THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable and deposis can be invesed and wihdrawn a any ime. However, cusomer behaviour is no fully raional and wihdrawals of balances are ofen performed wih a delay. This paper focuses on measuring he ineres rae risk of variable rae savings accouns on a value basis (duraion) and analyzes he problem how o hedge hese accouns. In order o model he embedded opions and he cusomer behaviour we implemen a parial adjusmen specificaion. The ineres rae policy of he bank is described in an errorcorrecion model. KEYWORDS Term srucure, duraion, uncerain cash flow, variable raes of reurn JEL codes: C33, E43 1. INTRODUCTION A major par of privae savings is deposied in variable rae saving accouns, in he US also known as demand deposis. Typically, deposis can be invesed and wihdrawn a any ime a no cos, which makes a savings accoun look similar o a money marke accoun. However, he ineres rae paid on savings accouns is ofen differen from he money marke rae. In Europe, he ineres rae paid on he savings accoun can acually be higher or lower han he money marke rae. Even when hese ineres raes differ, deposiors do no immediaely wihdraw heir money from savings accouns when raes on * We hank Dennis Bams, Joos Driessen, D. Wilkie, paricipans a he AFIR colloquium, and wo anonymous referees for commens on previous versions of he paper. The usual disclaimer applies. 1 Universiy of Amserdam ING Group and CenER, Tilburg Universiy ASTIN BULLETIN, Vol. 33, No., 3, pp. 383-397
384 FRANK DE JONG AND JACCO WIELHOUWER alernaive invesmens are higher. Whaever he causes of his behaviour (marke imperfecions, ransacion coss or oher), hese characerisics imply ha he value of he savings accouns from he poin of view of he issuing bank may be differen from he nominal value of he deposis. In he lieraure, he valuaion of savings accouns is well sudied. For example, Huchison and Pennacchi (1996), Jarrow and Van Devener (1998) and Selvaggio (1996) provide models for he valuaion of such producs. The firs wo papers build on he (exended) Vasicek (1977) model, whereas he laer paper uses a more radiional Ne Presen Value approach. In all hese papers here is lile explici modeling of he dynamic evoluion of he ineres rae paid on he accoun and he balance, and how his evoluion depends on changes in he erm srucure of marke ineres raes. For example, Jarrow and van Devener s (1998) model is compleely saic in he sense ha he ineres rae paid on he accoun and he balance are linear funcions of he curren spo rae. In pracice, i is well known ha ineres raes and balances are raher sluggish and ofen do no respond immediaely o changes in he reurn on alernaive invesmens, such as he money marke rae. Typically, he ineres rae paid on he accoun is se by he bank and he balance is deermined by clien behaviour. The balance depends, among oher hings, on he ineres rae bu also on he reurn on alernaive invesmens. Because he pahs of fuure ineres raes and he adjusmen of he balance deermine he value of he savings accouns, an analysis of dynamic adjusmen paerns is imporan. In his paper, we analyze he valuaion and hedging of savings deposis wih an explici model for he adjusmen of ineres raes and balances o changes in he money marke rae. A recen paper by Janosi, Jarrow and Zullo (JJZ, 1999) presens an empirical analysis of he Jarrow and van Devener (1998) model. They exend he saic heoreical model o a dynamic empirical model, ha akes he gradual adjusmen of ineres raes and balance ino accoun. Our approach differs from he JJZ paper in several respecs. Firsly, we rea he erm srucure of discoun raes as exogenous and calculae he value of he savings accoun by a simple Ne Presen Value equaion. This approach, suggesed by Selvaggio (1996) leads o simple valuaion and duraion formulas, and is applicable wihou assuming a paricular erm srucure model. The drawback of he NPV approach is ha we have o assume ha he risk premium implici in he discoun facor is consan, bu his may be a good firs approximaion because we wan o concenrae on he effecs of he dynamic adjusmen of he ineres rae paid on he accoun and balance and no on erm srucure effecs. Secondly, a difference beween he JJZ model and ours is he modeling of he long run effecs of discoun rae shocks. In our model, here is a long run equilibrium, in which he difference beween he ineres rae paid on he accoun and he money marke rae is consan, and he balance of he savings accoun is also consan (possibly around a rend). Shor erm deviaions from hese long run relaions are correced a a consan rae. This model srucure is known in he empirical ime series lieraure as an error correcion
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS 385 model 3. This model has some aracive properies, such as convergence of he effecs of shocks o a long-run mean. The ineres rae sensiiviy is quanified in a duraion measure. We demonsrae ha he duraion depends on he adjusmen paerns of ineres rae paid on he accoun and balance. We pay paricular aenion o he implicaions of he model for he hedging of ineres rae risk on savings deposis. We illusrae how o fund he savings deposis by a mix of long and shor insrumens ha maches he duraion of he savings accoun s liabiliies. The paper is organized as follows. Firs he valuaion of he savings accouns is deal wih in. In 3 he models on he pricing policy and he cusomer behaviour are presened, and a discree ime version of he model is esimaed for he Duch savings accouns marke. 4 deals wih he duraion of his produc and 5 wih hedging decisions. The paper is concluded in 6.. VALUATION OF VARIABLE RATE SAVINGS ACCOUNTS The valuaion problem of savings accouns and similar producs was analyzed by Selvaggio (1996) and Jarrow and Van Devener (1998). Their approach is o acknowledge ha he liabiliy of he bank equals he presen value of fuure cash ouflows (ineres paymens and changes in he balance). The presen value of hese flows does no necessarily equal he marke value of he money deposied, and herefore he deposis may have some ne asse value. Jarrow and Van Devener (1998) rea he valuaion of savings accouns in a no-arbirage framework and derive he ne asse value under a risk-neural probabiliy measure. However, in our paper we wan o implemen an empirical model for he savings rae and he balance, and herefore we need a valuaion formula based on he empirical probabiliy measure. We herefore adop he approach proposed by Selvaggio (1996), who calculaes he value of he liabiliies as he expeced presen value of fuure cash flows, discouned a a discoun rae which is equal o he risk free rae plus a risk premium 4. Hence, he discoun rae R() can be wrien as R ] g= r ] g +c, (1) where r() is he money marke rae and g is he risk premium. We can inerpre his discoun rae as he hurdle rae of he invesmen, ha incorporaes he riskiness of he liabiliies, as in a radiional Ne Presen Value calculaion. The main assumpion in his paper is ha his risk premium is consan over ime and does no depend on he level of he money marke rae. This assumpion is obviously a simplificaion. Any underlying formal erm srucure model, such as he Ho and Lee (1984) model, implies ha risk premia depend on he 3 We refer o Davidson e al. (1978) for an inroducion o error correcion models. 4 Selvaggio (1996) calls he risk premium he Opion Adjused Spread
386 FRANK DE JONG AND JACCO WIELHOUWER money marke rae. However, he risk premia are ypically small and since he focus of he paper is on modeling he dynamic adjusmen of ineres raes and balances, we ignore he variaion in he risk premium and focus on he effec of shocks o he money marke rae. Wih his srucure, he marke value of liabiliies is he expeced discouned value of fuure cash ouflows, i.e. ineres paymens on he accoun i() and changes in he balance D() 5 Rs L () = E ; # 3 e - 6i] sgd] sg-dl] sg@ dse. () D Noice ha in his seup reinvesmens of ineres paymens are couned as a par of deposi inflow D (). Working ou he inegral over D (s) by parial inegraion we find ha he value of he liabiliies equals Rs L () = E ; # 3 e - 6i ] sg- R] sg@ D] sgdse + D(). (3) D Since he marke value of he asses is equal o he iniial balance, D(), he ne asse value (i.e., he marke value of he savings produc from he poin of view of he bank) is Rs V () = D() - L () = E ; # 3 e - 6R] sg-i] sg@ D] sgdse. (4) D D For an inerpreaion of his equaion, noice ha R() i() is he difference beween he bank s discoun rae and he ineres paid on he accoun. Addiional savings generae value wih reurn R(). The coss of hese addiional savings are i(), however. The difference R() i() herefore can be inerpreed as a profi margin. The ne asse value is simply he presen value of fuure profis (balance imes profi margin). Therefore, he ne asse value is posiive if he ineres rae paid on he accoun is on average below he discoun rae. Obviously, he ne asse value is zero if he ineres rae paid on he accoun always equals he discoun rae. As an example, consider he siuaion where he ineres rae paid on he accoun is always equal o he discoun rae minus a fixed margin, i()=r() m, and he discoun rae is consan over ime. 6 Moreover, assume ha he balance is consan a he level D *. In ha case, he ne asse value of he savings accouns is ) V = n D R D ). (5) 5 For noaional clariy, he ime variaion in he discoun rae R is suppressed. If he discoun rae is - Rudu () ime varying, he exac expression for he discoun facor is e # 3. 6 This is a special case of he Jarrow and Van Devener (1998) model.
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS 387 Inuiively, his is he value of a perpeuiy wih coupon rae m and face value D ). Figure 1 graphs he ne asse value for differen values of R and m. For large profi margins and low discoun raes, he ne asse value can be a subsanial fracion of he marke value of he savings deposis. FIGURE 1: Ne asse value This figure shows he ne asse value of a deposi of 1, as a funcion of he discoun rae R and he profi margin m Obviously, his example describes he value in a saic seing. For he ineres rae sensiiviy of he ne asse value, we have o ake ino accoun ha afer a shock in ineres raes, he ineres rae paid on he accoun and he balance only gradually adjus o heir new equilibrium values. In he nex secion we herefore presen a model for he adjusmen paerns of ineres rae and balance afer shocks o he discoun rae. In he subsequen secion we presen discoun rae sensiiviy measures based on hese adjusmen paerns. 3. CLIENT AND BANK BEHAVIOUR The analysis in he previous secion shows ha he ne asse value of savings accouns depends on he specific paern of he expeced fuure ineres raes and balances. The main difference beween money marke accouns and savings accouns is he sluggish adjusmen of ineres raes and balance o changes in he discoun rae. In his secion we model hese adjusmen processes. The
388 FRANK DE JONG AND JACCO WIELHOUWER models highligh he parial adjusmen oward he long run equilibrium values of ineres raes and balances. In he analysis, we ake as given he pah of he money marke rae r]g and hence he pah of he discoun rae R] g= r] g +c. We describe he sochasic evoluion of he ineres rae paid on savings deposis, i(), and he balance, D(), condiional on he pah of he discoun rae. For he ineres rae paid on savings accouns, we propose he following sochasic error correcion specificaion di] g= l6 R] g-n- i] g@ d + v dw ] g (6) 1 1 where W 1 () is a sandard Brownian moion. This equaion saes ha he ineres rae adjuss o deviaions beween he long run value R() m and he curren rae. We see his as he arge policy rule of he bank ha ses he ineres rae. Deviaions are correced a speed k >, and in he long run, expeced ineres raes are a margin m below he discoun rae R(). The sochasic erm W 1 () models he deviaions from he arge policy rule. Such deviaions could be due o sudden demand shocks, compeiion from oher banks and he like. For he balance we propose a parial adjusmen specificaion ) dd] g=-m8d] g-d B d -h6r] g-n- i] g@ d + v dw ] g (7) This specificaion has hree componens. Firsly, here is an auonomous convergence o a long run mean D *, which is deermined by a radeoff by he cliens beween savings deposis and money marke accouns. Secondly, here is an ouflow of funds proporional o he excess of he discoun rae over he savings rae. Thirdly, here is an unpredicable sochasic componen. This descripion wih an auonomous convergence is especially suiable for a derended ime series. An auonomous convergence o a long run mean is expeced in a derended series for he balance. We derend by defining he variable D() as he fracion of oal shor erm savings ha is invesed in variable rae savings accouns. In his case D * is he long run fracion of oal shor erm savings ha is invesed in variable rae savings accouns. In his way, he rend growh of he oal savings marke doesn affec he empirical esimaion and he duraion analysis. Working ou he sochasic differenial equaions (6) and (7) gives: # # 1 - s - s - i] g= e l i] g+ l e l ] g6 R] sg - n@ ds + v e l ] gdw ] sg, (8a) ) -m ) m s - D ] g= D+ e _ D] g-di -h e ] g6 Rs ] g-n-is ] g@ ds # m s - + v # e ] gdw ] sg. (8b) To inerpre hese equaions, le s consider he siuaion where he discoun rae R is consan over ime. I is fairly easy o show ha he effec of a
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS 389 change in he discoun rae in his siuaion is given by he following parial derivaives i ] g l s - -l = l # e ] g ds= 1-e, (9a) D ] g m s - i s =-h # e ] g ] g < 1- Fds -m -l m s - ls - h e e ds=-h e - e # ] g d n. (9b) l- m The long run derivaive of he ineres rae paid on he accoun is one, bu in he shor run he effec is less han one. If j > and k > l (which we show laer is clearly he case empirically), he parial derivaive of he balance is negaive, and converges o zero in he long run. These parial derivaives can be used o sudy he effecs of a once-andfor-all shock o he discoun rae, a kind of impulse response analysis. Saring from he equilibrium siuaion D]g = D ) and i() = R- n, he expeced adjusmen paerns are illusraed in Figure for an increase in he discoun rae by 1%. The parameer values are picked from he empirical esimaes o be discussed shorly, and are equal o k =.79, l =.48 and j =.43 for he base case. We see ha he ineres rae doesn follow he jump in he discoun rae immediaely bu gradually adjuss o is new equilibrium value. The adjusmen of he balance is more complex. Iniially, he balance decreases because of wihdrawals caused by he relaively low ineres rae paid on he accoun. Bu as he ineres rae increases, his effec becomes smaller and evenually he auonomous convergence of he balance o is long run level dominaes. One inerpreaion of his is ha cliens who iniially preferred he variable rae savings accoun o he money marke accoun will reurn o variable rae savings accouns when he difference beween he ineres rae paid on he savings accoun and he money marke rae revers o he iniial level. Equaions (9a) and (9b) also highligh he effecs of he model parameers on he adjusmen of ineres raes and balance o a shock in he discoun rae. The effec of j is obvious, i increases he impac of an ineres rae shock. This effec may be imporan in he curren marke, as he increase in he use of inerne for banking services and he resuling lower ransacion and search coss will probably increase he ineres rae sensiiviy of he cusomers. The effec of he mean-reversion parameers k and l is more complicaed. A higher value of l speeds up he adjusmen of he balance iself, bu doesn affec he ineres raes. Wih a lower value of k, boh he adjusmen of he ineres rae and he balance are slower. The effec of he balance is a resul of he dependence of he balance on he ineres rae. These effecs are illusraed in Figure, where he dashed line gives he adjusmen paern for a lower value of k, and he doed line he paern wih a higher value of l.
39 FRANK DE JONG AND JACCO WIELHOUWER FIGURE : Adjusmen of ineres rae and balance of savings accouns This figure shows he adjusmen of ineres rae (op panel) end balance (boom panes) o a 1% shock in he discoun rae. The solid line is he base case. The dashed line is for a smaller value of k, he doed line for a larger value of l. The scale of he horizonal axis is years. We now presen some indicaive esimaes of he model parameers. This exercise is no mean o be a horough empirical invesigaion of he adjusmen paern bu merely serves as an illusraion of he model. In order o ranslae he coninuous ime parameers o a discree ime seing, we use he following approximae 7 discreizaion of he coninuous ime model Di = l7r -n- i A D+e, (1a) -1-1 1 ) - 1 DD =-m_ D -D id-h7r -n- ia D+e. (1b) The discoun rae is no direcly observed in he daa. Since a savings accoun shares characerisics of boh a money marke accoun and a long erm deposi, is required rae of reurn (or discoun rae) is proxied by a weighed average of he money marke rae (r ) and he long erm bond yield (y ). 8 7 This approximaion is quie accurae. For example, he exac mean reversion parameer for he ineres rae equaion is 1 - exp (- l D), which for small values of l or D is close o l D. 8 An alernaive bu equivalen way o jusify his proxy is o assume ha he risk premium of he savings deposi is a fracion of he risk premium on long erm bonds.
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS 391 We rea he weigh d as an unknown parameer which is esimaed from he daa. This leads o he following empirical model Di = a + a Dr + a 7i -" dr + ^ - dhy, A+ e, (11a) 1-1 -1 1-1 1, D = b + b D + b 7i -" dr -^ - dhy, A + e. (11b) 1-1 -1-1 1-1, This model is slighly more general han he heoreical model because i conains an immediae, discree adjusmen of he ineres rae o he money marke rae. Afer his iniial jump, he adjusmen o he new equilibrium is gradual. This effec urned ou o be so imporan empirically ha we included i in he empirical model. 9 The parameers of he coninuous ime model can be solved from he following equaions (wih D = 11 / for monhly daa) l=- a / D, m= ^ -bh/ D, h= 1 1 b / D. In fac, he long run deposi level and he average spread of he ineres raes over he esimaed discoun rae could be unraveled from he consan erms of he model. These are no very accuraely esimaed however and we refrain from drawing inferences abou hese parameers from he esimaes. We use monhly daa on ineres raes and deposis from he Duch savings accoun marke. The ineres rae paid on he accoun is aken from one of he price seers in he Duch marke. The sample period is 198:1 o 1999:1, spanning 17 years which is slighly longer han he samples of Huchison and Pennacchi (1996) or JJZ. To remove rends in he oal savings volume, we define he balance D as he fracion of variable rae savings accouns o oal savings. The following empirical esimaes are obained using leas squares: Di=-. 84 +. 7 Dr-. 667i -1-" dr -1 + ^1 - dhy -1, A + e1,, (1a) D=. 1 +. 996D -1+. 397i -1-" dr -1 + ^1 - dhy -1, A + e,. (1b) The esimae of d is around.. These esimaes imply he following annualized values for he coninuous ime parameers: k =.79, l =.48, and j =.43. Using hese parameers we can solve he second equaion for he seady sae value of he fracion of variable rae savings deposis o oal savings, ) D = 58.. 1 9 Noice ha including his erm does no invalidae he duraion analysis of he model, which is based on he gradual adjusmen paerns only. 1 The empirical average of D is.51
39 FRANK DE JONG AND JACCO WIELHOUWER 4. DURATION The previous secion showed ha he ineres rae paid on he accoun and he balance of savings accouns are relaed o he discoun rae. Therefore, he discoun rae sensiiviy of savings deposis will be differen from he discoun rae sensiiviy of a money marke accoun (which has a duraion of zero). In his secion, we sudy he sensiiviy of he ne asse value of a savings accoun o a parallel shif in he pah of he discoun raes. We sudy a shif from he original pah R() o R () + DR, and evaluae he derivaive in DR =. Wih some abuse of noaion, we will wrie he resuling expressions as V/ bu i should be kep in mind ha his refers o a parallel shif in he pah of discoun raes. This approach is close o a radiional duraion analysis, see e.g. Bierwag (1987), bu we ake ino accoun he dependence of fuure cash flows on discoun raes. In he iniial siuaion, he deposis are a heir equilibrium value D ). Differeniaion of he ne asse value wih respec he discoun rae gives V () 3 3 = E ;-# se 6R() s - i() s @ D() s ds + # e D -Rs -Rs 6Rs ()- is ()@ Dsds () 3 Rs Ds + e - ] g # 6 R() s -i] sg@ R ds F (13) The hree componens of his expression can be inerpreed as follows: 1. he ineres rae sensiiviy of he expeced discouned profis;. he change in he margin on he expeced fuure balances; 3. he expeced margin imes increases or decreases in he balance of he deposi. Noice ha if he fuure balances do no change as a resul of he ineres rae change, and if he margin is consan, only he firs erm (he sensiiviy of he presen value of he profis) remains. The second and hird erm are specific for savings accouns wih heir slow adjusmen of he ineres rae and balance, and are herefore he mos ineresing for our analysis. We shall now discuss he duraion of he accouns given he specific model for he evoluion of ineres raes and balances. Assume again ha R]g s = Ris consan, and ha he iniial siuaion is in equilibrium, D]g = D ) and i() = R- n. Under hese iniial condiions, he developmen of he ineres raes and he balance can be derived from equaions (8a) and (8b): -l l( s - ) l( s - ) R- i() = R-e i( )-l# e 6 R- n@ ds-v # e dw() s ( ) 1 # 1 l s - = n-v e dw(), s 1 1 # # ) s m( s ) ( u s) ( s ) hv1 - l - m # 1 v - # # ) m( s - ) m( s - ) D () = D + h e 6 R] sg- n- i] sg@ ds+ v e dw] sg (14a) = D + e e dw]g u ds+ e dw (). s (14b)
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS 393 Assuming ha he sochasic pars of he ineres rae and he balance are uncorrelaed, i.e. Cov(dW 1 (), dw ()) =, and noicing ha he parial derivaives (9a) and (9b) are non-sochasic, we can work ou he parial derivaive of he value: V () 3 D Rs =- se - E R -i() s D s ds # " 6 @ ]g, Rs R i() s Rs Ds + 3-6 - @ e E Ds () ds+ 3 - ] g # e E R-i s ds 6 @ # 6 ] g@ Rs Rs s Rs =- 3 ) se nd ds + 3 ) e e D ds - 3 -m -l - - -l - e nh e - e # # # d nds l- m n () (). R D 1 hn =- R D 1 1 + + l - b - l - m R + m R + l l (15) Wih an increase in he discoun rae, he firs erm reflecs he loss of value of he (perpeual) profi margin, he second erm he discouned value of he ineres paymens no made on he original balance during he ime he ineres rae paid on he accoun ^i ] gh is below he discoun rae minus he profi margin ^R - nh, and he hird erm he discouned value of he profi foregone on he balance ouflows. We can ransform his change of value o a duraion measure if we assume ha iniially, he ne asse value equals V D () = D() n R VD () Dur 1 1 1. R V () R R R h =- = - 1 R D() R R R n + l + - l- m b + m + l l (16) D The firs erm reflecs he duraion of a perpeuiy, and is deermined by he presen value of he profis in he seady sae. The second erm reflecs he value of he lower ineres raes paid on he exising balance, and is always negaive. The hird erm is he duraion of he profis on he addiional balance ouflows, and is posiive under he assumpion l> m. Especially when he margin m is hin and he ne asse value is low, he second erm may dominae he oher erms, leading o a negaive duraion for he ne asse value of a savings accoun. In ha case, an increase in he discoun rae will increase he ne asse value because for some ime he ineres rae paid on he savings accoun is lower han reurn on he asses deposied. As an illusraion Figure 3 shows he duraions as a funcion of he discoun rae R and he margin m (he oher parameers are pu equal o he esimaes of he previous secion). We see ha he duraion is ypically posiive, excep for low values of m, and declines wih he discoun rae. Mos of his effec is due o he duraion of he discouned profi margin, 1/R. Leaving ou his erm, we find he exra duraion of he ne asse value induced by he sluggish adjusmen paern. Figure 4 shows hese measures. Ineresingly, he exra duraion is always negaive, bu converges o zero for relaively big profi margins m.
394 FRANK DE JONG AND JACCO WIELHOUWER FIGURE 3: Duraion of savings deposis This figure shows he duraion (in years) of savings deposis as a funcion of he discoun rae (R) and he profi margin m. Figure 4: Duraion of savings deposis (excluding profi margin) This figure shows he exra duraion (in years) of savings deposis, in excess of he duraion of a perpeuiy (1/R), as a funcion of he discoun rae (R) and he profi margin m.
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS 395 5. HEDGING In his secion we consider he problem of hedging he ne asse value. Given he liabiliy value L D of he variable rae savings accouns, one can hedge he ne asse value by immunizaion. For simpliciy we assume he money deposied can be invesed in wo insrumens, Long Invesmens (LI) and Shor Invesmens (SI). The balance shee of he bank hen becomes V LI V SI L D V D where V D denoes he Ne Asse Value. We now consider he consrucion of an invesmen porfolio where he ineres rae risk on he ne asse value is fully hedged, i.e. he ne asse value V D is no sensiive o he parallel shifs in he discoun rae. From he balance shee we see ha his requires V V L + =. (17) SI LI D Of course, he soluion o his equaion, and hence he composiion of he invesmen porfolio, depends on he duraions of he shor and long invesmens. As a simple example, consider he case where he shor insrumen has zero duraion. In ha case he invesmen in he long insrumen is deermined by VLI LD =. (18) L We can find D from equaions (4) and (13). As an illusraion, Figure 5 graphs he required posiion in long (1 year mauriy) bonds in he hedge porfolio for differen value of R and m. As seen before, he duraion of variable rae savings accouns may be negaive, in paricular when he profi margin m is fairly small. In ha case he bank can hedge he accouns by aking a long posiion in long invesmens. Bu if Dur is posiive, which happens for example when he profi margin m is fairly high, one should ake a shor posiion in he long asse. Alernaively, if one does no like o ake shor posiions in bonds, one could use derivaive insrumens such as caps, which ypically have a negaive duraion, or forward conracs. 6. CONCLUSION This paper focuses on he valuaion and ineres rae sensiiviy of variable rae savings accouns. The duraion can be spli in hree differen effecs: he duraion of he expeced discouned profis; he change in margin on expeced fuure balances due o a change in ineres rae; he expeced margin imes increases or decreases in he balance of he accoun.
396 FRANK DE JONG AND JACCO WIELHOUWER Figure 5: Hedge porfolio This figure shows he posiion in long bonds (duraion 1 years) in he hedge porfolio of a 1 deposi, as a funcion of he discoun rae R and he profi margin m. The firs elemen is he sandard duraion for producs wihou embedded opions. The second and hird erm are non-sandard (for example, hey are zero for a money marke accoun) and arise due o he variable ineres rae paid on he accoun and he opion of he cliens o wihdraw and inves in he accoun a any ime. The duraion herefore crucially depends on he rapidness of he adjusmen of he ineres rae paid on he accoun o discoun rae changes and on he reacions of he cliens. These reacions will principally be deermined by he cliens ineres rae sensiiviy and by he marke efficiency. The models are esimaed for he Duch savings accoun marke. Duraion curves are given for differen margins. When hedging he savings deposis, one can consruc a porfolio wih he same duraion as he variable rae savings accouns. However, when one does no wan o go shor ino a cerain asse class, one migh need o include derivaives (for example caps) o hedge hese producs, since i is possible o have negaive duraions. The inuiion is ha an ineres rae increase migh lead o a fligh of cliens o money marke accouns. So buy insurance when money marke accouns are less aracive, which resul in profis when ineres raes spike up (he insurance pays ou). The gain due o he caps in an increasing ineres rae environmen hen offses he loss in he savings accouns. Hedging in his way cerainly smoohens he resuls on hese producs. Of course his can be achieved by going shor in long asses as well.
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS 397 For fuure research i migh be ineresing o analyze he second order effecs. Then muliple immunizaion can be achieved wih a porfolio wih hree asse classes. Finally, i is possible o make he discoun rae a funcion of a number of ineres raes wih differen mauriies. This will of course increase he complexiy of he model bu allows for he calculaion of key-rae duraions. REFERENCES BIERWAG, G.O. (1987) Duraion Analysis, Ballinger, Cambridge MA. DAVIDSON, J., HENDRY, D.F., SRBA, F. and YEO, S. (1978) Economeric Modelling of he Aggregae Time Series Relaionship beween Consumer Expendiure and Income in he Unied Kingdom, Economic Journal, 88, 661-69. HEATH, D., JARROW, R. and MORTON, A. (199) Bond pricing and he erm srucure of ineres raes: A new mehodology for coningen claims valuaion, Economerica 6, 77-16. HO, T.S.Y. and LEE, S.-B. (1986) Term srucure movemens and he pricing of ineres rae coningen claims, Journal of Finance 41, 111-19. HULL, J. (1993) Opions, Fuures and oher Derivaive Securiies, second ediion, Prenice-Hall. HUTCHISON, D.E. and PENNACCHI, G.G. (1996) Measuring Rens and Ineres Rae Risk in Imperfec Financial Markes: The Case of Raail Bank Deposis, Journal of Financial and Quaniaive Analysis 31, 41-417. JANOSI, T., JARROW, R. and ZULLO, F. (1999) An Empirical Analysis of he Jarrow-van Devener Model for Valuing Non-Mauriy Demand Deposis, Journal of Derivaives, Fall 1999, 8-31. JARROW, R.A., and VAN DEVENTER, D.R. (1998) The arbiarge-free valuaion and hedging of savings accouns and credi card loans, Journal of Banking and Finance, 49-7. SELVAGGIO, R.D. (1996) Using he OAS Mehodology o Value and Hedge Commercial Bank Reail Demand Deposi Premiums, Chaper 1 363-373. VASICEK, O. (1977) An equilibrium characerizaion of he erm srucure, Journal of Financial Economics 5, 177-188. FRANK DE JONG Finance Group Universiei van Amserdam Roeerssraa 11 118 WB, Amserdam he Neherlands Phone: +31--555815 Fax: +31--55585 Email: f.c.j.m.dejong@uva.nl