1FI3300Corporate Finance 010Spring Semester 2010Dr. Isabel TkatchAssistant Professor of Finance1Learning Objectives☺ Calculate the PV and FV in multi-period multi-CF time-value-of-money problems: ☺ General case☺ Perpetuity☺ Annuity2☺ 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 scheduleExampleYou 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 3(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?Present Value (PV) of a CF StreamCF1CF2 … CFt … CFT|------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T 12tTCFCF CF CFPV+++++4CFt = 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) 1212... ...(1 ) (1 ) (1 ) (1 )tTtTPVrr r r= + ++ ++++ + +Future Value (FV) of a CF StreamCF1CF2 … CFt… CFT|------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t TStep 1: calculate the present value of the CF stream5PV FV|------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t TStep 2: use the PV-FV formula to calculate the future value of the CF stream:(1 )TFV PV r=×+PerpetuityYou invest in a project that is expected to pay $1,200 a month, at the end of the month, forever.6The monthly cost of capital is 1%. What is the present value of this CF stream?2Present Value (PV) of a PerpetuityCF CF … CF … |------------|-----------|--------- … -----|----- … --------> time0 1 2 t CF = the SAMECF at the end of EVERY period (year)7First 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=Perpetuity 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?82. An asset that generates $890 a year forever is priced at $6,000. What is the required rate of return?AnnuityYou 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. 9Your required rate of return is 12% per annum. What is the maximum amount you’d pay for this CF stream?Present Value (PV) of an AnnuityCF CF … CF … CF |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t TCF = the SAME CF at the end of EVERY period (year)First CF (start date): end of the first period (date 1)10First 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)111TCFPVrr⎡⎤⎛⎞=−⎢⎥⎜⎟+⎝⎠⎢⎥⎣⎦Annuity, 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. 11The annual interest rate is 7%. What will be the account balance 15 years from now?Annuity, find the PMTYou are trying to borrow $200,000 to buy a house on a conventional 30-year mortgage with monthly payments.12The monthly interest rate on this loan is 0.70%. What is the monthly payment on the loan?3Annuity, 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 13gpy $, , gyour 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?Adjusting 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 14annual 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?21(1 ) (2 )11year yearFV PV r PV r−−⎡⎤⎡⎤=×+ =×+⎣⎦⎣⎦[]2(,1) (,2)2(,2)(,2)$1 1 $1 1 1 0.08 1 1.16641.16effe ctive year ef fe ctive yea rettective yearettective yearFV r rrr−−−−⎡⎤⎡⎤=×+ =×+⎣⎦⎣⎦⎡⎤+=+=⎣⎦= 64 1 0.1664 16.64%−= =Effective to Effective: Formular(effective, 1-period) = 1-period effective rateThe return on $1 invested for 1 periodr(effective, n-period) = n-period effective rateThe return on $1 invested for n periodsThe return on $1 invested for n periods(, ) (,1)11neffective n pe riod effective periodrr−−⎡⎤⎡ ⎤+=+⎣⎦⎣ ⎦Effective to Effective: n>1The effective monthly rate is 1%, what is the effective annual rate?Since the effective monthly rate is known and there are n=12 months in one year1% 0 01[][](,)12(,)12(,)12(,12 ) (,1)11% 0.01110.011 0.01 1 0.1268 12.681%effective months effective meffective monthlyeffective annualeffective annuaotlnhrrrrr−−⎡⎤+=+⎣⎦==+=+=+ −≅ =Effective to Effective: n>1The effective monthly rate is 1%, what is the effective quarterly