Macaulay durations for nonparallel shifts

Transcription

1 Ann Oper Res (007) 151: DOI /s Macaulay duratons for nonparallel shfts Harry Zheng Publshed onlne: 10 November 006 C Sprnger Scence + Busness Meda, LLC 007 Abstract Macaulay duraton s a well-known and wdely used nterest rate rsk measure. It s commonly beleved that t only works for parallel shfts of nterest rates. We show n ths paper that ths lmtaton s largely due to the tradtonal parametrc modellng and the dervatve approach, the Macaulay duraton works for non-parallel shfts as well when the non-parametrc modellng and the equvalent zero coupon bond approach are used. We show that the Macaulay duraton provdes the best one-number senstvty nformaton for non-parallel nterest rate changes and that a Macaulay duraton matched portfolo s least vulnerable to the downsde rsk caused by non-parallel rate changes under some verfable condtons. Keywords Macaulay duraton. Non-parallel shfts. Immunzaton. Lnear programmng AMS Classfcaton 65K10. 90C90 Macaulay duraton s a well-known nterest rate rsk measure for a portfolo of regular bonds (optons-free and default-free). It measures the percentage change n portfolo value due to the nstantaneous change n nterest rates. It also defnes a portfolo mmunzaton horzon over whch the portfolo value remans mmunzed from an nstantaneous shock n nterest rates. A smple strategy n mmunzaton s to match the present value and the Macaulay duraton of an asset portfolo wth those of the lablty. The portfolo s then mmunzed to nstantaneous small parallel movements n yelds (the whole yeld curve s shfted up or down by the same ncrement). A dynamc strategy (contnuous rebalancng) s needed to keep the portfolo mmunzed aganst small parallel shfts. The man crtcsm to the Macaulay duraton s that t s vald only to small parallel shfts. Much effort has been made to extend t to non-parallel shfts. Some other duratons have been suggested, for example, log-stochastc process duraton (Khang, 1979) when shortterm rates are more volatle than long-term rates, key-rate duraton (Ho, 199) when there H. Zheng ( ) Department of Mathematcs, Imperal College, London SW7 BZ, Unted Kngdom e-mal: h.zheng@mperal.ac.uk Sprnger

2 180 Ann Oper Res (007) 151: are several key rates whose changes determne changes of other rates, multdmensonal duraton (Rzadkowsk and Zaremba, 000) when several factors affect changes of the term structure, etc. These duratons are defned for more general patterns of rate changes and one would expect they perform better n mmunzaton than the Macaulay duraton does. However, ths s not the case n many emprcal tests, as ponted out by Joron and Khoury (1996, page 108): The Macaulay duraton was also compared to addtve, multplcatve, and log-multplcatve process duraton. Somewhat unexpectedly, they reported lttle dfference among duraton strateges and concluded that the smplest Macaulay duraton provdes the most cost-effectve mmunzaton method. The common feature of those mentoned duratons s that nterest rate changes are specfed exogenously by some parametrc models. Duratons are defned smply as dervatves of bond prce wth respect to underlyng factors. The mmunzaton based on these duratons works well f rate changes do follow specfed models, but may perform poorly f they do not. Ths ntutvely explans why those more recent duratons do not have consstent better performance than the smple Macaulay duraton smply because t s unlkely one can accurately predct rate changes n practce. Fong and Vascek (1984) defne a measure of mmunzaton rsk, called M-squared, and show that a Macaulay duraton matched portfolo structured wth mnmum M-squared s less vulnerable to any nterest rate movements wth bounded slopes. Two questons reman unanswered n Fong and Vascek (1984): 1. How to choose an mmunzaton portfolo when a Macaulay duraton matched portfolo does not exst, and. Is a Macaulay duraton matched mnmum M-squared portfolo least vulnerable to non-parallel nterest rate changes? The reason for these questons s that n Fong and Vascek (1984) a portfolo s assumed to be Macaulay duraton matched and therefore no further effort s made on other possbltes. Nawalkha and Chambers (1996) suggest another mmunzaton rsk measure, called M- absolute, whch s vald for any portfolos wthout duraton constrants, and show that a portfolo structured wth mnmum M-absolute s less vulnerable to any bounded nterest rate changes. M-absolute of a bond s the weghted sum of tme dfferences of cash flows wth a specfed holdng perod. The optmal soluton of a mnmum M-absolute portfolo s unstable n the sense that t can easly choose bonds of very dfferent duratons wth a small change of data. Zheng, Thomas, and Allen (003) suggest an alternatve nterest rate rsk measure, called approxmate duraton, for regular bonds n order to address the model msspecfcaton rsk. The approxmate duraton measures the bond prce senstvty to rate changes wthout assumng pror any partcular patterns of rate changes. The mmunzaton based on approxmate duraton does not completely remove the nterest rate rsk, but t mnmzes the overall downsde rsk to any patterns of rate changes, whch s n the same sprt as that of Nawalkha and Chambers (1996). Vnter and Zheng (003) extend the approxmate duraton further to nstantaneous forward rates wth bounded measurable rate changes. The approxmate duraton s characterzed as the medan tme of the dscounted cash flows wth nonsmooth optmzaton. Ths paper dscusses the measurement and mmunzaton of a bond (or a portfolo of bonds) wth respect to Lpschtz (non-parallel) changes of nterest rates,.e., the rate of change s bounded. In practce rate changes are always Lpschtz when yeld curves are calbrated wth lnear splnes (Ho, 1997) or other smooth splnes from the observed market data. The obectve of ths paper s to answer the followng two questons: 1. Is Macaulay duraton the best one-number senstvty measure for Lpschtz changes of nterest rates? and. Is the Macaulay duraton matched mnmum M-squared portfolo least vulnerable to Lpschtz changes of nterest rates? The man contrbuton of the paper s to gve postve answers to Sprnger

3 Ann Oper Res (007) 151: both questons (under some condtons for the second one), whch establshes the Macaulay duraton as a good nterest rate rsk measure for non-parallel shfts as well and explans partly many apparent paradoxes n the emprcal lterature on duratons. The paper s organzed as follows: Secton revews the equvalent ZCB approach to studyng the senstvty of regular bonds and shows the Macaulay duraton provdes the best one-number senstvty nformaton when rate changes are Lpschtz. Secton 3 dscusses the mmunzaton wth the mnmum downsde rsk and characterzes the condtons under whch the Macaulay duraton matched mnmum M-squared portfolo s least vulnerable to non-parallel rate changes. The conclusons summarzes the man results and open questons. The appendx contans the proofs of theorems. 1 Nonparametrc duraton measures Assume that the term structure of nterest rates s flat wth rate r (contnuous compoundng) and that a bond has cash flows c at tme t, = 1,...,N, untl maturty t N = T. The bond prce at tme 0 s equal to P = N =1 c e rt and the Macaulay duraton s defned by D = N =1 t c e rt /P. 1 A standard method to derve the Macaulay duraton s to set D = P /P where P s the dervatve of P wth respect to r. Ths approach gves clear fnancal nterpretaton to the Macaulay duraton,.e., t ndcates the magntude of percentage prce changes to yeld changes. An alternatve method to derve the Macaulay duraton s to fnd an equvalent zero-coupon bond wth the same present value and nterest rate senstvty as the gven bond,.e., the face value F and the maturty D of the equvalent ZCB are determned from equatons P = P 0 := Fe rd and P = P 0. The soluton D s agan the Macaulay duraton. When there are several factors affectng rate changes, the two methods produce dvergent results. The dervatve method changes to the partal dervatve method and the duraton becomes a vector, whch mples the change of the term structure must be specfed parametrcally and the model rsk s nherent. The equvalent zero coupon bond method keeps the same sprt wth a generalzed equvalence relaton and the duraton s stll a sngle number. The frst approach s dscussed by Ho (1997), Rzadkowsk and Zaremba (000), etc. The second approach s nvestgated by Zheng, Thomas, and Allen (003), Vnter and Zheng (003). We frst revew the dea of the second approach to general rate changes. Assume the ntal term structure of forward rates f s gven, that s, f (t) s the nstantaneous forward rate (contnuous compoundng) at tme t, seen at tme 0. The present value of a bond s gven by P( f ) = c v(t ) where v(t) = e t 0 f (u)du s the dscount factor at tme t. If there s an nstantaneous shft of forward rates from f to f + g, where g s a functon defned on a space S wth a norm, then the new bond prce P( f + g) can be approxmated (the frst order Taylor expanson) by P( f ) + P ( f ; g) where P ( f ; g) = c v(t ) ( ) t g(u)du 0 Sprnger

4 18 Ann Oper Res (007) 151: s the drectonal dervatve of P at f n the drecton g. Consder now a ZCB wth face value F and maturty D. Its present value s and ts drectonal dervatve s P 0 ( f ; g) = P 0( f ) P 0 ( f ) = Fv(D) The ZCB s sad to be equvalent to the gven bond f and Dvde () wth (1) and set to get ( ) D g(u)du. 0 P( f ) = P 0 ( f ) (1) P ( f ; g) = P 0 ( f ; g) for all g. () w = c v(t )/P( f ) (3) t w g(u)du = D 0 0 g(u)du for all g whch s equvalent to equaton H(D) = 0 where t H(D) = max w g(u)du g 1 D (4) Note H(D) s the normalzed maxmum devaton of nterest rate senstvtes of the two bonds. Once D s derved F s computed from (1). If rate changes are parallel,.e., g s a constant functon, then H(D) = w t D and the soluton to H(D) = 0 s the Macaulay duraton D = w t. However, f rate changes are not parallel then there s no soluton to H(D) = 0 unless the coupon bond tself has only one cash flow. We need to generalze the equvalence from the relaton H(D) = 0to somethng else. A natural canddate s to make H(D) as close to zero as possble. We therefore call a ZCB wth face value F and maturty D equvalent to the gven bond f D s the optmal soluton to the problem: mnmze H(D) subect to D 0 (5) and F satsfes (1). Vnter and Zheng (003) dscuss, among some other fnance problems, the equvalence relaton n the space S = L [0, ) wth the norm g =sup x g(x). The optmal soluton D a to (5) s the medan tme of the dscounted cash flows,.e., D a = t 0 f < 0 w 0 w Sprnger

5 Ann Oper Res (007) 151: and 0 w > 0 w for some nteger 0 and D a s any number n the nterval [t 0, t 0+1] f 0 w = 0+1 w. The result mples that f we know nothng about rate changes (always measurably bounded) then the ZCB that best approxmates a coupon bond should have the maturty equal to the medan tme of the dscounted cash flows of the gven bond. However, the resultng ZCB s unstable n the sense that ts maturty can vary greatly even there are only small changes of cash flows. For example, f a bond has two cash flows of equal present values, one n one year and the other n ten years, then the equvalent ZCB has duraton of ether one year or ten years wth any slght tppng of balance of cash flows. Note that D a s ntrnscally equal to one of cash flow dates t 0. It s undecded only when 0 w = 0+1 w, but that relaton s transent snce the weghts w change contnuously as tme t passes by. The optmal soluton D a s the same as the approxmate duraton, an nterest rate rsk measure dscussed n Zheng, Thomas, and Allen (003) n comparng duraton-based mmunzaton strateges. In ths paper we study the equvalent ZCB when S s a Lpschtz functonal space,.e., f g S then there exsts a K > 0, called Lpschtz constant, such that g(x) g(y) K x y for all x, y 0. If g s contnuously dfferentable wth bounded dervatves, then g s Lpschtz wth K = g. The converse s not true, for example, lnear splnes are Lpschtz, but are not dfferentable. The next result shows that the Macaulay duraton s a stable nterest rate senstvty measure vald not only for parallel shfts but also for non-parallel shfts. Theorem 1. Let the change of the term structure g be a Lpschtz functon wth a D-dependent norm g D defned by g D = max{ g(d), K }. Let all cash flows of the underlyng bond be nonnegatve. Then the maxmum devaton H(D) s characterzed by H(D) = 1 w (t D) + w t D and the optmal soluton to (5) s the mean tme of the dscounted cash flows,.e., the Macaulay duraton D m = w t. Theorem 1 shows that the Macaulay duraton D m not only gves exact senstvty nformaton when rate changes are parallel but also provdes the best approxmaton of that nformaton when rate changes are not parallel as long as the slope of rate changes s not too steep. The Macaulay duraton D m s also stable,.e., small changes of cash flows result n small changes of the Macaulay duraton, ths s because D m s the mean tme, not the medan tme, of the dscounted cash flows. In practce we do not observe the nstantaneous forward rate curve f drectly, but we can easly construct t based on the observed market data to any degree of accuracy (dependng on the avalablty of the data). Let R(0) be the short rate at tme t 0 = 0 and R(t )bethe zero rate at tme t, = 1,...,N. 3 The relaton between R(t) and f (u), 0 u t,sr(t) = (1/t) t 0 f (u)du wth R(0) = lm t 0 R(t) = f (0). Assume a lnear splne (the same technque apples to other splnes) s used to construct the nstantaneous forward rate curve f from rates f (t ) at tme t, = 0, 1,...,N. We can set f (0) = R(0) and compute f (t ) from the recursve formula f (t ) = (t R(t ) t 1 R(t 1 ))/(t t 1 ) f 1 for = 1,...,N. The curve f and the change g constructed ths way are Lpschtz. Heath et al. (199) dscuss the rsk-neutral forward rate process modellng and ts applcatons n prcng nterest rate dervatves. They specfy the whole term structure of f (s, t), the nstantaneous forward rates at tme t seen at tme s t, and assume f (, t) s drven by (6) Sprnger

6 184 Ann Oper Res (007) 151: some dffuson processes for fxed t. Therefore wth probablty one the realzed path f (, t) s nowhere dfferentable, certanly not Lpschtz. However, the HJM forward rate model can not be drectly appled n mmunzaton context because mmunzaton s concerned wth the mpact of an unpredctable regme change from f to f + g at a fxed tme s, not the dynamc process f tself over the whole perod. Note that the tradtonal Macaulay duraton s well-defned for bonds wth postve or negatve cash flows but the Macaulay duraton n Theorem 1 s only defned for bonds wth nonnegatve cash flows. Ths dsparty s due to the dfferent dervaton technques: one wth dervatve (parallel shft, vald n two drectons) and the other wth mnmum devaton (any pattern, vald only n one drecton). Ths observaton mples that the short-sellng (negatve cash flows) s not allowed when the Macaulay duraton s used for Lpschtz (non-parallel) rate changes. Downsde rsk mnmzng mmunzaton Suppose a bond (or a portfolo of bonds) s to be held for a perod D. The value of the bond at tme D s FV( f ) = D t f c e (u)du = P( f )e D 0 f (u)du. In practce FV( f ) may represent the value of the lablty at tme D to be pad. If forward rates change from f to f + g, then the value of the bond changes to FV( f + g). The mmunzaton s to choose a bond (or a portfolo of bonds) such that FV( f + g) FV( f ) for any rate changes g. The next result provdes a lower bound for FV( f + g) and shows that the Macaulay duraton maxmzes the lower bound. Theorem. Let the change of forward rates g be a Lpschtz functon (non-parallel shfts). Then for any holdng perod D > 0 FV( f + g) FV( f ) g D H(D)FV( f ) for all g (7) where H(D) s gven by (6). Furthermore, f the change of forward rates g s a constant functon (parallel shfts) and D s the Macaulay duraton, then FV( f + g) FV( f ) for all g. Theorem shows that a Macaulay duraton matched bond mmunzes the nterest rate rsk only when rate changes are parallel. In general t s mpossble to have a bond (or a portfolo of bonds) such that FV( f + g) FV( f ) for all rate changes g. The best one can hope for s to make the loss (the downsde rsk), when t occurs, as small as possble. (7) shows that the percentage loss s bounded by g D H(D), the product of the magntude of nterest rate changes and the maxmum devaton of a bond wth a holdng perod D. To mnmze the downsde rsk one should choose a bond (or a portfolo of bonds) wth mnmum H(D) as g D s uncontrollable. If the holdng perod D s a decson varable, then the Macaulay duraton D m mnmzes H(D). If the holdng perod D s fxed as n mmunzaton, then a bond wth the smallest H(D) n comparson wth other bonds s lkely to be least vulnerable to the loss caused by non-parallel rate changes. Ths rases a natural queston as whether a Sprnger

7 Ann Oper Res (007) 151: Macaulay duraton matched bond (or a portfolo of bonds) s stll a good choce n controllng the downsde rsk. Fong and Vascek (1984) defne M = w (t D) as the tme varance of a bond wth a holdng perod D. They clam a Macaulay duraton matched portfolo has the mnmum exposure to nterest rate changes when M s mnmzed. We can recover the same concluson from Theorem under the weak assumpton (rate changes are Lpschtz, not contnuously dfferentable) and provde a better obectve functon H(D) than M n choosng a bond portfolo wth the mnmum downsde rsk. Assume that the lablty portfolo s a ZCB wth the present value V and the duraton D, and that the asset portfolo s made of N bonds wth a holdng perod D. Assume that there are x unts of bond whch has the present value P, the Macaulay duraton D, and the tme varance M, = 1,...,N. Then the present value V (x), the Macaulay duraton D(x), and the maxmum devaton H(x) of the asset portfolo are gven by V (x) = P x D(x) = H(x) = (1/) (P x /V (x))d (8) (P x /V (x))m + (P x /V (x))d D. To match the present values of the asset and lablty portfolos decson varables x must satsfy the relaton V (x) = V. Denote y = (x P )/V the proporton of bond n the whole portfolo for = 1,...,N. From (8) we have y = 1 and y 0 for all. The Macaulay duraton and the maxmum devaton of the portfolo are gven by D(y) = y D and H(y) = (1/) y M + y D D, respectvely. We can set up an optmal portfolo (n the sense of mnmum downsde rsk) by solvng the followng optmzaton problem mnmze subect to (1/) y M + y D D y = 1 and y 0,. (9) (9) can be easly formulated by an equvalent lnear programmng problem. Fong and Vascek (1984) choose an optmal portfolo by mnmzng the M of the portfolo,.e., mnmze (1/) y M subect to y D = D (10) y = 1 and y 0,. (9) s a better formulated optmzaton problem than (10) n two aspects: 1. The mnmum value of (9) s less than or equal to that of (10) because any feasble soluton to (10) s a Sprnger

8 186 Ann Oper Res (007) 151: feasble soluton to (9) wth the same obectve functon value, whch mples that a portfolo chosen wth (9) has smaller bound for the downsde rsk than that chosen wth (10), and the downsde rsk tself s lkely to be smaller as a result.. The exstence of the optmal soluton to (9) s guaranteed because the feasble regon s a nonempty compact convex set n R N, however, the feasble regon to (10) can be an empty set, whch mples that a portfolo can always be chosen wth (9), but not necessarly wth (10). The example after Theorem 3 llustrates these ponts. Next we specfy the condtons under whch optmzaton problems (9) and (10) are equvalent,.e., a Macaulay duraton matched mnmum M portfolo s the same as a mnmum maxmum devaton portfolo whch s least vulnerable to non-parallel rate changes. Theorem 3. The optmzaton problems (9) and (10) have the same optmal soluton f and only f the followng two condtons are satsfed: () f (1/)M k D k = mn{(1/)m D : = 1,...,N} then D k D; () f (1/)M k + D k = mn{(1/)m + D : = 1,...,N} then D k D. Furthermore, f ether (1/)M k D k (1/)M D for all and D k < Dor(1/)M k + D k (1/)M + D for all and D k > D, then the optmal soluton to (9) s y k = 1 and y = 0 for k, and the mnmum value of (9) s strctly less than that of (10). Example. Consder an asset portfolo of 3 bonds: bond 1 has 50% cash flow at tme 4 and the rest at tme 8 (n terms of present values), bond has 50% cash flow at tme 10 and the rest at tme 14, bond 3 has only one cash flow at tme 8. (a) Assume the holdng perod s D = 10. The Macaulay duratons and the tme varances of these bonds are gven by D 1 = 6, M1 = 0, D = 1, M = 8, and D 3 = 8, M3 = 4. We can check that mn{(1/)m D }=(1/)M D and D > D, and that mn{(1/)m + D }=(1/)M3 + D 3 and D 3 < D, therefore condtons () and () n Theorem 3 are satsfed. Problems (9) and (10) have the same optmal soluton y 1 = 0, y = 0.5, y 3 = 0.5 and the optmal value 3. In ths case a Macaulay duraton matched mnmum M portfolo s the same as a mnmum maxmum devaton portfolo. (b) Bonds 1 and are the same as those n (a) but bond 3 s replaced by a cash flow at tme 11. Assume the holdng perod s stll D = 10. We have D 3 = 11 and M3 = 1. We can check that condton () s not satsfed because mn{(1/)m + D }=(1/)M3 + D 3 and D 3 > D. Problem (9) has the optmal soluton y 1 = 0, y = 0, y 3 = 1 and the optmal value 1.5. Problem (10) has the optmal soluton y 1 = 0., y = 0, y 3 = 0.8 and the optmal value.4. In ths case a Macaulay duraton matched mnmum M portfolo s not as good as a mnmum maxmum devaton portfolo. (c) All three bonds are the same as those n (b) but the holdng perod s replaced by D = 14. The tme varances of these bonds are changed to M1 = 68, M = 8, and M 3 = 9. We can check that condton () s not satsfed because mn{(1/)m D }=(1/)M D and D < D. Problem (9) has the optmal soluton y 1 = 0, y = 1, y 3 = 0 and the optmal value 6. Problem (10) has no feasble soluton due to D < D for all. In ths case a Macaulay duraton matched mnmum M portfolo smply does not exst whereas a mnmum maxmum devaton portfolo s stll well defned. An mmunzed portfolo requres contnuous rebalancng to keep t mmunzed. Such a strategy s untenable when there are transacton costs n tradng bonds. One has to strke a balance between two conflctng obectves of mnmzng the transacton cost and of Sprnger

9 Ann Oper Res (007) 151: Table 1 Profts/losses (Volumes) of mmunzaton strateges Macaulay (6.60) (4.67) (8.30) (11.7) (6.97) (10.15) (13.6) (0.00) M-Absolute (6.55) (3.73) (1.99) (0.55) (0.76) (0.80) (0.84) (0.00) mnmzng the maxmum devaton. We can acheve ths by solvng the followng LP: mnmze subect to (1 λ) ( a y + λ (1/) y = 1 and y 0, ) y M + y D D where a s the transacton cost assocated wth bond and 0 λ 1 s a preference parameter. If λ = 0 then the obectve s to mnmze the transacton cost. If λ = 1 then the obectve s to mnmze the maxmum devaton. A famly of optmal solutons can be constructed by varyng parameter λ, whch s smlar to the Markowtz s mean-varance effcent fronter. We perform a smple emprcal test to compare the Macaulay duraton strategy and the M-absolute strategy wth the obectve of mnmzng the downsde rsk. The data used are the observed US Treasury bonds and STRIPS rates. The data source s the Wall Street Journal (NY edton) around February 15 from 1994 to 001. Each year sx new Treasury bonds (maturty n one, two, three, fve, ten, and twenty-fve years) are added to the selecton unverse. All bonds are optons free wth face value 100. Coupons are assumed to be pad annually for ease of calculaton. STRIPS rates are used as zero rates. Assume the holdng perod s seven years from February 1994 (maturty n February 001) and the target value s one mllon dollars. The portfolo s rebalanced n every February. Table 1 dsplays the profts/losses (000 s) and the number of bonds traded (000 s, n parentheses) of the portfolo each year. The Macaulay and M-absolute strateges are stable and have smlar performances. Note that the M-absolute strategy has much fewer transactons than that by the Macaulay duraton strategy from The reason s that n 1996 the lablty has fve years to maturty and there are fve year bonds avalable n the asset portfolo, the M-absolute strategy swtches ts bond holdng from bonds of other maturtes to those of fve years. After that only small adustment s needed at rebalancng tme. The portfolo has a bullet structure and can sgnfcantly save the cost f transacton costs are not neglgble or f the portfolo s frequently rebalanced. On the other hand, such a concentraton may have adverse effect f there s default rsk of underlyng bonds (a topc not dscussed n ths paper). 3 Conclusons In ths paper we show that the Macaulay duraton works well for non-parallel shfts of nterest rates. It provdes the best one-number nterest rate senstvty nformaton for Lpschtz rate changes. A Macaulay duraton matched mnmum M-squared portfolo has the mnmum downsde rsk under some easly-verfed condtons. These results are vald for regular bonds,.e., there s no uncertanty n tmng and amount of cash flows, whch mples that Sprnger

10 188 Ann Oper Res (007) 151: the Macaulay duraton may work well for regular Treasury bonds (optons-free and defaultfree). However, f a bond has an embedded opton (callable bond, etc) or has credt rsk (corporate bond, etc.) then one should be cautous n applyng the results dscussed n ths paper, especally n mmunzaton. Ths s because the optmal portfolo set up wth the help of (9) contans only one or two bonds and s subect to severe credt/opton rsk. More research s needed on the role of the Macaulay duraton (or effectve duraton) for opton-embedded credt rsky bonds. Appendx Proof of Theorem 1. Notce frst that t (g(u) g(d))du D t The above calculaton s vald for both cases D t,ord > t. We can now show nequalty of (6) as follows. D K (u D)du K (t D). (11) ( t w g(u)du D = ) t w (g(u) g(d) du + w g(d)(t D) D t w (g(u) g(d))du D + w g(d)(t D) K w (t D) + w t D g(d) 1 w (t D) + w t D. (1) The last nequalty s due to g D 1. To show nequalty of (6) we defne g(d) = sgn( w t D), where sgn(x) = 1f x > 0 and 1 fx < 0, and g(u) = (u D) + g(d). Then g s Lpschtz wth g D = 1. Now compute t D g(u)du to get t D g(u)du = 1 (t D) + g(d)(t D). Therefore Sprnger H(D) 1 w (t D) + w (t D)g(D) = 1 w (t D) + w t D. (13)

11 Ann Oper Res (007) 151: (1) and (13) mply (6). Note that H(D) n (6) s a nonsmooth convex functon and, n general, nonsmooth optmzaton s needed to fnd the mnmum soluton. However, t s easy to solve ths partcular problem. The frst term of (6) reaches the mnmum when D = w t and the second term s zero wth ths choce of D. Therefore the Macaulay duraton D m = w t mnmzes H(D). Proof of Theorem. A smple calculaton shows that FV( f + g) FV( f ) = w e D t g(u)du 1 + D w g(u)du. (14) t The last nequalty s due to e x 1 + x for any real number x and w 0 and w = 1 (see (3)). If the change of forward rates g s a constant functon,.e., g(u) = c for some constant c, then (14) mples FV( f + g) FV( f ) ( 1 + c D w t ). If the holdng perod D s chosen to be the Macaulay duraton D = w t then we have FV( f + g) FV( f ) for all constant functons g. If the change of forward rates g s a Lpschtz functon, then (14) and (11) mply that FV( f + g) FV( f ) 1 + w (D t )g(d) K w (t D) 1 w t D g(d) K w (t D) 1 g D H(D) where H(D) s gven by (6). Proof of Theorem 3. Denote C = (1/)M, = 1,...,N. Note that (9) s equvalent to the followng LP: mnmze y C + z + + z subect to y D z + + z = D (15) y = 1 and z +, z, y 0,. Note also that one of z + and z must be zero for the optmal soluton of (15). If both z + and z are zero then problems (10) and (15) are equvalent. Assume condtons () and () are satsfed but problems (9) and (10) are not equvalent. Then ether z + or z s postve. If z + > 0 then z + s a basc varable of LP (15) and z = 0. Snce there are two basc varables for two equalty constrants we conclude that the other Sprnger

12 190 Ann Oper Res (007) 151: basc varable must be one of y, = 1,...,N, say y k. The optmal basc feasble soluton s gven by y k = 1, y = 0, k, and z + = D k D, z = 0. The obectve functon can be wrtten n terms of non-basc varables as C y + z + + z = (C + D )y + z D = k(c + D C k D k )y + z + (C k + D k D) The optmalty mples that all coeffcents of non-basc varables are non-negatve,.e., C + D C k D k 0 for all k. On the other hand, z + > 0 mples D k > D. We have arrved at a contradcton to condton (). Therefore z + > 0 s mpossble for the optmal soluton of (15) under condton (). Smlarly, we can show z > 0 s mpossble under condton (). We have proved condtons () and () mply the equvalence of (9) and (10). Assume problems (9) and (10) are equvalent but condtons () and () are not both satsfed. If condton () s not satsfed then C k D k C D for all and D k < D.We can now estmate the obectve functon of (15) wth the help of ts constrants: C y + z + + z = (C D )y + z + + D (C k D k )y + z + + D C k D k + D The lower bound C k D k + D s acheved when y k = 1, y = 0 for k, and z + = 0, z = D D k, whch s a basc feasble soluton as D k < D. However, z > 0 contradcts the equvalence of (10) and (15). Condton () s therefore satsfed. Note we have explctly constructed the optmal soluton to problem (15) when condton () s not satsfed. Condton () can be shown satsfed n the same way. Notes The author would lke to thank the anonymous referees for ther helpful comments and suggestons on earler versons. 1. If bond prce P s expressed n terms of ts yeld y by P = c (1 + y) t, then ts Macaulay duraton s defned by D mac = t c (1 + y) t /P, and ts modfed duraton s defned by D mod = (1/P)dP/dy = D mac /(1 + y). However, n contnuous compoundng case two duratons are the same D mac = D mod.. The norm g D s equvalent to the standard norm g =max( g(0), K ) due to the Lpschtz property of g. The beneft of usng g D nstead of g s that an equalty relaton (6) s establshed. If g s used n the defnton of H(D) nstead of g D, then H(D) s bounded above by 1 w (t D) + w t D (1 + D) and the Macaulay duraton D m provdes the least upper bound. Sprnger

13 Ann Oper Res (007) 151: Zero rates can be extracted from the observed coupon bonds wth the standard bootstrappng technque (Hull (00)) or the LP method (Allen, Thomas, and Zheng, 000). References Allen, D.E., L.C. Thomas, and H. Zheng. (000). Strppng Coupons wth Lnear Programmng. Journal of Fxed Income, 10(Sept), Fong, H.G. and O.A. Vascek. (1984). A Rsk Mnmzng Strategy for Portfolo Immunzaton. Journal of Fnance, 39, Ho, T. (199). Key Rate Duratons: Measures of Interest Rate Rsks. Journal of Fxed Income, (Sept), Hull, J.C. (00). Optons, Futures, & Other Dervatves. Prentce-Hall Internatonal. Joron, P. and S.J. Khoury. (1996). Fnancal Rsk Management. Blackwell. Khang, C. (1979). Bond Immunzaton when Short-term Rates Fluctuate More Than Long-term Rates. Journal of Fnancal and Quanttatve Analyss, 14, Nawalkha, S.K. and D.R. Chambers. (1996). An Improved Immunzaton Strategy: M-absolute. Fnancal Analysts Journal, 5(Sept/Oct), Rzadkowsk, G. and L.S. Zaremba. (000). New Formulas for Immunzng Duratons. Journal of Dervatves, 8(Wnter), Vnter, R.B. and H. Zheng. (003). Some Fnance Problems Solved wth Nonsmooth Optmzaton Technques. Journal Optmzaton Theory Applcatons, 119, Zheng, H., L.C. Thomas, and D.E. Allen. (003). The Duraton Derby: A Comparson of Duraton Based Strateges n Asset Lablty Management. Journal of Bond Tradng and Management, 1, Sprnger