Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions

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1 Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isk-etu tadeoff ad time value of moey? Risk-avese ivestos i fiacial makets equie highe ates of etu fo ivestig i isky secuities to compesate them fo thei isk. Ivestos who wat to avoid isk ca ivest i US Teasuy bills i which the US govemet fully guaatees futue paymet. Sice they ae essetially isk fee, US Teasuy bills offe low ates of etu. The secuities issued by copoatios such as stocks ad bods cotai isk because ivestos do ot kow with cetaity the futue etu they will ea. Ivestos equie highe etus o these secuities to compesate fo the additioal isk, establishig a isk-etu tadeoff i the fiacial makets. A maage would ot coside a $20 millio et cash iflow that she expects to eceive 20 yeas fom ow to have the same value as a $20 millio cash iflow today. If the maage eceived the $20 millio today, she could ivest it at a positive ate of iteest ad have a much lage amout i 20 yeas. Whe makig fiacial decisios that ivolve cash flows at vaious times i the futue, fiacial maages must take ito accout the time value of moey. Thus, time value of moey efes to the pocess of quatifyig the elatio betwee cash flows at diffeet poits i time. 2. Give seveal examples whe the fiacial maage would use the time value of moey. Whe thei fim issues bods to ivestos, fiacial maages should compae the value of futue cash paymets owed to the bodholdes i exchage fo the cash eceived today. Similaly, whe makig capital ivestmets, fiacial maages eed to compae the value of expected cash flows i the futue to the peset cash outlay eeded to udetake the ivestmet. Cocept Check 4.2. What is meat by the tem compoudig of iteest? Whe compoudig iteest, the iteest eaed ad paid i futue peiods depeds ot oly o the iitial peset value amout, but also o ay pevious iteest eaed (that

2 has ot bee withdaw). Thus, with compoudig, iteest eas iteest, ad the futue value amout iceases geometically ove time. 2. How do the legth of the compoudig tem ad the iteest ate affect futue values? Futue values icease as the level of the iteest ate ad legth of the compoudig tem icease. Fo example, a fim will ea moe o a $ millio ivested ove a 5-yea peiod at 8 pecet tha at 6 pecet. Similaly, holdig the iteest ate costat at 7 pecet, the fim will eceive a highe futue value by compoudig a $ millio ove 20 yeas istead of 5 yeas. Cocept Check 4.3. What steps ae eeded to solve fo the peset value of a futue amout? The fist step ivolved i solvig fo the peset value of a futue amout is to idetify the futue value amout (FV), the iteest ate pe peiod (), ad the legth of the discoutig peiod (). The secod step is to substitute these amouts ito the peset value of a futue amout fomula to solve fo the peset value (PV): PV FV =. ) 2. What is meat by the tems discoutig ad discout ate? Discoutig efes to the pocess of computig the peset value of a futue amout. The discout ate is the iteest ate used to compute the peset value amout. Othe tems fo the discout ate iclude the oppotuity cost, equied ate of etu, hudle ate, ad cost of capital. 3. How do the legth of the compoudig tem ad the iteest ate affect peset values? A ivese elatioship exists betwee peset values ad iteest ates. To illustate, suppose a fim wats to set aside eough moey today (a peset value amout) i ode to have $20 millio to expad its poductio facilities i thee yeas. By eaig a highe iteest ate o these ivested fuds, the fim could set aside a lowe amout today to achieve its desied goal of $20 millio i thee yeas. A ivese elatioship also exists betwee peset values ad the compoudig tem. I the pevious example, if the fim had five yeas, istead of thee yeas, to each its ivestmet goal, the fim could set aside a smalle peset value amout today. 2

3 Cocept Check 4.4. What steps ae ivolved i solvig fo the futue value of a auity? The fist step i solvig fo the futue value of a auity is to idetify the auity amout (PMT), the iteest ate pe peiod (), ad the umbe of paymets () i the auity. The secod step is to substitute these amouts ito the futue value of a auity fomula to solve fo the futue value (FV): FV = PMT ). 2. Whe usig the futue value of a auity fomula (Equatio 4.3), at what poit i time is the futue value computed? Whe usig the futue value of a auity fomula, the futue value is obtaied at the time of the last auity paymet. Cocept Check 4.5. What ae the steps ivolved i solvig fo the peset value of a auity? The fist step i solvig fo the peset value of a auity is to idetify the followig: the auity amout (PMT), the iteest ate pe peiod (), ad the umbe of paymets () i the auity. The secod step is to substitute these amouts ito the peset value of a auity fomula to solve fo the peset value (PV): PV = PMT ). 2. What is the diffeece betwee a odiay auity ad a auity due? With a odiay auity, cash flows occu at the ed of each peiod. With a auity due, cash flows occu at the begiig of each peiod. The peset value fomulas diffe slightly fo these two types of auity. 3

4 Cocept Check 4.6. What is a pepetuity? A pepetuity is a auity with a ifiite life. Pefeed stock is a example of a pepetuity. 2. What ae the steps ivolved i solvig fo the peset value of a pepetuity? Solvig fo the peset value of a pepetuity ivolves dividig the pepetuity paymet (PMT) by the discout ate. Thee ae o additioal steps. Cocept Check 4.7. What adjustmets eed to be made to the basic time value of moey equatios whe iteest is compouded semi-aually, quately, mothly, o daily? Compoudig moe fequetly tha oce a yea equies adjustig the iteest ate ad umbe of peiods to eflect the compoudig basis. Fo example, suppose a ivesto eas iteest at a ate of 0 pecet pe yea, compouded semiaually ove a 7-yea peiod. The adjustmets to the time value of moey fomulas ivolve halvig the iteest ate (0%/2 = 5% pe peiod) ad doublig the umbe of peiods (2 x 7 = 4 peiods). 2. What is cotiuous compoudig? What steps ae ivolved i computig the futue value of a peset amout whe iteest is cotiuously compouded? With cotiuous compoudig, iteest is compouded evey ifiitesimal pat of a secod. The fist step is to idetify the peset value amout (PV), the iteest ate pe yea (), ad the umbe of yeas (). The secod step is to substitute these amouts ito the equatio FV = PVe. I this equatio, e is the base of atual logaithm, appoximately equal to Cocept Check 4.8. How do you compute the effective aual iteest ate, give a omial ate? The effective aual iteest ate is computed by solvig the equatio: eff m om = + m. 4

5 I this equatio, eff is the effective aual iteest ate; om is the omial aual iteest ate; ad m is the umbe of compoudig peiods pe yea. 2. Will the diffeece betwee the effective aual iteest ate ad the omial ate icease o decease as the compoudig fequecy (m) iceases? Why? The diffeece betwee the effective aual iteest ate ad the omial ate will icease as the compoudig fequecy iceases. Fo ay give iteest ate, the geate the compoudig fequecy, the highe is the futue value afte oe yea fo a give peset value amout. The geate the futue value afte oe yea, the highe is the effective aual ate that would be eeded to covet the peset value amout ito that futue value. 5

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