Lecture 16 Ivestmet, Time, ad Risk (Basic issues i Fiace) 1. Itertemporal Ivestmet Decisios: The Importace o Time ad Discoutig 1) Time as oe o the most importat actors aectig irm s ivestmet decisios: A irm s decisio to purchase a capital good, such as a ew plat or a ew piece o heavy machiery, diers rom its decisio to hire more labor or purchase additioal raw materials, i large part because o very dieret time horizos or these decisios. Oce a irm purchases a ew capital good, the irm is stuck with that decisio or years to come, while labor ca be laid o o a short otice, ad raw material orders ca be quickly cacelled. ) Preset Value ad Discoutig dollars ivested today at a iterest rate r would icrease i value to ( 1 r) dollars i oe t year, ( 1 r)(1 r) = dollars i two years, ad ( 1 r) dollars i t years. The t preset value o ( 1 r) dollars received t years rom today equals t t ( 1 r) / = dollars. By similar reasoig, the promise to pay dollars t years rom today has a preset value o: PV = t ( 1 r) Preset Value o Proit: Suppose π t represets the irm s proit i time period t, ad ( 1 r) represets the rate at which the irm is willig to trade uture or preset icome just as it represeted the rate at which a idividual would trade uture or preset cosumptio. π 1 π π π π PV = L (1) ( 1 r ) The maximizatio o log-ru proit implies the maximizatio oπ PV. Equatio (1) shows that the maximizatio o log-ru proit depeds ot oly oe the aual low o ecoomic proit, π t, but also critically depeds o the discout rate, r. Firms with short time horizos place a low value o uture proits ad are willig to trade curret proit or uture proit oly a at high iterest rate, ad thereore have high discout rates. These irms wat to ear high proits i early periods eve i it meas sacriicig large uture proits. Firms with loger time horizos are willig to trade curret proits or uture proits at a low iterest rate, ad thereore have lower discout rates. These irms are willig to sacriice curret proits i order to ear higher proits i the uture. Assumig that goes to iiity adπ t is costat ad equal toπ or all values o t, Equatio (1) ca be rewritte as: = π π π 1 π π π PV L = π L () ( 1 r) Itermediate Microecoomics 90
1 π PV = ( π π PV ), π PV ( 1 r) = π π PV, π π PV = () r Geeralized Preset Value Calculatios Equatio () ca be geeralized or ay sum begiig at ay time t, ot just or t = 1. π π π π π PV = = L (4) t c c 1 c t= c where c is ay umber less tha, equal to, or greater tha zero. I Equatio (4), c represets the time whe the low o proit begis. I the proit low begis i the curret time period, c = 0. Rewritig Equatio (4) will yield: 1 π π π 1 π π π PV = 1 L = = c c 1 1 (1 ) (1 ) (1 ) (1 ) (1 ) c (5) r r r r r r r(1 r) Ex) Suppose that proit low is $0. Ad discout rate is 0. with c = 4 ad -4. The use Equatio (5) to calculate preset value o respective cases. $0 $0 π PV = = $751.88. Ad π 4 1 PV = $1, 6 5 0.1(1 0.1) 0.1(1 0.1) = ) The Value o a Bod or Perpetuity Oe importat use o discoutig is to determie the value o bods ad perpetuities. * Bod: A debt security issued by a govermet or a irm where the purchaser leds the issuer a lump sum amout o moey today i retur or a promise rom the issuer to pay a iite stream o uture paymets. * Coupo: The aual iterest paymet o a bod. * Perpetuity: A debt security where the purchaser leds the issuer a lump sum amout o moey today i retur or a promise rom the issuer to pay a ixed amout o icome to the purchaser orever. To determie the value o a bod to its buyer, we calculate the preset value o the paymets stream. Suppose, or example, a bod is sold or $1,000 ad pays a coupo o $0 per year or each o the ext years ad a pricipal repaymet i the teth year o $1,000. $0 $0 $0 $0 PV = L $1,000 Itermediate Microecoomics 91
At a discout rate o zero, the preset value o the bod will be equal to PV = ($0) $1,000 = $,000. As the discout rate icreases above zero, the preset value will cotiuously declie. Usig Equatios () ad (5), we ca simpliy the calculatio o the preset value o the bod as ollows: $0 $0 $1,000 PV = r r(1 r) $0 : PV o a aual paymet o $0 orever with the irst paymet beig made i oe r year (rom Eq.()) $0 : PV o the paymets that would be made rom t = 11 to t = o a bod that r pays a $0 coupo orever. So the irst two terms o the r.h.s. o the above calculatio measure the preset value o the years o coupo paymets o the bod. Ad the third term o the above calculatio is the PV o the pricipal repaymet that is received i years. Now, i the purchaser o the bod has a 5% discout rate, the preset value o the bod is: $0 $0 PV =.05 (.05)(1.05) $1,000 (1.05) = $,000 $1,7.8 $61.91 = $1,86.08 To id the preset value o a perpetuity, which, begiig i year 1, pays $1,000 per year orever, use Equatio () to calculate: $1,000 $1,000 $1,000 $1,000 PV = L = r Govermet ad corporate bods are sold i the bod market i the same way that shares o stock are sold i the stock market. Buyers ad sellers agree to exchage a bod at a equilibrium price p. The yield o a bod is simply the rate o retur o the market price (market value) o the bod. The ace value o the bod equals the pricipal paymet to the ower whe the bod reaches maturity. To determie the yield or rate o retur o a bod, we rewrite Equatio (1) i the ollowig orm: x1 x x pricipal p = L (6) ( 1 rr) (1 rr) (1 rr) (1 rr) Buyer kows bod price (p), the coupo paymets (the values o x i or all i), ad the pricipal paymet; thereore, the oly ukow is the yield or rate o retur o the bod. Itermediate Microecoomics 9
. The Firm s Ivestmet Decisio 1) Net preset Value Criterio I the PV o the expected cash low geerated by a ivestmet is greater tha the PV o the cost o the ivestmet, the a irm should make the ivestmet. NPV C 1 = L (7) ( 1 ) r I NPV > 0, the irm udertakes the ivestmet. I NPV < 0, the irm does ot. ) Firm s Discout Rate The discout rate o a particular project is the rate o retur o the irm s ext best alterative ivestmet project with the same risk. I other words, the discout rate is the opportuity cost o capital ivested i projects with the same risk. It is ecessary or the irm to put projects i dieret risk categories, because ivestors isist o earig higher rates o retur o riskier ivestmets. For sae or low-risk ivestmets, a commo measure o the opportuity cost o capital is the rate o retur o a -year U.S. Treasury Bod. Because there is essetially o risk o the U.S. govermet deaultig o its bods, the retur o a govermet bod is cosidered a risk-ree retur. Firms ad idividuals always have the opportuity to ivest i risk-ree U.S. govermet bods. Ex) Suppose that prior to August 1, 008, Disey is cosiderig buildig a Phatasmic theme restaurat at its MGM Studios them park at Disey World i Orlado, FL. The restaurat will cost $6millio to build ad Disey expects it to geerate a cash low o $1 millio per year or each o the ext years. Ater years, Disey plas to completely chage the theme o the restaurat ad expects to sell the salvageable equipmet or $1. millio. Based o the estimates, Disey eeds to determie whether it should build the ew restaurat. Now, usig dieret discout rates o 5%, %, ad 15%, calculate the respective NPVs. Ad i the Disey s expected cash low rom the restaurat decreases by 5% due to uexpected iacial turmoil, how do the NPVs chage? ) Iteral Rate o Retur Criterio A alterative to the et preset value criterio or evaluatig ivestmets is the iteral rate o retur (IRR) o a ivestmet, which is the discout rate or which the et preset value o the ivestmet equals zero. He IRR criterio or a ivestmet states that i the IRR is greater tha the irm s required rate o retur o ivestmets, the irm should udertake the ivestmets; i the IRR is less tha the irm s required rate o retur, the irm should ot udertake the ivestmet. What is the irm s required rate o retur? I the irm is borrowig Itermediate Microecoomics 9
to pay or the ivestmets, the required rate o retur will be the cost o borrowig. The irm the udertakes the ivestmet i ad oly i the IRR is greater tha the irm s cost o borrowig. For ay ivestmet, there always exists at least oe IRR. Cosider the Disey example (page 9). The IRR is calculated by settig the NPV equal to zero as ollows: $1 $1 $1 $1. NPV = $6 L = 0 1 IRR (1 IRR) (1 IRR) (1 IRR) $1 $1 NPV = $6 IRR ( IRR)(1 IRR) Solvig by trial ad error, IRR=0.11: $1. (1 IRR) $1 $1 $1. $6 = $6 $8.6.6 0.8 0.01 = 0 0.11 (0.11)(1.11) (1.11) I Disey s cost o borrowig is less tha 1.1 percet, Disey will build the restaurat. = 0 4) Real Versus Nomial Discout Rates ad Cash Flows Real values adjust or ilatio; omial values do ot. The real iterest rate o a bod measures the icrease i real purchasig power that the purchaser receives each year. The dierece betwee the real ad omial iterest rate ca be closely approximated by: r i π (8) For example, i a govermet bod pays a omial 7% each year ad the ilatio rate is %, the the real iterest rate is approximately 5%. The use o the omial iterest rate overstates the icrease i purchasig power to the leder each year. Suppose a idividual purchases a $,000 oe-year bod payig a omial 7% iterest rate. The bod holder receives a $700 iterest paymet plus a $,000 pricipal repaymet ext year. I omial terms, the bod holder sacriiced $,000 o purchasig power this year i retur or $,700 o purchasig power ext year. I real terms, however, i the price idex is 1.00 today ad ilatio is %, the the price idex ext year is 1.0. I retur or sacriicig $,000 i purchasig power today, the leder receives $,700/$1.0=$,490.0 i purchasig power ext year, or slightly uder a 5% real retur.. The Impact o Risk o Ivestmet Decisios 1) Diversiiable ad No-diversiiable Risk Diversiicatio requires a irm to reduce its risk by choosig a appropriate collectio o ivestmets, some o which are high-risk gambles ad some low-risk gambles. Risk that ca be elimiated through diversiicatio is called diversiiable risk. Risk that caot be elimiated through diversiicatio is called odiversiiable risk or market risk. It is Itermediate Microecoomics 94
sometimes useul to thik o diversiiable risk as risk that aects oly a small umber o assets, whereas odiversiiable risk aects a large umber o assets. The risk o ivestig i just a ew stocks is primarily diversiiable risk, because purchasig a broader base o stocks ca reduce it. Similarly, the risk udertake by lie ad automobile isurace compaies is also diversiiable risk, because by sellig isurace to may dieret idividuals, the compaies ca reduce their risk. What costitutes odiversiiable risk? Oe example is risk associated with luctuatios i geeral ecoomic coditios (ex. Ecoomic dowtur resultig rom 9.11 terrorist attack, uaticipated oil price shock, a iacial crisis i Southeast Asia, uexpected icrease i iterest rate, etc.) Diversiiable risk is a uctio o the umber o stocks owed. Ad odiversiiable risk is idepedet o the umber o stocks owed. risk NR Diversiiable risk Total risk Nodiversiiable risk # o stocks ) Capital Asset Pricig Model (CAPM) Need to examie how irms iclude risk i their et preset value calculatios. The most commoly used method is called the capital asset pricig model, or the CAPM. The CAPM measures the risk o a particular capital ivestmet by comparig that risk with the risk o ivestig i the etire stock market. Suppose a idividual ivested i the etire stock market; that is, bought every sigle stock oered or sale. The ivestor would bear o diversiiable risk, because she would be completely diversiied; she would, however, bear some odiversiiable risk. Because ivestmet i the etire market icludes some odiversiiable risk, she would demad a risk premium or ivestig i the market istead o ivestig i risk-ree govermet bods. I the retur o the etire market is r m ad the retur o risk-ree govermet bods is r, the the risk premium is r r ). ( m Itermediate Microecoomics 95
A Ivestmet s Asset Beta ( β ): I a ivestor is cosiderig ivestig i the stock o just oe compay, he ca use the capital asset pricig model to calculate the risk premium he would demad to ivest i that stock. Accordig to the CAPM, he would demad a risk premium o that stock that is proportioal to the risk premium o the etire market, so that: risk premium = r r = β ( r r ) (9) i or m r i = r β r r ) () ( m where r i is the expected retur o the stock, ad β, called the asset beta, is a measure o the odiversiiable risk associated with the ivestmet. The asset beta β is related oly to odiversiiable risk. Because borrowers kow that leders ca elimiate all o their diversiiable risk, they will ot pay a risk premium to leders to cover diversiiable risk. The asset beta measures the odiversiiable risk associated with a particular ivestmet compared to a ivestmet i the etire market. * For a ivestmet i the etire market, the risk premium is ( r r ), ad thereore β = 1. m Ivestmets that are riskier (less risky) tha ivestig i the etire market have asset betas greater (smaller) tha 1. Some ivestmets are more highly correlated with movemets i the stock market tha others. I a 1 percet decrease i the value o the market, r m, is associated with a percet decrease i the retur o a particular ivestmet, the β = or that ivestmet. This ivestmet has a very high positive correlatio with the etire market ad is quite risky. Proits i the airlie ad steel idustries are very sesitive to macroecoomic shocks, ad thereore have relatively high asset betas. Proits i the ready-to-eat cereal, soaps ad detergets, ad ood-processig idustries are much less sesitive to macroecoomic shocks, ad thereore have relatively low asset betas. I the β associated with a particular stock ivestmet is kow, it is possible to calculate the correct discout rate to use i calculatig the NPV o that ivestmet. The discout rate is: * I statistics, β is techically deied as: cov( ri, rm ) β =, var( rm ) where cov( ri, rm ) is the covariace o ri ad r m, which is a measure o the liear associatio betwee them; ad var( r m ) is the variace o r m, which measures the dispersio o r m aroud its mea value. This ratio, ad thereore, β, measures the volatility o the ivestmet i divided by the volatility o the etire market. Itermediate Microecoomics 96
discout rate = r β ( r r ) (11) For a stock ivestmet, calculatig Equatio (11) is straightorward. Over may years, the risk premium o the etire stock market, m m r r, has averaged approximately 8 percet. I the curret risk-ree retur o govermet bods is 5 percet, the: discout rate = 0.05 β (0.08) By ollowig past treds i a stock s value, it is possible to calculate the stock s β. Firms use the CAPM to determie the discout rate to use whe calculatig the et preset value o a ivestmet. Accordig to the CAPM, the larger the irm s asset beta, the larger the irm s discout rate, ad the less likely the irm is to udertake a ivestmet. Itermediate Microecoomics 97