2013/4/9 Topics Covered Principles of Corporate Finance Present value and Future value By Zhang Xiaorong GNTY WR Net present value NPV Rule and rr rule 2: How to calculate Opportunity Cost of Capital Present values Valuing Long-Lived Assets PV Calculation Short Cuts Compound Interest Time Value of Money Time Value of Money Time is valuable for money Time I Dollar today is more valuable than I Now Inflation(or deflation? Uncertainty Present value Future value resent Current Future value Discount Future cash flow cash flow 1. Single period discount for future cash flow 2. Can get future value of today s cash flow
2013/4/9 1 Principles of Corporate Finance By Zhang Xiaorong 2: How to Calculate Present Values 2-1 — PresentValue and FutureValue — Net PresentValue — NPV Rule and IRR Rule — Opportunity Cost of Capital — Valuing Long-Lived Assets — PV Calculation Short Cuts — Compound Interest Topics Covered 2-2 — Time is valuable for money ◦ 1 Dollar today is more valuable than 1 Dollar “tomorrow” ◦ Consumption foregone ◦ Inflation (or deflation?) ◦ Timing ◦ Risk TimeValue of Money 2-3 TimeValue of Money — Time Now NextYear A：$1000 $0 B：$0 $1000 — Uncertainty A： $0 B： $1000 $2000 1-4 Present Value Now Tomorrow Present Value Future cash flow ??? CF1 Discount 1.Single period discount for future cash flow 2-5 FutureValue Now Tomorrow Current cash flow Future value Earning interest 2. Can get future value of today’s cash flow CF0 ??? 2-6
2013/4/9 Present and Future value Present and Future value Present value Amount to which an Amount to which an future cash investment will grow after earning interest after earning interest Present Value Discount rate value today of a Interest rate " used to compute present values future cash of future cash floy Discount Factors and rates Future values Future Value of $100= FV present values of future cash flows F=$100×(1+r) SI future payment. Future values Present value F=$l00×(1+r) Present value= Pv What is the future value of $400, 000 if interest is compounded annually at a rate of 5% for one year? PV=discount factor x C FV=$40000×(1+.05)=$420,000
2013/4/9 2 Present and FutureValue Present Value Value today of a future cash flow. Future Value Amount to which an investment will grow after earning interest 2-7 Present and FutureValue Present Value Value today of a future cash flow. Future Value Amount to which an investment will grow after earning interest Discount Rate “Interest rate” used to compute present values of future cash flows. 2-8 Discount Factors and Rates Discount Rate Interest rate used to compute present values of future cash flows. Discount Factor Present value of $1 future payment. 2-9 FutureValues FutureValue of $100 = FV FV r =$100´ + (1 ) 2-10 FutureValues FV r =$100´ + (1 ) Example - FV What is the future value of $400,000 if interest is compounded annually at a rate of 5% for one year? $400,000 (1 .05) $420,000 1 FV = ´ + = 2-11 Present Value 1 PV = discount factor Present Value = PV ´C 2-12
2013/4/9 Present value Valuing an Office Building Discount Factor= DF= PV of s Step I: Forecast cash fiows Cost of building Co 370 Sale price in Year I= CI=420 DE (1+r) SteP 2: Estimate opportunity cost of capital Discount Factors can be used to comp If equally risky investments in the capital market present value of any cash flow. ffer a return of 5%, then Cost of capi Valuing an Office Building Net present value Step 3: Discount future cash flow NPV=PV-required investment 420=400 Step 4: Go ahead if Py of payoff exceeds NPV=Co+ NPI=400-370=30 Valuing an Office Building Valuing an office Building Question-2: Where do we get the discount Cost of building: 320 rate of5‰? Cost of land: 50 scount rate is the market rate of return by the investments at the same risk level company does not have to of your investment. pay for the land. Why is 50 included as part of the cost? It considers both TIMING and RISK. (Take it as Compare apple to apple
2013/4/9 3 Present Value Discount Factor = DF = PV of $1 Discount Factors can be used to compute the present value of any cash flow. 1 DF (1 ) +r = 2-13 Valuing an Office Building Step 1: Forecast cash flows Cost of building = C0 = 370 Sale price inYear 1 = C1 = 420 Step 2: Estimate opportunity cost of capital If equally risky investments in the capital market offer a return of 5%, then Cost of capital = r = 5% 2-14 Valuing an Office Building Step 3: Discount future cash flows Step 4: Go ahead if PV of payoff exceeds investment 400 (1 .05) 420 (1 ) 1 = +r = + = C PV NPV = 400- = 370 30 2-15 Net Present Value r C + + 1 NPV = C NPV = PV - required investment 1 0 2-16 Valuing an Office Building — Decompose the cost ◦ Cost of building: 320 ◦ Cost of land: 50 Question-1: The company does not have to pay for the land. Why is 50 included as part of the cost? 2-17 Valuing an Office Building Question-2: Where do we get the discount rate of 5%? — The discount rate is the market rate of return given by the investments at the same risk level as that of your investment. — It considers bothTIMING and RISK.(Take it as given till chapter 7) — Compare apple to apple, not to orange. 2-18
2013/4/9 Risk and present value Risk and present value Higher risk projects require higher VofC,=$420at12% rates of return Higher required rates of return cause Pv=420 lower Pv: 1+.1=375 PV of C,=$420 at 5% P of C,=$420 at 5% 420 PV= =400 PV=420 1+.05 +05=400 Risk and Net Present value Decision Rules for Investment rules NPV=PV-required investment Accept investments that offer rates of return in excess of their opportunity cost NPV=375,000-370.000 ccept investments that have positive =$5,000 Rate of return rule Net Present value rule Accept investments that offer rates of return Accept investments that have positive net in excess of their opportunity cost of capital Example Example In the project listed the foregone investment Suppose we can invest $50 today and receive $60 ne year: Should we accept the project given return=-pot=4200370001350135% 370,000 NPV=-50+ 4.55 1.10
2013/4/9 4 Risk and Present Value — Higher risk projects require higher rates of return — Higher required rates of return cause lower PVs 400 1 .05 420 PV PV of C1 $420 at 5% = + = = 2-19 Risk and Present Value 400 1 .05 420 PV PV of C1 $420 at 5% = + = = 375 1 .12 420 PV PV of C1 $420 at 12% = + = = 2-20 Risk and Net Present Value $5,000 NPV = 375,000 - 370,000 NPV = PV - required investment = 2-21 — Two important rules ◦ Accept investments that offer rates of return in excess of their opportunity cost of capital ◦ Accept investments that have positive net present value Decision Rules for Investment 2-22 Rate of Return Rule — Accept investments that offer rates of return in excess of their opportunity cost of capital Example In the project listed below, the foregone investment opportunity is 12%. Should we do the project? .135 or 13.5% 370,000 420,000 370,000 investment profit Return = - = = 2-23 Net Present Value Rule — Accept investments that have positive net present value Example Suppose we can invest $50 today and receive $60 in one year. Should we accept the project given a 10% expected return? $4.55 1.10 60 NPV = -50+ = 2-24
2013/4/9 Opportunity Cost of Capital Opportunity Cost of Capital limited, so is capital. To invest in the specified project means You may invest $100, 000 today. Depending on giving up other opportunities of investment he ny, you may get one of The opportunity cost of capital is the three possible cash payoffs highest rate of return among the alternatives. Economy i Slump Normal Boom than the opportunity cost of capital, Payoff;s80,000110000140,000 you are using the capital in the most efficient NPV=$95,650-100,000=S-4,350 Opportunity Cost of Capital Opportunity Cost of Capital Example. continued Example.continued You notice that one stock in the market has the The stocks expected payoff leads to an expected ame risk as that of your investment. The stock my, is forecasted at S110 Expected return=expected profit The stocks expected payoff leads to an 110-95.65 expected return 95651r15% Opportunity Cost of Capital Opportunity Cost of Capital ounting the expected payoff at the expected Notice that you come to the same conclusion if leads to the Pl and NPVof the project you compare the expected project return with the cost of capital. l10.000 15s95.650 Expected retun=expectedI =.0or10% wesley NPV=$95650-100,000=$-4,350 Expected return on the investment is 10%, less than the expected return on the stock, the opportunity cost of capital 15%
2013/4/9 5 Opportunity Cost of Capital — Resources are limited, so is capital. — To invest in the specified project means giving up other opportunities of investment. — The opportunity cost of capital is the highest rate of return among the alternatives. — If your investment gives a higher rate of return than the opportunity cost of capital, you are using the capital in the most efficient way. 2-25 Opportunity Cost of Capital Example You may invest $100,000 today. Depending on the state of the economy, you may get one of three possible cash payoffs: Payoff $80,000 110,000 140,000 Economy Slump Normal Boom 110,000 PV $95,650 1.15 NPV $95,650 100,000 $ 4,350 = = = - = - 2-26 Opportunity Cost of Capital Example - continued You notice that one stock in the market has the same risk as that of your investment. The stock is trading for $95.65. Next year’s price, given a normal economy, is forecasted at $110. The stocks expected payoff leads to an expected return. 2-27 Opportunity Cost of Capital Example - continued The stocks expected payoff leads to an expected return. .15 or 15% 95.65 110 95.65 expected profit Expected return = - = = investment 2-28 Opportunity Cost of Capital Example - continued Discounting the expected payoff at the expected return leads to the PV and NPV of the project 110,000 PV $95,650 1.15 NPV $95,650 100,000 $ 4,350 = = = - = - 2-29 Opportunity Cost of Capital Example - continued Notice that you come to the same conclusion if you compare the expected project return with the cost of capital. expected profit 110,000 100,000 Expected return .10 or 10% 100,000 Expected return on the investment is 10%, less than the expected return on the stock, or the opportunity cost of capital 15%. investment - = = = 2-30
2013/4/9 Opportunity Cost of Capital PV of a Long-lived Asset Capital and Cost of Borrowing Tomorrow You are going to start your own business by CF2 parents, who are rich and generous and Present Discount do not require any return the market rate of return of such projects is I2% on average. Multi-period discount PV of a Long-lived Asset PV of a Long-lived Asset Discount Factor= DF Pv of $l P=DF×C、1+ DE DE Discount Factors can be used to compute Discount Factors can be used to compute the present value of any cash flow PV of a Long-lived Asset PV of a Long-lived Asset Example PV=DFC C The payment terms are 2 years same as cash Ifyou can earn 8% on your money how much (1+r) lacing"I"with"t"allows the formula used for cas nt In time
2013/4/9 6 Opportunity Cost of Capital — Distinguish between Opportunity Cost of Capital and Cost of Borrowing You are going to start your own business by taking an investing project.You can get bank loan at 8%, or you can get the seed fund from your parents, who are rich and generous and do not require any return.The market rate of return of such projects is 12% on average. What’s the discount rate of your project? 2-31 PV of a Long-lived Asset Now Tomorrow Present Value Future value CF0 CF2 Discount Multi-period discount The Day after Tomorrow Discount 2-32 PV of a Long-lived Asset Discount Factor = DF = PV of $1 Discount Factors can be used to compute the present value of any cash flow. DF r = t + 1 (1 ) 2-33 PV of a Long-lived Asset Discount Factors can be used to compute the present value of any cash flow. (1 ) 1 DF +r = 1 1 1 1 r C PV DF C + = ´ = 2-34 PV of a Long-lived Asset Replacing “1” with “t” allows the formula to be used for cash flows that exist at any point in time t t t r C PV DF C (1+ ) = ´ = 2-35 PV of a Long-lived Asset Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? 2-36
2013/4/9 PV of a Long-lived Asset PV of a Long-lived Asset PVs can be added together to evaluate multiple cash flows. The payment terms are 2 years same as cash. Ifyou can earn 8% on your money, how much oney should you set aside today in order te P=-+-C2+ 1+r)2 r)2 P==92.57202 PV of a Long-lived Asset PV of a Long-lived Asset PVs can be added together to evaluate 5200 multiple cash flows Presert value PV07(0m=265.88 s93.46 PV of a Long-lived Asset PV of a Long-lived Asset and the other two years from now, the value of Assume that the cash flows Assumer, =20% and r2=7%. follows. Given a 5% required tte of return, create a present alue worksheet and show the DFI DF2=02 Year0 Year 1 Year 2 -170.000-100.000+320.000
2013/4/9 7 PV of a Long-lived Asset Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? PV = = 3000 1 08 2 572 02 ( . ) $2, . 2-37 PV of a Long-lived Asset PVs can be added together to evaluate multiple cash flows. PV C r C r = + + + + 1 1 2 2 (1 ) (1 ) .... 2-38 PV of a Long-lived Asset PVs can be added together to evaluate multiple cash flows. 1 2 265.88 (1 077) 200 (1 .07) 100 = + = + + PV 2-39 PV of a Long-lived Asset Present Value Year 0 100/1.07 200/1.0772 Total = $93.46 = $172.42 = $265.88 $100 $200 Year 0 1 2 2-40 PV of a Long-lived Asset Given two dollars, one received a year from now and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%. .87 .83 2 1 (1 .07 ) 1.00 2 (1 .20 ) 1.00 1 = = = = + + DF DF 2-41 PV of a Long-lived Asset Example Assume that the cash flows from the construction and sale of an office building is as follows. Given a 5% required rate of return, create a present value worksheet and show the net present value. 170,000 100,000 320,000 Year 0 Year 1 Year 2 - - + 2-42
2013/4/9 PV of a Long-lived Asset PV of a Long-lived Asset Assume that the cash flows from the construction and sale Assume that the cash flows from the construction and sale f an office building is as follows. Given a 5% require rate of return, create a present value worksheet and show rate of, create a present value worksheet and show the net present ralue Discount s17,000 val T70000-170000 t=952 2a=907 10000105-95,238 NPV= Total= $25.0 3200001052=5907想 Total-NPV-$25,0 When Cash Flows Can Be Short cuts Added Up General Rules Sometimes there are shortcuts that make Present values can be added up it very easy to calculate the present value Future)cash flows at the same of an asset that pays off in different period can be added up periods. These tools allow us to cut But cash flows at different periods can through the calculations quickly. Short Cuts: Perpetuity Short Cuts: Perpetuity Perpetuity - Financial concept in which a Perpetuity -Financial concept in which a cash flow is theoretically received forever. cash flow is theoretically received forever. ash flow Return= nt val
2013/4/9 8 PV of a Long-lived Asset Example - continued Assume that the cash flows from the construction and sale of an office building is as follows. Given a 5% required rate of return, create a present value worksheet and show the net present value. ( ) $25,011 2 .907 320,000 290,249 1 .952 100,000 95,238 0 1.0 170,000 170,000 Value Present Flow Cash Factor Discount Period 2 1.05 1 1.05 1 = = = + + = - - - - NPV Total 2-43 PV of a Long-lived Asset Present Value Year 0 -170,000 -100,000/1.05 320,000/1.052 Total = NPV -$170,000 = -$170,000 = $95,238 = $290,249 = $25,011 -$100,000 +$320,000 Year 0 1 2 Example - continued Assume that the cash flows from the construction and sale of an office building is as follows. Given a 5% required rate of return, create a present value worksheet and show the net present value. 2-44 When Cash Flows Can Be Added Up — General Rules ◦ Present values can be added up; ◦ (Future) cash flows at the same period can be added up; ◦ But cash flows at different periods can not. 2-45 Short Cuts — Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tools allow us to cut through the calculations quickly. 2-46 Short Cuts: Perpetuity Perpetuity - Financial concept in which a cash flow is theoretically received forever. … 1 C 2 C 3 C 0 2-47 Short Cuts: Perpetuity Perpetuity - Financial concept in which a cash flow is theoretically received forever. PV C r = = present value cash flow Return 2-48
2013/4/9 Short Cuts: Perpetuity Short Cuts: Perpetuity Perpetuity -Financial concept in which a sh flow is theoretically received for fever of Cash Flow = cash flow discount rate cash flow C CE C PV of Cash Flow discount rate 1+r(1+r)2(1+r) +r) CC Example of Perpetuity Short Cuts: Growing Perpetuity Imagine you deposit US$100 Million in bank at the annual rate of 10%. At the end of every year, Growing Perpetuity-cash flow is you withdraw the interest of US$ IOMillion, and theoretically growing constantly at g(g<r) you go on with this withdrawing?(Suppose the rate Cc×(1+g)C×(1+g)2 Short Cuts: Growing Perpetuity Short Cuts: Growing Perpetuity Growing Perpetuity -cash flow is theoretically growing constantly at g(g<r) PV of Cash Flo PV of cash flow= 1+g
2013/4/9 9 Short Cuts: Perpetuity Perpetuity - Financial concept in which a cash flow is theoretically received forever. r C PV 1 0 discount rate cash flow PV of Cash Flow = = 2-49 Short Cuts: Perpetuity 1 1 1 1 2 3 1 1 1 cash flow PV of Cash Flow discount rate ... .. 1 (1 ) (1 ) (1 ) lim (1 ) n n n C C C C PV r r r r C C C ®¥ r r r r = = + + + + + + + + + é ù = - = ê ú ë û + 2-50 — Imagine you deposit US$100 Million in bank at the annual rate of 10%.At the end of every year, you withdraw the interest of US$10Million, and leave the principal in the bank. How long can you go on with this withdrawing? (Suppose the rate does not change) Forever! Example of Perpetuity 2-51 Growing Perpetuity - cash flow is theoretically growing constantly at g (g<r) and received forever. Short Cuts: Growing Perpetuity 0 … 1 C 2 C×(1+g) 3 C ×(1+g) 2 2-52 Growing Perpetuity - cash flow is theoretically growing constantly at g (g<r) and received forever. 1 cash flow PV of Cash Flow discount rate C PV r g = = - Short Cuts: Growing Perpetuity 2-53 Short Cuts: Growing Perpetuity 2 1 1 1 1 1 2 3 2 3 1 1 cash flow PV of Cash Flow discount rate (1 ) (1 ) (1 ) ... .. 1 (1 ) (1 ) (1 ) 1 1 1 1 1 1 ... lim 1 1 1 1 1 1 1 1 n n n n C C g C g C g PV r r r r g C g g g C C r r r r r r g r - ®¥ = + + + = + + + + + + + + + é ù æ ö + ê ú -ç ÷ é ù + æ + + ö æ ö + è ø = + + + = = ê ú ê ú ç ÷ ç ÷ + + è + ø è ø + + ê ú + ê ú ë û - ê ú + ë û 1 r g - 2-54
2013/4/9 Short Cuts: Annuity Short Cuts: Annuity Annuity- An asset that pays a fixed sum Annuity- An asset that pays a fixed sum each year for a specified number of years each year for a specified number of years Year of payment Present value Short Cuts: Annuity Short Cuts: Annuity Due Annuity-An asset that pays a fixed sum Annuity Due-An asset that pays a fixed each year for a specified number of years. sum each year for a specified number of years, the first cash flow occurring in ar o PV of annuity=Cx PⅤ of annuity due=Cx (+y(+n) Annuity Example Annuity Example is the ext 5 years, with no cash don. What EI car really costing you? is the car really costing you Cost=5.000
2013/4/9 10 Annuity - An asset that pays a fixed sum each year for a specified number of years. Short Cuts:Annuity 0 1 C 2 C 3 C T C L 2-55 Annuity - An asset that pays a fixed sum each year for a specified number of years. r Perpetuity (first C payment in year 1) Perpetuity (first payment in year t + 1) Annuity from year 1 to year t Asset Year of Payment 1 2…..t t + 1 Present Value t r r C (1 ) 1 + ÷ ø ö ç è æ ÷ ÷ ø ö ç ç è æ + ÷ ø ö ç è æ ÷ - ø ö ç è æ t r r C r C (1 ) 1 Short Cuts:Annuity 2-56 Annuity - An asset that pays a fixed sum each year for a specified number of years. Short Cuts:Annuity ( ) 1 1 PV of annuity 1 C t r r r é ù = ´ - ê ú ê ú + ë û 2-57 Annuity Due - An asset that pays a fixed sum each year for a specified number of years, the first cash flow occurring in year 0. ( ) 1 1 1 PV of annuity due (1 ) 1 1 1 (1 ) t t C r r r r C C r r r - é ù = ´ ê ú - ´ + ê ú + ë û é ù = + ´ - ê ú ë û + Short Cuts:Annuity Due 2-58 Annuity Example Example Tiburon Autos offers an “easy payment” scheme on a new Toyota of $5,000 a year, paid at the end of each of the next 5 years, with no cash down. What is the car really costing you? 2-59 Annuity Example Tiburon Autos offers an “easy payment” scheme on a new Toyota of $5,000 a year, paid at the end of each of the next 5 years, with no cash down. What is the car really costing you? ( ) 5,000 4.100 $20,501 .07 1 .07 1 .07 1 Cost 5,000 5 = ´ = ú û ù ê ë é + = ´ - 2-60