INSTANTANEOUS INTEREST RATES AND HAZARD RATES

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 6, Number /25, pp. - INSTANTANEOUS INTEREST RATES AND HAZARD RATES Gheorghiţă ZBĂGANU Gheorghe Mihoc Caius Iacob Isiue of Mahemaical Saisics ad Applied Mahemaics of he Romaia Academy, Casa Academiei Româe, Calea 3 Sepembrie o. 3, 57 Buchares, Romaia. zbagag@csm.ro A isaaeous ieres rae (IIR) is a CADLAG fucio δ:[,),) which has he meaig ha for moeary ui (MU) borrowed a ime = coss σ( ) = exp x )dx MU a mome =. The mappig = /σ( is he survival fucio of some lifeime τ. I his framework, δ is he failure rae (FR) of τ. We ivesigae he aalogy IIR FR i he case of credi reimburseme. We say ha a IIR δ is of posiive ype if for ay cosa cash flow o he ierval [,T] he flow of pricipals is o-egaive. We prove ha δ is of posiive ype iff τ is a lifeime wih decreasig mea rezidual life (DMRL).. CREDIT REIMBURSEMENT. DISCRETE TIME We deal wih wo parers: he credior C ad he debior D. A mome =, C leds o D a cash amou C. Afer a deal, he wo parers agree o a reimburseme schedule. They agree ha he isaaeous ieres rae is δ. Defiiio.. A isaaeous ieres rae (IIR) is ay fucio δ:[,) [,) which is righcoiuous ad has limi o he lef. Is meaig is ha MU borrowed a ime = coss σ( ) = exp x )dx MU a ime =. We call he fucio σ he frucificaio facor. For ay fucio which is righ coiuous ad wih fiie limis o he lef we shall use he abbreviaio CADLAG. Defiiio.2. A reimburseme schedule of he credi C i isallmes o he ierval [,T] wih IIR δ is ay sysem (D,R,C,δ) where D ={ = < < < = T }, R = ( ), 2 ),, )). The umber j is he mome of he jh payme ad he quaiy R j is he value of he jh payme. The reimburseme codiio is ) exp u)du + 2 ) exp 2 u)du ) exp u)du = C. (.) The moivaio of (.) is ha a amou of R( j ) MU paid a mome j has he same value as a j amou of j ) exp u) du MU a mome =. Noice ha he firs payme is made a mome : we deal wih posicipaed paymes. Reccomeded bymarius IOSIFESCU, member of he Romaia Academy

2 Gheorghiţă Zbăgau 2 Defiiio.3. The fucio F:[,) [,] defied by = e u) du = is called he σ( acualizaio facor. Usig his oaio, he reimburseme codiio becomes ) ) + 2 ) 2 ) + + ) ) = C (.2) Noice ha we acceped ha δ. I seems aural o be so: a egaive ieres rae has o ecoomic meaig. I seems also aural o cosider Defiiio.4. A IIR δ is called aural iff u)du =. Thus, for a aural IIR he fucio F has he followig properies: ) =, F is o-icreasig ad ) =. I ha case he fucio = is o-decreasig, coiuous, ) = ad ) =. So, F is a disribuio fucio of some o-egaive radom variable τ. This radom variable ca be ierpreed as beig a lifeime so, F is a life disribuio. Moreover, sice he mappig! u)du is righdiffereiable, i is absoluely coiuous. I has a desiy f( =. I his ierpreaio we ca wrie + h) P( τ h τ > = lim = lim. (.3) h hf ( ) h h I his form, δ has bee iesively sudied i reliabiliy heory uder he ame of failure rae ([],[2],[3], [5]) or hazard rae ([2], [6[) ad i demography ad acuaries uder he ame of moraliy rae ([4] or eve moraliy force ([7]). Coclusio: oe may hik of δ as beig he hazard rae of a lifeime τ. If u)du <, he his lifeime τ may also assume he value + wih probabiliy ) = exp δ ( u )du. Usig his similariy, i he case of aural IIR s, he reimburseme codiio (.2) has a probabilisic ierpreaio: i is he expecaio of some discree radom variable cosruced usig D. Namely, le us add j o D he poi + =. Le also R( ) =, R( j ) = R( j- ) + j ) = k ) Cosider he discreizaio of τ deoed by τ D give by τ D = { τ [ j, j+ )} = { τ < } + { τ< 2 } + 2 { 2 τ< 3 } + + { τ< + } (.4) j Proposiio.. If a IIR δ is aural, he he reimburseme codiio (.2) is equivale o he fac ha ER(τ D ) = C. Proof. We have ER(τ D ) = R( j ) P( j τ < j+ ) = R( j ) ( j ) j+ )) = R( j ) j ) + = R( j- ) j )) = R( j ) j ) (R( j ) R( j- ) j ) = k= R( j- ) j )) (as + ) = ) = R( ) =!) j ) j ) ad by (.2) he las sum is equally o C.

3 3 Isaaeus ieres raes ad hazard raes A payme R( j ) has wo compoes: he pricipal ad he ieres. The pricipal, deoed by a( j ) is he fracio of he deb C which is paid by he isallme R( j ) while he ieres, deoed by d( j ), is a exra pay meaig he cos of he credi. I is acceped ha if we deoe S( ) = C, S( ) = C a( ),, S( j ) = S( j- ) a( j ), (hus S( ) =! ) he remaiig deb afer he jh payme, he he ieres produced by S( j- ) o j he ierval bewee wo successive paymes [ j-, j ) is equal o S( j- ) exp δ ( u )du. j If we deoe, as usual i bakig ad accouig, k i k := δ F exp ( u)du = k ) ) ( k k he he coecio bewee isallmes, pricipals ad ieress is give by he well kow relaio (see, for isace [4] ) ) = Ci + a( ) = S( ) ( + i ) S( ),., k ) = S( k- )( + i k ) S( k ), k (.6) This meas ha if we kow he isallme j ) we ca compue he pricipal a( j ) ad coversely, if he credi reimburseme schedule coais he pricipals a( j ) oly, oe ca compue he isallmes. Equaio (.6) do o have a immediae probabilisic meaig. However, we ca sae Proposiio.2. If τ D has he same meaig as i Proposiio.. he (.6) becomes E(R(τ D ) ; τ D m- ) = C (S( m ) + m ))P(τ > m ), m (.7) Proof. Wrie (.6) uder he form (R( k ) R( k- ) k ) = S( k- ) k- ) S( k ) k ), k (.8) ad add hem for k = o k = m. As R( ) =, ) = ad S( ) = C, we ge R( )( ) 2 )) + R( 2 )( 2 ) 3 )) + + R( m- )( m ) m- )) + R( m ) m ) = C S( m ) m ) or R( )P(τ D = ) + R( 2 ) P(τ D = 2 ) + + R( m- ) P(τ D = m- ) = C S( m ) m ) R( m ) m ); his is (.7). k ) (.5) 2. CONTINUOUS CASH FLOW. ANALOGY IIR FAILURE RATE Now, we shall assume ha he reimburseme is made by a cash flow. Defiiio 2.. A cash flow is ay CADLAG fucio r: [,) [,). Noice ha he fucio R( = s)ds does exis ad is righ-differeiable. Moreover, if R is is righ derivaive, he R = r. The meaig is ha D ad C accep a coiuous reimburseme schedule usig r, give a IIR δ. If C leds o a D capial amou of C MU, he reimburseme codiio is ha s δ ( u)du rse () d s= C. (2.) Defiiio 2.2. We deoe such a reimburseme schedule by (r,c, δ). If R() < he r is called proper. If = for greaer ha some T, he (r,c, δ) will be called aural. If F ad τ have he same meaig as i he firs secio, he reimburseme codiio is The aalog of Proposiio.. is rsfs () ()d s= C. (2.2)

4 Gheorghiţă Zbăgau 4 Proposiio 2.. The reimburseme codiio (2.2) is equivale o ER(τ) = C. Proof. Remark ha R() = ad use iegraio by pars: ER(τ) = ER(τ) R() = R ' (s)p(τ > s) ds = s) s) ds The cash flow has wo compoes: he flow of pricipals ad he flow of ieress. The firs oe will be deoed by a ad he secod by d. Mahemaically, = a( + d(, where d( is he ieres paid for he remaiig deb ad a( is he flow of pricipals. The codiio for a o be a flow of pricipals for he credi C is ha a() s ds = C. Le, as before, S( = () = () d = () S a s C ads deoe he remaiig deb a mome. We wa o fid he relaioship bewee r ad a. Le us accep ha a deb of S( MU lef upaid i he + h ierval [,+h) yields a ieres d(,+h) = exp u)du S( MU. If we le h ad use he righ coiuiy of δ we ifer ha d(, + h) lim = δ () S(). h h (2.3) Usig his fac we ge he followig resul. Proposiio 2.2. Suppose ha a is a CADLAG flow of pricipals for he deb C ad δ is a aural IIR. The he reimburseme schedule is = a( + S( (2.4) Therefore he aalog of (.7) is E(R(τ); τ < = C (R( + S() P(τ > (2.) Moreover, he mappig (R( + S() P(τ > is o-icreasig. Proof. We have o check ha he reimburseme codiio ( a( + S()d = C holds. Bu, by our assumpios, S ad F are righ-differeiable ad S = - a, F = - δf. This meas ha ( a( + S()d = ( S ( + S(F ()d = ( SF) (d = )S() )S(). As ) =, S() = C ad S() =, i follows ha d = C. Moreover, replacig he iegraio limis by ad 2, < 2, we ge he formula 2 d = )S( ) 2 )S( 2 ), (2.6) which implies i paricular ha

5 5 Isaaeus ieres raes ad hazard raes d = C S(. (2.7) If we use agai he iegraio by pars formula i he form Ef(τ) = f() + d, (2.8) which holds for ay coiuous righ-differeiable fucio f (see, for isace, [8]), for he paricular fucio f( = R(x equaio (2.7) becomes E(R(τ = C S( P(τ >, (2.9) which is he same as he claim (2.5). Fially, he las claim is obvious from (2.6) : he cash flow r is o-egaive. The above resul ca be used i wo ways: he firs problem is o fid r kowig a while he secod oe is o fid a kowig r. Proposiio 2.3. Suppose oe has a reimburseme schedule for he pricipals, i.e. a CADLAG mappig a from [,) o [,) such ha a(d <. Le S( = ( ) ( τ ) Proof. By (2.5), ( ) = () + ) () d = + ) () () S() δ d r <. Sice δ = S( ( ) ( τ ) d. The r is proper if S τ E < F. (2.) R a S r C S dr. Thus, R() < is he same as ( τ ) ( τ ) f, where f = - F is he desiy of F, F S δ () S() d = f ( dx= E F We proved (2.) ad, moreover, he equaliy S τ R( ) = C+ E. F I he secod case, oe kows r ad was o fid a. If we suppose ha R ad δ are differeiable, he (2.5) ivolves a iegral equaio wih oe ukow fucio a. I is possible ha his equaio have o accepable soluio. Defiiio 2.3. Call a reimburseme schedule (r,c,δ) realisic if he iegral equaio (2.4.) has a o-egaive soluio, a, wih he propery ha S() = (i) (ii) a(s)ds = C. Proposiio 2.4. If r is coiuous ad δ is differeiable, he a formal soluio of (2.4.) is a( = The equaio (C - s)s)ds) = s)s)ds = - S( (2.) S( = E(R(τ) R( τ > (2.2)

6 Gheorghiţă Zbăgau 6 always holds. Moreover, if r is proper or if lim (iii) = he S() = hus S( = a(s)ds. If r is proper or if lim =, he (r,c,δ) is realisic iff he map! E(R(τ) R( τ > is o icreasig. A equivale codiio is ha dx. (2.3) Proof. (i) By our assumpios, a is coiuous, hece S is differeiable. Moreover, S = -a hece (2.5) becomes S ( = S( wih he iiial codiio S() = C. This is a liear differeial equaio. If oe (ii). solves i usig he mehod of variaio of cosas, oe ges S( = implies ha S() = C. Takig he derivaive of S oe ge (2.). The iegral equaio a( = - a(s)ds, S() = C, bu hey are o equivale. Remark ha dx which, by (2.2), a(s)ds = C implies ha S ( = S(, a(s)ds = S() S(). If we wa a o be a real dx reimburseme schedule he we should add he codiio S() =. The equaliy S( = always holds. By L Hospial rule, S() = lim = lim provided ha he las limi exis. F'( Thus a codiio i order ha S() = would be lim =. However, i is possible ha his las limi does o exis ad sill S() =. Tha migh happe if r is proper. I ha case we eed aoher proof. Firs, we check (2.2). Remark ha E(R(τ) R( τ > = ( (, τ) ( dλ( ) ( τ ) > dx dp (here λ is Lebesgue measure) = ( R( τ) R( ) ( τ ) > ( ) ( [, ) ( ( τ> dp dλ( dp = (by Fubii!)=, hece we checked he claimed equaliy. If r is proper, he R() < so S() = lim E(R(τ) R( τ > lim E(R() R( τ > = lim (R() R() =, hece S() =. (iii) We wa ha he fucio S( = E(R(τ) R( τ > = Bu he codiio S is exacly (2.3.) dx be o icreasig S.

7 7 Isaaeus ieres raes ad hazard raes Example 2.. Suppose ha = δ = cos. I his case = e - δ τ Expoeial(δ). If a is kow, he = δs(; by (2.) we ge R() = C + E(e δτ S(τ)) = C + δ S(d. If r is kow ad r is proper or lim =, he he flow of pricipals is give by a( = δs(. The schedule ( r,c,δ) is realisic iff S( / δ. Couerexample 2.. Cosider he same IIR as before. Suppose ha = Cδ, hus R( = Cδ. This is o a proper schedule. As R() = CδEτ = C, he reimburseme codiio (2.2) is fulfilled. However, his is o a realisic schedule: S( = CδE(τ τ > is always equal o C, implyig ha a =. No maer how much D pays o C he deb remais he same! O he corary, if = 2m [,T] ( hece R( = m( T) 2 wih some cosa m such ha me(τ T) = C, he a( = δs( implies a() = δs() = δc <. Now, r is proper, bu o realisic. Example 2.. Cosider he same IIR. Le r : [,) [,) be o icreasig ad suppose ha ) =. The (r,c,δ) is realisic. Ideed, we check ha a( δs(. Ideed, δs( = δ e e δ δx dx = δ r ( e x dx δ e x dx (sice x!) =. Defiiio 2.4. (see [],[2],[5]). A lifeime τ is called a DMRL (Decreasig Mea Rezidual Life) iff he mappig E( := E(τ τ > is o icreasig. If is failure rae δ τ = f τ / F τ is odecreasig, he τ is called a IFR (Icreasig failure rae i). I is easy o see ha if τ is a IFR, he τ also is a DMRL (see for isace []). Defiiio 2.5. Le δ be a IIR. We call δ of posiive ype iff for ay o-icreasig r such ha lim = ad for ay credi C he reimburseme schedule (r,c,δ) is realisic. Now, we characerize posiive ype IIR s. Proposiio 2.5. Le δ be a IIR ad le τ be a lifeime wih he propery ha he failure rae of τ is δ. The (i) δ is of posiive ype iff τ is a DMRL; (ii) If δ is o decreasig he δ is of posiive ype; (iii) If δ is periodic he δ is of posiive ype if ad oly if i is cosa. Proof. (i) Suppose ha τ is a DMRL. The E( = oe fids he equivale codiio τ is a DMRL dx is o icreasig E (. By differeiaig dx. (2.4) We wa o prove ha δ is a IIR of posiive ype. Le r:[,) [,) be o icreasig ad lim =. Our ask is o prove ha a( S(. As S( = dx his is he same as

8 Gheorghiţă Zbăgau 8 dx. (2.5) Bu x (2.4). We checked (2.5). T dx dx = dx because of Coversely, suppose ha δ is of posiive ype. Choose = r [,T] (. The R = C/I, where I = d. We kow ha a( = C/I (C C/I posiive ype he (I s)ds) T - s)ds) C,T >. I follows ha if δ is of T s)ds T >. Leig T, (2.4) follows. (ii) Obvious. Ay IFR is a DMRL. (iii) If δ is periodic (say, + p) = for some p > ) he E( is periodic, oo, sice he failure rae of he rezidual lifeime (τ- τ> is δ ( = +. The E( should be some cosa: E( = α for some α. Thus dx As a byproduc we oice = α α F ( dx = α τ Expoeial(α). Corollary 2.6. If τ is a DMRL ad R:[,) [,) is cocave ad icreasig he R(τ) is a DMRL, oo. Proof. As R is coiuous ad oe-o-oe, we jus have o remark ha E(R(τ) R( R(τ) > R() = E(R(τ) R( τ > ad apply Proposiio 2.5 (i). Examples 2.2. The cosa simple ieres rae ( i.e. = disribuio) ad he usual oe (i.e. =(+i) [] i + i{ } i + i, correspodig o a Pareo wih [] ad {} deoig he ieger ad he fracioary pars of are o of posiive ype. Here i is he yearly ieres rae, supposed o be cosa. For he firs case he compuaios are easy ad lef o he reader. For he secod oe apply Proposiio 2.5 (iii). REFERENCES. AHMAD, I.A., A class of saisics useful i esig icreasig failure rae average ad ew beer ha used life disribuios, J. Saisical Plaig Iferece, 4, pp. 4-49, BARLOW, R. E., PROSCHAN, F., Saisical Theory of Reliabiliy ad Life Tesig. To begi wih, Silver Sprig, MD, BLOCK, H., SAVITS, T. H., The IFRA closure problem, A. Probab., 4,pp. 3-32, BURLACU, V., CENUŞĂ, G., Bazele maemaice ale eoriei asigurărilor, Buchares, ASE, CAO, J.H., WANG. Y.D., The NBUC ad NWUC classes of life disribuios, J. Appl. Probab.,28, pp , GNEDENKO, B., BELEAEV Y.,SOLOVIEV A., Mehodes mahemaiques das la heorie de fiabilie, Moscow, Mir, GOOVAERTS, M.J., KAAS R., HEERWAARDEN, A.E. BAUWELINCKX, T., Effecive Acuarial Mehods,Amserdam, Norh Hollad, ZBĂGANU, G., Meode maemaice î eoria riscului şi acuaria. Buchares, Uiversiy Press, 24. Received February 5, 25