2013/4/9. Topics Covered. Principles of Corporate Finance. Time Value of Money. Time Value of Money. Future Value

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1 3/4/9 Priciples of orporate Fiace By Zhag Xiaorog : How to alculate s Topics overed ad Future Value Net NPV Rule ad IRR Rule Opportuity ost of apital Valuig Log-Lived Assets PV alculatio Short uts ompoud Iterest - - Time Value of Moey Time is valuable for moey Dollar today is more valuable tha Dollar tomorrow osumptio foregoe Iflatio (or deflatio?) Timig Risk Time Value of Moey Time Now Next Year A:$ $ B:$ $ Ucertaity A: $ B: $ $ -3-4 Future Value Now Tomorrow Now Tomorrow??? F F??? Preset Value Discout Future cash flow urret cash flow Earig iterest Future value.sigle period discout for future cash flow. a get future value of today s cash flow -5-6

3/4/9 Preset ad Future Value Preset ad Future Value Future Value Amout to which a ivestmet will grow after earig iterest Value today of a future cash flow.

2 3/4/9 Preset ad Future Value Preset ad Future Value Future Value Amout to which a ivestmet will grow after earig iterest Value today of a future cash flow. Future Value Amout to which a ivestmet will grow after earig iterest Value today of a future cash flow. Discout Rate Iterest rate used to compute preset values of future cash flows Discout Factors ad Rates Future Values Discout Rate Iterest rate used to compute preset values of future cash flows. Discout Factor Preset value of $ future paymet. Future Value of $ FV FV $ ( + r) -9 - Future Values FV $ ( + r) -FV What is the future value of $4, if iterest is compouded aually at a rate of 5% for oe year? FV $4, ( +.5) $4, PV PV discout factor - -

3 3/4/9 Discout Factor DF PV of $ DF ( +r) Discout Factors ca be used to compute the preset value of ay cash flow. Valuig a Office Buildig Step : Forecast cash flows ost of buildig 37 Sale price i Year 4 Step : Estimate opportuity cost of capital If equally risky ivestmets i the capital market offer a retur of 5%, the ost of capital r 5% -3-4 Valuig a Office Buildig Net Step 3: Discout future cash flows PV 4 (+r ) (+.5) 4 Step 4: Go ahead if PV of payoff exceeds ivestmet NPV NPV PV - required ivestmet NPV + + r -5-6 Valuig a Office Buildig Decompose the cost ost of buildig: 3 ost of lad: 5 Questio-: The compay does ot have to pay for the lad. Why is 5 icluded as part of the cost? Valuig a Office Buildig Questio-: Where do we get the discout rate of 5%? The discout rate is the market rate of retur give by the ivestmets at the same risk level as that of your ivestmet. It cosiders both TIMING ad RISK.(Take it as give till chapter 7) ompare apple to apple, ot to orage

4 3/4/9 Risk ad Higher risk projects require higher rates of retur Higher required rates of retur cause lower PVs PV of $4 at 5% 4 PV Risk ad PV of $4 at % 4 PV PV of $4 at 5% 4 PV Risk ad Net Decisio Rules for Ivestmet NPV PV - required ivestmet NPV 375, - 37, $5, Two importat rules Accept ivestmets that offer rates of retur i excess of their opportuity cost of capital Accept ivestmets that have positive et preset value - - Rate of Retur Rule Accept ivestmets that offer rates of retur i excess of their opportuity cost of capital Net Rule Accept ivestmets that have positive et preset value I the project listed below, the foregoe ivestmet opportuity is %. Should we do the project? profit 4, 37, Retur.35 or 3.5% ivestmet 37, Suppose we ca ivest $5 today ad receive $6 i oe year. Should we accept the project give a % expected retur? 6 NPV -5 + $

5 3/4/9 Opportuity ost of apital Resources are limited, so is capital. To ivest i the specified project meas givig up other opportuities of ivestmet. The opportuity cost of capital is the highest rate of retur amog the alteratives. If your ivestmet gives a higher rate of retur tha the opportuity cost of capital, you are usig the capital i the most efficiet way. -5 Opportuity ost of apital You may ivest $, today. Depedig o the state of the ecoomy, you may get oe of three possible cash payoffs: Ecoomy Payoff Slump $8, Normal,, PV $95,65.5 NPV $95,65, $ 4,35 Boom 4, -6 Opportuity ost of apital -cotiued You otice that oe stock i the market has the same risk as that of your ivestmet. The stock is tradig for $ Next year s price, give a ormal ecoomy, is forecasted at $. The stocks expected payoff leads to a expected retur. Opportuity ost of apital -cotiued The stocks expected payoff leads to a expected retur. expected profit Expected retur ivestmet or 5% Opportuity ost of apital -cotiued Discoutig the expected payoff at the expected retur leads to the PV ad NPV of the project, PV $95,65.5 NPV $95,65, $ 4,35 Opportuity ost of apital -cotiued Notice that you come to the same coclusio if you compare the expected project retur with the cost of capital. expected profit,, Expected retur. or % ivestmet, Expected retur o the ivestmet is %, less tha the expected retur o the stock, or the opportuity cost of capital 5%

6 3/4/9 Opportuity ost of apital Distiguish betwee Opportuity ost of apital ad ost of Borrowig You are goig to start your ow busiess by takig a ivestig project. You ca get bak loa at 8%, or you ca get the seed fud from your parets, who are rich ad geerous ad do ot require ay retur. The market rate of retur of such projects is % o average. What s the discout rate of your project? Now Preset Value Tomorrow F F Discout Multi-period discout Discout The Day after Tomorrow Future value -3-3 Discout Factor DF PV of $ DF ( + r ) t Discout Factors ca be used to compute the preset value of ay cash flow. PV DF + r DF (+r ) Discout Factors ca be used to compute the preset value of ay cash flow PV DF t t t ( + r) You just bought a ew computer for $3,. The paymet terms are years same as cash. If you ca ear 8% o your moey, how much moey should you set aside today i order to make the paymet whe due i two years? Replacig with t allows the formula to be used for cash flows that exist at ay poit i time

7 3/4/9 You just bought a ew computer for $3,. The paymet terms are years same as cash. If you ca ear 8% o your moey, how much moey should you set aside today i order to make the paymet whe due i two years? 3 PV $, 57. (. 8) PVs ca be added together to evaluate multiple cash flows. PV + + ( + r) ( + r) PVs ca be added together to evaluate multiple cash flows. $ $ PV + (+.7) (+ 77) Year Year /.7 $93.46 /.77 $7.4 Total $ Give two dollars, oe received a year from ow ad the other two years from ow, the value of each is commoly called the Discout Factor. Assume r % ad r 7%. DF DF. (+. ). (+.7 ) Assume that the cash flows from the costructio ad sale of a office buildig is as follows. Give a 5% required rate of retur, create a preset value worksheet ad show the et preset value. Year 7, Year, Year + 3,

8 3/4/9 - cotiued Assume that the cash flows from the costructio ad sale of a office buildig is as follows. Give a 5% required rate of retur, create a preset value worksheet ad show the et preset value. Period Discout Factor (.5) ash Flow 7,, + 3, NPV Total Preset Value 7, 95,38 + 9,49 $5, cotiued Assume that the cash flows from the costructio ad sale of a office buildig is as follows. Give a 5% required rate of retur, create a preset value worksheet ad show the et preset value. +$3, Year -7, -$7, -$7, -,/.5 $95,38 3,/.5 $9,49 Total NPV $5, -$, Year -44 Whe ash Flows a Be Added Up Geeral Rules Preset values ca be added up; (Future) cash flows at the same period ca be added up; But cash flows at differet periods ca ot. Short uts Sometimes there are shortcuts that make it very easy to calculate the preset value of a asset that pays off i differet periods. These tools allow us to cut through the calculatios quickly Short uts: Perpetuity Perpetuity -Fiacial cocept i which a cash flow is theoretically received forever. Short uts: Perpetuity Perpetuity -Fiacial cocept i which a cash flow is theoretically received forever. 3 Retur r cash flow preset value PV

9 3/4/9 Short uts: Perpetuity Perpetuity -Fiacial cocept i which a cash flow is theoretically received forever. cash flow PV of ash Flow discout rate PV r Short uts: Perpetuity cash flow PV of ash Flow discout rate PV r ( + r) ( + r) ( + r) lim r r( r) + r of Perpetuity Imagie you deposit US$ Millio i bak at the aual rate of %. At the ed of every year, you withdraw the iterest of US$Millio, ad leave the pricipal i the bak. How log ca you go o with this withdrawig? (Suppose the rate does ot chage) Forever! Short uts: Growig Perpetuity Growig Perpetuity -cash flow is theoretically growig costatly at g (g<r) ad received forever. (+g) (+g) Short uts: Growig Perpetuity Growig Perpetuity -cash flow is theoretically growig costatly at g (g<r) ad received forever. cash flow PV of ash Flow discout rate PV r g Short uts: Growig Perpetuity cash flow PV of ash Flow discout rate ( + g) ( + g) ( + g) PV r ( + r) ( + r) ( + r) + g g g g... lim + r r r r r + g r g + r r

10 3/4/9 Short uts: Auity Auity -A asset that pays a fixed sum each year for a specified umber of years. 3 L T Short uts: Auity Auity -A asset that pays a fixed sum each year for a specified umber of years. Asset Perpetuity (first paymet i year ) Perpetuity (first paymet i year t + ) Year of Paymet..t t + r t r ( + r) Auity from year to year t t r r ( + r) Short uts: Auity Auity -A asset that pays a fixed sum each year for a specified umber of years. PV of auity r r r t ( + ) Short uts: Auity Due Auity Due -A asset that pays a fixed sum each year for a specified umber of years, the first cash flow occurrig i year. PV of auity due ( r) t + r r( + r) + t r r( + r) Auity Tiburo Autos offers a easy paymet scheme o a ew Toyota of $5, a year, paid at the ed of each of the ext 5 years, with o cash dow. What is the car really costig you? Auity Tiburo Autos offers a easy paymet scheme o a ew Toyota of $5, a year, paid at the ed of each of the ext 5 years, with o cash dow. What is the car really costig you? ost 5, 5.7.7( +.7) 5, 4. $,

3/4/9 Auity You agree to lease a car for 4 years at $3 per moth. You are ot required to pay ay moey up frot or at the ed of your agreemet. If your opportuity cost of capital is.

11 3/4/9 Auity You agree to lease a car for 4 years at $3 per moth. You are ot required to pay ay moey up frot or at the ed of your agreemet. If your opportuity cost of capital is.5% per moth, what is the cost of the lease? Auity -cotiued You agree to lease a car for 4 years at $3 per moth. You are ot required to pay ay moey up frot or at the ed of your agreemet. If your opportuity cost of capital is.5% per moth, what is the cost of the lease? Lease ost ( +.5) ost $, Auity : Amortized Loas Auity : Amortized Loas 本金 P:, 年利率 :.% 还款年限 N: 年份本期期初贷款余额本期还款额本期偿还利息本期偿还本金本期期末余额, 6,75, 6,75 93,75 93,75 6,75 9,373 6,9 86, ,83 6,75 8,68 7,59 79,3 4 79,3 6,75 7,93 8,35 7,88 5 7,88 6,75 7,88 9,87 6, ,693 6,75 6,69,5 5, ,588 6,75 5,59,6 4,47 8 4,47 6,75 4,47,7 8,45 9 8,45 6,75,85 3,45 4,795 4,795 6,75,48 4,795 年利率 %, 第 6 年起年利率下跌至 8% 年份本期期初贷款余额本期还款额本期偿还利息本期偿还本金本期期末余额, 6,75, 6,75 93,75 93,75 6,75 9,373 6,9 86, ,83 6,75 8,68 7,59 79,3 4 79,3 6,75 7,93 8,35 7,88 5 7,88 6,75 7,88 9,87 6, ,693 5,45 4,935,56 5,77 7 5,77 5,45 4,94,357 39,8 8 39,8 5,45 3,86,66 7, ,554 5,45,4 3,47 4,37 4,37 5,45,45 4, Auity : Amortized Loas 年利率 %, 第 6 年起年利率上升至 % 年份本期期初贷款余额本期还款额本期偿还利息本期偿还本金本期期末余额, 6,75, 6,75 93,75 93,75 6,75 9,373 6,9 86, ,83 6,75 8,68 7,59 79,3 4 79,3 6,75 7,93 8,35 7,88 5 7,88 6,75 7,88 9,87 6, ,693 7,4 7,43 9,7 5,98 7 5,98 7,4 6,38,876 4,6 8 4,6 7,4 4,933,8 8,94 9 8,94 7,4 3,47 3,643 5,8 5,8 7,4,834 5,8 $97. Oe More Auity What is the preset value of a four-year auity of $ per year that makes its first paymet two years from today if the discout rate is 9%? $33.97 $ $ $ $

12 3/4/9 Oe More Auity What is the preset value of a four-year auity of $ per year that makes its first paymet two years from today if the discout rate is 9%? 4 $ $ $ $ $ PV t 3 4 (.9) (.9) (.9) (.9) (.9) t $37.97 $37.97 PV $97..9 ompoud Iterest vs. Simple Iterest of Simple Iterest Aual iterest rate of time deposit i Bak of hia: -year 3.% -years 3.75% 3-years 4.5% 5-years 4.75% ompoud Iterest vs. Simple Iterest ompoud iterest Iterest of this year is accrued to the pricipal o which iterest is calculated ext year. FV (+r) Simple iterest Iterest of this year is ot accrued to the pricipal. FV (+ r) ompoud Iterest Suppose that Jay Ritter ivested i the iitial public offerig of the Modigliai compay. Modigliai pays a curret divided of $., which is expected to grow at 4-percet per year for the ext five years. What will the dividedbe i five years? FV ( + r) T $5.9 $. (.4) ompoud Iterest ompoud Iterest $. (.4) 4 $. (.4) $. (.4) 3 $. (.4) $. (.4) $. $.54 $.6 $3. $4.3 $ FV of $ % Simple % ompoud Number of Years -7-7

13 3/4/9 ompoud Iterest PV of $ Number of Years % 5% % 5% Simple Iterest Simple iterest whe time period is less tha oe year Aual iterest rate of time deposit i Bak of hia: 3-moth.6% 6-moth.8% -moth 3.% EAR ad APR APR (Aual Percetage Rate) Stated iterest rate i fiacial istitutios What ivestors really ear depeds o the frequecy of iterest paymets, i.e., semi-aually, quarterly, mothly or cotiuously. EAR (Effective Aual Rate) Rate that ivestors actually ear their iterests. EAR differs uder the same APR if iterest paymet frequecy differs. EAR ad APR From APR to EAR APR EAR ( + ) From EAR to APR / [( + EAR) ]* APR ompoud Iterest Periods Iterest Value Aually per per APR after compouded year period oe year iterest rate 6% 6%.6 6.% ompoud Iterest Suppose you are offered a automobile loa at a APR of 6% per year. What does that mea, ad what is the true rate of iterest, give mothly paymets?

3/4/9 ompoud Iterest EAR 6.678% -cotiued Suppose you are offered a automobile loa at a APR of 6% per year. What does that mea, ad what is the true rate of iterest, give mothly paymets?

14 3/4/9 ompoud Iterest EAR 6.678% -cotiued Suppose you are offered a automobile loa at a APR of 6% per year. What does that mea, ad what is the true rate of iterest, give mothly paymets? Assume $, loa amout. Loa Pmt, (.5), otiuous ompoudig At a give aual rate (APR), if cash flow comes i evely, what s the EAR? APR EAR ( + ) As approaches ifiity, APR EAR ( + ) APR lim( + ) > APR e -8 otiuous ompoudig otiuous ompoudig Your are ivestig $ o a project which is cotiuously compouded at a aual rate of 5%. What s the EAR ad FV of cash flow i the ed of year ad year? APR EAR e e FV * e.5.5 FV *( + 5.7%) 5.7% FV *( + 5.7%),5.7,5.7-8 ) What s the preset value of aperpetuity ($ i each year ed) at a aually compoudedrateof8.5%? PV /.85 $54.54 ) What s the PV of aperpetuity for which each year $ comes i but cotiuously ad evely at a aually compouded rateof8.5%? EAR e APR, APR l( EAR + ) APR l( + 8.5%) 6.97% PV /6.97% $ otiuous ompoudig What s the PV of adoatio that $ billio comes i evely i each year for years? The aually compouded rateis %. EAR e APR, APR l( EAR + ) APR l( + %) 9.53% PV of Perpetuityat year /9.53%.493 PV of Perpetuity at year /9.53%.493 PV of Perpetuity at year.493/( + %) PV of Auity at year

Chapter 4: Time Value of Money

Chapter 4: Time Value of Money FIN 301 Class Notes Chapter 4: Time Value of Moey The cocept of Time Value of Moey: A amout of moey received today is worth more tha the same dollar value received a year from ow. Why? Do you prefer a