Discounted Cash Flow ValuationFINC 3610 ‐ YostDiscounted Cash Flow Valuation103Future Value of Multiple Cash Flows() ()ttttCFrCFrCFFV +++×++×=−...11110• You open a bank account today with $500. You expect to deposit $1,000 at the end of each of the next three years. Interest rates are 5%, compounded annually. How much will you have in your account in three years?104Present Value of Multiple Cash Flows()() ()ttrCFrCFrCFCFPV+++++++=1...11221100• You just inherited some money from now dead Uncle Fred. You plan to use the money for a vacation, but know you first need to put aside some to cover your books and supplies over the next two years. You expect to need $4,000 in each of the next two years. Interest rates are 10%, compounded annually. How much of now dead Uncle Fred’s money do you need to put aside today?105Discounted Cash Flow ValuationFINC 3610 ‐ YostValuing Perpetuities• Perpetuity: A level stream of cash flows which continue forever (sometimes called consols)consols).• Present Value of a Perpetuity: 106Valuing Perpetuities• Assuming that interest rates are 10%, what is the value today of a perpetuity paying $500 per year, with the firstpaying $500 per year, with the first payment one year from today?• Would you be willing to pay $6,500 for the same perpetuity if interest rates were 8%?107Growing Perpetuities• Present Value of a Growing Perpetuity:• Suppose you own a perpetuity that promises to pay $1 next year, after which the payment is expected to grow at 5% per year forever. If interest rates are 10%, what is the value of the perpetuity?108Discounted Cash Flow ValuationFINC 3610 ‐ YostGrowing Perpetuities• Assume a growing perpetuity just made a payment of $120 yesterday. If the cash flow is expectedto grow at 5%the cash flow is expected to grow at 5% and interest rates are still 10%, what is the price of the perpetuity today?109Present Value of an Annuity• Annuity: A level stream of cash flows for a fixed period of time.• Present Value of an Annuity:110()⎥⎦⎤⎢⎣⎡+−×=trrCF111PV10Present Value of an Annuity• We can rearrange the equation to the following:• Present Value of an Annuity:111()rrCFt⎥⎦⎤⎢⎣⎡+−×=111PV10Discounted Cash Flow ValuationFINC 3610 ‐ YostPresent Value of an AnnuityLet’s return to our earlier example:•You just inherited some money from now deadYou just inherited some money from now dead Uncle Fred. You plan to use the money for a vacation, but know you first need to put aside some to cover your books and supplies over the next two years. You expect to need $4,000 in each of the next two years. Interest rates are 10%. How much of now dead Uncle Fred’s money do you need to put aside today?112• Future Value of an Annuity:Future Value of an Annuity()[]11FVtCF• This, of cour se, can also be rearranged…113()[]11FVt−+×=trr()[]rrCFt11FVt−+×=Future Value of an Annuity• What is the future value (at year 2) of the previous example?114Discounted Cash Flow ValuationFINC 3610 ‐ YostAnnuities: A Real‐Life Example• Books and beer are expensive! You now have a balance of $2,000 on your VISA card. The interest rate on that card is 2% perThe interest rate on that card is 2% per month. However, in an attempt to not let your debt stifle your social life, you pay only the $50 minimum payment each month (starting next month) and make no more charges on that card. How long will it tak e you to pay off the balance?115Annuities: A Real‐Life Example• How much would you have to pay each month if you wanted to pay off the balance in 3 years?116Growing Annuities• Present Value of a Growing Annuity:⎤⎡⎞⎛t117⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛++−×−=trggrCF111PV10Discounted Cash Flow ValuationFINC 3610 ‐ YostAnnuities Due• Annuity Due: An annuity for which the cash flows occur at the __________ of the periodperiod.• PV Annuity Due = (PV Ordinary Annuity) x (1 + r)118The Effect of Compounding• Annual Percentage Rate (APR): The nominal, stated annual interest rate that ignores the effect of compound interest within the year. p yThe APR is the periodic rate (r) times the number of compoundings per year (m).• Effective Annual Rate (EAR): The effective annual interest rate, which tak es into account the effect of compound interest.119APR and EAR• Example: A bank loan is quoted as 10% APR, compounded semiannually. What is the EAR?p y12011 −⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+=mmAPREARDiscounted Cash Flow ValuationFINC 3610 ‐ YostAPR and EAR: An Example• Which loan would you choose?–Bank A: 15% compounded daily–Bank B: 15.5% compounded quarterly–Bank C: 16% compounded annually121Amortization• What is an amortized loan?• You plan to buy a $200,000 house. You will put 10% down and finance the rest with a 30 year mortgage at 6% APR, compounded monthly. What are the monthly payments?122Beg. Bal PMT Interest Principal End. Bal.Amortization Schedule1234,263.34 21.32 1,057.87 3,205.463,205.46 16.03 1,063.16 2,142.302,142.30 10.71 1,068.48 1,073.821,073.82 5.37 1,073.82 0.00Discounted Cash Flow ValuationFINC 3610 ‐ YostSuggested Problems• Concepts Review and Critical Thinking Questions2through 8124–2 through 8• Questions and Problems:–1, 3, 4, 5, 7, 10, 12, 20, 21, 24, 26, 28, 36, 41, 43, 45, and 54Additional Practice• You want to buy a new, fully‐loaded Ford Explorer. You have managed to talk the salesman down to $40 000 You plan onsalesman down to $40,000. You plan on putting a 10% down payment on it and have secured a 60 month loan at 9% APR, compounded monthly, for the balance. How much are your monthly payments?125Additional Practice• Assuming a 10% interest rate, compounded annually, what is the value today of$1,000 per year forever,value today of $1,000 per year forever, with the first payment starting one year from today?• What if the first payment was in 5 years?126Discounted Cash Flow ValuationFINC 3610 ‐ YostAdditional Practice• Given an interest rate of 10% APR, compounded annually, what is the value in five years of a perpetual stream of $120 annual payments starting in nine years?127Additional Practice• You have just read an advertisement that says, “Pay us $100 a year for 10 years, starting next year, and we will pay you (and your heirs) $100 a year thereafter in perpetuity.” At what range of interest rates would you accept this