unexpeced losses, and exreme evens. Expeced losses ha are long-erm average losses are covered by proper credi risk pricing and profiabiliy managemen. Possible unexpeced losses ha are higher han long-erm average losses are covered by capial reserves up o cerain level of severiy. Exreme unexpeced losses above he severiy level shall be rare by definiion and hence i is no efficien o keep reserves for hem coninuously. However, he insiuion should foresee impac of exreme evens which could evolve from curren siuaion and i has o have a plan for each such even o miigae is impac if i evolves. This ask is suppored by sress esing. Sress esing is, in oher words, invesigaion of an impac of meaningfully defined scenarios of fuure developmen, exreme developmen in paricular. In credi risk sress esing, an impac of hese scenarios is measured in erms of losses o be suffered on accoun of credi risk or in erms of change in credi risk parameers. This paper focuses Komerèní banka, a.s., Praha; The Insiue of Economic Sudies of he Charles Universiy, Praha (perkaderabek@seznam.cz); Komerèní banka, a.s., Praha (ales_slaby@kb.cz); Komerèní banka, a.s., Praha; Sociéé Genérale, Paris (Josef.Vodicka@socgen.com). 340 PRA GUE ECO NO MIC PA PERS, 4, 2008

2 on impac of adverse macroeconomic developmen on probabiliy of defaul (PD) of individuals. In line wih Vasicek (2002), he background for Basel II model of unexpeced losses, we consider wo ypes of credi risk facors affecing probabiliy of defaul: global (sysemic) and individual (idiosyncraic). Individual facors are specific for each ransacion and are averaged in diversified porfolio. Global facors are common o all ransacions in he porfolio and can be inerpreed as he macroeconomic condiions. Hence he macroeconomic variables are in he cener of our ineres and are viewed as drivers of key individual facors, such as income and level of savings. In lieraure, an impac of he macroeconomic environmen on credi risk has been sudied from wo perspecives. A boom-up approach is based on daa of individual cliens while a op-down approach relies on aggregae daa. This paper adops he boom-up approach similarly as Gross and Souleles (2001), who inspec sabiliy of he credi risk models of he US households using he panel daa se of credi cards. A op-down approach is more common in he lieraure as he aggregae daa are more easily available: The ime-series approach o modelling he arrears of UK individuals using he macroeconomic daa is adoped by Whiley, Windram and Cox (2004). A laen-facor model is developed by Jakubík (2007), which is hen applied o he Czech banking secor. When modelling credi sress in he Czech Republic he following issues are faced regarding macroeconomic variables: The ransiive economy has no experienced regular economic cycle. The economic downurn in he 1990s was driven by specific economic condiions ha are no likely o be repeaed. In las en years he economic condiions have gradually improved. Reail banking business has been dynamically developing wihin a relaively shor ime from almos zero o is conemporary complexiy. Moreover, he porfolio is usually closely conrolled by he Risk Managemen and hence we will develop a heoreical macroeconomic model raher han any saisical model. A clear sory behind he resuls is also an advanage of his approach so ha i can beer serve as a diagnosic ool o improve he insiuion s undersanding of is risk profile. I is anoher imporan reason for sress esing as poined ou by he Commiee of European Banking Supervisors (2006). The fundamenal paper in he field of corporae defaul rae modelling and bond pricing is by Meron (1974). This paper deals wih modelling defaul rae of individual cliens, which is no a opic frequenly discussed in lieraure. We adop he approach by Meron (1974) and modify i for he loans of individuals. Heømánek e al. (2007) refer o differen sensiiviies of corporaions and households o he macroeconomic environmen which suppors he need o build he model for individual cliens differenly. Meron (1974) saes ha on he mauriy dae, he firm mus eiher pay he promised paymen o he bondholders or else he curren equiy will be valueless. He concludes ha he firm will defaul if he value of equiy is lower han he promised paymen. This conclusion is based on he assumpion ha he firm canno issue any PRA GUE ECO NO MIC PA PERS, 4,

3 new senior (or of equivalen rank) claims on he firm nor i canno pay cash dividends or repurchase shares prior o he mauriy dae of he deb. If we adoped his approach o he case of individuals, we would sae ha he clien defauls if and only if he value of his propery (incl. he ne presen value of his fuure income) is lower han he promised paymen. However, an assumpion ha he clien canno issue any claims on his propery is no as realisic as i is in he case of firms. Hence, we will absrac of he clien s propery and focus solely on his income and savings. The paper is organized as follows: Secion 2 inroduces he underlying dynamic model of clien s income and savings. In Secion 3, he defaul is defined and sensiiviy of probabiliy of defaul o changes of macroeconomic variables is invesigaed. Secion 4 deals wih issues concerning implemenaion of he model. In Secion 5, an example of resuls of he model is presened. Secion 6 concludes. 2. Model of In come and Sav ings In his secion, we will develop a dynamic model of individual s income and savings. The model will be used laer for compuaion and sress of PD. The following noaion will be used hroughou he paper: Macro variables are denoed by capial leers while he variables concerned wih an individual clien or a loan are denoed in lower case. The nominal income i is divided beween he consumpion c, sum of all annuiy paymens a and deposi s s -1 (1 + r s -1), s i c a [ s s ( 1 r )], (1) 1 1 where s is he amoun of savings and r s -1 is an ineres rae on savings per one ineres period. To simplify formulae, refer o savings accrued in ime as s s s ( 1 r ). (2) 1 1 We conservaively assume ha each clien who is borrowing money has no available savings in he ime of loan graning, i.e. s 0 0. (3) If he clien has savings a he ime of loan graning he will spend hem and borrow only he res of money needed. Le consumpion c depend linearly on disposable income plus savings coming from he previous period, namely, c P [ s i a P ], (4) where is he clien s minimum level of real consumpion and i is assumed o be consan over ime, P is a price level in ime. Hence, P is he clien s minimum nominal consumpion. The linear relaionship was used due o he algebraical simpliciy of he resuls. 342 PRA GUE ECO NO MIC PA PERS, 4, 2008

4 The coefficien [0,1] is a marginal propensiy o consumpion. When he clien s income (or savings coming from he previous period) increases by 1 CZK his nominal consumpion increases by CZK. Dynamics of savings can be described using (1) and (4) as s s i a c s i a P s i a P ( 1 ) [ s i a P ], s ( 1 ) [ s 1 ( 1 r 1 ) i a P ], [ ]. (5) Variables a and P are given exogenously, dynamics of income i is specified hereunder. Denoe i 0 he equilibrium clien s income in ime 0 and I he nominal per capia income in he economy in ime. Assume ha ln (i / I ) is he AR(1) process wih inercep. The inercep erm is given by he raio of he clien s equilibrium income o he per capia income in he economy. This raio is assumed o be consan over ime. Hence, we can wrie he income equaion as i i0 1 1n 1n + ( 1 ) 1n i, (6) I I I 0 where (1 ) is an auoregression coefficien such ha 0 < < 1, and assume ha has a disribuion wih zero mean. Logarihm of income in ime can be expressed as 3. Prob a bil iy of De faul 3.1 Definiion of Defaul ni 1n i i ( 1 ) 1n 1nI. (7) I I 0 For he sake of definiion of defaul le us assume ha he clien canno borrow addiional money or resrucure he deb. Hence defaul occurs when clien s income plus savings coming from he previous period do no cover he insallmens plus minimum consumpion, i.e. s + i < a + P. By equaion (5), his is equivalen o he condiion when he curren volume of savings falls o negaive values, s < 0. Le us define he defaul formally. Le D be he random even ha he clien is in defaul in ime. Then D 1 s 0. (8) D will be sudied in sensiiviy analysis. Analogically le D 1, be he even ha he clien defauled in ime period from 1 o, D k,... s D 1, will be sudied in PD dynamics analysis. min 1 { } 0. (9) PRA GUE ECO NO MIC PA PERS, 4,

5 3.2 Sensiiviy Analysis We will deermine he ransacion-specific facors affecing sensiiviy of PD o he sress of macroeconomic condiions. I will allow idenifying poenially risky pars of reail porfolios. When performing he sensiiviy analysis, we will sar wih a given (ypically expeced) PD in ime under he given loan parameers and macroeconomic condiions in ime and 1. We will inspec change of PD caused by a change (sress) in macroeconomic condiions in ime having all he parameers in ime 1 consan. The variables concerned wih he changed condiions will be marked by a prime, e.g. P, while he original variables remain denoed in he simple way, e.g. P. Denoe p he probabiliy ha he clien is in defaul in ime, p P[D ]. (10) Our goal is o express he sressed PD p using p and he macroeconomic variables, boh original and sressed. For expecaion of ln i given by (7) we have condiionally on informaion prior o : 1n î E 1 [1n i ]. 0 = 1n i + ( 1 ) 1n 1 1nI. (11) I 0 I 1 Since values of variables from ime 1 are known in ime and I is exogenous, all he variables involved in equaion (11) are deerminisic and herefore î I = î I Definiion of defaul (8), model specificaion (5) and (7), and equaion (11) imply ha p P[ s 0] P[i P a s ] i (12) P i P a s (13) îi î i Pexp( ) P a s î i F P a s. iî where F is he CDF of exp( ). Le us assume ha p 0. (14) 344 PRA GUE ECO NO MIC PA PERS, 4, 2008

6 When F is increasing, we can uniquely define F 1 [p ] and express he consumpion level under zero disposable income as i F [ p ] a s. P 1 (15) I is assumed o be posiive, which is equivalen o he condiion p F a s. (16) Using he sressed values (marked by prime) we obain an equaion analogical o (13) î p P a s F. (17) î Afer subsiuion of equaion (15) ino (17) and due o (12), p F I P I P F p a a / P s a P 1 P [ ] 1 1. (18) / P î î The raio a / î will be called insallmen o income raio (IIR), s / î is referred o as he savings o income raio (SIR) in he nex paragraphs. Discussion of he heoreical findings derived can be found in secion Force of Habi: An Alernaive Approach In his secion, we presen a less radiional approach. I leads o simpler resuls and may be more realisic for some economiss. I is presened as a possible alernaive only and will no be discussed in he nex secions. We can use he per capia consumpion C insead of he price index, i.e. se P C. (19) Jusificaion of his approach is in he work of Abel (1990), who argues ha he individual s uiliy depends on he raio of his consumpion o he lagged 1 cross-secional average level of consumpion. This approach is now popular in financial heory; see for example Campbell and Cochrane (1999). To illusrae is reasonabiliy on he pracical example, consider a morgage lasing 20 years. In he year 1988, which is exacly 20 years ago, people had no cell phones. In he year 2008, a cell phone belongs o he necessary equipmen and people mus buy i for heir income. This rise of expendiures is no conained in he consumer price index. 1 We will use he non-lagged average level of consumpion because i resuls in a simpler formula bu he lagged consumpion could be employed as well. PRA GUE ECO NO MIC PA PERS, 4,

7 When cell phones (similarly as any oher goods) enered CPI, only weighs of he goods changed and i caused no immediae inflaion. On he oher hand, i was immediaely refleced in he rise of per capia consumpion. Adoping his approach o our model, he former minimum real consumpion would hen express he minimum consumpion relaive o he per capia consumpion in he economy. Loosely speaking, is he minimum living sandard of he clien relaive o his neighbours. Le us assume ha on macroeconomic level, per capia consumpion is he consan proporion of per capia income I, C I. (20) Hypohesis of I / C consancy on he U.S. economy daa spanning from Q o Q is discussed by Hayashi (2000), p The coinegraion es of ln I and ln C (wihou he inercep erm) is performed wih he null hypohesis of no coinegraion. His resuls rejec he null hypohesis of no coinegraion on he subsample from Q o Q4 1986, hence suppor coinegraion. On he whole sample, he null hypohesis of no coinegraion is no rejeced. The reason is he declining personal saving rae from he mid 1980s. However, for our purposes, assumpion of he consan personal saving rae is realisic enough. By equaions (19) and (20), P is defined as P I, (21) hence he price level is no longer necessary. Then, equaion (5) becomes s ( 1 ) [ s i a I ]. s ( 1 ) [ s ( 1 r ) i a I ]. 1 1 Assumpions abou i dynamics remain unchanged and hence equaions (7) and (11) hold. Now we can provide he sensiiviy analysis under he new assumpions. By (21), equaion (18) becomes 1 a p F F [ p ] î (22) a I s I a I 1 I 1 î. (23) We can see ha raio of he changed and original real income does no affec p anymore. Jusificaion is ha boh he income and average consumpion (used insead of he price level) changes proporionally wih he per capia income. In oher aspecs, conclusions are similar o hose of equaion (18). 4. Model Im ple men a ion This par of he paper deals wih issues concerning calibraion of he heoreical models esablished hereofore. Tenaive scenarios of fuure developmen of he Czech 346 PRA GUE ECO NO MIC PA PERS, 4, 2008

8 economy are given exogenously and referred o in Secion 4.1. Secion 4.2 discusses relaionship beween he loan ineres rae and ineres rae in he economy. Annuiy dynamics in case of re-fixes is inspeced in Secion 4.3. Disribuion of income change is seleced in Secion Sress Tesing Scenarios Trajecories for developmen of he economy are exogenous o our model. Several enaive scenarios are provided, covering beside he expeced developmen he following unfavourable ypes of evoluion: recession and growing ineres raes. Expeced scenario is he poin predicion of macroeconomic developmen. Recession will negaively affec incomes, i.e. wages decline and unemploymen increases. Scenario wih growh of ineres raes is crucial for morgage loans sress esing. 4.2 Ineres Raes For he loans ha are assumed o be re-fixed in he fuure, he new loan nominal ineres rae per one period r mus be compued. Denoe R ineres rae per one period in he economy. For simpliciy we will assume a consan addiive gross margin. Hence if he loan is re-fixed in ime > 0, is new ineres rae will be where R is a nominal ineres rae in he economy in ime. 4.3 Annuiy Dynamics r r0 R0 R, (24) For simpliciy le us assume in his secion ha he clien has only one loan subjec o ineres rae refix. Le a and v denoe annuiy and residual ousanding amoun of he loan as a ime. As refixes are usual only for morgages his assumpion is realisic, moreover, i is only a echnical complicaion o generalize he analysis for more loans wih annuiy change. For consumer loans, annuiy is consan over he whole life of he loan. For vanilla morgage loans, annuiy changes ake place in agreed ime of conrac re-fix and depend on he ineres rae as a he ime of refix. Le us assume ha re-fix akes place in ime. Mauriy n, ypically in monhs, will be expeced o be fixed, i.e. only annuiy, no ime o mauriy, changes in he ime of re-fix. Le us denoe ime of previous ineres rae fixaion or loan graning. Then he residual ousanding amoun in ime can be compued as he amoun ha remains o be paid wih annuiy a and nominal ineres rae r for n monhs, namely, v a n ( 1 r ) 1. (25) n r ( 1 r ) PRA GUE ECO NO MIC PA PERS, 4,

9 New annuiy afer re-fix in ime can be derived from v as a v n r ( 1 r ), (26) n ( 1 r ) 1 and afer subsiuion from equaion (25) as a a r r ( 1 r ) ( 1 r ) n n n ( 1 r ) 1. (27) n ( 1 r ) 1 For he sensiiviy analysis, we need he raio of he changed and original annuiy. Informaion prior o ime 1 is known and hence a ô and r ô remain unchanged. Raio of he changed and original annuiy can be hen expressed as a r ( 1 r ) a r ( 1 r ) n n n ( 1 r ) 1. (28) n ( 1 r ) Disribuion of Income Change Loans can be paid off from various sources of income, such as wage, ren, ineress, dividends ec. However, wage srongly prevails for mos cliens and hence we will focus on his source of income in our analysis. Income should reflec also he possibiliy of being unemployed. When losing a job he clien receives social aid from he sae. Oher sources of income in such siuaion may be: wihdrawal of savings, work in he grey economy or help of clien s family or friends. We will assume ha possible unemploymen is already incorporaed in he process of income. In equaion (7), is assumed o have a disribuion wih zero mean. Here enaively assume ha / has a Suden disribuion wih d > 1 degrees of freedom, is CDF will be denoed G. (Noe ha mean of he Suden disribuion wih d > 1 degrees of freedom is 0.) Then, where > 0 is he scaling facor. 5. Resuls 5.1 Sensiiviy Analysis F[ x] G[( 1n x) / ; d], (29) Sensiiviy o he sress varies across cliens. We will deermine is main drivers in his secion using he heoreical findings from Secion 3.2 wih deails abou annuiy and income change disribuion from secions 4.3 and 4.4. Remind ha he sensiiviy analysis sudies one-period changes and period of annuiy paymen is usually one monh. Parameers of he model will be chosen wih 348 PRA GUE ECO NO MIC PA PERS, 4, 2008

10 respec o his fac. They will reflec he scenario of growing ineres raes, inflaion and real income, which will urn ou o be he riskies one for morgages. The sressed annual loan ineres rae will be probably 0.25% or a mos 0.5% above he expeced rae. For he annual expeced ineres rae of 5.5%, he expeced monhly rae is r = 0.055/12 and he sressed will be chosen r = /12. Figure 1 (in appendix) shows dependence of he annuiy sress on he sress of ineres raes. The sensiiviy increases wih he growing ime o mauriy. For he ypical morgage loan 240 monhs o mauriy and discussed r, r we have a / a This value is used in he graph of equaion (18), see Figure 2. Growing ineres raes are usually accompanied by he increase of inflaion. We will assume a monhly price level sress by 0.5%. The sressed real income is assumed o increase also by 0.5% and so he nominal income grows by 1%. Clien s IIR is chosen o be 0.6 and SIR 0.2. Parameers of he income disribuion are se o be d = 4 and = Impac of IIR and SIR is depiced in he upper wo graphs of Figure 2. IIR has clearly a greaer effec on he PD sress because real annuiy change is higher han he price level sress. Sensiiviy o he choice of he disribuion parameers is presened in he middle par of he figure, deails of he wo graphs for p < 0.1 are ploed below. While he degrees of freedom d impac he sress significanly less han he scale parameer for p 0.5, he siuaion is exacly opposie for p 0 (he case of mos loans). To sum up, he following ransacion-specific drivers of sensiiviy o sress were idenified: IIR, SIR and he ime o mauriy. IIR deermines he sensiiviy of PD change caused by he sress of real annuiy. The nominal annuiy is eiher consan (consumer loans) or may be changed (morgages). Is change is hen caused by he sress of loan ineres rae and he exen is deermined by he ime o mauriy. Anoher facor conribuing o real annuiy and he only one for consumer loans is he price level. I is relaively less volaile han he nominal annuiy in ime of re-fix. SIR expresses sensiiviy o he sress of price level. The purchasing power of savings depends on he price level, hence cliens wih relaively high savings are subjec o risk of price level sress bu, on he oher hand, hey are generally less risky han hose wih no savings. However, level of savings depends on clien s decision and canno be direcly influenced by he financial insiuion. Hence, he mos risky segmen is morgages wih principal risk drivers being large IIR and long ime o mauriy. 5.2 Dynamics of PD We will demonsrae he dynamic model on he example in his secion. The exogenous scenarios of macroeconomic developmen are presened in Figure 3. They specify he annual ineres rae 12R, index of per capia income I (wages adjused for he income of he unemployed) and price level index P. The ime period is equal o one monh. Realizaion of clien s savings and income developmen was performed via equaions (5) and (7) using error erms of he income change generaed from he disribuion (29). This sep was repeaed imes and raes of defaul compued. PRA GUE ECO NO MIC PA PERS, 4,

11 The following loan and clien properies were used in he simulaion: a CZK i0 i CZK P 5000 CZK n 240 r / 12 r s d 4 In he firs example, loan ineres rae is re-fixed each 12 monhs using he algorihm described in secion 4.2. This is he case of a morgage loan. Resuls can be seen in Figure 4. In he firs graph here is he moraliy funcion from he ime of loan graning ill he ime. The second graph describes probabiliy of defaul in ime of he loans performing ill ime 1. The hird and fourh graph depics developmen of annuiy and loan ineres rae in ime. In he second example, parameers of he loan are he same bu annuiy is consan over he whole life of he loan (or a leas for he observed period). This may be he case of consumer loan or morgage loan wih a long fixaion period. Noe ha mauriy of he loan does no maer in his case. Defaul raes are depiced in Figure 5. In boh examples bu especially in he second one, we can observe peak in P[D D 1, 1 ] in ime = 1. I is due o he fa-ailed disribuion of income change: The peak is even more significan for d = 1 and no presen for d =. The gradual increase of P[D D 1, 1 ] afer ime = 2 is caused by decreasing income of par of he cliens. Conrary o his acs he effec of increasing average savings of hose who did no defaul, which prevails in our example afer he ime of 10, when PD sars o decline. While in he second example no serious impac of differen scenarios on PD is observed we have idenified a serious risk of ineres rae increase in he firs example. I is caused by re-fixaion of morgage loans, when annuiy subsanially increases and causes he immediae growh of he probabiliy of defaul of he non-defauled loans in he nex monhs. 6. Conclusion We proposed a dynamic macroeconomic model of individual clien s PD. Scenarios of macroeconomic developmen and evenual fixaion periods were specified exogenously. The ransacion-specific drivers of sensiiviy o he sress were idenified o be IIR, SIR and for morgages also ime o mauriy. IIR deermines sensiiviy o he sress in real annuiy, which is eiher given solely by he price level sress for he fixed nominal annuiy or conains also nominal annuiy change depending on he ineres rae sress and ime o mauriy. SIR expresses sensiiviy o he sress of price level. Alhough 350 PRA GUE ECO NO MIC PA PERS, 4, 2008

12 cliens wih high SIR are more sensiive o he sress, hey are generally less risky. However, level of savings canno be direcly influenced by he financial insiuions. In he empirical simulaion, an imporan risk was found in he segmen of morgages under he scenario of growing ineres raes. IIR and ime o mauriy are he key risk drivers in his case. This conclusion sends also anoher clear message: Poenial allowing for higher IIR and mauriy lengh for morgages in he graning process should be reaed carefully. An alernaive approach based on he habi formaion was presened. I does no require he exogenous price level and yields simpler resuls. Conclusions under his approach are similar. References Abel, A. (1990), Asse Prices under Habi Formaion and Caching Up wih he Joneses. American Economic Review, 1990, 80, pp Campbell, J. Y., Cochrane, J. H. (1999), By Force of Habi: A Consumpion-Based Explanaion of Aggregae Sock Marke Behavior. The Journal of Poliical Economy, 1999, 107 (2), pp Commiee of European Banking Supervisors (2006), Technical Aspecs of Sress Tesing under he Supervisory Review Process. CEBS Consulaion Paper 12, Gross, D. B., Souleles, N. S. (2001), An Empirical Analysis of Personal Bankrupcy and Delinquency. Naional Bureau of Economic Research Working Paper 8409, Hayashi, F. (2000), Economerics. Princeon: Princeon Universiy Press, Heømánek, J., Hlaváèek, M., Jakubík, P. (2007), Credi Risk, Credi Growh Models and Sress Tesing. CNB Economic Research Bullein, 2007, 1, pp Jakubík, P. (2007), Macroeconomic Environmen and Credi Risk. Czech Journal of Economics and Finance (Finance a úvìr), 2007, 57, pp Meron, R. (1974), On he Pricing of Corporae Deb: The Risk Srucure of Ineres Raes. Journal of Finance, 1974, 29, pp Vasicek, O. (2002), The Disribuion of Loan Porfolio Value. Risk, 2002, 15, pp Whiley, J., Windram, R., Cox, P. (2004), An Empirical Model of Household Arrears. Bank of England Working Paper 214, APPENDIX Figure 1 Resuls of Equaion (28) for r = 0.055/12, i.e. 5.5% p.a. PRA GUE ECO NO MIC PA PERS, 4,

13 Figure 2 Sensiiviy of PD o he Change of Parameers. Noe: The reference values are IIR a / î = 0.6, SIR s / î = 0.2, price level sress P / P = 1.005, per capia income sress I / I = 1.01, annuiy sress a / a = 1.02, degrees of freedom d = 4, scaling parameer = 0.02, if no specified differenly in he graph. 352 PRA GUE ECO NO MIC PA PERS, 4, 2008

14 Figure 3 Scenarios of he Macroeconomic Developmen: Ineres Rae p.a. 12R, Index of per capia Income I and Price Level Index P. PRA GUE ECO NO MIC PA PERS, 4,

15 Figure 4 Annuiies and PD s under Differen Scenarios wih Re-fix Each 12 Periods (monhs) 354 PRA GUE ECO NO MIC PA PERS, 4, 2008

16 Figure 5 PD s under Differen Scenarios wih Consan Annuiy. PRA GUE ECO NO MIC PA PERS, 4,