Derivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity)

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Ralph Craig
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1 Aity Deivatios 4/4/ Deivatio of Aity ad Pepetity Fomlae A. Peset Vale of a Aity (Defeed Paymet o Odiay Aity 3 4 We have i the show i the lecte otes ad i ompodi ad Discoti that the peset vale of a set of eeal cash flows is: t t t ( ( o t t t The cash flow diaam above shows a set of cash flows betwee t ad t whee each cash flow is idetical. This is called a aity. Sice thee is o cash flow at t ad the est of the cash flows ae idetical, that is: ad 3.., we ca simplify a bit, ad i paticla, facto ot the costat cash flow : t ( t We ca also wite this as follows: ( (... ( The objective is to combie this set of fiite tems ito oe compact eqatio. We will make a sbstittio fo the qatity /( to simplify the otatio that follows: Let, the... Mltiply each side by : Steve Fed

flow diaam above shows a set of cash flows betwee t ad t whee each cash flow is idetical. This is called a aity.

2 Aity Deivatios 4/4/ Sbtact the secod eqatio fom the fist: Notice that i the sbtactio all the middle tems wee elimiated, leavi s with oly two tems istead of tems. Solvi fo : ( ( Sbstitte back : ( ( Mltiply the meato ad the deomiato by : { ( } o ( The mbe i the backets is called the aity discot facto. The deivatio above has show that the peset vale of a aity ca be calclated as: t ( t o as ( Bt the fist fom has tems, while the secod is vey compact ad is the podct of the cash flow ad the aity discot fctio fomla. Fom the deivatio it is also vey clea that the aity discot facto is the sm of the idividal discot factos fo each cash flow. It shold be easy to show that the aity facto mst be a mbe less tha. 4 Steve Fed

The deivatio above has show that the peset vale of a aity ca be calclated as: t ( t o as ( Bt the fist fom has tems, while the secod is vey compact ad is the podct of the cash flow

3 Aity Deivatios 4/4/ B. Fte Vale of a Aity Sice the fte vale (FV fo a amot woth at the peset time is: FV ( ad fo a aity, the peset vale was jst show i Pat A to be: ( The we ca combie these two fomlas to ive s the fte vale of a aity: ( FV ( Simplifyi: FV ( ( ( FV ( 4 Steve Fed 3

4 Aity Deivatios 4/4/. Peset Vale of a Pepetity A aity which has ifiite tems ( is called a pepetity. To deive the fomla fo a pepetity, we se the followi popety: Whe > lim( We ca demostate this si a calclato. Tyi, ad. we et.757. Whe we icease to be, we et x -4, which is a vey small mbe. Ty this yoself! Now o back to the peset vale fomla fo the aity: Aity ( Fo a pepetity, we se the same fomla bt will appoach ifiity! Pepetity ( lim Usi the limit eqatio we have demostated above, the tem with the eative expoet oes to zeo as oes to ifiity, ad o pepetity eqatio is edced to a vey simple: Pepetity 4 Steve Fed 4

Whe we icease to be, we et 4.489 x -4, which is a vey small mbe. Ty this yoself!

5 Aity Deivatios 4/4/ D. Peset Vale of a Aity with Gowi Paymets Occasioally, we eed to calclate the peset vale of a set of paymets whee is ot costat. If ows at a costat ate, sch that i i(, we wold call it a owi paymet aity. A example of this is if divideds at time t ae $, ad ow at a ate of 5% evey peiod. That is, D $, D (.5 $.5 ad D (.5 $.5. If this coties fo peiods, we ca et a expessio fo the peset vale, ad it is also vey simple to calclate if appoaches ifiity. To show this, we se some of the eslts we obtaied i o aity deivatio fo a costat paymet. Usi the sbstittio, we wee able to show that... ad that: ( If we have a owi paymet aity whee i i(, ad ate of owth, o peset vale will be: ( ( ( ( (... ( 3 Notice that o fist cash flow is at t althoh we se i o fomla. That is, (, ( o ( etc. O sbstittio fo the owi paymet aity will be familia eqatio:, which yields the... 4 Steve Fed 5

evey peiod. That is, D $, D (.5 $.5 ad D (.5 $.5. If this coties fo peiods, we ca et a expessio fo the peset vale, ad it is also vey simple to calclate if appoaches ifiity.

6 Aity Deivatios 4/4/ 4 Steve Fed 6 Sice this eqatio is exactly what we had befoe, we ca se o pio eslt that this will eqal: ( The oly diffeece is that we eed to sbstitte back o ew defiitio fo :. The maiplatio to simplify is a bit complicated, so I will povide the eslts ad the show the details: Details: ( Now mltiply both the meato ad the deomiato by : ( ( ( ( Sice (

7 Aity Deivatios 4/4/ Oe applicatio fo the costat owth aity fomla is whe divideds ow at a costat ate fo peiods. Eve moe popla is the assmptio that divideds cotie to ow foeve at this ate. It is assmed that yo fist divided is at t. Yo wat to calclate the fai pice fo the stock at t, o P. This is the peset vale of all fte divideds which will stat to ow at t si ate, bt yo fist divided is D. Sice stocks ae isky, yo will discot si a ate hihe tha the isk fee ate. It is commo to se the expected ate of et fo the stock fo this discot ate, ad we will deote this ate hee si k. P D k k If appoaches ifiity, as lo as the discot ate is eate tha the owth ate o < k: lim k To show this ty.5, k., ad, o yo calclato. The above Divided Discot Model will the edce to the compact eqatio show below: P D k This is also called the Godo Gowth Model. 4 Steve Fed 7

This is the peset vale of all fte divideds which will stat to ow at t si ate, bt yo fist divided is D. Sice stocks ae isky, yo will discot si a ate hihe tha the isk fee ate.

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