FI3300 Corporate Finance

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1 Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multi-peiod multi-cf time-value-of-money poblems: Geneal case Pepetuity Annuity Find the ate of etun in multi-peiod (multi-cf time-value-of-money poblems Adjusting the ate of etun: The fequency of compounding Inflation Loan amotization schedule 1 2 Example You plan to spend the next fou summes aboad. The fist summe tip, which is exactly one yea away, will cost you $22,000, the second, $27,500, the thid, $,000 and the fouth $5, How much should you deposit in you account today (pays 6% inteest pe annum so that you will have exactly enough to finance all the tips? 2. If you boow the money to finance those tips (at 6% inteest pe annum and plan to epay it in 5 yeas when you get you tust fund, how much do you expect to pay? Pesent Value (PV of a CF Steam CF 1 CF 2 CF t CF T > time t T CF CF CF CF PV = (1 + (1 + (1 + ( t T t T CF t = the cash flow on date t (end of yea t = the cost of capital fo one peiod (one yea t = date index, t = 1,2,,,T T = the numbe of peiods (numbe of yeas 4 Futue Value (FV of a CF Steam Pepetuity CF 1 CF 2 CF t CF T > time t T Step 1: calculate the pesent value of the CF steam PV FV > time t T Step 2: use the PV-FV fomula to calculate the futue value of the CF steam: FV = PV (1 + T You invest in a poject that is expected to pay $1,200 a month, at the end of the month, foeve. The monthly cost of capital is 1%. What is the pesent value of this CF steam? 5 6 1

2 Pesent Value (PV of a Pepetuity CF CF CF > time t CF = the SAME CF at the end of EVERY peiod (yea Fist CF (stat date: end of the fist peiod (date 1 We get the same CF FOREVER (T =, infinity = the cost of capital fo one peiod (one yea CF PV = Pepetuity examples 1. Suppose the value of a pepetuity is $8,900 and the discount ate is % pe annum. What must be the annual cash flow fom this pepetuity? 2. An asset that geneates $890 a yea foeve is piced at $6,000. What is the equied ate of etun? 7 8 Annuity You conside investing in eal estate. You expect the popety to yield (i.e., geneate ent CFs of $18,000 a yea fo the next twenty yeas, afte which you will be able to sell it fo $250,000. You equied ate of etun is % pe annum. What is the maximum amount you d pay fo this CF steam? 9 Pesent Value (PV of an Annuity CF CF CF CF > time t T CF = the SAME CF at the end of EVERY peiod (yea Fist CF (stat date: end of the fist peiod (date 1 Last CF (end date: end of the last peiod (date T T = the numbe of peiods (numbe of yeas = the cost of capital fo one peiod (one yea T CF 1 PV = Annuity, find the FV You open a savings account and deposit $20,000 today. At the end of each of the next 15 yeas, you deposit $2,500. The annual inteest ate is 7%. What will be the account balance 15 yeas fom now? Annuity, find the PMT You ae tying to boow $200,000 to buy a house on a conventional 0-yea motgage with monthly payments. The monthly inteest ate on this loan is 0.70%. What is the monthly payment on the loan? 11 2

3 Annuity, find the PMT: Challenge You plan to etie in 0 yeas. Then you will need $200,000 a yea fo 10 yeas (fist withdawal at t=1. Ten yeas late you expect to go to a etiement home whee you will stay fo the est of you life. To ente the etiement home, you will have to make a single payment of $1,000,000., You can stat saving fo you etiement in an account that pays 9% inteest a yea. Theefoe, stating one yea fom now (end of the fist yea: t =1, you will make equal yealy deposits into this account fo 0 yeas. In 0 yeas (on date t=0, you expect a deposit of $500,000 to you etiement account fom you cash value insuance policy. What should be you yealy deposit into the etiement account? Adjusting the ate of etun The fequency of compounding: Quoted (stated ate Effective ate Always use the effective annual ate to discount annual CFs, effective monthly ate to discount monthly CFs etc. The case of inflation: Nominal ate Real ate Always use the nominal ate to discount nominal CFs and the eal ate to discount eal CFs Effective to Effective: Example The annual inteest ate is 8%. What is the 2-yea ate of etun on $1? 2 1 (1 yea (2 yea FV = PV 1+ = PV 1+ ( ettective,2 yea 2 ( effective,1 yea ( effective,2 yea FV = $1 1 + = $ [ ] = 1 + ( ettective,2 yea = = = = 16.64% Effective to Effective: Fomula (effective, 1-peiod = 1-peiod effective ate The etun on $1 invested fo 1 peiod (effective, n-peiod = n-peiod effective ate The etun on $1 invested fo n peiods n 1+ ( effective, n peiod = 1+ ( effective,1 peiod Effective to Effective: n>1 The effective monthly ate is 1%, what is the effective annual ate? Since the effective monthly ate is known and thee ae n= months in one yea = 1% = ( effective, monthly ( effective, months 1 ( effective,1 mot nh 1+ = + ( effective, annual ( effective, annual 1+ = = =.68% Effective to Effective: n>1 The effective monthly ate is 1%, what is the effective quately ate? Since the effective monthly ate is known and thee ae n= months in one quate ( effective, monthly = 1% = ( effective, months ( effective,1 month 1+ = 1+ ( effective, quately ( effective, quately 1+ = = =.0%

4 Effective to Effective: n<1 The effective monthly ate is 1%, what is the effective weekly ate? Since the effective monthly ate is known and thee ae 4 weeks in one month o (1/4=0.25 months in one week ( effective, monthly = 1% = = 1 + ( effective,4 weeks ( effective,1 week 1 4 ( effective, weekly 1 ( effective, monthly 1 + = + ( effective, weekly 0.25 = = % 4 Effective to Effective: n<1 The effective annual ate is %, what is the effective ate fo 10 months? Since the effective annual ate is known and thee ae (10/=0.8 yeas in a peiod of 10 months we get ( effective, annual = 10% = = 1 + ( effective,10 months ( effective, months ( effective,10 months ( effective, 10 months 1+ = = = 8.27% Quoted to Effective: Example The offe: a cedit cad with 9% APR (annual pecentage ate. 9% is the quoted (stated annual inteest ate. The convention: since cedit cad payments ae monthly, the fequency of compounding is monthly o m= times in one yea. The teminology: 9% a yea, compounded monthly. The poblem: 9% is NOT the effective annual inteest ate (9% is not the annual ate of etun. To compae this ate to othe offes o discount annual CFs we need the effective ate. Example Continued Since the fequency of compounding is monthly, stat by finding the effective monthly ate: ( quoted, annual = ( quoted, month = 9% = 0.09 ( effective,1 peiod ( effective,1 month ( effective,1 month ( quoted, m peiods = m ( quoted, months = 0.09 = = = 0.75% Example Continued Now, use the effective-to-effective fomula to find any othe effective inteest ate. What is the effective annual inteest ate? (Remembe: thee ae n= months in one yea ( effective, monthly ( effective, annual ( effective, annual = = 075% 0.75% 1+ ( effective, annual = 1+ ( effective, monthly 1+ = = = 9.8% Example Continued What is the effective quately inteest ate? (Remembe: thee ae n= months in one quate ( effective, monthly = = 0.75% 1+ ( effective, quately = 1+ ( effective, monthly ( effective, quately ( effective, annual 1+ = = = 2.27% 4

5 Example Continued What is the effective weekly inteest ate? (Remembe: thee ae 4 weeks in one month o n=1/4=0.25 months in one week ( effective, monthly = = 0.75% 1+ = 1+ ( effective, weekly ( effective, monthly ( effective, weekly ( effectiveann, ual = = = 0.19% Quoted to Effective: Fomula (quoted, m-peiod = quoted ate, compounded m times (effective, 1-peiod = 1-peiod effective ate ( effective,1 peiod = ( quoted, m peiods m Quoted to Effective: Example Quoted to Effective: Example You plan to buy a ca fo $45,000. The deale offes to finance the entie amount and equies 60 monthly payments of $ What is the effective monthly inteest ate? 2. What annual inteest ate will the deale state (quote?. What is the effective annual inteest ate? You bank states that the inteest ate on a thee month cetificate of deposit (CD is 4.68% pe annum. 1. What is the quoted (stated inteest ate? 2. What is the fequency of compounding?. What is the effective annual inteest ate? Quoted to Effective: Example You ae tying to boow $200,000 to buy a house on a conventional 0-yea motgage with monthly payments. You bank is asking fo 8.4% a yea. 1. What is the quoted (stated inteest ate? 2. What is the fequency of compounding?. What is the effective annual inteest ate? 4. What is the monthly payment on the loan? Inflation The inflation ate (i: the ate of a geneal ise in pices ove time. If i>0 then the same commodity becomes moe expensive ove time. If you could buy a poduct fo $100 in 2005, in 2006 you had to pay $10.2 fo the same poduct. In 2007 you had to pay $ and in 2008, $ The implied inflation ates ae: Date Pice Inflation ate - i 2005 $ $ / =.2% 2007 $ / = 2.85% 2008 $ / =.8%

6 Inteest ates and inflation The eal inteest ate ( eal : the etun (compensation you demand fo lending someone money and thus postponing consumption. The nominal inteest ate ( nominal : inflation-adjusted adjusted inteest ate that epesents compensation fo both: inflation and postponing consumption. In the eal wold, all the quoted ates ae nominal ates (e.g., ca loan, house loan, student loan. Nominal and eal Inteest ates The nominal annual ate of etun in 2006 was 4.91% (0 yea US-Teasuy bond, what is the eal annual ate of etun? (1 + nominal l = (1 + eal l x (1 + i i = inflation ate nominal = nominal inteest ate eal = eal inteest ate Note: all ates must be fo the same peiod (say, one yea. 1 2 Inteest ates and inflation Given the nominal (annual ate of etun of a 0 yea US-Teasuy bond, find its eal (annual ate of etun Yea i nominal eal % 4.91% (1.049 / = % % 4.84% % 4.28% Examples 1. If the eal inteest ate is 8% and the inflation ate is 4%, what is the nominal inteest ate? 2. If the nominal inteest t ate is.2% 2% and that inflation ate is.6%, what is the eal inteest ate? (1 + nominal = (1 + eal x (1 + i FI 00 - Copoate Finance Zinat Alam 4 Textbook Example: Annuity due An annuity pays $00 a yea fo thee yeas You own a popety that you want to ent fo 10 yeas. Pospective tenant A pomises to pay $,000 pe yea with payments made at the end of each yea. Pospective tenant B pomises to pay $,000 pe yea with payments made at the beginning i of each yea. Which is the bette deal if the appopiate annual discount ate is 10%? (odinay annuity: $00 $00 $00 T = 0 T = 1 T = 2 T = annuity due: $00 $00 $00 T = 0 T = 1 T = 2 T = 5 6 6

7 The Relation between (odinay annuity and annuity due PV(annuity due = PV(odinay annuity x (1 + FV(annuity due = FV(odinay annuity x (1 + Textbook example: loan amotization You boowed $8,000 fom a bank and pomised to epay the loan in five equal annual payments. The fist payment is at the end of the fist yea. The annual inteest ate is 10%. Wite down the amotization schedule fo this loan. Compute the annual payment ($2, Textbook example: loan amotization We sepaate each payment into two pats: Inteest payment Repayment of pincipal Fo a fixed payment loan: Total payment is fixed Inteest payment deceases ove time Pincipal epayment inceases ove time Amotization schedule table Date Beginning Total Payment Inteest (10% Pincipal End 0 8, , , , , , , loan amotization: solution Fist yea: Beginning balance = 8,000 Inteest payment = 8,000 x 0.1 = 800 Pincipal epayment = 2, = 1,10.8 New pincipal balance = 8,000 1,10.8 = 6, Amotization schedule Date Beginning Total Payment Inteest (10% Pincipal End 0 8, , , ,10.8 6, , , , , ,

8 loan amotization: solution Second yea: Beginning balance = 6, Inteest payment = 6, x 0.1 = Pincipal epayment = 2, = 1, New pincipal balance = 6, , = 5, Amotization schedule Date Beginning Total Payment Inteest (10% Pincipal End 0 8, , , ,10.8 6, , , , , , , , , Amotization schedule Date Beginning Total Payment Inteest (10% Pincipal End 0 8, , , ,10.8 6, , , , , , , ,585.56, , , ,744. 1, , , ,

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