# HOW TO CALCULATE PRESENT VALUES

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1 Chapter 2 HOW TO CALCULATE PRESENT VALUES Brealey, Myers, and Allen Principles of Corporate Finance 11th Edition McGraw-Hill/Irwin Copyright 2014 by The McGraw-Hill Companies, Inc. All rights reserved.

2 2-1 FUTURE VALUES AND PRESENT VALUES Calculating Future Values Future Value Amount to which investment will grow after earning interest Present Value Value today of future cash flow 2-2

3 2-1 FUTURE VALUES AND PRESENT VALUES Future Value of \$100 = FV \$100 (1 t r) Example: FV What is the future value of \$100 if interest is compounded annually at a rate of 7% for two years? FV \$100 (1.07) (1.07) FV \$100 (1.07) 2 \$ \$

4 FİGURE 2.1 FUTURE VALUES WİTH COMPOUNDİNG 2-4

5 2-1 FUTURE VALUES AND PRESENT VALUES Present value = PV PV = discount factor C 1 2-5

6 2-1 FUTURE VALUES AND PRESENT VALUES Discount factor = DF = PV of \$1 Discount factors can be used to compute present value of any cash flow 2-6

7 2-1 FUTURE VALUES AND PRESENT VALUES Given any variables in the equation, one can solve for the remaining variable Prior example can be reversed PV DF 2 C 2 PV 1 (1.07)

8 FİGURE 2.2 PRESENT VALUES WİTH COMPOUNDİNG 2-8

9 2-1 FUTURE VALUES AND PRESENT VALUES Valuing an Office Building Step 1: Forecast Cash Flows Cost of building = C 0 = \$700,000 Sale price in year 1 = C 1 = \$800,000 Step 2: Estimate Opportunity Cost of Capital If equally risky investments in the capital market offer a return of 7%, then cost of capital = r = 7% 2-9

10 2-1 FUTURE VALUES AND PRESENT VALUES Valuing an Office Building Step 3: Discount future cash flows C \$800,000 PV 1 (1r ) (1.07) \$747,664 Step 4: Go ahead if PV of payoff exceeds investment NPV \$747,664 \$700, 000 \$47,

11 2-1 FUTURE VALUES AND PRESENT VALUES Net Present Value NPV = PV required NPV = C 0 C1 1 r investment 2-11

12 2-1 FUTURE VALUES AND PRESENT VALUES Risk and Present Value Higher risk projects require a higher rate of return Higher required rates of return cause lower PVs PV of C1 \$800,000 at 7% \$800,000 PV \$747,

13 2-1 FUTURE VALUES AND PRESENT VALUES Risk and Net Present Value NPV = PV required investment NPV = \$747,664 \$700,000 \$47,664 NPV at 5%= \$761,000 - \$700,00 = \$61,904 NPV at 12%= \$714, ,00 = \$14,

14 2-1 FUTURE VALUES AND PRESENT VALUES Net Present Value Rule Accept investments that have positive net present value Using the original example: Should one accept the project given a 12% opportunity cost? NPV = \$700,000 + \$800, \$14,

15 2-1 FUTURE VALUES AND PRESENT VALUES Rate of Return Rule Accept investments that offer rates of return in excess of their opportunity cost of capital In the project listed below, the opportunity cost of capital is 12%. Is the project a wise investment? Return profit investment \$800,000 \$700,000 \$700, , or 14.3% 2-15

16 FİNDİNG THE DİSCOUNT RATE Often we will want to know what the implied interest rate is in an investment FV = PV(1 + r) t => FV / PV = (1 + r) t r = (FV / PV) 1/t 1 Ex.: You are looking at an investment that will pay \$1.200 in 5 years if you invest \$1.000 today. What is the implied rate of interest? r = (1200 / 1000) 1/5 1 = = 3.714%

17 FİNDİNG THE NUMBER OF PERİODS Solve for t: FV = PV(1 + r) t FV / PV = (1 + r) t ln(fv / PV) = t ln(1 + r) t = ln(fv / PV) / ln(1 + r)

18 NUMBER OF PERİODS EXAMPLE 1 You want to purchase a new car and you are willing to pay \$ If you can invest at 10% per year and you currently have \$15.000, how long will it be before you have enough money to pay cash for the car - without additional savings? t = ln(fv / PV) / ln(1 + r) t = ln(20,000 / 15,000) / ln(1.1) = t = / = 3.02 years

19 2-1 FUTURE VALUES AND PRESENT VALUES Multiple Cash Flows Discounted Cash Flow (DCF) formula: C PV (1 r) C (1 r) C t (1 r) t 2-19

20 FİGURE 2.5 NET PRESENT VALUES 2-20

21 2-2 PERPETUİTİES AND ANNUİTİES Perpetuity Financial concept in which cash flow is theoretically received forever PV of cash flow cash flow discount rate PV 0 C 1 r 2-21

22 2-2 PERPETUİTİES AND ANNUİTİES Perpetuity Return r cash flow present value C PV 2-22

23 2-2 PERPETUİTİES AND ANNUİTİES Present Value of Perpetuities What is the present value of \$1 billion every year, for eternity, if the perpetual discount rate is 10%? 2-23

24 2-2 PERPETUİTİES AND ANNUİTİES Present Value of Perpetuities What if the investment does not start making money for 3 years? 2-24

25 2-2 PERPETUİTİES AND ANNUİTİES Annuity Asset that pays fixed sum each year for specified number of years Asset Perpetuity (first payment in year 1) Year of Payment 1 2..t t + 1 Present Value Perpetuity (first payment in year t + 1) Annuity from year 1 to year t C C 1 t r r (1 r) 2-25

26 2-2 PERPETUİTİES AND ANNUİTİES PV of annuity C 1 r r 1 1 t r 2-26

27 2-2 PERPETUİTİES AND ANNUİTİES Example: Tiburon Autos offers payments of \$5,000 per year, at the end of each year for 5 years. If interest rates are 7%, per year, what is the cost of the car? 2-27

28 2-2 PERPETUİTİES AND ANNUİTİES Annuity Example: The state lottery advertises a jackpot prize of \$365 million, paid in 30 yearly installments of \$ million, at the end of each year. Find the true value of the lottery prize if interest rates are 6%. 1 1 Lottery Value Value \$167,500,

29 2-2 PERPETUİTİES AND ANNUİTİES Suppose you take a 4-year car loan from bank. You borrow \$20,000 and will make monthly payments. Bank charges you 8% per year, compounded monthly. What is your monthly payment? What is monthly interest rate? 8%/12 = 0.667% per month or 0.08/12= T (1 r) 1 C T r(1 r) (1.0067) 1 C (1.0067) C C / /29/

30 2-2 PERPETUİTİES AND ANNUİTİES Future Value of an Annuity FVof annuity C 1 r r t

31 2-2 PERPETUİTİES AND ANNUİTİES Future Value of an Annuity What is the future value of \$20,000 paid at the end of each of the following 5 years, assuming investment returns of 8% per year? FV 20, \$117,

32 2-3 GROWİNG PERPETUİTİES AND ANNUİTİES Constant Growth Perpetuity PV 0 r C 1 g g = the annual growth rate of the cash flow This formula can be used to value a perpetuity at any point in time PV t C t 1 r g 2-32

33 2-3 GROWİNG PERPETUİTİES AND ANNUİTİES Constant Growth Perpetuity What is the present value of \$1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and constant growth rate of 4%? PV \$ billion 2-33

34 2-3 GROWİNG PERPETUİTİES AND ANNUİTİES Growing Annuities Golf club membership is \$5,000 for 1 year, or \$12,750 for three years. Find the better deal given payment due at the end of the year and 6% expected annual price increase, discount rate 10%. 2-34

35 ANNUITY EXAMPLE: Jacqui wants to retire twenty years from now. She wants to save enough so that she can have a pension of \$10,000 a year for ten years. How much must she save each year, denoted S furing the the first 20 years to achieve her goal if the interest rate is 5%? Assume all cash flows ocur at the end of the year. The first date at which S is saved is date 1 and the first date at which the \$10,000 pension is received is date

36 2-4 HOW INTEREST İS PAİD AND QUOTED Effective Annual Interest Rate (EAR) Interest rate annualized using compound interest Annual Percentage Rate (APR) Also called: Quoted rate Interest rate annualized using simple interest 2-36

37 2-4 HOW INTEREST İS PAİD AND QUOTED Invest \$1000 for 1 year. Bank pays 12% interest rate compounded semiannually. What is APR? Semiannual interest rate= 6% APR = 2 x 6% = 12% What is EAR? 9/29/

38 2-4 HOW INTEREST İS PAİD AND QUOTED EAR: Bank pays 12% int rate compounded semiannually. Semi-annual rate = 0.10/2 = FV = \$1000(1.06)(1.06)=\$1,123.6 Rate of return = ( ) / 1000= % (1 EAR EAR) 1 APR m /29/ m

39 2-4 HOW INTEREST İS PAİD AND OUTLAİD Given a monthly rate of 1%, what is the (EAR)? What is the (APR)? EAR = (1+.01) EAR = (1+.01) = r 1 =.1268, or 12.68% APR =.0112 =.12, or 12.00% 2-39

40 2-4 How Interest is Paid and Quoted Quoted Rate % Compoundin g Freq., (m) Periodica l Interest Rate %, Invest (\$) FV, after a year: Rate of Return % 12 Annual, (1.12)= Semiannual, (1.06) 2 = Quarterly, (1.03) 4 = Monthly, (1.01) 12 = Daily, 365 0, ( ) 365 = /29/ ?

41 AMORTİZATİON LOANS & SCHEDULES Amortized loans: Borrower repays the principal over the life of the loan (ex. Home mortgages, car loans, etc). Two methods: Fixed amount of principal to be repaid each period, that results in uneven payments. Fixed payments, results in uneven principal reduction. Tables showing how do you amortize your loan to a bank are called amortization schedules. 9/29/

42 Amortization Construct an amortization schedule for a car loan of 20,000 TL, at 1.2% monthly rate, (APR= 1.2x12= 14.4%per year). Loan to be paid back in 3 years monthly payments % 1 t 36-20,000 PMT... PMT 9/29/

43 DEVELOPİNG THE AMORTİZATİON SCHEDULE FİXED PAYMENTS 1. We start by finding the required payments. 20,000 = PMT (1/ /0.012(1.012) 36 ) PMT = 20,000 / ( ) = TL 2. Find interest charge for month 1: Int = Beginning balance * periodical int. rate Int = 20,000 * = 240 TL 9/29/

44 Developing the amortization schedule 3. Find repayment of principal in month 1: Repmt = PMT Interest Repmt = = TL 4. Find ending balance after month 1: Ending Balance = Beginning Balance Repmt End. Balance = 20, = 19, TL Let us see the rest of it on excel. 9/29/

45 AMORTİZED LOAN WİTH FİXED PRİNCİPAL PAYMENT - EXAMPLE Consider a \$50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay \$5,000 in principal each year plus interest for that year. Note that pmt period is annual in this example. Click on the Excel icon to see the amortization table 9/29/

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