FI3300 Corporate FinanceLearning ObjectivesExamplePresent Value (PV) of a CF StreamFuture Value (FV) of a CF StreamPerpetuityPresent Value (PV) of a PerpetuityPerpetuity examplesAnnuityPresent Value (PV) of an AnnuityAnnuity, find the FVAnnuity, find the PMTAnnuity, find the PMT: ChallengeAdjusting the rate of returnEffective to Effective: ExampleEffective to Effective: FormulaEffective to Effective: n>1Slide 18Effective to Effective: n<1Slide 20Quoted to Effective: ExampleExample ContinuedSlide 23Slide 24Slide 25Quoted to Effective: FormulaSlide 27Slide 28Slide 29InflationInterest rates and inflationNominal and real Interest ratesSlide 33ExamplesTextbook Example: Annuity dueAn annuity pays $300 a year for three yearsThe Relation between (ordinary) annuity and annuity dueTextbook example: loan amortizationSlide 39Amortization schedule tableloan amortization: solutionAmortization scheduleSlide 43Slide 44Slide 45FI3300Corporate FinanceSpring Semester 2010Dr. Isabel TkatchAssistant Professor of Finance12Learning Objectives☺Calculate the PV and FV in multi-period multi-CF time-value-of-money problems: ☺General case☺Perpetuity☺Annuity☺Find the rate of return in multi-period (multi-CF) time-value-of-money problems☺Adjusting the rate of return:☺The frequency of compounding☺Inflation☺Loan amortization schedule3ExampleYou plan to spend the next four summers abroad. The first summer trip, which is exactly one year away, will cost you $22,000, the second, $27,500, the third, $33,000 and the fourth $35,000. 1. How much should you deposit in your account today (pays 6% interest per annum) so that you will have exactly enough to finance all the trips?2. If you borrow the money to finance those trips (at 6% interest per annum) and plan to repay it in 5 years when you get your trust fund, how much do you expect to pay?4Present Value (PV) of a CF Stream CF1 CF2 … CFt … CFT |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T CFt = the cash flow on date t (end of year t)r = the cost of capital for one period (one year)t = date index, t = 1,2,3,…,TT = the number of periods (number of years) 1 21 2... ...(1 ) (1 ) (1 ) (1 )tTt TCFCF CF CFPVr r r r= + + + + ++ + + +5Future Value (FV) of a CF Stream CF1 CF2 … CFt … CFT |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T Step 1: calculate the present value of the CF streamPV FV |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T Step 2: use the PV-FV formula to calculate the future value of the CF stream:(1 )TFV PV r= � +6PerpetuityYou invest in a project that is expected to pay $1,200 a month, at the end of the month, forever. The monthly cost of capital is 1%. What is the present value of this CF stream?7Present Value (PV) of a Perpetuity CF CF … CF … |------------|-----------|--------- … -----|----- … --------> time0 1 2 t CF = the SAME CF at the end of EVERY period (year)First CF (start date): end of the first period (date 1)We get the same CF FOREVER (T = , infinity)r = the cost of capital for one period (one year) CFPVr=8Perpetuity examples1. Suppose the value of a perpetuity is $38,900 and the discount rate is 12% per annum. What must be the annual cash flow from this perpetuity?2. An asset that generates $890 a year forever is priced at $6,000. What is the required rate of return?9AnnuityYou consider investing in real estate. You expect the property to yield (i.e., generate) rent CFs of $18,000 a year for the next twenty years, after which you will be able to sell it for $250,000. Your required rate of return is 12% per annum. What is the maximum amount you’d pay for this CF stream?10Present Value (PV) of an Annuity CF CF … CF … CF |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T CF = the SAME CF at the end of EVERY period (year)First CF (start date): end of the first period (date 1)Last CF (end date): end of the last period (date T)T = the number of periods (number of years)r = the cost of capital for one period (one year) 111TCFPVr r� �� �= -� �� �+� �� �� �11Annuity, find the FVYou open a savings account and deposit $20,000 today. At the end of each of the next 15 years, you deposit $2,500. The annual interest rate is 7%. What will be the account balance 15 years from now?12Annuity, find the PMTYou are trying to borrow $200,000 to buy a house on a conventional 30-year mortgage with monthly payments. The monthly interest rate on this loan is 0.70%. What is the monthly payment on the loan?13Annuity, find the PMT: ChallengeYou plan to retire in 30 years. Then you will need $200,000 a year for 10 years (first withdrawal at t=31). Ten years later you expect to go to a retirement home where you will stay for the rest of your life. To enter the retirement home, you will have to make a single payment of $1,000,000. You can start saving for your retirement in an account that pays 9% interest a year. Therefore, starting one year from now (end of the first year: t =1), you will make equal yearly deposits into this account for 30 years. In 30 years (on date t=30), you expect a deposit of $500,000 to your retirement account from your cash value insurance policy. What should be your yearly deposit into the retirement account?14Adjusting the rate of return☺The frequency of compounding:☺Quoted (stated) rate☺Effective rate☺Always use the effective annual rate to discount annual CFs, effective monthly rate to discount monthly CFs etc.☺The case of inflation:☺Nominal rate☺Real rate☺Always use the nominal rate to discount nominal CFs and the real rate to discount real CFs.Effective to Effective: ExampleThe annual interest rate is 8%.What is the 2-year rate of return on $1?[ ]2 1(1 ) (2 )2( ,1 ) ( ,2 )2( ,2 )( ,2 )1 1$1 1 $1 1 1 0.08 1 1.16641.16year yeareffective year effective yearettective yearettective yearFV PV r PV rFV r rrr- -- ---� � � �= � + = � +� � � �� � � �= � + = � +� � � �� �+ = + =� �= 64 1 0.1664 16.64%- = =Effective to Effective: Formular(effective, 1-period) = 1-period effective