Chapter 11 Appedices: Review of Topics from Foudatios i Fiace ad Tables A: INTRODUCTION The expressio Time is moey certaily applies i fiace. People ad istitutios are impatiet; they wat moey ow ad are geerally willig to pay (or impose a charge) for havig moey ow (or havig to wait). The time value of moey is certaily amog the most importat cocepts i fiace. Iterest is a charge imposed o borrowers for the use of leders' moey. The iterest cost is usually expressed as a percetage of the pricipal (the sum borrowed). Whe a loa matures, the pricipal must be repaid alog with ay upaid accumulated iterest. I a free market ecoomy, iterest rates are determied joitly by the supply of ad demad for moey. Thus, leders will usually attempt to impose as high a iterest rate as possible o the moey they led; borrowers will attempt to obtai the use of moey at the lowest iterest rates available to them. Factors affectig the levels of iterest rates will do so by affectig supply ad demad coditios for moey. Amog these factors are: 1. Iflatio: Because of dimiished purchasig power, moey received i the future by leders is worth less tha the moey they led ow. Leders will require a premium (iterest) i additio to the pricipal to compesate them for this loss of purchase power. Furthermore, iflatio makes curret moey balaces more attractive to borrowers. Thus, iflatio decreases the supply of ad icreases the demad for moey. Iterest rates will icrease as the rate of iflatio icreases. 2. Risk or Ucertaity: Creditors aturally prefer to kow with certaity that the moey they loa will be repaid i its etirety. If leders are ucertai as to whether their loas will be repaid, they will require premiums to compesate them for this risk. Higher iterest rates will result from icreased ucertaity. 3. Itertemporal Moetary Prefereces: I geeral, cosumers (ad corporatios) will prefer to have moey ow rather tha be forced to wait for it. If cosumers have moey ow, they ca choose to sped it ow or sped it at some later date. However, if cosumers must wait for their moey, they do ot have the optio to sped it ow; they must wait for some later date to sped it. If cosumers icrease their desire to sped more ow rather tha later, iterest rates will icrease. 4. Govermet Policy: Govermetal moetary policy will affect both supply ad demad coditios for moey. Through moetary policy, the govermet ca directly cotrol the supply of moey; ad through its participatio i bod markets, it ca ifluece the demad for moey. Govermetal fiscal policy (spedig ad tax programs) have a sigificat effect o the demad for moey. 5. Costs of Extedig Credit: Both leders ad borrowers face various egotiatig ad admiistrative costs whe a loa is exteded. Most of these costs ca be categorized as trasactios costs. Leders will require iitiatio fees such as "poits" or higher iterest paymets as compesatio for these costs. B: CALCULATION OF SIMPLE INTEREST Iterest is computed o a simple basis if it is paid oly o the pricipal of the loa.
Compoud iterest is paid o accumulated loa iterest as well as o the pricipal. Thus, if a sum of moey (X 0 ) were borrowed at a aual iterest rate (i) ad repaid at the ed of () years with accumulated iterest, the total sum repaid (FV or Future Value at the ed of Year ) is determied as follows: (1) FV X 0 (1 + i) The subscripts () ad (0) merely desigate time; they do ot imply ay arithmetic fuctio. The product ( i) whe multiplied by X 0 reflects the value of iterest paymets to be made o the loa; the value (1) accouts for the fact that the pricipal of the loa must be repaid. If the loa duratio icludes some fractio of a year, the value of () will be fractioal; e.g., if the loa duratio were oe year ad three moths, () would be 1.25. The total amout paid (or, the Future Value of the loa) will be a icreasig fuctio of the legth of time the loa is outstadig () ad the iterest rate (i) charged o the loa. For example, if a cosumer borrowed $1000 at a iterest rate of 10% for oe year, his total repaymet would be $1100, determied from Equatio 1 as follows: FV 1 $1000(1 + 1.1) $1000*1.1 $1100 If the loa were to be repaid i two years, its future value would be determied as follows: FV 2 $1000(1 + 2.1) $1000*1.2 $1200 Cotiuig our example, if the loa were to be repaid i five years, its future value would be: FV 5 $1000(1 + 5.1) $1000*1.5 $1500 The loger the duratio of a loa, the higher will be its future value. Thus, the loger leders must wait to have their moey repaid, the greater will be the total iterest paymets made by borrowers. C: CALCULATION OF COMPOUND INTEREST Iterest is computed o a compoud basis whe a borrower must pay iterest o ot oly the loa pricipal, but o accumulated iterest as well. If iterest must accumulate for a full year before it is compouded, the Future Value of such a loa is determied with Equatio (2): (2) FV X 0 (1 + i) For example, if a idividual were to deposit $1000 ito a savigs accout payig aually compouded iterest at a rate of 10% (here, the bak is borrowig moey), the future value of the accout after five years would be $1610.51, determied by Equatio 2 as follows: FV 5 $1000(1+.1) 5 $1000 x 1.1 5 $1000 x 1.61051 $1610.51 Notice that this sum is greater tha the future value of the loa ($1500) whe iterest is ot 10
compouded. The compoud iterest formula ca be derived ituitively from the simple iterest formula. If iterest must accumulate for a full year before it is compouded, the the future value of the loa after oe year is $1100, exactly the same sum as if iterest had bee computed o a simple basis: (3) FV X 0 (1+i) X 0 (1+ 1i) X 0 (1+i) 1 $1000(1+.1) $1100 The future values of loas where iterest is compouded aually ad whe iterest is computed o a aual basis will be idetical oly whe () equals oe. Sice the value of this loa is $1100 after oe year ad iterest is to be compouded, iterest ad future value for the secod year will be computed o the ew balace of $1100: (4) FV 2 X 0 (1+1i)(1+1i) X 0 (1+i)(1+i) X 0 (1+i) 2, FV 2 $1000 (1 +.1)(1 +.1) $1000 (1 +.1) 2 $1210 This process ca be cotiued for five years: FV 5 $1000(1+.1)(1+.1)(1+.1)(1+.1)(1+.1) $1000(1+.1) 5 $1610.51 More geerally, the process ca be applied for a loa of ay maturity. Therefore: FV X 0 (1+i)(1+i)... (1+i) X 0 (1+i), FV $1000 (1 +.1)(1 +.1)... (1 +.1) $1000 (1 +.1) D. FRACTIONAL PERIOD COMPOUNDING OF INTEREST I the previous examples, iterest is compouded aually; that is, iterest must accumulate at the stated rate i for a etire year before it ca be compouded or re-compouded. I may savigs accouts ad other ivestmets, iterest ca be compouded semiaually, quarterly or eve daily. If iterest is to be compouded more tha oce per year (or oce every fractioal part of a year), the future value of such a ivestmet will be determied as follows: (6) FV X 0 (1 + i/m) m, where iterest is compouded (m) times per year. The iterpretatio of this formula is fairly straightforward. For example, if (m) is 2, the iterest is compouded o a semiaual basis. The semiaual iterest rate is simply (i/m) or (i/2). If the ivestmet is held for () periods, the it is held for (2) semiaual periods. Thus, we compute a semiaual iterest rate (i/2) ad the umber of semiaual periods the ivestmet is held (2*). If $1000 were deposited ito a savigs accout payig iterest at a aual rate of 10% compouded semiaually, its future value after five years would be $1628.89, determied as follows: 11
FV 5 $1000(1 +.1/2) 2x5 $1000(1.05) 10 $1000(1.62889) $1628.89 Notice that the semiaual iterest rate is five percet ad that the accout is outstadig for te six-moth periods. This sum ($1628.89) exceeds the future value of the accout if iterest is compouded oly oce aually ($1610.51). I fact, the more times per year iterest is compouded, the higher will be the future value of the accout. For example if iterest o the same accout were compouded mothly (twelve times per year), the accout's future value would be $1645.31: FV 5 $1000(1 +.1/12) 12x5 $1000(1.008333) 60 $1645.31 The mothly iterest rate is.008333 ad the accout is ope for (m*) or 60 moths. With daily compoudig, the accout's value would be $1648.60: FV 5 $1000(1 +.1/365) 365x 5 $1648.60 Therefore, as (m) icreases, future value icreases. However, this rate of icrease i future value becomes smaller with larger values for (m); that is, the icreases i (FV ) iduced by icreases i (m) evetually become quite small. Thus, the differece i the future values of two accouts where iterest is compouded hourly i oe ad every miute i the other may actually be rather trivial. E. CONTINUOUS COMPOUNDING OF INTEREST If iterest were to be compouded a ifiite umber of times per period, we would say that iterest is compouded cotiuously. However, we caot obtai a umerical solutio for future value by merely "pluggig" i ifiity for m i Equatio 6 - calculators have o ifiity key. I the previous sectio, we saw that icreases i (m) cause the future value of a ivestmet to icrease. As (m) approaches ifiity, (FV ) cotiues to icrease, however at decreasig rates. More precisely, as (m) approaches ifiity, the future value of a ivestmet ca be defied as follows: (7) FV X 0 e i where (e) is the atural log whose value ca be approximated at 2.718 or derived from the followig: (8) 1 e lim (1 + ) m m m That is, as (m) approaches ifiity, the value of the limit i expressio (8) approaches the umber (e). Notice the similarity betwee Equatios (6), (8) ad (9). I fact, Equatio (7) ca be derived easily from Equatios (6) ad (9) which defies e i as follows: i i m (9) e lim (1 + ) m m 12
I may calculatios ivolvig cotiuous compoudig of iterest, the value 2.718 serves as a approximatio for the umber (e). If a ivestor were to deposit $1000 ito a accout payig iterest at a rate of 10%, cotiuously compouded (or compouded a ifiite umber of times per year), the accout's future value would be approximately $1648.72: FV 5 $1000 * e.1 5 $1000 * 2.718.5 $1648.72 The future value of this accout exceeds oly slightly the value of the accout if iterest were compouded daily. Also ote that cotiuous compoudig simply meas that iterest is compouded a ifiite umber of times per time period. Years to maturity () Future Value Simple Iterest Future Value Compouded Aually Future Value Compouded Mothly Future Value Compouded Daily Future Value Compouded Cotiuously 1 110 110 110.47 110.52 110.52 2 120 121 122.04 122.14 122.14 3 130 133.31 134.81 134.98 134.99 4 140 146.41 148.94 149.17 149.18 5 150 161.05 164.53 164.86 164.87 10 200 259.37 270.70 271.79 271.83 20 300 672.75 732.81 738.70 738.91 30 400 1,744.94 1,983.74 2,007.73 2,008.57 50 600 11,739.09 14,536.99 14,831.16 14,841.40 Aual Percetage Yield varies with.100000.104713.1051557.1051709 TABLE 1: Future Values ad Aual Percetage Yields of accouts with iitial $100 deposits at 10% iterest 13
QUESTIONS AND PROBLEMS 1. Why do iterest rates charged by baks for the purchase of automobiles ted to exceed iterest rates paid o savigs accouts? Why are home loa (mortgage) iterest rates usually lower tha iterest rates charged credit card customers? 2. The Williams Compay has borrowed $10,500 at a aual iterest rate of ie percet. How much will be a sigle lump sum repaymet i eight years icludig both pricipal ad iterest accumulated o a simple basis? That is, what is the future value of this loa? 3. The Cobb Compay has issued te millio dollars i te percet coupo bods maturig i five years. Iterest paymets o these bods will be made semi-aually. a. How much are Cobb's semi-aual iterest paymets? b. What will be the total paymet made by Cobb o the bods i each of the first four years? c. What will be the total paymet made by Cobb o the bods i the fifth year? 4. What would be the lump sum loa repaymet made by the Williams Compay i Problem 2 if iterest were compouded: a. aually? b. Semiaually? c. mothly? d. daily? e. cotiuously? 5. The Speaker Compay has the opportuity to purchase a five-year $1000 certificate of deposit (C.D.) payig iterest at a aual rate of 12%, compouded aually. The compay will ot withdraw early ay of the moey i its C.D. accout. Will this accout have a greater future value tha a five-year $1000 C.D. payig a aual iterest rate of 10%, compouded daily? 6. The Waer Compay eeds to set aside a sum of moey today for the purpose of purchasig for $10,000 a ew machie i three years. Moey used to fiace this purchase will be placed i a savigs accout payig iterest at a rate of eight percet. How much moey must be placed i this accout ow to assure the Waer compay $10,000 i three years if iterest is compouded yearly? 7. Assumig o withdrawals or additioal deposits, how much time is required for $1000 to double if placed i a savigs accout payig a aual iterest rate of 10% if iterest were: a. computed o a simple basis? b. compouded aually? c. compouded mothly? d. compouded cotiuously? Solutios 2. FV 8 10,500 (1 + 8 x.09) 10,500 x 1.72 18,060 14
3. a. [10% x $10,000,000] / 2 $500,000 b. 10% x $10,000,000 2 x $500,000 $1,000,000 c. $10,000,000 + $1,000,000 Pricipal + iterest i year five $11,000,000 4. a. FV 8 10,500 (1 +.09) 8 10,500 x 1.99256 20,921.908 b. FV 8 10,500 (1 +.09) 2x8 10,500 x 2.0223702 2 21,234.887 c. FV 8 10,500 (1 +.09) 12x8 10,500 x 2.0489212 12 21,513.673 d. FV 8 10,500 (1 +.09) 365x8 10,500 x 2.0542506 365 21,569.632 e. FV 8 10,500 e.09x8 10,500 x 2.0544332 21,571.549 5. For example, let X 0 $1000 i each case For CD 1 : FV 5 1000(1 +.12) 5 1,762.3417 For CD 2 : FV 5 1000(1.10) 365x5 1,648.6005 365 6. Solve for X 0 : X 0 FV 10,000 7938.322 (1+i) (1+.08) 3 7. I all cases here, FV 2X 0. Thus, let FV 2000 ad X 0 1000 a. 2000 1000 (1 + x.1); 2(1 + x.1); 1.1; 10 years b. 2000 1000 (1.1) ; usig logs: log 2000 (log 1000) + *log (1.1) 3.30103 3 + *(.04139);.30103 (.04139); 7.2725 years c. 2000 1000 (1 +.10) 12 ; 12 log 2000 (log 1000) / [12 log(1.008333)] 6.9603407 years d. 2000 1000 e.1 ; use atural logs: l 2000 (l 1000) +.1; 6.931478 years 15
F. INTRODUCTION Cash flows realized at the preset time have a greater value to ivestors tha cash flows realized later for the followig reasos: 1. Iflatio: The purchasig power of moey teds to declie over time. 2. Risk: Oe ever kows for sure whether he will actually realize the cash flow that he expects. 3. The optio to either sped moey ow or defer spedig it is likely to be worth more tha beig forced to defer spedig the moey. The purpose of the Preset Value cocept is to provide a meas of expressig the value of a future cash flow i terms of curret cash flows. That is, the Preset Value cocept is used to determie how much a ivestor would pay ow for the promise of some cash flow to be received at a later date. The preset value of this cash flow would be a fuctio of iflatio, the legth of wait before the cash flow is received, its risk ad the time value a ivestor associates with moey (how much he eeds moey ow as opposed to later). Perhaps the easiest way to accout for these factors whe evaluatig a future cash flow is to discout it i the followig maer: CF (1) PV ( 1+ k) where (CF ) is the cash flow to be received i year (), (k) is a appropriate discout rate accoutig for risk, iflatio, ad the ivestor's time value associated with moey, ad PV is the preset value of that cash flow. The discout rate eables us to evaluate a future cash flow i terms of cash flows realized today. Thus, the maximum a ratioal ivestor would be willig to pay for a ivestmet yieldig a $9000 cash flow i six years assumig a discout rate of 15% would be $3891, determied as follows: PV $9000 (1 + 15) $9000 2.31306 6 $3890.95 I the above example, we simply assumed a fiftee percet discout rate. Realistically, perhaps the easiest value to substitute for (k) is the curret iterest or retur rate o loas or other ivestmets of similar duratio ad risk. However, this market determied iterest rate may ot cosider the idividual ivestor's time prefereces for moey. Furthermore, the ivestor may fid difficulty i locatig a loa (or other ivestmet) of similar duratio ad risk. For these reasos, more scietific methods for determiig appropriate discout rates will be discussed later. I ay case, the discout rate should accout for iflatio, the ivestmet risk ad the ivestor's time value of moey. G. DERIVING THE PRESENT VALUE FORMULA The preset value formula ca be derived easily from the compoud iterest formula. Assume a ivestor wishes to deposit a sum of moey ito a savigs accout payig iterest at a 16
rate of fiftee percet, compouded aually. If the ivestor wishes to withdraw from his accout $9,000 i six years, how much must he deposit ow? This aswer ca be determied by solvig the compoud iterest formula for X 0 : FV $9000 $9000 FV X 0 (1 + i) ; X 0 $3890. 95 6 (1 + i) (1 + 15) 2.31306 Therefore, the ivestor must deposit $3890.95 ow i order to withdraw $9,000 i six years at fiftee percet. Notice that the preset value formula (3.1) is almost idetical to the compoud iterest formula where we solve for the pricipal (X 0 ): CF FV PV ; X 0 ( 1+ k) (1 + i) Mathematically, these formulas are the same; however, there are some differeces i their ecoomic iterpretatios. I the iterest formulas, iterest rates are determied by market supply ad demad coditios whereas discout rates are idividually determied by ivestors themselves (although their calculatios may be iflueced by market iterest rates). I the preset value formula, we wish to determie how much some future cash flow is worth ow; i the iterest formula above, we wish to determie how much moey must be deposited ow to attai some give future value. H: PRESENT VALUE OF A SERIES OF CASH FLOWS If a ivestor wishes to evaluate a series of cash flows, he eeds oly to discout each separately ad the sum the preset values of each of the cash flows. Thus, the preset value of a series of cash flows (CF t ) received i time period (t) ca be determied by the followig expressio: CFt (2) PV t (1 k) t 1 + For example, if a ivestmet were expected to yield aual cash flows of $200 for each of the ext five years, assumig a discout rate of 5%, its preset value would be $865.90: 200 (1 +.05) 200 (1 +.05) 200 (1 +.05) 200 (1 +.05) 200 + 865.90 (1 +.05) PV + + + 1 2 3 4 5 Therefore, the maximum price a idividual should pay for this ivestmet is $865.90 eve though the cash flows yielded by the ivestmet total $1000. Because the idividual must wait up to five years before receivig the $1000, the ivestmet is worth oly $865.90. Use of the preset value series formula does ot require that cash flows (CF t ) i each year be idetical, as does the auity model preseted i the ext sectio. 17
I: ANNUITY MODELS The expressio for determiig the preset value of a series of cash flows ca be quite cumbersome, particularly whe the paymets exted over a log period of time. This formula requires that () cash flows be discouted separately ad the summed. Whe () is large, this task may be rather time-cosumig. If the aual cash flows are idetical ad are to be discouted at the same rate, a auity formula ca be a useful time-savig device. The same problem discussed i the previous sectio ca be solved usig the followig auity formula: CF 1 (3) PVA 1 K (1 + k) where (CF) is the level of the aual cash flow geerated by the auity (or series). Use of this formula does require that all of the aual cash flows be idetical. Thus, the preset value of the cash flows i the problem discussed i the previous sectio is $865.90, determied as follows: PV A 200 (.05) 1 1 4000(.216475) $865.90 5 (1 +.05) As () becomes larger, this formula becomes more useful relative to the preset value series formula discussed i the previous sectio. However, the auity formula requires that all cash flows be idetical ad be paid at the ed of each year. The preset value auity formula ca be derived easily from the perpetuity formula discussed i the ext sectio or from the geometric expasio procedure described i the derivatio box. Note that each of the above calculatios assumes that cash flows are paid at the ed of each period. If, istead, cash flows were realized at the begiig of each period, the auity would be referred to as a auity due. Each cash flow geerated by the auity due would, i effect, be received oe year earlier tha if cash flows were realized at the ed of each year. Hece, the preset value of a auity due is determied by simply multiplyig the preset value auity formula by (1+k): CF 1 (4) PVdue 1 (1 + k) K (1 + k) The preset value of the five-year auity due discouted at five percet is determied: PV A 200 (.05) 1 1 (1 +.05) 4000(.2164738)(1.05) $909.19 5 (1 +.05) J: BOND VALUATION Because the preset value of a series of cash flows is simply the sum of the preset values of the cash flows, the auity formula ca be combied with other preset value formulas to evaluate ivestmets. Cosider, for example, a 7% coupo bod makig aual iterest paymets for 9 years. If this bod has a $1,000 face (or par) value, ad its cash flows are 18
discouted at 6%, its value ca be determied as follows: 70 1 1000 $1000 PV 1 $1166.67(.4081) 9 + + 9 (.06) (1 +.06) (1 +.06) 1.689479 $476.118 + 587.406 $1068.774 Thus, the value of a bod is simply the sum of the preset values of the cash flow streams resultig from iterest paymets ad from pricipal repaymet. Now, let us revise the above example to value aother 7% coupo bod. This bod will make semiaual (twice yearly) iterest paymets for 9 years. If this bod has a $1,000 face (or par) value, ad its cash flows are discouted at the stated aual rate of 6%, its value ca be determied as follows: 35 1 1000 $1000 PV 1 $1166.67(.4126) $481.368 587.406 $1068.774 18 + + + 18 (.03) (1 +.03) (1 +.03) 1.7024 Agai, the value of the bod is the sum of the preset values of the cash flow streams resultig from iterest paymets ad from the pricipal repaymet. However, the semi-aual discout rate equals 3% ad paymets are made to bodholders i each of eightee semi-aual periods. K: PERPETUITY MODELS As the value of () approaches ifiity i the auity formula, the value of the right had side term i the brackets: 1 (1 + k) approaches zero. That is, the cash flows associated with the auity are paid each year for a period approachig "forever." Therefore, as () approaches ifiity, the value of the ifiite time horizo auity approaches: (5) CF PV p k The perpetuity model is useful i the evaluatio of a umber of ivestmets. Ay ivestmet with a idefiite or perpetual life expectacy ca be evaluated with the perpetuity model. For example, the preset value of a stock, if its divided paymets are projected to be stable, will be equal to the amout of the aual divided (cash flow) geerated by the stock divided by a appropriate discout rate. I Europea fiacial markets, a umber of perpetual bods have bee traded for several ceturies. I may regios i the Uited States, groud rets (perpetual leases o lad) are traded. The proper evaluatio of these ad may other ivestmets requires the use of perpetuity models. PV p $200 $4000.05 The maximum price a ivestor would be willig to pay for a perpetual bod geeratig a aual cash flow of $200, each discouted at a rate of 5% ca be determied from Equatio (5): 19
L: GROWING PERPETUITY AND ANNUITY MODELS If the cash flow associated with a ivestmet were expected to grow at a costat aual rate of (g), the amout of the cash flow geerated by that ivestmet i year (t) would be: (6) CF t CF 1 (1+g) t-1, where (CF 1 ) is the cash flow geerated by the ivestmet i year oe. Thus, if a stock payig a divided of $100 i year oe were expected to icrease its divided paymet by 10% each year thereafter, the divided paymet i the fourth year would be $133.10: CF 4 CF 1 (1 +.10) 4-1 Similarly, the cash flow geerated by the ivestmet i the followig year (t+1) will be: (7) CF t+1 CF 1 (1 + g) t The stock's divided i the fifth year will be $146.41: CF 4+1 CF 1 (1+.10) 4 $146.41 If the stock had a ifiite life expectacy (as most stocks might be expected to), ad its divided paymets were discouted at a rate of 13%, the value of the stock would be determied by: PV gp $100.13.10 $100 $3333.33.03 This expressio is called the Gordo Stock Pricig Model. It assumes that the cash flows (divideds) associated with the stock are kow i the first period ad will grow at a costat compoud rate i subsequet periods. More geerally, this growig perpetuity expressio ca be writte as follows: (8) PV gp CF1 k g The growig perpetuity expressio simply subtracts the growth rate from the discout rate; the growth i cash flows helps to "cover" the time value of moey. This formula for evaluatig growig perpetuities ca be used oly whe (k) > (g). If (g) > (K), either the growth rate or discout rate has probably bee calculated improperly. Otherwise, the ivestmet would have a ifiite value (eve though the formula would geerate a egative value). CF1 (1 + g) (9) PVgp 1 k g (1 + k) 20
Cash flows geerated by may ivestmets will grow at the rate of iflatio. For example, cosider a project udertake by a corporatio whose cash flow i year oe is expected to be $10,000. If cash flows were expected to grow at the iflatio rate of six percet each year util year six, the termiate, the project's preset value would be $48,320.35, assumig a discout rate of 11%: PV gp $10,000 (1 +.06) 1.11.06 (1 +.11) 6 6 $200,000(1.7584) $48320.45 Cash flows are geerated by this ivestmet through the ed of the sixth year. No cash flow was geerated i the seveth year. Verify that the amout of cash flow that would have bee geerated by the ivestmet i the seveth year if it had cotiued to grow would have bee $10,000(1.06) 6 $14,185. M: STOCK VALUATION Cosider a stock whose aual divided ext year is projected to be $50. This paymet is expected to grow at a aual rate of 5% i subsequet years. A ivestor has determied that the appropriate discout rate for this stock is 10%. The curret value of this stock is $1000, determied by the growig perpetuity model: PV gp $50 $1000.10.05 This model is ofte referred to as the Gordo Stock Pricig Model. It may seem that this model assumes that the stock will be held by the ivestor forever. But what if the ivestor iteds to sell the stock i five years? Its value would be determied by the sum of the preset values of cash flows the ivestor expects to receive: PV GA DIV1 (1 + g) 1 k g (1 + k) where (P ) is the price the ivestor expects to receive whe he sells the stock i year (); ad (DIV 1 ) is the divided paymet the ivestor expects to receive i year oe. The preset value of the divideds the ivestor expects to receive is $207.53: PVGA $50 (1 +.05) 1.10.05 (1 +.10) 5 5 207.53 The sellig price of the stock i year five will be a fuctio of the divided paymets the prospective purchaser expects to receive begiig i year six. Thus, i year five, the prospective purchaser will pay $1276.28 for the stock, based o his iitial divided paymet of $63.81, determied by the followig equatios: 21
DIV 6 DIV 1 (1+.05) 6-1 $63.81 Stock value i year five 63.81/(.10-.05) $1276.28 The preset value of the $1276.28 the ivestor will receive whe he sells the stock at the ed of the fifth year is $792.47: $1276.28 PV $792.57 5 (1 +.1) The total stock value will be the sum of the preset values of the divideds received by the ivestor ad his cash flows received from the sale of the stock. Thus, the curret value of the stock is $207.53 plus $792.47, or $1000. This is exactly the same sum determied by the growig perpetuity model earlier; therefore, the growig perpetuity model ca be used to evaluate a stock eve whe the ivestor expects to sell it. 22
QUESTIONS AND PROBLEMS 1. What is the preset value of a security promisig to pay $10,000 i five years if its associated discout rate is: a. twety percet? b. te percet? c. oe percet? d. zero percet? 2. What is the preset value of a security to be discouted at a te percet rate promisig to pay $10,000 i: a. twety years? b. te years? c. oe year? d. six moths? e. sevety three days? 3. The Gehrig Compay is cosiderig a ivestmet that will result i a $2000 cash flow i oe year, a $3000 cash flow i two years ad a $7000 cash flow i three years. What is the preset value of this ivestmet if all cash flows are to be discouted at a eight percet rate? Should Gehrig Compay maagemet be willig to pay $10,000 for this ivestmet? 4. The Horsby Compay has the opportuity to pay $10,000 for a ivestmet payig $2000 i each of the ext ie years. Would this be a wise ivestmet if the appropriate discout rate were: a. five percet? b. te percet? c. twety percet? 5. The Foxx Compay is sellig preferred stock which is expected to pay a fifty dollar aual divided per share. What is the preset value of divideds associated with each share of stock if the appropriate discout rate were eight percet ad its life expectacy were ifiite? 6. The Evers Compay is cosiderig the purchase of a machie whose output will result i a te thousad dollar cash flow ext year. This cash flow is projected to grow at the aual te percet rate of iflatio over each of the ext te years. What will be the cash flow geerated by this machie i: a. its secod year of operatio? b. its third year of operatio? c. its fifth year of operatio? d. its teth year of operatio? 7. The Wager Compay is cosiderig the purchase of a asset that will result i a $5000 cash flow i its first year of operatio. Aual cash flows are projected to grow at the 10% aual rate of iflatio i subsequet years. The life expectacy of this asset is seve years, ad the appropriate discout rate for all cash flows is twelve percet. What is the maximum price 23
Wager should be willig to pay for this asset? 8. What is the preset value of a stock whose $100 divided paymet ext year is projected to grow at a aual rate of five percet? Assume a ifiite life expectacy ad a twelve percet discout rate. 9. Which of the followig series of cash flows has the highest preset value at a five percet discout rate: a. $500,000 ow b. $100,000 per year for eight years c. $60,000 per year for twety years d. $30,000 each year forever 10. Which of the cash flow series i Problem 9 has the highest preset value at a twety percet discout rate? 11. What discout rate i Problem 4 will reder the Horsby Compay idifferet as to its decisio to ivest $10,000 for the ie year series of cash flows? That is, what discout rate will result i a $10,000 preset value for the series? 12. What would be the preset value of $10,000 to be received i twety years if the appropriate discout rate of 10% were compouded: a. aually? b. mothly? c. daily? d. cotiuously? 12.a. What would be the preset value of a thirty year auity if the $1000 periodic cash flow were paid mothly? Assume a discout rate of 10% per year. b. Should a ivestor be willig to pay $100,000 for this auity? c. What would be the highest applicable discout rate for a ivestor to be willig to pay $100,000 for this auity? Solutios 1 a. PV CF 10,000 10,000 10,000 4018.775 (1+k) (1+.20) 5 1.2 5 2.48832 b. PV 10,000 10,000 6209.213 1.10 5 1.61051 c. PV 10,000 10,000 9514.656 1.01 5 1.0510101 d. PV 10,000 10,000 10,000 1.0 5 1 2. a. PV 10,000 10,000 1486.436 1.1 20 6.7275 b. PV 10,000 10,000 3855.432 1.1 10 2.5937425 c. PV 10,000 10,000 9090.909 1.1 1 1.1 24
d. PV 10,000 10,000 10,000 9534.625; 1.1.5 1.1.5 1.0488088 Note: 6 moths is.5 of oe year e. PV 10,000 10,000 9811.184; 1.1.2 1.0192449 Note: 73 days is.2 of oe year 3. PV Σ CF t 2000 + 3000 + 7000 t1 (1 + k) t 1.08 1 1.08 2 1.08 3 PV 1851.85 + 2572.02 5556.83 9980.70; 10,000 > 9980.70 Sice P 0 > PV, the ivestmet should ot be purchased. 4. PV CF [ 1-1 ] k k(1+k) a. PV A 2000[ 1-1 ] 2000[20-12.892178] 14,215.643.05.05(1.05) 9 b. PV A 2000[ 1-1 ] 2000[ 10-4.2409762 ] 11,518.048.10.10(1.10) 9 c. PV A [ 1-1 ] 2000 [ 5-.9690335] 8,061.933.2.2(1.2) 9 5. PV p CF 50 625 k.08 6. CF CF 1 (1 + g) -1 a. CF 2 10,000 (1 +.1) 2-1 10,000 (1 +.1) 10,000 x 1.1 11,000 b. CF 3 10,000 (1 +.1) 3-1 10,000 x 1.21 12,100 c. CF 5 10,000 (1 +.1) 5-1 10,000 x 1.4641 14,641 d. CF 10 10,000 (1 +.1) 10-1 10,000 x 2.3579477 23, 579.477 7. PV ga CF 1 x 1 _ (1 + g) k-g (k-g)(1+k) 5000 x 1 _ (1 +.10) 7.02 (.12-.10)(1 +.12) 7 PV ga 5000 x [50-44.075033] 29,624.837 8. PV gp CF 1 100 1428.5714 k-g.12-.05 9. $60,000 per year for 20 years a. PV 500,000 b. PV 100,000 [ 1 _ 1 ] 646,321.27.05.05(1.05) 8 c. PV 60,000 [ 1 _ 1 ] 747,73262.05.05(1.05) 20 d. PV 30,000 600,000.05 Series (c) has the highest preset value. 10. a. PV 500,000 b. PV 100,000 [ 1 _ 1 ] 383,715.98.2.2(1.2) 8 c. PV 60,000 [ 1 _ 1 ] 292,174.78.2.2(1.2) 20 d. PV 30,000 150,000.2 11. Plug discout rates ito the preset value auity fuctio util you fid oe that sets PV equal to the purchase price. Try 15%: PV 9543.1685 < 10,000 Try 13%: PV 10,803.31 > 10,000 Try 14%: PV 9,892.8294 < 10,000 Try 13.7%: PV 10,001.638 > 10,000 Try 13.71% PV 9,997.977 < 10,000 Try 13.704%: PV 10,000.174 > 10,000 25
Thus, K is approximately 13.704% 12. a. PV 10,000 10,000 1486.436 1.1 20 6.7275 b. PV 10,000 10,000 1364.615 (1+.1/12) 12*20 7.328074 c. PV 10,000 10,000 1353.7236 (1+.1/365) 365*20 7.3870321 d. PV 10,000 * e -.1*20 1353.3528 13. a. First, the mothly discout rate is.1 12.008333 PV 1,000 * [ 1 _ 1 ].008333.008333(1+.008333) 360 1,000 * 113.95082 $113,950.82 b. Yes, sice the PV exceeds the $100,000 price c. 100,000 1,000 * [ 1 _ 1 ] (k/12) (k/12)(1+k/12) 360 Solve for k; by process of substitutio, we fid that k.11627. 26
N: INTRODUCTION TO RETURNS The purpose of measurig ivestmet returs is simply to determie the ecoomic efficiecy of a ivestmet. Thus, a ivestmet's retur will express the profits geerated by a iitial cash outlay relative to the amout of that outlay. There exist a umber of methods for determiig the retur of a ivestmet. The measures preseted i this chapter are retur o ivestmet ad iteral rate of retur. Arithmetic ad geometric mea rates of retur o ivestmet will be discussed alog with iteral rate of retur ad bod retur measures. These methods differ i their ease of computatio ad how they accout for the timeliess ad compoudig of cash flows. O: RETURN ON INVESTMENT: ARITHMETIC MEAN Perhaps the easiest method to determie the ecoomic efficiecy of a ivestmet is to add all of its profits ( t ) accruig at each time period (t) ad dividig this sum by the amout of the iitial cash outlay (P 0 ). This measure is called a holdig period retur. To ease comparisos betwee ivestmets with differet life expectacies, oe ca compute a arithmetic mea retur o ivestmet (ROI) by dividig the holdig period retur by the life expectacy of the ivestmet () as follows: π t (1) t 1 ROI A P 0 The subscript (A) after (ROI) desigates that the retur value expressed is a arithmetic mea retur ad the variable ( t ) is the profit geerated by the ivestmet i year (t). Sice it is ot always clear exactly what the profit o a ivestmet is i a give year, oe ca compute a retur based o periodic cash flows. Therefore, this arithmetic mea rate of retur formula ca be writte: (2) ROI A t 0 CF P 0 t t 1 CF P 0 t 1 Notice that the summatio i the first expressio begis at time zero, esurig that the iitial cash outlay is deducted from the umerator. (The cash flow [CF 0 ] associated with ay iitial cash outlay or ivestmet will be egative.) The primary advatage of Equatio (2) over (1) is that a profit level eed ot be determied each year for the ivestmet; that is, the aual cash flows geerated by a ivestmet do ot have to be classified as to whether they are profits or merely retur of capital. Multiplyig (P 0 ) by () i the deomiator of (2) to aualize the retur has the same effect as dividig the etire fractio by () as i (1). I the secod expressio, the summatio begis at time oe. The iitial outlay is recogized by subtractig oe at the ed of the computatio. For example, cosider a stock whose purchase price three years ago was $100. This stock paid a divided of $10 i each of the three years ad was sold for $130. If time zero is the stock's date of purchase, its arithmetic mea aual retur is 20%: 27
28 ROI A 100 + 10 + 10 + 10 + 130 3 100 60 300.20 Idetically, the stock's aual retur is determied by (3): (3) ROI A t 1 DIV P 0 t P P0 + P 0 where (DIV t ) is the divided paymet for the stock i time (t), (P 0 ) is the purchase price of the stock ad (P ) is the sellig price of the stock. The differece (P - P 0 ) is the capital gai realized from the sale of the stock. Cosider a secod stock held over the same period whose purchase price was also $100. If this stock paid o divideds ad was sold for $160, its aual retur would also be 20%: ROI A 160 100 60 0 +.20 3 100 300 Therefore, both the first ad secod stocks have realized arithmetic mea returs of 20%. The total cash flows geerated by each, et of their origial $100 ivestmets, is $60. Yet, the first stock must be preferred to the secod sice its cash flows are realized sooer. The arithmetic mea retur (ROI a ) does ot accout for the timig of these cash flows. Therefore, it evaluates the two stocks idetically eve though the first should be preferred to the secod. Because this measure of ecoomic efficiecy does ot accout for the timeliess of cash flows, aother measure must be developed. P: RETURN MEASUREMENT: GEOMETRIC MEAN The arithmetic mea retur o ivestmet does ot accout for ay differece betwee divideds (itermediate cash flows) ad capital gais (profits realized at the ed of the ivestmet holdig period). That is, ROI A does ot accout for the time value of moey or the ability to re-ivest cash flows received prior to the ed of the ivestmet's life. I reality, if a ivestor receives profits i the form of divideds, he has the optio to re-ivest them as they are received. If profits are received i the form of capital gais, the ivestor must wait util the ed of his ivestmet holdig period to re-ivest them. The differece betwee these two forms of profits ca be accouted for by expressig compouded returs. That is, the geometric mea retur o ivestmet will accout for the fact that ay earigs that are retaied by the firm will be automatically re-ivested, thus compouded. If returs are realized oly i the form of capital gais, the geometric mea rate of retur is computed as follows: (4) ROI P / P 1 g o 28
29 For example, the geometric mea retur o the secod stock whose priced icreased from $100 to $160 discussed i Sectio B is 16.96%, determied as follows: ROI g 3 160 /100 1 1.6 1 3.1696 If divideds or itermediate cash flows from the security are realized before the ed of the holdig period, returs r t should be computed for each period t ad the averaged as follows: r t P t P P + t 1 t 1 (5) ROI Π(1 + r ) 1 g t 1 t DIV t Suppose that the stock i our previous example paid $20 i divideds i each of the three years of the holdig period rather tha geeratig a $60 capital gai over the three year period. The retur r t for each period would be 20% ad the geometric mea retur for the stock would be 20% computed as follows: ROI Π(1 + r ) 1 3 (1 + r ) 1 3 (1 +.2)(1 +.2)(1 +.2) 1. 20 g t 1 t Note that the geometric mea retur is higher if profits ca be withdraw from the ivestmet durig the holdig period. 3 Π t 1 t Q: INTERNAL RATE OF RETURN The primary stregth of the iteral rate of retur (IRR) as a measure of the ecoomic efficiecy of a ivestmet is that it accouts for the timeliess of all cash flows geerated by that ivestmet. The IRR of a ivestmet is calculated by usig a model similar to the preset value series model discussed i Sectio C: CFt PV P0 t 1 (1 + r) or, CFt (6) NPV 0 (1 r t t 0 + ) t where et preset value (NPV) is the preset value of the series et of the iitial cash outlay, ad (r) is the retur (or discout rate) that sets the ivestmet's NPV equal to zero. The ivestmet's iteral rate of retur is that value for (r) that equates NPV with zero. There exists o geeral format allowig us to solve for the iteral rate of retur (r) i terms of the other variables i Equatio (6); therefore, we must substitute values for (r) util we 29
30 fid oe that works (uless a computer or calculator with a built-i algorithm for solvig such problems ca be accessed). Ofte, this substitutio process is very time-cosumig, but with experiece calculatig iteral rates of retur, oe ca fid shortcuts to solutios i various types of problems. Perhaps, the most importat shortcut will be to fid a easy method for derivig a iitial value to substitute for (r) resultig i a NPV fairly close to zero. Oe easy method for geeratig a iitial value to substitute for (r) is by first calculatig the ivestmet's retur o ivestmet. If a ivestor wated to calculate the iteral rate of retur for the first stock preseted i Sectio B, he may wish to first substitute for (r) the stock's 20% retur o ivestmet: 100 10 NPV + + 0 (1 + r) (1 + r) 1 10 (1 r) + 2 10 + 130 10 + -100+ + 3 (1 + r) (1+ r) 10 (1 r) + 2 + 140 3-3.7 (1.2) Sice this NPV is less tha zero, a smaller (r) value should be substituted. A smaller (r) value will decrease the right-had side deomiators, icreasig the size of the fractios ad NPV. Perhaps a feasible value to substitute for (r) is 10%. The same calculatios will be repeated with the ew (r) value of 10%: 10 NPV 100 + + 1.1 10 2 (1.1) + 10 3 + 22.54 (1.1) Sice the ew NPV exceeds zero, the (r) value of 10% is too small. However, because -3.70 is closer to zero tha 22.54, the ext value to substitute for (r) might be closer to 20% tha to 10%. Perhaps a better estimate for the IRR will be 18%. Substitutig this value for (r) results i a NPV of.86: 10 NPV 100 + + 1.18 10 2 (1.18) + 140 3-0.86 (1.18) This NPV is quite close to zero; i fact further substitutios will idicate that the true stock iteral rate of retur is approximately 18.369%. These iteratios have a patter: whe NPV is less tha zero, decrease (r) for the ext substitutio; whe NPV exceeds zero, icrease (r) for the ext substitutio. This process of iteratios eed oly be repeated util the desired accuracy of calculatios is reached. The primary advatage of the iteral rate of retur over retur o ivestmet is that it accouts for the timeliess of all cash flows geerated by that ivestmet. However, IRR does have three major weakesses: 1. As we have see, IRR takes cosiderably loger to calculate tha does ROI. Therefore, if ease of calculatio is of primary importace i a situatio, the ivestor may prefer to use ROI as his measure of efficiecy. As discussed i the appedix to this chapter, there do exist calculators ad computer programs that will compute IRR very quickly. 30
31 2. Sometimes a ivestmet will geerate multiple rates of retur; that is, more tha oe (r) value will equate NPV with zero. This will occur whe that ivestmet has associated with it more tha oe egative cash flow. Whe multiple rates are geerated, there is ofte o method to determie which is the true IRR. I fact, oe of the rates geerated may make ay sese. Whe the IRR is ifeasible as a method for comparig two ivestmets, ad the ivestor still wishes to cosider the time value of moey i his calculatios, he may simply compare the preset values of the ivestmets. This approach ad its weakesses will be discussed i later chapters. 3. The iteral rate of retur is based o the assumptio that cash flows received prior to the expiratio of the ivestmet will be re-ivested at the iteral rate of retur. That is, it is assumed that future ivestmet rates are costat ad equal to the IRR. Obviously, this assumptio may ot hold i reality. R: BOND YIELDS By covetio, rates of retur o bods are ofte expressed i terms somewhat differet from those of other ivestmets. For example, the coupo rate of a bod is the aual iterest paymet associated with the bod divided by the bod's face value. Thus, a four-year $1000 corporate bod makig $60 aual iterest paymets has a coupo rate of 6%. However, the coupo rate does ot accout for the actual purchase price of the bod. Corporate bods are usually traded at prices that differ from their face values. The bod's curret yield accouts for the actual purchase price of the bod: (7) cy INT P 0 If the above 6% bod were purchased for $800, its curret yield would be 7.5%. The formula for curret yield, while easy to work with, does ot accout for ay capital gais (or losses) that may be realized whe the bod matures. Furthermore, curret yields do ot accout for the timeliess of cash flows associated with bods. The bod's yield to maturity, which is essetially its iteral rate of retur does accout for ay capital gais (or losses) that may be realized at maturity i additio to the timeliess of all associated cash flows: CFt INT F (8) NPV 0 P t 0 t + t 0 (1 + y) t 1 (1 + y) (1 + y) The yield to maturity (y) of the above bod would be (6% ). Thus, (y) is idetical to the bod's iteral rate of retur. (However, i most istaces, the bod's yield to maturity will ot equal its coupo rate.) If the bod makes semiaual iterest paymets, its yield to maturity ca be more accurately expressed: 31
2 CFt INT / 2 F (9) NPV 0 P t 0 t + t 0 (1 + y) t 1 (1 + y / 2) (1 + y) Here, we are cocered with semiaual iterest paymets ad (2) six-moth time periods where () is the umber of years to the bod's maturity. While yield to maturity is perhaps the most widely-used of the bod retur measures, it still assumes a flat yield curve. This meas that coupo paymets received prior to bod maturity will be ivested at the same rate as the bod s yield, a urealistic assumptio whe iterest rates are expected to chage sigificatly over time. S: INTRODUCTION TO RISK AMD EXPECTED RETURN Whe a firm ivests, it subjects itself to at least some degree of ucertaity regardig future cash flows. Maagers caot kow with certaity what ivestmet payoffs will be. This chapter is cocered with forecastig ivestmet payoffs ad returs ad the ucertaity associated with these forecasts. We will defie expected retur i this chapter, focusig o it as a retur forecast. This expected retur will be expressed as a fuctio of the ivestmet's potetial retur outcomes ad associated probabilities. The riskiess of a ivestmet is simply the potetial for deviatio from the ivestmet's expected retur. The risk of a ivestmet is defied here as the ucertaity associated with returs o that ivestmet. Although other defiitios for risk such as the probability of losig moey or goig bakrupt ca be very useful, they are ofte less complete or more difficult to measure. Our defiitio of risk does have some drawbacks as well. For example, a ivestmet which is certai to be a complete loss is ot regarded here to be risky sice its retur is kow to be -100% (though we ote that it probably would ot be regarded to be a particularly good ivestmet). Cosider a ecoomy with three potetial states of ature i the ext year ad Stock A whose retur is depedet o these states. If the ecoomy performs well, state oe is realized ad the stock ears a retur of 25%. If the ecoomy performs oly satisfactorily, state two is realized ad the stock ears a retur of 10%. If the ecoomy performs poorly, state three is realized ad the stock achieves a retur of -10%. Assume that there is a twety percet chace that state oe will occur, a fifty percet chace that state two will occur ad a thirty percet chace that state three will occur. The expected retur o the stock will be 7%, determied by Equatio (10): (10) E[ RA ] i 1 R Ai P i E[R] (.25 *.20) + (.10 *.50) + (-.10 *.30).07, where (R i ) is retur outcome (i) ad (P i ) is the probability associated with that outcome. Therefore, our forecasted retur is 7%. The expected retur cosiders all potetial returs ad weights more heavily those returs that are more likely to actually occur. Although our forecasted retur level is seve percet, it is obvious that there is potetial for the actual retur outcome to deviate from this figure. This potetial for deviatio (variatio) will be measured i the followig sectio. 32
T: VARIANCE AND STANDARD DEVIATION The statistical cocept of variace is a useful measure of risk. Variace accouts for the likelihood that the actual retur outcome will vary from its expected value; furthermore, it accouts for the magitude of the differece betwee potetial retur outcomes ad the expected retur. Variace ca be computed with Equatio (11): 2 (11) σ ( R E[ R] i 1 ) 2 i P i Figure 2: Expected Retur, Variace ad Stadard Deviatio of Returs for Stock A i R i P i R i P i R i - E[R a ] (R i - E[R a ]) 2 (R i - E[R a ]) 2 P i 1.25.20.05.18.0324.00648 2.10.50.05.03.0009.00045 3 -.10.30 -.03 -.17.0289.00867 E[R a ].07 2 a.01560 a.1249 The variace of stock returs preseted i Sectio G is.0156: 2 (.25-.07) 2 *.2 + (.10-.07) 2 *.5 + (-.10-.07) 2 *.3.0156 The statistical cocept of stadard deviatio is also a useful measure of risk. The stadard deviatio of a stock's returs is simply the square root of its variace: σ i 1 ( R E [ R] ) 2 i P i Thus, the stadard deviatio of returs o the stock described i Sectio F is 12.49%. Cosider a secod security, Stock B whose retur outcomes are also depedet o ecoomy outcomes oe, two ad three. If outcome oe is realized, Stock B attais a retur of 45%; i outcomes two ad three, the stock attais returs of 5% ad -15%, respectively. From Figure 11, we see that the expected retur o Stock B is seve percet, the same as for Stock A. However, the actual retur outcome of Stock B is subject to more ucertaity. Stock B has the potetial of receivig either a much higher or much lower actual retur tha does Stock A. For example, a ivestmet i Stock B could lose as much as fiftee percet, whereas a equal ivestmet i Stock A caot lose more tha te percet. A ivestmet i Stock B also has the potetial of attaiig a much higher retur tha a idetical ivestmet i Stock A. Therefore, returs o Stock B are subject to greater variability (or risk) tha returs o Stock A. The cocept of variace (or stadard deviatio) accouts for this icreased variability. The variace of Stock B (.0436) exceeds that of Stock A (.0156), idicatig that Stock B is riskier tha Stock A. 33
Figure 3: Expected Retur, Variace, ad Stadard Deviatio of Returs for Stock B i R i P i R i P i R i - E[R b ] (R i - E[R b ]) 2 (R i - E[R b ]) 2 P i 1.45.20.09.38.1444.02888 2.05.50.025.02.0004.00020 3 -.15.30 -.045.22.0484.01452 E[R b ].070 2.04360 b.2088 With the expected retur ad stadard deviatio of returs of a ivestmet, we ca establish rages of potetial returs ad probabilities that actual returs will fall withi these rages if it appears that potetial returs for that ivestmet are ormally distributed. For example, cosider a third stock with ormally distributed returs with a expected level of 7% ad a stadard deviatio of 10%. From Table V i the text appedix, we see that there is a 68% probability that the actual retur outcome o this stock will fall betwee -.03 ad.17: E[R] - 1 < R i < E[R] + 1 (.07 -.10) < R i < (.07 +.10) A similar aalysis idicates a 95% probability that the actual retur outcome will fall betwee -.13 ad.27: E[R] - 2 < R i < E[R] + 2 (.07 -.20) < R i < (.07 +.20) Obviously, a smaller stadard deviatio of returs will lead to a arrower rage of potetial outcomes give ay level of probability. If a security has a stadard deviatio of returs equal to zero, it has o risk. Such a security is referred to as the risk-free security with a retur of (r f ). Therefore, the oly potetial retur level of the risk-free security is (r f ). No such security exists i reality; however, short-term Uited States treasury bills are quite close. The U.S. govermet has prove to be a extremely reliable debtor. Whe ivestors purchase treasury bills ad hold them to maturity, they do receive their expected returs. Therefore, short-term treasury bills are probably the safest of all securities. For this reaso, fiacial aalysts ofte use the treasury bill rate (of retur) as their estimate for (r f ) i may importat calculatios. U: HISTORICAL VARIANCE AND STANDARD DEVIATION Empirical evidece suggests that historical stock retur variaces (stadard deviatios) are excellet idicators of future variaces (stadard deviatios). That is, a stock whose previous returs have bee subject to substatial variability probably will cotiue to realize returs of a highly volatile ature. Therefore, past riskiess is ofte a good idicator of future riskiess. A stock's historical retur variability ca be measured with a historical variace: 2 2 1 (12) σ h ( Rt R ) t 1 34