A REVIEW OF ELEMENTARY MATHEMATICS: FUNCTIONS AND OPERATIONS

Transcription

1 QRMC02 9/7/0 4:39 PM Page 7 CHAPTER TWO A REVIEW OF ELEMENTARY MATHEMATICS: FUNCTIONS AND OPERATIONS 2. INTRODUCTION This chapter cotais a review of some of the most basic mathematics fuctios ad operatios required for fiacial aalysis. Ayoe iterested i pursuig study i fiace should be very comfortable workig through the material i this chapter (ad chapter 3) prior to erollig i a itroductory fiace course. Most of the cocepts preseted i the early part of this chapter will likely have bee covered durig the reader s high-school (or juior high-school) math sequece; however, may studets will beefit from a review. Some readers will wish to merely skim this material before progressig to subsequet chapters. Readers with weaker mathematics preparatio may fid sectios ad applicatios followed by a asterisk (*) somewhat more difficult. Such readers may prefer to work through relevat exercises at the ed of the chapter before spedig too much time with this material. I some cases, the more difficult material may be skipped etirely. Studets i ay itroductory or itermediate fiace course must be comfortable workig with material i sectios 2.2 through 2.8. Hece, studets should be certai that they are comfortable with this material before attemptig ay study i fiace. Other sectios may be importat as well, eve i may itroductory fiace courses, depedig o the reader s iterests or exact course or program cotet. For example, material i sectio 2.9 will be eeded by studets plaig study of material related to portfolio maagemet, sectio 2.0 will be eeded by studets plaig study i iterest rate theory or fixed icome maagemet, ad the remaider of the chapter will be eeded by studets iterested i the pricig of optios cotracts ad other derivatives. 2.2 VARIABLES, EQUATIONS, AND INEQUALITIES Mathematics is the primary laguage of fiacial model-buildig, while variables, fuctios, equatios, ad iequalities are the buildig blocks of fiacial models. A

2 QRMC02 9/7/0 4:39 PM Page 8 8 Elemetary mathematics: fuctios ad operatios variable, usually represeted as a letter or group of letters, deotes a quatity that ca be assumed from a give set of umbers. A fuctio is a rule that assigs to each elemet x of a set X a uique elemet y of a set Y. For example, the followig reads y is a fuctio of x : y = f(x), (2.) where y is a depedet variable ad x is a idepedet variable. The variable y may also be expressed as a fuctio of a series of idepedet variables. Such a fuctio might be described more explicitly by specifyig a particular mathematical operatio to defie the rule relatig the sets. For example, we could write oe example of the above fuctioal relatioship as y = 3x + 4. I this equatio, we used two arithmetic operatios, multiplicatio ad additio. A equatio such as the oe we just cosidered is a statemet that two mathematical expressios are equal, while a iequality states that oe expressio is greater tha (>) or less tha (<) aother. 2.3 EXPONENTS After the four basic arithmetic operatios, expoetiatio is the most commoly used operatio i fiace. Expoets eable us to express a repetitive series of multiplicatios or divisios i a cocise form. They are particularly useful i fiace where we frequetly eed to evaluate a large umber of paymets extedig over may years. Expoets are also useful for expressig various types of roots of umbers or variables. Expoets are sometimes referred to as powers. Cosider the fuctio y = 3 2, which reads y equals three squared or y equals three to the power two. The value of y is 3 3 = 9. We refer to the power 2 as the expoet of 3. More geerally, we may write fuctios of the followig form: y = x, (2.2) where represets the expoet for variable x. If is a positive iteger (a whole umber greater tha zero), it is a expoet referrig to the umber of times mius oe the variable x is to be multiplied by itself. Thus, for example, if were to equal 4, y would equal x x x x = x 4 ; that is, x is multiplied by itself three times. Clearly, it is more efficiet to write x 4 tha x x x x. Where y = x ad x is 3 ad is 4, y = 3 4 = 8. The expoet ca also be a fractioal or egative umber. A egative expoet associated with x meas that y is the iverse of x ; that is, y = x, which is the iverse of x : y = x = x. (2.3)

3 QRMC02 9/7/0 4:39 PM Page 9 Expoets 9 Thus, x = /x ad x 2 = /x 2. If y = 4 2, the y = /4 2 = /6. Note that oe may multiply ay term by its iverse to obtai. For example, x 2 x 2 = x 2 /x 2 =. If is a fractioal value, y requires fidig the th root of x: / y= x = x. (2.4) Thus, x /2 = x (the square root of x equals x /2 ) ad x /3 = x. For example, 4 /2 = 2 ad 64x /3 = 4. Furthermore, if y = x m/, the y = x m. For example, if y = 4 3/2, y = 8. If y = x m/, the y = x m. Whe the expoet of a real umber equals zero, the fuctio has a value equal to oe; that is, x 0 = for ay real x. The product of ay base x with expoets ad m equals the base x with a expoet equal to the sum of the origial expoets; that is, x x m = x +m. For example, y = = =,024. Fially, (x ) m equals x m. 3 APPLICATION 2.: INTEREST AND FUTURE VALUE (A more complete presetatio of this material is provided i chapter 4) Suppose we deposit $00 ito a bak accout payig iterest at a aual rate of 5% compouded aually. The total amout i our accout after oe year will be $05, or.05 times our origial deposit amout of $00: $00.05 = $05. If we left our $05 i the accout for a secod year, our accout would have a future value of $05.05 = $0.25. Thus, our accout will have a future value of $ : $ = $ = $0.25. More geerally, our accout will have a future value of $00.05, where is the umber of years that the accout will accumulate iterest. Agai, the cocepts of iterest ad future value are dealt with more extesively i chapter 4. We ca use egative expoets to determie how much we would have to deposit ow to obtai a give future value i the future. Suppose we wished to make a deposit ow ito a savigs accout payig 5% iterest such that the amout i the accout i two years were $00. Thus, how much would we have to deposit ow to obtai a give future value i two years? We fid the solutio to this problem as follows: $ = $ This problem ca be writte $ 00 2 $ = = $ Compoud iterest meas that iterest accrues o accumulated iterest o loas as well as pricipal.

4 QRMC02 9/7/0 4:39 PM Page 0 0 Elemetary mathematics: fuctios ad operatios 2.4 THE ORDER OF ARITHMETIC OPERATIONS AND THE RULES OF ALGEBRA (Backgroud readig: sectio 2.3) May equatios require combiatios of differet types ad series of operatios. Mathematicias use well-established rules for sequecig operatios i more complicated equatios, which may be summarized as follows: A B Operatios withi iermost sets of paretheses (or brackets) are performed first. After operatios withi iermost sets of paretheses are performed, operatios may be performed withi sets movig to outer levels, util operatios withi the outermost level are performed. For example, cosider the expressio: 0 {[3 + (6 4)] [5 (7 + )]}. This expressio simplifies to 0 {[3 + (24)] [5 (8)]}, ad further simplifies to 0 {[27] [90]} = 0 {2,430} = 24,300. Varyig types of operatios withi a give level of paretheses are performed i the followig order: Expoets. 2 Multiplicatio ad divisio (do these i ay order). 3 Additio ad subtractio (do these i ay order). Cosider the equatio y = This equatio simplifies to y = = = 44. The followig equatio may be simplified usig a combiatio of rules A ad B above: y = 3 + {4[3(2 6 + ) + 2( (+2) ) + 7] 2 } + 3{2 + [4 2 3 ( )(2 + 4)]}, y = 3 + {4[3(2 + ) + 2( (3) ) + 7] 2 } + 3{2 + [4 2 3 (3 + 25)(6)]}, y = 3 + {4[3(3) + 2( ) + 7] 2 } + 3{2 + [4 8 (28)(6)]}, y = 3 + {4[39 + 2(4 + 92) + 7] 2 } + 3{2 + [5,376]}, y = 3 + {4[39 + 2(96) + 7] 2 } + 3{5,378}, y = 3 + {4[ ] 2 } + 6,34, y = 3 + {4[438] 2 } + 6,34 = 3 + {4 9,844} + 6,34 = , ,34, y = 783,53. Notice that i the earlier formulatios of the equatios, higher levels of paretheses are distiguished with brackets [ ] ad braces { }. Now, cosider the followig equatio: PVA = $, ( ) 5

5 QRMC02 9/7/0 4:39 PM Page The umber e This equatio, which is used for evaluatig series of cash flows over time (see sectio 4.9 for more details o its applicatio) ca be simplified ad solved as follows: PVA = $, ( ) = $, 000 = $, = $, = $, 000 [ ] 005. = $, = $4, = $, ( 05. ) 5 5 APPLICATION 2.2: INITIAL DEPOSIT AMOUNTS (A more complete presetatio of this material is provided i chapter 4) Suppose that you wished to have $,000 i your bak accout i five years. Your bak accout pays iterest at a aually compouded rate of 0% or 0.0. How much would you eed to deposit i your accout today for your balace to be $,000 i five years? First, we ote that the accout balace ( + 0.0) times as great at ay poit i time as it was oe year earlier because of the 0% iterest rate. For example, the accout will have a balace of $,000 after five years; i four years, the accout will have a balace of $909.9 (which is $,000 [ + 0.]). Furthermore, the accout balace will be ( + 0.) times as large as years earlier. We kow that the future value of the accout will be $000; what will be the iitial deposit amout? Let X 0 represet the iitial deposit amout of the accout, solved for as follows: $, 000 ( ) X 0 = 5 5 $, 000 $, 000 = = = ( 0. ). 605 Note the order of arithmetic operatios i the above equatio. 2.5 THE NUMBER E The umber e, also kow as the base of the atural expoetial fuctio, has a orepeatig, otermiatig decimal value of approximately The umber e may be computed more precisely as follows: e = (2.5)

6 QRMC02 9/7/0 4:39 PM Page 2 2 Elemetary mathematics: fuctios ad operatios Table 2. Computig the umber e m y = + m m y y = ( + /2) y = ( + /3) y = ( + /4) y = ( + /) y = ( + /5) y = ( + /6) y = ( + /7) y = ( + /8) y = ( + /9) y = ( + /0) y = ( + /00) ,000 y = ( + /,000), ,000 y = ( + /00,000) 00, ,000,000 y = ( + /0,000,000) 0,000, Obviously, oe caot use the arithmetic operatios described thus far to divide by ifiity or raise the sum to the power ifiity. However, oe ca replace the value for ifiity with the variable m ad observe what happes to the value of the fuctio y i the followig as m is replaced by itegers of icreasig value: y = + m As m approaches ifiity, y approaches the value of the umber e. Table 2. provides computatios to demostrate how y approaches e as m approaches ifiity. The umber e is defied agai i chapter 8. This umber e is used as the base for the atural log fuctio (discussed i the ext sectio) ad is particularly useful for computatios ivolvig cotiuous growth or compoudig (see chapter 4). m. 2.6 LOGARITHMS (Backgroud readig: sectios 2.2 ad 2.5) A logarithm is a power to which aother umber (called the base) must be raised to obtai a give value. For example, suppose we wish to fid the power to which 0 must be raised i order to obtai 00; that is, we wish to solve 00 = 0 for. The umber 2 is the appropriate power or expoet such that 00 = 0 2. The umber 0 is our base ad 2 is the logarithm (or log) of 00. Thus we say that 2 is the base te log of 00. The base te logs are the secod most commoly used i fiace.

7 QRMC02 9/7/0 4:39 PM Page 3 Logarithms 3 The most commoly used log i fiace is the atural log (also called a Napieria log). The base of the atural log is the umber e, which as we observed i sectio 2.5, has a orepeatig, otermiatig decimal value of approximately The atural log is particularly useful for certai types of computatios ivolvig cotiuous growth or compoudig (see chapter 4). If x = l(y), we say that x is the atural log of y ad y = e x 2.78 x. For example, the atural log of 22 is approximately 3.09 (i.e., l(22) 3.09), which meas that e or The followig rules of operatio apply to operatios ivolvig logs of a give base: log(x y) = log(x) + log(y), log(x y) = log(x) log(y), log(x y ) = y log(x). Logs are useful whe solvig for i expoetial fuctios such as y = cx. If y = cx, the from the rules of operatio above, the followig must be true: l(y) = l(c) + l(x). Suppose, for example, that we wished to solve for i the followig: 50 = 3 7. We may do so by fidig the logs of both sides of the equatio: l(50) = l(3) + l(7). Solvig, we obtai 3.92 = ; = APPLICATION 2.3: THE TIME NEEDED TO DOUBLE ONE S MONEY (You may also wish to read related material i chapter 4) How log would it take to double the balace of a bak accout payig iterest compouded at a aual rate of 5%? For example, how log would it take $,000 to double to $2,000 i a 5% bak accout? We determie this by solvig the followig for : $2,000 = $,000( ). First, divide both sides by,000 ad fid the atural logs of both sides, to obtai We rewrite this equatio to obtai 2 l, 000 l[( 0. 05) ] l(. 05)., 000 = + = l(2) = l(.05), = We solve for, the umber of years required to double our moey i the bak accout, as follows: = =

8 QRMC02 9/7/0 4:39 PM Page 4 4 Elemetary mathematics: fuctios ad operatios $,000 must be left i this 5% bak accout for years to double. A simple rule of thumb based o this aalysis is sometimes used by practitioers kow as the Rule of 72. If oe divides 72 (it is slightly higher tha 00 times the atural log of 2) by the iterest rate of the accout expressed as a percetage (i this case, 5% which is slightly higher tha 00 times the atural log of plus the iterest rate), oe ca approximate the legth of time required for a bak accout to double. I this case, our approximatio would hold that the accout would require 72/5 = 4.4 years to double. 2.7 SUBSCRIPTS Cosider the expressio x i, which might be read x subscript i or x sub i. Subscripts are variables or umbers that serve to idetify the variables with which they are associated. For example, i the expressio x + x 2 + x 3, which reads x sub oe plus x sub two plus x sub three, the subscripts distiguish the x variables from each other. Sometimes it is more practical to idetify variables with subscripts tha with differet letters, especially whe there are may variables or there is a well-defied order to them. Frequetly, the subscript t is used to idetify time periods. For example, suppose that a ivestmet will pay its ower $00 i oe year, $200 i two years, $300 i three years, ad $400 i four years. Let the variable CF t refer to the cash flow (paymet) geerated by the ivestmet for a give year t. The followig table associates each cash flow with its appropriate idetifyig subscript: t CF t CF = $00 2 CF 2 = $200 3 CF 3 = $300 4 CF 4 = $ SUMMATIONS (Backgroud readig: sectios 2.3 ad 2.7) The summatio, or sigma, otatio ( ) is a shorthad meas of expressig our itet to add a series of variables. Cosider the followig expressio: y= i= x i, (2.6) which reads y equals the sum from i equals to of x sub i. This expressio orders values of x from oe (the first x) to (the last x) ad sums them. The subscript i serves as the couter (its value icreases by oe as we cotiue the operatio) ad the iteger uder the uppercase sigma ( ) is the startig value for the order. The stoppig poit for

9 QRMC02 9/7/0 4:39 PM Page 5 Summatios 5 the order is the superscript. The sigma meas that we will add all of the x values withi the stated rage. Suppose that we wish to sum the followig ordered series of variables: x, x 2, x 3, ad x 4. There are two ways to represet this operatio: 4 x i i = = x + x 2 + x 3 + x 4. If the series is log or ivolves may variables, the summatio otatio is a more efficiet meas of represetig the summatio operatio. For most fuctios ivolvig the summatio otatio, it will be ecessary to perform aother series of operatios before addig. For example, cosider the followig: 3 i 2 i = z = x = x + x + x, which reads z equals the sum from i equals oe to three of x squared. Each of the x values must be squared before the summatio takes place. Suppose, for example that x = 3, x 2 = 6, x 3 = 9, ad x 4 = 2. The z would equal = = 270. APPLICATION 2.4: MEAN VALUES (You may also wish to read related material i sectio 5.) A stock geerated the followig profits over a five-year period: Year Profit ($) Let the subscript t represet time (years through 5) ad let π t represet profit for a give year. The stoppig value for our summatio will equal 5. We ca determie the average or mea (E) profit over the five-year period as follows: 5 πt πt t = t = E = = = = The value that we have computed here is ofte referred to as the arithmetic mea. We simply add the observed values ad divide by the umber of observatios.

10 QRMC02 9/7/0 4:39 PM Page 6 6 Elemetary mathematics: fuctios ad operatios 2.9 DOUBLE SUMMATIONS (Backgroud readig: sectio 2.8) Double summatios are very useful whe workig with variables that are to be paired ad ordered for operatios. The double summatio otatio is very useful for cocisely represetig may of the types of log ad repetitive series of operatios that arise i fiace. The subscript otatio ca be geeralized to deote a variety of characteristics associated with a variable. For example, cosider the followig array or matrix of umbers o the left, which ca be idetified by row ad colum umbers give i the array o the right: a a a a a a a a a, 2, 3, 2, 22, 23, 3, 32, 33, Each of the umbers i the first matrix is associated with a variable a i,j, where the first subscript i represets the row of elemet a i,j ad j represets the colum of elemet a i,j. Each of the three elemets i a give row will have the same i subscript; each of the elemets i a give colum will have the same j subscript. For example, sice the umber 2 is i the secod row ad third colum, it is idetified with variable a 2,3. Suppose that we wished to add the elemets i the three rows ad colums. We could add elemets across the top row, the across the secod row ad the across the bottom row ad fially add the three row totals. This operatio is equivalet to y= y= y= 3 3 a ij, i= j= 3 3 a ij, i= j= 3 3 a ij, i= j= = [a, + a,2 + a,3 ] + [a 2, + a 2,2 + a 2,3 ] + [a 3, + a 3,2 + a 3,3 ], = [ ] + [ ] + [ ], = [20] + [22] + [22] = 64. We started with row (i = ) ad summig the three variables i row. With i =, we set j =, j = 2, ad j = 3 i order. We summed the three variables a,, a,2, ad a,3 to obtai 20. Whe we completed the summatio i the first row, we moved to the secod row, settig i equal to 2. With i = 2, we set j =, j = 2, ad j = 3 i order. We summed the three variables a 2,, a 2,2, ad a 2,3 to obtai 23. We the repeated this operatio for the third row. Fially, we summed the three row totals to obtai 64.

11 QRMC02 9/7/0 4:39 PM Page 7 Products PRODUCTS (Backgroud readig: sectio 2.8) The product, or, otatio provides a shorthad expressio multiplyig together a series of variables. Cosider the followig expressio: which reads y equals the product from i equals to of terms labeled x sub i. This expressio orders values of x from oe to ad multiplies them. The subscript i serves as the couter (its value icreases by oe as we cotiue the operatio) ad the iteger uder the upper-case pi ( ) is the startig value for the order. The stoppig poit for the order is the superscript. The pi meas that we will multiply together all of the x values. Suppose that we wish to multiply the followig ordered series of variables: x, x 2, x 3, ad x 4. There are two ways to represet this operatio: 4 xi = x x x x i= y= If the series is log or ivolves may variables, the product otatio is a more efficiet meas of represetig the multiplicatio operatios. For may fuctios ivolvig the product otatio, it will be ecessary to perform aother series of operatios before multiplyig. For example, cosider the followig: 3 i 2 i= x i i= z = x = x x x, which reads z equals the product from i equals oe to three of x squared. Each of the x values must be squared before the multiplicatio takes place., APPLICATION 2.5: GEOMETRIC MEANS (Backgroud readig: applicatio 2.4) Geometric meas are particularly useful for situatios ivolvig compoud iterest, returs, or growth. The geometric mea of a series is computed as the th root of the product of the series C g = i= x. i (2.7)

12 QRMC02 9/7/0 4:39 PM Page 8 8 Elemetary mathematics: fuctios ad operatios For example, suppose the value of a $,000 security held by a ivestor grew by 0% i its first year ad 20% i its secod year, but lost 30% i its third year. Its fial value would be determied as follows: V 3 =,000 [( + 0.0) ( ) ( 0.30)] = 924. The arithmetic mea retur o this ivestmet is computed to be 0 ([ ] 3 = 0), although we saw i the above equatio that the ivestmet actually lost value. This discrepacy is due to the fact that the arithmetic mea retur does ot accout for the compoudig of growth before the year it lost value. The geometric mea retur, computed as follows, will accout for the compoudig of growth: C g = ( ) ( ) ( 0. 30) = = = We see that, o average, the ivestmet will be as large i each of the three years tha i the previous year; thus, the geometric mea growth rate is computed as = APPLICATION 2.6: THE TERM STRUCTURE OF INTEREST RATES (Backgroud readig: applicatio 2.5) The term structure of iterest rates is cocered with the chage i iterest rates o debt securities resultig from varyig times to maturity o the debt. For example, it may be cocered with explaiig why the iterest rate o debt maturig i oe year might be 4% versus 7% for debt maturig i 20 years. Geerally, at a give poit i time, we observe loger-term iterest rates exceedig shorter-term rates, although this is ot always the case (for example, the years 980 3). There are several theories which attempt to explai relatioship betwee log- ad short-term iterest rates. Amog the best kow is the Pure Expectatios Theory, which states that log-term spot rates (iterest rates o loas origiatig ow) ca be explaied as a product of short-term spot rates ad short-term forward rates (iterest rates o loas committed to ow but actually origiatig at later dates). For example, if we borrow ow for three years at 5%, the three-year spot rate equals 5%. If we commit ow with a bak to borrow i oe year at 8% for oe year, the oe year forward rate for oe year equals 8%. Thus, we may write y 0,3 = 0.05 ad y,2 = 0.08, where the first subscript represets the year i which the loa is origiated ad the secod subscript represets the year i which the loa is repaid. Where y t,m is the rate o a loa origiated at time t to be repaid at time m, the Pure Expectatios Theory defies the relatioship betwee log- ad short-term iterest rates as follows: ( + y0, ) = ( + yt, t). t= (2.8) Thus, the log-term spot rate y 0,t is defied as the th root of the product of the oeperiod spot rate y 0, ad a series of oe-period forward rates y t,t mius oe. That is,

13 QRMC02 9/7/0 4:39 PM Page 9 Factorial products 9 the log-term spot rate equals the geometric mea of the short-term (oe-year) spot ad forward rates mius oe. Cosider a example where we ca borrow moey today for oe year at 4%; y 0, = Suppose that we are able to obtai a commitmet to obtai a oe-year loa oe year from ow at a iterest rate of 8%. Thus, the oe-year forward rate o a loa origiated i oe year equals 8%. Accordig to the Pure Expectatios Theory, we compute the two-year spot rate as follows: 2 ( + y02, ) 2 = ( + y t, t ) = ( )( ) =. 34, t= y 0,2 = ( )( ) /2 =. 34 = FACTORIAL PRODUCTS (Backgroud readig: sectio 2.0) Factorial products are very useful for determiig umbers of possible combiatios of objects ad for workig through differet types of probability distributios. A factorial product (!) is defied as follows:! = ( ) ( 2) ( 3)... 2 = ( i). (2.9) For example, 5! = = 20. Suppose that we wished to purchase five give stocks oe at a time. How may possible orderigs of trasactios are there? There are five differet stocks that we could purchase first. Give our first purchase, there are four differet stocks that we could purchase secod. Thus, there are 5 4 = 20 possible orderigs for the first two purchases. The, there are three possible orderigs for the third purchase, give our first two purchases. This process cotiues util we otice that there are 5! = = 20 possible orderigs for the five stock purchases. Thus,! represets the umber of potetial orderigs or permutatios of a series of evets. We will make extesive use of factorial products later whe we discuss probability ad optio pricig. We also ote that 0! = ad that 69! i= 0 APPLICATION 2.7: DERIVING THE NUMBER E (Backgroud readig: sectio 2.5) The umber e was defied i sectio 2.5 ad approximated at A secod defiitio which is ofte quite useful i fiace ad probability computatios is give as follows: e = = ! i= 0 (2.0)

14 QRMC02 9/7/0 4:39 PM Page Elemetary mathematics: fuctios ad operatios 2.2 PERMUTATIONS AND COMBINATIONS (Backgroud readig: sectio 2.) A permutatio of a set of objects is a ordered arragemet of some or all of the objects i the set. I sectio 2., we demostrated that a set of objects ca be ordered! ways. More geerally, a set of r objects take from a set of objects ca be ordered P r ways where P r is defied as follows:! P r = = ( )( 2) ( r + ). (2.) ( r)! For example, suppose that a ivestor wishes to purchase 5 stocks from 2 i a particular idustry. Suppose that she will select 5 stocks from the 2 ad the eter purchases i sequece oe at a time. How may possible orderigs (permutatios) of 5 purchases from a idustry of 2 must she select from? This umber is determied as follows: 2! 479, 00600, Pr = 2P5 = = = 95, 040. ( 2 5)! 5, 040 There are 2 possible selectios for the first stock, for the secod, 0 for the third, ad so o: P r = 2 P 5 = = 95,040. We have determied the possible umber of orderigs of stock purchases; ow we will determie the umber of possible combiatios of stocks. We will ow assume that the purchase order for the five stocks combies all five stocks, rather tha assume five separate sequetial purchase orders. Where P r represeted the umber of orderigs of sequetial purchases, C r (which reads choose r ) will represet the umber of possible combiatios irrespective of order: Cr =! P r = r r r r =!( )!!. (2.2) I our example, there will be 792 possible combiatios of 2 stocks from which the ivestor ca select 5: C 2 5 C ! = 5 = = 52!( 5)! ( ) ( ) , 040 = = = =

15 QRMC02 9/7/0 4:39 PM Page 2 Exercises Solve each of the followig for y: EXERCISES (a) y = 5 2 ; (e) y = 5 2 ; (i) y = 5 /2 ; (m) y = 5 3/2 ; (b) y = 5 3 ; (f) y = 5 3 ; (j) y = 5 /3 ; () y = 5 /3 ; (c) y = 5 4 ; (g) y = 5 4 ; (k) y = 5 /4 ; (o) y = 5 3/4 ; (d) y = 25 2 ; (h) y = 25 2 ; (l) y = 25 /2 ; (p) y = 25 3/ Solve each of the followig for y: (a) y = ; (e) y = 5 /2 5 3/2 ; (b) y = ; (f) y = 5 /3 5 3 ; (c) y = ; (g) y = 5 /4 5 3/4 ; (d) y = ; (h) y = 25 /4 25 3/ Solve the followig for FV: FV = 50( + 0.3) Solve the followig for PV: PV = 80( + 0.2) Simplify the followig: (a) y = 4x 3 6(5x + 7x); (b) y = 7(2x 2 + 3x 5x 2 ) + 4x; (c) y = 25 + {2x[3x + 4(x + 5) 2 6] 5} 7x Solve each of the followig for y: (a) y = 9(4( ) 2 ) (2 + 7); (b) y = (( )( ) 3 ) Solve the followig for PVA: 2.8. Fid the followig atural logs: PVA = $ ( ) 7 (a) l( ); (d) l(.); (g) l(3); (j) l(,000). (b) l(0.5); (e) l(2.7); (h) l(0); (c) l(); (f) l(2.78); (i) l(00); 2.9. Determie how log it takes to double the moey i a bak payig iterest compouded aually at each of the followig rates:

16 QRMC02 9/7/0 4:39 PM Page Elemetary mathematics: fuctios ad operatios (a) 0.03; (c) 0.07; (e) 0.2; (b) 0.05; (d) 0.0; (f ) Work through each of the parts of problem 2.8 usig the Rule of The future value of a accout whose iterest is compouded ca be approximated with the followig future-value formula: FV = PV e m. A simple rule used by practitioers holds that the legth of time that it takes for a accout to double whe iterest is compouded equals 0.72 divided by the iterest rate (expressed as a decimal). Demostrate this rule to be approximately correct usig the future-value equatio expressed above Demostrate that the umber e is the future value of a accout with a iitial deposit of $ ad a iterest rate of 0.0 compouded cotiuously Cosider the followig series: t X (a) Idetify each of the variables X with a subscript t. (b) What is the value of 4 t=x t? (c) What is the value of 4 t=3x t? (d) What is the value of 4 t=3x 2 t? (e) What is the value of t=3x 4 2 t? (f ) What is the arithmetic mea of X t? (g) What is the geometric mea of X t? 2.4. Let w = 0.4, w 2 = 0.5, ad w 3 = 0.. Furthermore, the followig table (or matrix ) provides values for x i,j : (a) Compute 3 i=w i. (b) Compute 3 i= 3 j=w i w j x i,j

17 QRMC02 9/7/0 4:39 PM Page 23 Appedix 2.A The Excel spreadsheet Suppose that the oe-year spot rate y 0, of iterest is 5%. Ivestors are expectig that the oe-year spot rate oe year from ow will icrease to 6%; thus, the oe-year forward rate y,2 o a loa origiated i oe year is 6%. Furthermore, assume that ivestors are expectig that the oe-year spot rate two years from ow will icrease to 7%; thus, the oe-year forward rate y 2,3 o a loa origiated i two years is 7%. Based o the pure expectatios hypothesis, what is the three-year spot rate? 2.6. Solve the followig for FV: FV = 4 t=25( + 0.0) t Solve each of the followig for x: (a) x = 8!; (b) x = 9!; (c) x = 0! Solve each of the followig: (a) 6P 4 ; (d) 6C 4 ; (b) 4P 4 ; (e) 4C 4 ; (c) 8P 2 ; (f) 8C 2. APPENDIX 2.A AN INTRODUCTION TO THE EXCEL SPREADSHEET This appedix is iteded to itroduce the reader to a few of the bare essetials for workig with spreadsheets such as Excel, Lotus, ad Quattro Pro. We will emphasize Excel here, because it is curretly the best-sellig spreadsheet program, but most of these basics will also apply to other spreadsheet programs as well. Excel is a electroic spreadsheet program desiged to perform mathematics, statistics, ad fiacial computatios as iteded by the user. The spreadsheet is a rectagular array comprisig cells that are orgaized ito rows idetified by umbers ad colums idetified by letters. Normally, whe oe loads a ew Excel file, the cursor is iitiated i cell A, the uppermost ad leftmost cell o the spreadsheet. Arrow keys o the keyboard ca be used to move the cursor about the scree. The user creates a spreadsheet file by movig the cursor to appropriate cells ad makig etries with the keyboard or mouse. Three of the most importat types of etries are values, labels, ad formulas. Suppose, for example, that I wished to compose a simple icome statemet for a firm. The firm had $,000 i sales ad $600 i costs. I could eter labels i colum A to idetify the umbers ad formula that I will eter i colum B. As we see i table 2.A., I eter labels i cells A, A2, ad A3 for Sales, Costs ad Profits. We eter umbers i cells B ad B2 for Sales ad Costs. I cell B3, I eter i the formula =b2-b3 to calculate profits. The user should be aware of a few spreadsheet covetios. Label etries either begi with a letter or a apostrophe. Number etries begi with a umber, decimal sig, plus

18 QRMC02 9/7/0 4:39 PM Page Elemetary mathematics: fuctios ad operatios Table 2.A. A simple icome statemet example 2 3 A B Sales 000 Costs 600 Profits =b2-b What you eter 2 3 A B Sales 000 Costs 600 Profits 400 What appears o your scree. Table 2.A.2 A simple formula example A B 2 Coefficiet Base Expoet 2 4 Result =b*b2^b3 What you eter A B 2 Coefficiet Base Expoet 2 4 Result 48 What appears o your scree sig, or mius sig. Formula etries ca begi with a equals sig or a umber. The sig for the multiplicatio operatio is * ad the expoet operatio is ^. Thus, 3 times 4 squared would be etered i the spreadsheet as 3*4^2. If the user wished to allow the coefficiet, base ad expoet to vary, he could create a spreadsheet like that i table 2.A.2. I eter umerical values for my coefficiet, base, ad expoet i cells B, B2, ad B3. The result to my equatio is obtaied by typig the formula =b*b2^b3 ito cell B4. Now, the user may chage values i cells B, B2, ad B3 to obtai the result for ay equatio of the form y = cx. To edit a formula or other etry, the user may move the cursor to the appropriate cell ad left click, the move the cursor to the Formula Bar ear the top of the scree (the etry i the cell should appear there). Oe ca edit the cell i the Formula Bar. To save a worksheet file, the user may either click the diskette ico ear the top of the scree or click File i the top left-had corer, the Save As ad follow istructios. To prit a file, the user may either click the priter ico ear the top of the scree or click File i the top left-had corer, the Prit ad follow istructios. Oe exits by clickig the X butto i the top right-had corer of the scree. The Escape key (Esc) i the upper left-had corer of the keyboard may sometimes be used to block a actio that has bee iitiated. Amog the most useful fuctios for fiace applicatios i Excel is the Paste Fuctio ( f x ) o the toolbar ear the top of the scree. This fuctio will retur a meu of categories of formulas icludig fiacial, statistical, math, ad trig fuctios. Work through the meu ad follow istructios to use a particular fuctio. Fuctio defiitios ad help ca be obtaied by clickig Help or a questio mark butto o a page or pallette. Perhaps the best way to lear how to use a spreadsheet is simply to experimet. Moder electroic spreadsheets are meu-drive. Meus ad toolbars ear the top of the scree will provide may optios ad categories of commads ad fuctios. Oe of the most importat thigs to remember while workig is to save frequetly ad to back up your spreadsheet files, because accidets frequetly occur that ca cause you to lose work.