DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES

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1 DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES Icreasig ad Decreasig Auities ad Time Reversal by Jim Farmer Research Paper No. 2000/02 November 2000 Divisio of Ecoomic ad Fiacial Studies Macquarie Uiversity Sydey NSW 2109 Australia

2 The Macquarie Uiversity Actuarial Studies Research Papers are writte by members or affiliates of the Departmet of Actuarial Studies, Macquarie Uiversity. Although urefereed, the papers are uder the review ad supervisio of a editorial board. Editorial Board: Sue Clarke Jim Farmer Copies of the research papers are available from the World Wide Web at: Views expressed i this paper are those of the author ad ot ecessarily those of the Departmet of Actuarial Studies.

3 Icreasig ad Decreasig Auities ad Time Reversal by Jim Farmer. BEc, FIAA. Abstract Stadard formulae for decreasig auities are derived from those for icreasig auities by reversig the directio of the time axis. Keywords: Decreasig auities, icreasig auities, time reversal. 1. Itroductio The stadard formulae for decreasig auities ca all be derived from first priciples. They ca also be derived from the formulae for icreasig auities by usig the fact that the sum of correspodig icreasig ad decreasig auities produces some multiple of a level auity. For example: ( ) ( ) ( ) Ia + Da = + a (1.1) 1 ( ) ( ) ( ) Ia + Da = + a (1.2) 1 ( Ia) ( Da) a + = (1.3) These methods of derivatio ca be foud i may mathematics of fiace textbooks, such as Broverma (1996). If time-lies showig the paymet streams of a icreasig auity ad its correspodig decreasig auity are examied, the obvious major differece betwee the diagrams is that for the icreasig auity the paymets icrease as time moves forward while for the decreasig auity the paymets icrease as time moves backwards. This suggests that it may also be possible to derive the formulae for decreasig auities from those for icreasig auities by reversig the directio of the time axis. This paper shows how this may be doe. 2. Basic compoud iterest fuctios I will use the ormal compoud iterest symbols whe the time axis rus i the usual directio ad starred symbols to deote the correspodig quatities whe the time axis has bee reversed. If the axis is reversed, the the value of a ivestmet declies as time icreases. Accumulatig with compoud iterest for t years o the reversed time axis is equivalet to discoutig for t years o the ormal time axis. Hece: Rearragig (2.1) gives 1+ i = v (2.1) i v 1 d = = (2.2) Note that i the usual situatio where i is positive, i will be egative. Ivertig (2.1) gives The v = 1+ i (2.3) 1

4 Takig the logarithm of (2.1) gives d 1 v i = = (2.4) δ = δ (2.5) 3. Level Auities I this sectio we derive some time reversal relatioships ivolvig level auities. It is ot beig suggested that these relatioships are a efficiet way of derivig formulae for level auities. These relatioships are developed solely because they become useful i the ext sectio whe we derive formulae for decreasig auities. For ease of explaatio, the followig derivatios will cosider paymets occurrig aually ad assume we are dealig with effective aual iterest rates. However, the methods will work for ay arbitrary time period desired. I the followig diagram the first time-lie shows paymets of 1 pa for years. The positios of the stadard discrete auities are idicated. I the secod time-lie, the paymets have ot bee moved but the directio of the time axis has bee reversed. That is, moey still icreases i value as it moves towards the right, but this directio is ow regarded as movig backwards i time. The auity symbols have agai bee placed i their correct positios o the diagram, this time with starred symbols. For example, s gives the value of the paymets as at the date of the last paymet, which is ow the left-most paymet sice the time axis is reversed. a a&& s s&& s&& s a&& a Comparig the two diagrams gives the followig relatioships. a =&& s (3.1) a&& = s (3.2) s = a&& (3.3) && s = a (3.4) Here are the correspodig diagrams for cotiuous level auities. The shaded bar idicates cotiuous paymets at the rate of 1 pa for years. 2

5 a s s a These diagrams give the followig relatioships. a = s (3.5) s = a (3.6) If you fid these equatios ucovicig, you ca always verify them usig the appropriate equatios selected from (2.1) to (2.5). For example, equatio (3.1) ca be verified as follows. a ( i) ( i) 1 v && s i d d However, I would ot wat to ecourage that this form of verificatio be doe routiely sice it would mea that the time reversal process for derivig decreasig auity formulae would be grossly iefficiet compared to the techiques give i sectio Decreasig Auities I this sectio we are assumig that we have derived the formulae for icreasig auities from first priciples ad ow wish to derive those for decreasig auities. The followig diagram shows paymets of, 1, 2,...,1 spaced oe year apart. The positios of the stadard discrete decreasig auities are idicated. (I all the remaiig diagrams, the vertical scale has bee compressed relative to the horizotal scale to save space.) As before, the secod time-lie does ot move the paymets but the directio of the time axis has bee reversed. Note that o this time-lie as we move left time is icreasig ad the size of the paymets icreases. That is, this time-lie will show us icreasig auities. The icreasig auity symbols have bee placed i their correct positios o the diagram, with starred symbols beig used due to the time axis havig bee reversed. 3

6 ( Da ) ( Da&& ) ( Ds ) ( Ds&& ) ( Is&& ) ( Is ) ( Ia&& ) ( Ia) This diagram gives the followig relatioships. ( Da) ( Is) = && (4.1) ( Da) = ( Is) && (4.2) ( Ds) ( Ia) ( Ds ) = ( Ia) = && (4.3) && (4.4) Here are the correspodig diagrams for the step decreasig cotiuous auities. The shaded regios idicate paymets at the rate of pa for the first year, 1 pa for the secod year, ad so o for years. ( Da ) ( Ds ) ( Is ) ( Ia) This diagram gives the followig relatioships. 4

7 ( Da ) ( I s ) ( Ds ) ( I a ) = (4.5) = (4.6) Here are the diagrams for the cotiuously decreasig cotiuous auities. ( Da ) ( Ds ) ( Is ) ( Ia) This diagram gives the fial two required formulae. ( Da ) ( I s ) = (4.7) ( Ds ) ( I a) = (4.8) All the coceptually difficult work is ow over; all that remais is the algebra. We develop equatios (4.1) to (4.8), substitutig i where ecessary from (2.1) to (2.5) ad from (3.1) to (3.6). Here are the results. ( Ds) ( Ia&& ) ( Da) ( Is &&) ( Da&& ) ( Is) && s a a d i i && s a a i d d ( 1 ) ( 1 ) + + a&& v s i i s d i i ( Ds &&) ( Ia) ( 1 ) ( 1 ) + + a&& v s i i s i d d ( Da ) ( I s ) && s a a δ δ δ 5

8 ( Ds ) ( Ia) ( 1 ) ( 1 ) + + a&& v s i i s δ δ δ ( Ds ) ( I a) ( Da ) ( I s ) s a a δ δ δ ( 1 ) ( 1 ) + + a v s i i s δ δ δ 5. Efficiecy of the Derivatio I practice, we would ot derive all 8 decreasig auity formulae usig time reversal. This was doe above merely to demostrate that the process does work i all 8 cases. Whe derivig the formulae for icreasig auities, we would usually oly derive those for ( Ia ) ad ( Ia ) from first priciples. All the other formulae ca be derived from these usig appropriate accumulatio terms. For example: ( Is ) = ( Ia ) ( 1+ i ) ( Ia &&) = ( Ia ) ( 1+ i ) ( Ia) ( Ia) i = δ Similar commets apply to decreasig auities. We really oly eed use the time reversal approach twice, to determie the formulae for ( Da ) ad ( Da ). All other decreasig auity formulae ca the be derived usig appropriate accumulatio terms. The relevat questio to ask here is whether the time reversal derivatio is more efficiet tha the more traditioal methods outlied briefly i the itroductio. The aswer to this is probably subjective ad will deped o what types of derivatio the reader feels most comfortable with. It seems to me that the algebra ivolved i the time reversal method is slightly simpler tha the traditioal method. However, the time reversal method is coceptually cosiderably more difficult to uderstad; reversig the time axis is ot a cocept that comes aturally. (Perhaps it comes more aturally to teachers who are used to metally reversig the horizotal axis of ay graph so that they ca face their audiece ad poit i the directios that match the graph beig projected oto the wall behid them!) Hece I do ot expect the time reversal techique to ever supplat the traditioal derivatio. Still, I hope that at least some readers will share my view that it is a elegat piece of mathematics. Bibliography Broverma, Samuel A Mathematics of Ivestmet ad Credit. Secod Editio. Actex Publicatios Ic. 6