Chapter One BASIC MATHEMATICAL TOOLS


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1 Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is to be raised is called a expoet. For istace, the expressio 4 meas four to the third power or 4 x 4 x I eeral, whe we say Y we mea multiply Y by itself umber of times. Because of the exteded use of expoets, we will briefly review the rules for deali with powers. Rule : Y 0 Rule : Y m Y Y m Rule : Y m /Y Y m Rule 4: (Y m Y m Rule 5: Y / Y Rule 6: Y  /Y Ay umber to the zero power is equal to oe. The product of a commo umber or variable with differet expoets is just that umber with a power equal to the sum of the expoets. This rule is also ow as the product rule. The quotiet of a commo umber or variable with differet expoets is just that umber with a power equal to the differece of the expoets. This rule is also called the quotiet rule. A variable or umber with a expoet that is raised to aother power is equal to that umber with a power equal to the product of the expoets. This rule also carries the ame power rule. A variable or umber that has a expoet of the form / is just the th root of that umber. For istace, 6 / is equivalet to the square root of 6 or A umber or variable raised to a eative power is just the reciprocal of that umber raised to the positive power. This last rule will come i very hady whe wori problems i the text. Few, if ay, calculators will tae the eative root of a umber. However, by owi that 5  is the same as /5 we ca easily use a calculator to fid that 5 5 ad /5 equals Aru J Praash /
2 INTRODUCTION TO GEOMETRIC ERIE A mathematical series is the sum of a sequece of real umbers. A series ca be fiite, with limited umber of terms, or ifiite, with ulimited umber of terms. For a fiite series, let be the umber of terms i a series ad a i be the i th term of the series, the a fiite series ca be expressed as a a a  a a i i Cosider the followi example: obtai the sum of 4 By DIRECT METHOD, we have However, cosider the followi: 4. (I Multiplyi both sides by, we et 4 5 (II Now we subtract equatio (II (I, we et 5 What is uique about the above series? For uiqueess, cosider the followi: You pic ay two cosecutive terms, let us say rd ad 4 th, the third term is term is. Defie the ratio, ad the fourth succeediterm R precediterm You ca see that ay succeedi term over precedi term remais costat ad is equal to. Now, we itroduce the followi defiitio Aru J Praash /
3 I ay series if the ratio of succeedi term over precedi term remais costat, the the series is ow as eometric series ad the ratio is ow as COMMON RATIO. With the help of the above arumet, we ca defie a eometric series as follows A series is said to be eometric if the ratio of each succeedi term to the precedi term always remais a costat. This costat is called the commo ratio. For example, a series such as where the ratio a, ar, ar ar,..., ar, ar,... ar ar i i r remais costat for ay two successive terms is a eometric series. If the umber of terms i the series is fixed, for example, say te or twety, the the series is fiite. If there are a ifiite umber of terms, the series is called a ifiite series. A series is coveret if its sum equals a fiite real umber. I a fiite series, a expressio ca always be obtaied for the sum of the series, a ar ar... ar  reardless of whether r is reater or less tha oe (i a ifiite series the thi will chae see below. A ifiite series will oly be coveret if the commo ratio is less tha oe. imple formulas for the sum of each series follow. um of a Fiite Geometric eries The sum of a fiite series,, ca be obtaied as follows: a ar ar.... ar  (. Multiplyi both sides of this expressio by r, r ar ar ar... ar (. ubtracti equatio (. from (., ad caceli terms, we et (r  ar  a Aru J Praash /
4 or ar a r (. Thus we have the followi result The sum of a fiite eometric series with commo ratio r is ive by ar a r a ar or r a( r or r Note that this formula fails whe r. I fact, whe r a a a to terms a (.4 Thus, oe does ot eed ay alebraic formula to obtai this sum. I the above series, the umber of terms is fiite. What happes whe the umber of terms is ifiite? um of a Ifiite Geometric eries Cosider the followi: We have the ifiite series ive by where r is assumed to be less tha oe. a ar ar... up to. (.5 The commo ratio is r. If r, the a a a... up to. Addi ay costat up to ifiity will always be ifiity. Furthermore, if r >, the what happes? Every succeedi term will be reater tha the precedi term. Aru J Praash 4/
5 Hece, the value of the sum will aai be ifiity. Therefore, it is obvious that the sum of the ifiite eometric series ca exist oly if the commo ratio r < Multiplyi both sides by r, as i the previous case equatio.5, we obtai r ar ar ar... up to. (.6 ubtracti equatio (.5 from (.6 we et or (r  a a s (.7 r Thus we have the followi result The sum of a ifiite eometric series with commo ratio r is ive by a for r < r for r The simple examples that follow illustrate the usae of these formulas. Example.: Obtai the sum of 4. We ca obtai the sum directly as I order to apply formula (., ote that a, ad the commo ratio is foud as r 4, a costat The total umber of terms,, is 5. Thus Aru J Praash 5/
6 a( r r 5 ( which is the same result if the sum is calculated directly. I order to obtai the sum usi formula (., ote that the sum ca be writte as x x x with a, a commo ratio of, ad 4. Therefore, a( r r 4 ( 0 Example.: Obtai the sum of.06 (.06 ( (.06 0 Thus a r ( ( ( Aru J Praash 6/
7 Example.: Obtai the sum of L up to.06 (.06 (.06 We ca rewrite this as L up to.06 (.06 (.06 (.06 (.06 with a r ad <. Thus applyi formula (.7, we have 0 a.06 r Example.4: Obtai the sum of d d( d( L up to ( ( ( where >, d > 0. d a Rewriti this series, we obtai d d d d L Aru J Praash 7/
8 It is easy to see that the commo ratio r < sice >. Hece, by formula (.7, d a d d. r People familiar with basic fiacial maaemet will recoize that this is the famous Gordo's equity valuatio formula, where d divided to be paid ext period, cost of equity capital, ad earis rowth rate. To use this equity valuatio formula suppose the BP Corporatio iteds to pay aual divideds of $ ext year, ad the earis of BP are rowi at a aual rate of 6 percet. The cost of capital to the BP Corporatio is assumed to be 8 percet. If these values are expected to cotiue ito the future, how much would a share of stoc i the BP Corporatio be worth? Applyi the above equity valuatio formula, we have d $ $50 that is, a share of stoc is worth $50. Formulas such as these are used extesively i the developmet of fiacial models ad, as ca be see from the examples, ca simplify reatly the tass at had. INTRODUCTION TO ARITHMETIC ERIE A arithmetic series is a mathematical series that the differece of ay two successive members of the sequece is a costat. For istace, the sequece, 5, 7, 9,,... is a arithmetic proressio with commo differece. The costat differece is called commo differece, deoted by d. Mathematically, ai ai d, for all i (.8 Applyi equatio (.8 recursively, the terms i a arithmetic series ca expressed as follows: Aru J Praash 8/
9 a a a d a a d a d M a M a i a a i d d a a ( i d ( d For a arithmetic series with terms, usi the a i s computed above, the value (sum of the series ca be expressed as ( a d ( a d [ a ( d] [ a ( d] a L (.9 imilarly, the riht had side of equatio (.9 ca be writte iversely as [ a ( d] [ a ( d] ( a d ( a d a L (.0 Add both sides of equatios (.9 ad (.0. All terms ivolvi d cacel, ad so we are left with: [ a ( d] Rearrai ad rememberi that a a ( d, we et: [ a ( d] ( a a (. Ituitively, this formula ca be derived by realizi that the sum of the first ad last terms i the series is the same as the sum of the secod ad secod to last terms, ad so forth, ad that there are rouhly / such sums i the series. Aother way to loo at this is that the value of the arithmetic series is the umber of terms i the series times the averae value of the terms. The averae must be (a a /, sice the values appear evely spaced out aroud this poit o the real umber lie. Example.5: Fid the value of the followi series: Aswer: Note that is a arithmetic series with commo differece d ad we have 00, a ad a Usi equatio (., Aru J Praash 9/
10 00 ( ,050 Example.6: Fid the value of the followi series: Aswer: Note that is a arithmetic series with commo differece d 4 ad we have a 7 ad a 49, but we do ot ow the value of. Usi the fact that a a ( d, we et: 49 7 ( 4 olvi the above equatio for we obtai 59. Usi equatio (., ( ,847 Cosider the expressio THE MEANING OF THE NUMBER e m m ad let m tae differet values (e..,,000, 00,000, 00,000,000, etc.. The values of the expressios for the differet values of m are as follows. For, m,000,000, For, m 00,000 00,000 00, For, m 00,000,000 00,000,000 00,000, Thus, i eeral, we ca write Aru J Praash 0/
11 m Lim e m.7888 m (. This limit value yields the irratioal umber ow as e. Now cosider m to be the umber of periods duri the year that $ is compouded whe the iterest rate is 00 percet. The whe m, the case of cotiuous compoudi, that dollar would be worth approximately $.7 at the ed of a year. Further, if we use ay iterest rate, a, istead of 00 percet, it ca be easily show that, Lim a a e (. where a is ay costat. Thus $0 ivested today at 8 percet would be worth ( l0e (.08( $0.8 at the ed of a year if the dollar were to be compouded cotiuously. For those of you who are some owlede of calculus, we ca mathematically prove equatio (.. EXPONENTIAL FUNCTION AND LOGARITHMIC FUNCTION Both expoetial fuctio ad its iverse, amely the loarithmic fuctio, are widely used i the mathematics of time value of moey. The basic expoetial fuctio is defied by y f(x b x (.4 where b, a real umber, is the base such that b > 0 ad b ot equal to. The domai of f is the set of all real umbers. Example: x a. y b. y 4 x c. y 0.4 x d. y. x The loarithmic fuctio is the mathematical operatio that is the iverse of expoetial fuctio. The iverse of the expoetial fuctio i expressio (.4 is x lo b (y (.5 Aru J Praash /
12 The domai of loarithm is the set of positive real umbers, that is, y > 0. Note that expressio reads as: x equals to lo to the base b of y. Loarithms are useful i solvi equatios i which expoets are uow. The most widely used bases for loarithms are 0, the mathematical costat e.788 ad. Whe "lo" is writte without a base (b missi from lo b, the itet ca usually be determied from cotext: atural loarithm (loe or l i mathematical aalysis commo loarithm (lo 0 i eieeri ad whe loarithm tables are used to simplify had calculatios biary loarithm (lo i iformatio theory ad musical itervals Here we review some of the rules for loarithmic operatios: let b be the base (b > 0 ad b, c be a costat ad x, y, z be positive real variables. Rule : (Product Rule lo b ( x y lob ( x lo ( y Lo of a product is the sum of two los. b Rule : (Quotiet Rule x lob lob ( x lo y b ( y Lo of a ratio is the differece of two los. Rule : (Power Rule c lo ( x c lo b b ( x Rule 4: lo b b ice b b, this rule follows from the defiitio of loarithm i equatio (.5. Rule 5: lo b 0 ice b 0, this rule follows from the defiitio of loarithm i equatio (.5. Rule 6: (Chae of Base lo ( x lo ( c lo ( x b b c For variable x, this rule chaes the base of loarithm from b to c. Rule 7: (Iverse of Base lo b ( c This rule allows oe to swap base ad arumet of a lo c ( b loarithm. The first five rules are selfexplaatory ad easy to apply. Rules 6 ad 7, o the other had, may become hady whe oe wats to compute the value of a loarithm su a oeieeri type of calculator. Most of the fiacial calculators oly have the atural lo ad/or Aru J Praash /
13 commo lo, it would be a problem if you eed to calculate a lo with a base differet tha e. For example, how do you wat to fid the value of lo. ( if oly atural lo is available o your calculator? You ca solve this problem by applyi both Rule 6 ad Rule 7 above: lo. ( lo. (e l( (by Chae of Base Rule l( (by Iverse of Base Rule l(..567 Chapter Homewor Problems Obtai the values of the followi: up to ( (.06 d d( d( 5. ( ( d d( d( 6. ( ( d( (... up to ( (.06 ( ( (.06 (.0 0 (.06 (.0 00 Aru J Praash /
14 ( ( ( up to 0 ( ( (.06 (.0 0 (.06 ( ( ( ( ( (.0 olutios to Chapter Problems I alebra this id of problem is ow as the sum of the first umbers. The formula for this problem is ( Where umber of umbers. I this case, 00. Therefore, 00( , Note that i the preset form it is ot a eometric series because 00 However, ecod Term First Term 4 Third Term ecod Term Fourth Term Third Term Therefore, the series is a eometric series oly from Here, First Term a, commo ratio r is ad the umber of terms is 99 (cout repeatedly. Thus, the sum of the eometric series is 00 Aru J Praash 4/
15 a( r r ( 99 (. Therefore, the aswer will be ( 99 Remember, i was ot part of the eometric series. olutio (. is the sum of oly therefore you have to add to arrive the correct aswer up to By careful examiatio we ote that (a This is a ifiite eometric series. (b The commo ratio, which is more tha. ice we ow that if i ifiite eometric series its commo ratio is equal to or reater tha, the sum is always. Hece, the aswer is ifiity ( (.06 Note that (a This is a fiite series. (b The ratio of ay succeedi term over precedi term is costat. That is r (c Number of terms. (d First term a 0. Hece the aswer will be 0 ( Aru J Praash 5/
16 [ (.06 ] ( (.06 d d( d( 5. ( ( d( ( Note that (a This is a fiite series. (b The ratio of ay succeedi term over precedi term is costat. That is d ( d ( ( ( r d d( ( ( (c Number of terms is. d (d First term is a. Hece the sum of the eometric series is Aru J Praash 6/
17 d d d d ( ( ( ( ( ( ( 6. ( ( ( ( d d d... up to This problem is exactly lie problem umber 5 except here we are deali with a ifiite eometric series. The sum of the eometric series is r a for r < for r Therefore, for our problem Aru J Praash 7/
18 d d d for >. d ( ( Cautio: We ow that the commo ratio is, the., ad we also ow that if CR or CR > For CR to be equal to, must be equal to ; for CR to be reater tha, must be reater tha. o if or >, we will have ; ad for < (or >, d ( (.06 ( ( (.06 (.0 0 (.06 (.0 00 This problem loos siister, but i reality is ot so. Just rewrite the problem as follows, ( ( (.06.0 (.0 (.0 (.0 (. Note that is the sum of two eometric series. The first part (the top half of equatio (. has commo ratio of while the secod part (the bottom half of equatio (. has.06 commo ratio of..0 For the first part, the sum is Aru J Praash 8/
19 For the secod part, the sum is 00 ( ( (.0 ( (.06 ( The aswer will be the sum of these two parts, that is, the sum of equatios (. ad ( ( ( (. We rewrite 000 (. 50, where 90 with a, r ad is a eometric series. (. (. 50 Hece, the aswer will be [ (. ] (. ( up to 0 ( ( (.06 (.0 0 (.06 ( Aru J Praash 9/
20 imilar to Problem 7, we rewrite the problem ito two parts: ( (.06 0 up to (.5 00 (.06.0 (.0 (.0 Note that the first part of this problem (the top half of equatio (.5 is the same as the first part of equatio (. i Problem 7. Its sum is preseted i equatio (.. The secod part of this problem (the bottom part of equatio (.5 is a ifiite eometric series with commo ratio. Therefore, the aswer will be [ (.06 ] ( (.06 0 (.06 0 ( ( ( ( ( (.0 This is a fiite eometric series with a 0 (.0 r CR (.0 umber of terms 500 Therefore, the aswer will be Aru J Praash 0/
21 0 (.0 (.0 0 ( [ (.0 ] ( Aru J Praash /
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