Duration Notes 1. To motivate this measure, observe that the duration may also be expressed as. a a T a

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1 Duio Noes Mculy defied he duio of sse i Le he sem of pymes cosiuig he sse be,,..., d le /( + ) deoe he discou fco. he Mculy's defiiio of he duio of he sse is 3 2 D o moive his mesue, obseve h he duio my lso be expessed s D whee is he pese vlue of he sse, i.e., Weil seems o ibue he oigiio of he duio cocep o Mculy. he oio of vege peiod of poducio goes bck fhe. Fo exmple, Bohm-Bwek defied vege peiod of poducio which is ideicl o Hicks' vege peiod d so o Mculy's. Bohm-Bwek is ibued wih he ioducio of he cocep of he vege peiod of poducio d by his he me "... he vege ime disce bewee oigil ipus d he fil poduc" (See C. C. vo Weizscke, Sedy Se Cpil heoy, Spige-Velg 97, p. 33). Bohm-Bwek ssumed h oupu ws icesig fucio of ime, i.e., q() whee q' >. I simple poi ipu - poi oupu model, i is cle h he seleced is he vege peiod of poducio. Noe h usig Mculy's mesue of duio, we obi q e D () qe () Hece, we my ce he oio of vege peiod of poducio o equivlely duio bck les s f s Bohm-Bwek's Posiive heoy of Cpil. 2 Fedeick R. Mculy, Some heoeicl Poblems Suggesed by he Movemes of Iees Res, Bod Yields, d Sock Pices i he Uied Ses sice 856 (New Yok: Columbi Uivesiy Pess fo he Niol Bueu of Ecoomic Resech, 938). 3 I m o sue h Mculy llowed fo he de bu I m icludig i so h he compiso wih Hicks' elsiciy is pecise.

2 Sice he duio mesue c be udesood s he me peiod of he sse. I 939, Hicks oed h he vege peiod of sse, i.e., is he elsiciy of he sse vlue wih espec o he discou fco. Hick oed h his elsiciy "... is he vege legh of ime fo which he vious pymes e e defeed fom he pese, whe he imes of defeme e weighed by he discoued vlues of he pymes." 4 o see his, oe h he elsiciy of he sse vlue wih espec o he discou fco is he pecege chge i sse vlue pe pecege chge i he discou fco, i.e., 4 See J. R. Hicks, Vlue d Cpil, (Oxfod: Cledo Pess, 939), p

3 d ε d d d D Hece Mculy's duio d Hicks' vege peiod of sse e equivle coceps. leively, he duio my be expessed usig he elsiciy of he sse vlue wih espec o he e of iees. Diec clculio shows d ε d d - D, o equivlely, D ε whee ε is he elsiciy of he sse vlue wih espec o he iees e. Noe h he e of chge i he vlue of he sse wih espec o he iees e is d ε ( D) - D () he duio of pepeuiy wih oe doll pyoff i ech peiod begiig wih is 3

4 D ( ) ε ( ) + d so iees e of 5% yields duio of 2. I my be oed h his duio is lso he miimum umbe of peiods equied o cquie he pese vlue of he pepeuiy i csh, i.e., sice he pese vlue of he pepeuiy is 2. I his cse, he e of chge i he pese vlue of he pepeuiy is d 2 D 2 d he iees elsiciy of he pepeuiy is ε D + + Equivlely, d d so oe pece icese i he iees e yields oe pece decese i he vlue of he pepeuiy. Weil sys h he elies ppe he is we of o immuizio is llig C. Koopms' wok, he Risk of Iees Flucuios i Life Isuce Compies. 5 Howeve, 5 his ws wie while Koopms woked fo Pe Muul Life Isuce Compy d hs o bee published. See R. L. Weil, "Mculy's Duio: ppeciio," Joul of Busiess, (Ocobe 973). 4

5 Weil lso oes h i 952, Redigo defied he me em of sse sem d libiliy sem, d showed h he pyoff of isuce compy is immue o smll chges i he iees e if he me em of he sse equls h of he libiliy. 6 Redigo's defiiio of me em is Leig l, l,..., l l deoe he sequece of libiliies d,,..., he sequece of sse pyoffs, i follows h Redigo showed h he isuce compy is immuized if l l (2) Now le L deoe he pese vlue of he libiliy sem, i.e., l L l d le D L d D deoe he duio of he libiliy d sse, especively. he, sig Redigo's lysis i ems of duio, oe h (2) is equivle o he followig equio (3), D D L L. (3) 6 See F. M. Redigo, "Review of he Piciples of Life-Office Vluios," Joul of he Isiue of cuies, (952). 5

6 Fom () i is lso cle h (3) my be equivlely expessed s d dl d dl (4) Hece, i is quie cle h if Redigo's codiio (2) is sisfied he he fim is immuized. leively, le he vlue of he isiuio o copoio be expessed s V, whee V - L. he, usig (), i follows h dv ( ) ( ) ( ) d dl D DLL DLL D heefoe, dv ( ) > < > < s DLL D lhough Smuelso ws ppely uwe of Mculy's wok, his is esseilly wh he showed i Smuelso showed h if he weighed vege ime peiod of he csh ouflow exceeds h of he csh iflow he he isiuio will pofi whe iees es ise. Smuelso's weighed vege ime peiod is equivle o D L L d D, fo ouflows d iflows, especively. I 957, Dud gued h he oly ficil sses wih log duios wee gowh socks. he he gued h isiuios wih log duio libiliies eed o hold gowh 7 See Pul. Smuelso, "he Effecs of iees Re Iceses o he Bkig Sysem," meic Ecoomic Review 35 (Mch 945), his icle is olie he followig ul: <hp://idex2.umdl.umich.edu/cgibi/so/viewiem.sble/28282/di9533/95p237/?fmefme&useidnouseid&dpi3&cofigs o> 6

7 socks o educe he isk of ficil loss fom iees e flucuios. 8 I is ppe h he isiuio c immuize by usig gowh sock. he duio mesue fo he gowh sock is ( + ) D g his duio is lge h h fo o gowh sock wih level pyme sem, i.e., ( + ) g > ( + ) I he bsece of immuizio, i is lso ppe h he vlue of he isiuio will oly fll if he iees e flls. Duio Mhemics Duio of uiy Fis coside he duio of uiy. Le fo,..., deoe he cos cshflow d le deoe he vlue of he uiy. he diec clculio shows h he duio of he uiy is D whee D ( ) ( ) 8 See Dvid Dud, "Gowh Socks d he Peesbug Pdox," Joul of Fice 2, (Sepembe 957),

8 Duio of Bod Le deoe he iees pe peiod d f deoe he fce vlue muiy. Similly le deoe he vlue of he iees pymes d F deoe he pese vlue of he fce vlue. Filly, le B deoe he bod vlue. he he bod duio D B is DB + B f B B f + F F B D B D F + F B Hece, he bod duio is he weighed vege of he duio of he uiy epeseig he iees pymes d he duio of he picipl pyme. Duio of Sum Le epese he sum of sse vlues, 2,...,. Similly, le epese he cshflow of he sum of he sses de, i.e., he duio of he sum is weighed vege of he sse duios sice 8

9 9 D D Similly, he duio of diffeece is he weighed sum of he diffeece of duio mesues. Duio of Pofolio Le α α epese he vlue of he pofolio α (α,..., α ) of sses. he he duio of he pofolio is weighed sum of he sse duios. D D α α α