Finance 2009 Spring Chapter 4 Discounted Cash Flow Valuation

4-2 Chapter Outline Valuation: The One-Period Case The Multiperiod Case Compounding Periods Simplifications (Annuities & Perpetuities)‏ What Is a Firm Worth?

4-3 Basic Definitions Time Line PV (Present Value): earlier money on a time line FV (Future Value): later money on a time line r (Interest rate): “exchange rate” between earlier money and later money  Discount rate, Cost of capital  Opportunity cost of capital, Required return 0 12t PVFV …

4-4 Future Values 01 $1,000 FV=? r = 5% Interest = 1000 x.05 = 50 Value in one year = principal + interest = 1, = 1,050 Future Value (FV) = 1,000 x (1 +.05) = 1,050 Example: 1 year Example: 2 year 02 $1,000 FV=? r = 5% 1 FV = 1,000 x 1.05 x 1.05 = 1,102.50

4-5 Future Values: General Formula FV = PV(1 + r) t  FV = future value  PV = present value  r = period interest rate, expressed as a decimal  t = number of periods Future value interest factor = (1 + r) t

4-6 Effects of Compounding Simple interest  FV with simple interest = = 1100 Compound interest  FV with compound interest = =  The extra 2.50 comes from the interest =.05 x 50 = $1,000 FV=? r = 5% 1

4-7 Future Values Example: 5 year FV = 1,000 x (1.05) 5 = 1, The effect of compounding  is small for a small number of periods  increases as the number of periods increases  FV with simple interest = $1, $1,000FV=? …r = 5%

4-8 Future Values: compound effect

4-9 Future Values Example: 200 year FV = 10 x (1.055) 200 = 447, $10FV=? …r = 5.5% The effect of compounding  Simple interest = (10)(.055) =  Compounding added $447, to the value of the investment

4-10 Future Values

4-11 FV as a General Growth Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years? FV = 3,000,000(1.15) 5 = 6,034, ,000,000FV=? …r = 15%

4-12 Present Values How much do I have to invest today to have some amount in the future?  FV = PV(1 + r) t  PV = FV / (1 + r) t Discounting  mean finding the present value of some future amount. Value  the present value unless we specifically indicate that we want the future value.

4-13 Present Values Example1: need $10,000 for a new car, 1 yr, 7% Example2: prepare daughter’s college tuition $150,000, 17 yr, 8% 01 PV=? $10,000 r = 7% PV = 10,000 / (1.07) 1 = 9, PV=?$150,000 …r = 8% PV = 150,000 / (1.08) 17 = 40,540.34

4-14 PV – Important Relationship For a given interest rate the longer the time period, the lower the present value For a given time period the higher the interest rate, the smaller the present value

4-15 Discount Rate What is the implied interest rate on an investment? Rearrange the basic PV equation and solve for r  FV = PV(1 + r) t  r = (FV / PV) 1/t – 1

4-16 Discount Rate Example1: invest $1000 today, pay $1200 in 5 years Example2: invest $5,000 today, double in 6 years 0 5 $1,000 $1,200 r = ? r = (1200 / 1000) 1/5 – 1 = = 3.714% 0 6 $5,000 $10,000 r = ? r = (10,000 / 5,000) 1/6 – 1 =.1225 = 12.25%

4-17 Number of Periods Start with basic equation and solve for t FV = PV(1 + r) t t = ln(FV / PV) / ln(1 + r)‏

4-18 Number of Periods Example: want to buy a new car (price $20,000)‏ have $15,000 today can invest (r = 10%)‏ How long? 0 t = ? $15,000 $20,000 r = 10% t = ln(20,000 / 15,000) / ln( ) = 3.02 years

4-19 Spreadsheet Example Use the following formulas for TVM calculations  FV(rate,nper,pmt,pv)‏  PV(rate,nper,pmt,fv)‏  RATE(nper,pmt,pv,fv)‏  NPER(rate,pmt,pv,fv)‏ Click on the Excel icon to open a spreadsheet containing four different examples.

4-20 Formula

4-21 FV - Multiple Cash Flows Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? 0 12 $500 r = 9% $600FV=? FV = 500(1.09) (1.09) = 1,248.05

4-22 FV - Multiple Cash Flows How much will you have in 5 years if you make no further deposits? $500 r = 9% 3 $600 4 FV=? FV = 500(1.09) (1.09) 4 = 1, Second way – use value at year 2: First way: FV = 1,248.05(1.09) 3 = 1,616.26

4-23 FV - Multiple Cash Flows ₤7,000 r = 8%3 ₤4,000 Today (year 0): FV = 7000(1.08) 3 = 8, Year 1: FV = 4,000(1.08) 2 = 4, Year 2: FV = 4,000(1.08) = 4,320 Year 3: value = 4,000 Total value in 3 years = = 21, the value at year 3 the value at year 4 Value at year 4 = 21,803.58(1.08) = 23, Example 6.1

4-24 FV - Multiple Cash Flows to invest $2,000 at the end of each of the next 5 years. (Figure 6.4)‏

4-25 PV - Multiple Cash Flows Example 6.3  Year 1 CF: 200 / (1.12) 1 =  Year 2 CF: 400 / (1.12) 2 =  Year 3 CF: 600 / (1.12) 3 =  Year 4 CF: 800 / (1.12) 4 =  Total PV =

4-26 PV - Multiple Cash Flows to pay $1,000 at the end of every year for the next 5 years. (Figure 6.5)‏

4-27 Using a Spreadsheet You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows Setting the data up is half the battle – if it is set up properly, then you can just copy the formulas Click on the Excel icon for an example

4-28 Net Present Value The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment. NPV = –Cost + PV The Net Present Value is positive, so the investment should be purchased.

4-29 NPV Decisions Your broker calls you and tells you a investment opportunity. invest $100 today, receive $40 in one year and $75 in two years. require a 15% return on investments of this risk Should you take the investment? $100 r = 15% $40$75 Year 1 CF: 40 / (1.15) 1 = Year 2 CF: 75 / (1.15) 2 = PV = = 91.49

4-30 Saving For Retirement You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? … … 0 25,000 25,000 25,000 25,000 25,000 Investment today = $1,084.71

4-31 Annuities & Perpetuities Annuity  finite series of equal payments that occur at regular intervals  ordinary annuity: the 1 st payment occurs at the end of the period  annuity due: the 1 st payment occurs at the beginning of the period Perpetuity  infinite series of equal payments  PV = C / r 0 12t … C …CC 0 12t … C …C C

4-32 Annuities & Perpetuities Example: Buying a new Italian sports car Ordinary annuity t = 48 months, r = 0.01 / month, C = $632 Formula: Annuity PV factor = (1 - PV factor) / r Spreadsheet

4-33 Growing Annuities & Perpetuities Growing Annuity  A growing stream of cash flows with a fixed maturity Growing Perpetuity  A growing stream of cash flows that lasts forever 0 12t … C … C(1+g)‏C(1+g) t … C … C(1+g)‏

4-34 Buying a House You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house? Bank loan  Monthly income = 36,000 / 12 = 3,000  Maximum payment =.28(3,000) = 840  PV = 840[1 – 1/ ] /.005 = 140,105 Total Price  Closing costs =.04(140,105) = 5,604  Down payment = 20,000 – 5,604 = 14,396  Total Price = 140, ,396 = 154,501

4-35 Finding the Payment Borrow $20,000 for a new car  r = 8% / year, compounded monthly  4 year loan  what is your monthly payment? Spreadsheet  PMT(rate,nper,pv,fv)‏

4-36 Finding the Number of Payments Example: spring break vacation  put $1,000 on your credit card  afford to $20 / month  r = 1.5% / month

4-37 Finding the Rate Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $ per month for 60 months. What is the monthly interest rate?  t = 60  PV = 10,000  C =

4-38 Without a Financial Calculator Trial and Error Process  Choose an interest rate and compute the PV of the payments based on this rate  Compare the computed PV with the actual loan amount  If the computed PV > loan amount, then the interest rate is too low  If the computed PV < loan amount, then the interest rate is too high  Adjust the rate and repeat the process until the computed PV and the loan amount are equal Finding the Rate

4-39 Annuity Due You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years?

4-40 Perpetuity Example:  stock price = $ 40  dividend = $1 / quarter  perpetuity formula: PV = C / r Current required return:  40 = 1 / r  r =.025 per quarter want to sell preferred stock at $100 per share, dividend ?  100 = C /.025  C = $2.50 per quarter

4-41 Formula

4-42 APR & EAR the actual rate paid use for comparison to alternatives the annual rate quoted by law APR = period rate x # of periods period rate = APR / # of periods NEVER divide the effective rate by the number of periods per year Effective Annual Rate (EAR)‏Annual Percentage Rate (APR)‏

4-43 Computing to earn 1% per month on $1 invested today. APR? 1 x 12 = 12% effectively earning? FV = 1(1.01) 12 = rate = ( – 1) / 1 =.1268 =12.68% to earn 3% per quarter. APR? 3 x 4 = 12% effectively earning? FV = 1(1.03) 4 = rate = ( – 1) / 1 =.1255 = 12.55% if the monthly rate is.5%, APR? 5 x 12 = 6% if the semiannual rate is.5%, APR?.5 x 2 = 1% if the APR is 12% with monthly compounding, monthly rate? 12 / 12 = 1% need to make sure that the interest rate and the time period match. Effective Annual Rate (EAR)‏Annual Percentage Rate (APR)‏

4-44 Decisions II Suppose you invest $100 in each account. How much will you have in each account in one year? pays 5.3%, with semiannual compounding EAR = ( /2) 2 – 1 = 5.37% Semiannual rate =.0539 / 2 =.0265 FV = 100(1.0265) 2 = pays 5.25%, with daily compounding EAR = ( /365) 365 – 1 = 5.39% Daily rate =.0525 / 365 = FV = 100( ) 365 = Saving account 2Saving account 1

4-45 Computing Payments with APRs Suppose you want to buy a new computer system price = $3,500, monthly payments, loan period = 2 years, r = 16.9% with monthly compounding. What is your monthly payment? Monthly rate = Number of months = 2 x 12

4-46 FV with Monthly Compounding Suppose you deposit $50 a month into an account APR = 9%, monthly compounding How much will you have in the account in 35 years?

4-47 PV with Daily Compounding need $15,000 in 3 years for a new car deposit money into an account, APR = 5.5%, daily compounding how much would you need to deposit?

4-48 Continuous Compounding Sometimes investments or loans are figured based on continuous compounding EAR = e q – 1  The e is the exponential function.  q = r x t Example: What is the effective annual rate of 7% compounded continuously?  EAR = e.07 – 1 =.0725 or 7.25%

4-49 What Is a Firm Worth? Conceptually, a firm should be worth the present value of the firm’s cash flows. The tricky part is determining the size, timing and risk of those cash flows.