GENERALIZED DURATION MEASURES IN A RISK IMMUNIZATION SETTING. IMPLEMENTATION OF THE HEATH-JARROW-MORTON MODEL. Alina Kondratiuk-Janyska 1

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1 GENERALIZED DURATION MEASURES IN A RISK IMMUNIZATION SETTING. IMPLEMENTATION OF THE HEATH-JARROW-MORTON MODEL Alina Kondraiuk-Janyska 1 Marek Kaluszka 2 Applicaiones Mahemaicae 33,2 26), Absrac. The aim of his paper is o se differen lower bounds on he change of he expeced ne cash flow value on ime H > in general erm srucure models referring o he sudies of Fong and Vasiček 1984), Nawalkha and Chambers 1996), and Balbás and Ibánez 1998) among ohers. New immunizaion sraegies are derived wih new risk measures: generalized duraion and M-Absolue of Nawalkha and Chambers and exponenial risk measure. Furhermore, examples of specific one-facor HJM models are provided and he problem of immunizaion is discussed. MSC: 62P2, 91B28 Keywords: Asse-liabiliy porfolio; Immunizaion; Duraion; M-Absolue; One-facor HJM models 1. Inroducion Bondholders are subjec o ineres risk caused by changes in ineres raes. Therefore he problem of bond invesmen immunizaion agains ineres risk is an imporan issue o bond porfolio managers and researchers. This problem has persised in he lieraure since he Macaulay s definiion of duraion 1938) and i is shown independenly by Samuelson 1945) and Redingon 1952) ha if he Macaulay duraion of asses and liabiliies are equal, he porfolio is proeced agains a local parallel change in he yield curve. Fisher and Weil 1971) formalize he radiional heory of immunizaion defining he condiions under which he value of an invesmen in a bond porfolio is hedged agains any parallel shifs in he forward raes. The main resul of his heory is ha Dae: 19h June Cener of Mahemaics and Physics Technical Universiy of Lodz, Al. Poliechniki 11, Lodz, Insiue of Mahemaics Technical Universiy of Lodz, ul. Wolczanska 215, 93-5 Lodz, Poland, E- mail: akondra@p.lodz.pl 2 Insiue of Mahemaics Technical Universiy of Lodz, ul. Wolczanska 215, 93-5 Lodz, Poland, kaluszka@p.lodz.pl

2 2 immunizaion is achieved if he Fisher-Weil duraion of he porfolio is equal o he lengh of he invesmen horizon see Rządkowski and Zaremba, 2). Unforunaely, his radiional approach has serious limiaion since i implies arbirage opporuniy inconsisen wih he rules of modern finance heory. To overcome i, he pioneer work of Fong and Vasiček 1984) indicaes new direcion in sudying immunizaion. They propose o deermine a lower bound of he changes in a porfolio value which leads o a risk conrolling sraegy. Nawalkha and Chambers 1996), Balbás and Ibáñez 1998), Balbás e al. 22), Nawalkha e al. 23) and Kaluszka and Kondraiuk-Janyska 24) follow heir approach immunizing a single liabiliy. However, in realiy invesors have o deal wih muliple liabiliies see Hürlimann, 22) under muliple shocks in he erm srucure of ineres raes TSIR for shor). Moreover, here is a demand for research considering porfolio immunizaion under sochasic duraion in he case of a sream of liabiliy ouflows, as over he las decade one of he cornersones of ineres risk managemen is modeling he sochasic behavior of ineres raes. Senay Agca 22) invesigaes empirically classical and sochasic duraions bu does no discuss any porfolio value in connecion e.g. wih a sochasic duraion. To our knowledge he firs o sudy he lower bounds of he ne presen porfolio value under he above condiions was Gajek 25) see also Gajek and Osaszewski, 24). He considered he hedging problem a ime under random changes of he basic TSIR corresponding o e.g. a supermaringale-like shif facors srucure for an insurance companies. The derived lower bounds include asse and liabiliy duraions as a risk measure. We do no generalize Gajek s 25) novel resuls bu inspired by his aricle we focus on immunizing muliple liabiliies formulaing he problem from a differen sandpoin see also Kondraiuk-Janyska and Kaluszka, 26), namely of proeced fixed income asse inflows bonds) and random liabiliy ouflows. The aim of his paper is o se differen lower bounds on he change of he expeced ne cash flow value difference beween asse and liabiliy sream) on ime H > which is called a rebalanced ime in general erm srucure models. New immunizaion sraegies are derived wih new risk measures like generalized duraions Proposiion 3) or M-Absolue of Nawalkha and Chambers Proposiion 1) and a compleely new risk measure Proposiion 2). The remainder of his paper is organized as follows. Secion 2 defines a generalized duraion measure and presens examples of he sochasic and polynomial duraions as paricular cases. Secion 3 gives he noaion and assumpions. Secions 4 presens immunizaion sraegies based on single-risk measure or muliple-risk measure models. Secion 5 provides examples of specific one-facor HJM models and discusses he problem of immunizaion.

3 3 2. Generalized duraion measures Duraion is unquesionably he mos widely used risk measure wih a long hisory. We inroduce a generalized duraion wih respec o a fixed funcion γ = γ), where paricular γ funcions yield well-known duraions ge from differen models of ineres raes behavior, eiher deerminisic or sochasic. Define D A γ) = γs)dsda), 1) where A) is an accumulaed value of asses described precisely in he nex Secion. In he case γ) 1, generalized duraions become a classical duraion. When γ) = k, k 2, we obain higher order duraion risk measures derived from polynomial models;see e.g. Chambers e al. 1988), Prisman and Shores 1988), Rządkowski and Zaremba 2). On he oher hand, generalized duraions appear in sochasic models of insananeous forward raes behaviors wih appropriae γ funcions. The mos popular arbirage-free model for describing erm srucure is he Heah, Jarrow and Moron 1992) model. This is a very general and popular approach due o is flexibiliy o he number of random facors used and differen volailiy srucures ha can be assumed for differen mauriy forward raes. When here is one source of randomness, muli-facor HJM model becomes a one-facor. Mos of he available shor rae models are specific cases of one-facor HJM models. Given ha shor rae models are relaively simple compared o heir muli-facor counerpars and ha abou 9% of he variaion in he yield curve can be explained by only one facor see Lierman and Scheinkman, 1991), we focus on one-facor HJM models, where he evoluion of he insananeous forward rae on [, T ] is specified by he following sochasic process: df, T ) = α, T, ω)d + σ, T, ω)dw, where T, α, T, ω) is he insananeous forward rae drif funcion, σ, T, ω) is he insananeous forward rae volailiy funcion and W is he Brownian moion on a probabiliy space Ω, F, P) equipped wih a filraion F = F ) T. Au and Thurson 1995) and Munk 1999) derive he duraion measures of cerain coninuous ime onefacor HJM models generalizing he previous Cox, Ingersoll and Ross 1979) resul. Denoe by P, m) he ime- price of a zero-coupon bond mauring a ime m and paying one uni, m T ). By B, T ) we mean he ime- price of a bond porfolio wih coupons C 1,..., C n a daes 1,..., n, where i T for i = 1,..., n. In a classical approach duraion is a measure of he proporional percenage in a bond s price due o shifs in he erm srucure. Basing on i, Au and Thurson 1995) use he

4 4 definiion of duraion a ime which is a special case of 1) which yields D HJM = B, T ) f, ) /B, T ) ni=1 C i P, i ) i σ, s, ω)ds n D HJM = / C i P, i ). σ,, ω) i=1 Some examples are given below for differen volailiy funcions in one-facor HJM models: Consan volailiy σ, T, ω) = σ, Meron, 1973, Ho-Lee, 1986), ni=1 C i i )P, i ) D HJM = ni=1. C i P, i ) Exponenially decaying volailiy σ, T, ω) = σe bt ), Vasiček, 1977), ni=1 C i P, i ) 1 ) e b i ) D HJM = b n i=1 C i P, i ) Consan decay volailiy σ, T, ω) = 1+T, Au and Thurson, 1995), ni=1 C i P, i ) ln 1 + i ) D HJM = ni=1. C i P, i ) σ 1+T Consan mauriy σ, T, ω) =, Au and Thurson, 1995), ) ni=1 C i P, i ) ln 1+i 1+ D HJM = Ti=1. C i P, i ) σ. Sochasic volailiy Cox, Ingersoll and Ross, 1985), σ, s, ω) ds = 2σ f, ) sinhγt )) 2γ coshγt )) + b sinhγt )), σ,, ω) = σ f, ), D HJM = T 2C i P, i ) sinhγ i )) T 2γ coshγ i )) + b sinhγ i )) / C i P, i ), i=1 where 2γ = b 2 + 2σ 2 and b, σ >, sinhx) = ex e x 2, coshx) = ex +e x 2. Alhough a class of duraion measures for he HJM ineres rae model has been consruced Cox, Ingersoll and Ross, 1979, Au and Thurson, 1995, Munk, 1999), here are no sudies deermining he lower bound of he change in a bond porfolio value in his model. Therefore, he aim of his paper is o inroduce arbirage-free models seing differen lower bounds on he change of he expeced ne cash flow value difference beween asse and liabiliy sream) where a generalized duraion is an immunizaion measure. i=1

5 5 3. Preliminary noaions Denoe by [, T ] he ime inerval wih = he presen momen, and le H be an invesor planning horizon, < H < T, when he porfolio is rebalanced. The porfolio consiss of bond inflows A occurring a fixed ime T = 1, 2,..., d ) o cover muliple liabiliies L due a daes T = 1, 2,..., d, ), where d = T. This is a ypical siuaion e.g. when an insurance company has o discharge is random liabiliies and invess he money in acquiring an immunized bond porfolio. Denoe he se of available bonds by A. Generally, his is an arbirary subse of [, ) d ha migh be nonconvex since we do no assume ha he marke is complee and bonds are infiniely divisible. Addiionally, we assume ha liabiliies are nonnegaive random variables. Consequenly, N = A L is he ne cash flow a ime. By f, s) we mean an insananeous forward rae over he ime inerval [, s] and herefore we can wrie ha invesing 1 a ime in a zero coupon-bond we ge exp s f, u)du) a ime s. The se of insananeous forward raes {f, s) : < s} deermines a random erm srucure of ineres raes. Hence ) a = A exp H f, u)du is he ime-h value of A, ) l = L exp H f, u)du is he ime-h value of L, n = a l is he ime-h value of ne worh, A) = s a s is an accumulaed value of asses, L) = s l s is an accumulaed value of liabiliies, N) = A) L), V ) = E n = ENT ) is he ime-h average value of he porfolio of asse and liabiliy flows if forward raes equal fuure spo raes. A decision problem of an invesor is o design he sream of bonds o cover a sream of liabiliies. If among available bonds here are such ha N = for all, hen he porfolio is immunized. In realiy, he marke is incomplee, which excludes an ideal adjusmen asses o liabiliies. An invesor consrucing a bond porfolio mees wo kinds of risks: reinvesmen and price. The firs one is conneced wih he way of reinvesing coupons paid before an invesmen horizon. The oher appears by pricing bonds before heir expiry daes. Since a porfolio value a ime H depends on he reinvesmen sraegy, we require he following open-loop sraegy: a) If < H hen he value of N a ime H is equal o ) H N exp f, s)ds. Tha means ha if N = A L > for < < H, an invesor purchases H )-year srip bonds. Oherwise, he sells shor H )-year srip bonds.

6 6 b) If > H, he value of N a H equals ) ) H N exp fh, s)ds = N exp fh, s)ds, H which means ha a ime H he porfolio priced according o he TSIR is sold by he invesor. As a consequence, he value of he ne cash flow a H equals ) H N exp f H, s)ds = n exp k)), where k) = H [f H, s) f, s)] ds 2) is a shock in he insananeous forward rae and a b = mina, b). From he invesor s sandpoin, he average ime-h value of N under he scenario a)-b) is given by ) T V k) = E exp k)) dn). 3) The classical immunizaion problem is o find a porfolio such ha V k) V ) for all k K, where K sands for a feasible class of shocks. Our aim is o find a lower bound on inf k K V k) which is dependen only on bond porfolio proporions. Nex, we selec a = a porfolio among available bonds on he marke such ha his lower bound is maximal. M-Absolue as a Risk Measure 4. Risk measure models The linear cash flow dispersion measure he M-Absolue defined by Nawalkha and Chambers 1996) H da) M NCh = da) is an immunizaion risk measure designed o build immunized bond porfolios in he case of a single liabiliy. In he case of muliple liabiliies, define he generalized M- Absolue of Nawalkha and Chambers by M = A) AT ) + ELT ) L)) d. I is easily seen ha M = AT )M NCh in he case of a single nonrandom liabiliy a ime H. The following assumpions will be needed hroughou he paper: A1. A random variable l is independen from he TSIR for every >. A2. Ee k)) is coninuous on [, T ].

7 7 Define D A γ) = γs)dsda), D L γ) = E γs)dsdl) as he generalized duraions of asses and liabiliies, respecively, and γ = γ) is a fixed funcion. Proposiion 1. Under assumpions A1 A2, a lower bound on he pos-shifs change in he value of he ne cash flow a H is given by where K 1 = inf V k) V ) km + D A γ) D L γ), 4) k K 1 { } k ) : Ee k)) γ) k for all [, T ], wih k being a nonnegaive number. Proof. From assumpion A1, we ge V k) = [ E n e k)] = = Ee kt ) ENT ) = = + ENT ) EN)) Since ENT ) = V ), for all k ) K 1 we have V k) V ) k as desired. + Ee k) den) EN) Ee k)) d Ee k)) d + ENT ) ENT ) EN)) Ee k)) ) γ) d γ) ENT ) EN)) d + ENT ). EN) ENT ) d + γs)dsden), γs)ds ENT ) EN)) As a corollary of Proposiion 1 we ge he following immunizaion sraegy: min A) A AT ) A) + E L) LT )) d. subjec o D A γ) D L γ) = d, where d is a fixed nonnegaive value of a duraion gap. T

8 8 Example 1. Suppose ha hree kinds of zero coupon bonds are available on he marke. The face value of he bond a he mauriy dae = 1, 2, 4 is B and an invesor is o discharge fixed liabiliies P a = 3, 5. Take T = 5 and le he planning horizon be H = 3. The ime-3 value of B and P is denoed by b and p, respecively. Consider he siuaion when an expendiure-income plan is such ha N5) =, γ) γ and d is a nonnegaive real number. Denoing by x he amoun of purchased -year bond unis, he immunizaion problem should be solved due o he model: min x s b s p s ) d 5) x ) s subjec o x b = p, γ x b p ) = d, b for = 1, 2, 4. ) ) Solving problem 5) we obain x 1 =, x 2 = 1 2b 2 p 3 p 5 d γ, x 4 = 1 2b 4 p 3 + 3p 5 + d γ under he condiion p 3 p 5 + d γ. If p 3 < p 5 + d γ, hen he se of consrains is empy. ) If we ake γ) = γ, we ge x 1 =, x 2 = 1 7 b 2 12 p p 5 1 d 6 γ, x 4 = 1 5 b 4 12 p p when p 3 2 d 7 γ 9 7 p 5. Oherwise, he se of consrains is empy. Exponenial Risk Measure In his subsecion we presen a lower bound on he change of he expeced ne cash flow value based on an exponenial risk measure. Le us inroduce an enropy funcion Hf) = f) ln f)d f)d ln f)d). Proposiion 2. Le assumpions A1 A2 hold. Then ) T inf V k) V ) k 1 k 2 ln e ENT ) N)) d + D A γ) D L γ), 6) k K 2 where K 2 = { ) k ) : H Ee k )) γ ) k 1, k 2 are nonnegaive numbers. Proof. By he proof of Proposiion 1 we ge V k) V ) = + E NT ) N)) Ee k)) d } Ee k)) γ) d k 2, k 1 and E NT ) N)) Ee k)) γ) d γ) E NT ) N))). ) d γ

9 9 Applying he Young inequaliy we obain V k) V ) H Ee k )) ) γ ) ) Ee k)) T γ) d ln e ENT ) N)) d + γ) ENT ) EN)) d which complees he proof of 6). Inequaliy 6) implies he following immunizaion problem: find a porfolio which minimizes exp E NT ) N)) d 7) subjec o D A γ) D L γ) = d, where d is a fixed nonnegaive value of a duraion gap. Example 2. Under he assumpions like in Example 1 in he case γ) γ, sraegy 7) leads o he following opimizaion problem min exp x s b s p s ) d 8) x ) s subjec o x b = p, γ x b p ) = d, b for = 1, 2, 4. Solving problem 8) we obain he same resuls as in he previous Example, i.e x 1 =, ) ) x 2 = 1 2b 2 p 3 p 5 d γ, x 4 = 1 2b 4 p 3 + 3p 5 + d γ under he condiion p 3 p 5 + d γ. If p 3 < p 5 + d γ, hen he se of consrains is empy. Generalized Duraion as a Risk Measure The appropriae sraegy would be o hold a porfolio of asses whose schedule of cash flow covered he paern of liabiliies under a consan TSIR. Thus, i is worh considering immunizaion among porfolios saisfying weak version of he Axiom of Solvency Gajek, 25): A = {A ) : A) E NT ) + L)) for all [, T ]}. 9)

10 1 Proposiion 3. Under assumpions A1 A2 and for all A ) A, where K 4 = inf V k) V ) D A γ) D L γ), 1) k K 4 { } k ) : Ee k)) γ) for all [, T ]. Proof. By he proof of Proposiion 1 we ge V k) V ) = which complees he proof. ENT ) EN)) Ee k)) d ENT ) EN)) γ)d, As a consequence of Proposiion 3 we obain he sraegy: find a porfolio which maximizes D A γ) D L γ) subjec o A) E NT ) + L)) for all [, T ]. 5. Non-sandard selecion of γ funcions in sochasic models To apply Proposiions 1-3 we need a γ funcion. Obviously, one may ake γ) = 1 or γ) = 2 for T, which gives well-known radiional duraion or convexiy, respecively. Bu such a choice is jusified only if we canno modeled he TSIR because of he lack of daa or an unexpeced even. In he case when we use sochasic models of he TSIR, he main quesion is wha a shock concerns. We will say ha a shock appears when a model is incorrecly fied o he realiy or when he assumed model parameers differ from real ones. According o he above remark we require in Proposiion 1 his deviaion o be wihin a band of widh k. Hence and by he definiion of K 1 we conclude ha 1. If k =, hen a model is perfecly fied and Ee k)) = γ) for T. 2. If k >, he model under assumed parameers is unadjused and k measures he deviaion size of γ ) around he real unknown Ee k )). Therefore we propose o ake γ funcion such ha Ee k)) = γ) for T. In paricular, in he HJM model we see ha is given by Ee k)) which is no a consan funcion

11 in he Meron model, where f, ) = r + a + σw, and r, a, σ are posiive consans, we have Ee k)) = σ 2 2 σ H) 2 2 H) 2) exp σ 2 6 H) 2 + 2H H) 2) exp σ H) H) 2 + H 3 3)) for H H) 3 + 3H H) 2 + H 3 3)) for > H. in he he Vasiček model described by df, ) = a bf, ))d + σdw where a, b, σ are posiive consans, we ge σ 2 2b 3e bh ) + 2e 2bH ) e bh+) + 2e b) 2 )) exp σ 2 2b 3e bh ) + e 2bH ) + e bh+) + 2e bh e b ) e 2bH Ee k)) = σ 2 2b e b H) e bh+) + 2e b) 2 )) exp σ 2 2b e b H) + e bh+) + 2e bh e b ) e 2bH 1 3 In he above models W is a one-dimensional sandard Brownian moion under he spo maringale measure P. This is our suggesion of a γ selecion differen from sandard funcions derived in sochasic models see Au and Thurson, 1995, Munk, 1999). The comparison of heir effeciveness demands a huge empirical research which exceeds he scope of his paper. for H for > H. References Agca, S. 22) The performance of alernaive ineres rae risk measures and immunizaion sraegies under a Heah-Jarrow-Moron framework. Par of disseraion work available a hp://scholar.lib.v.edu/heses/available/ed / unresriced/senayagcadisseraion.pdf. Au, K. T., Thurson, D. C. 1995) A new class of duraion measures. Economics Leers 47, Balbás, A., Ibáñez, A. 1998) When can you immunize a bond porfolio? Journal of Banking & Finance 22, Balbás, A., Ibáñez, A., López, S. 22) Dispersion measures as immunizaion risk measures Journal of Banking & Finance 26,

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