# Chapter 4 Discounted Cash Flow Valuation

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1 University of Science and Technology Beijing Dongling School of Economics and management Chapter 4 Discounted Cash Flow Valuation Sep Dr. Xiao Ming USTB 1

2 Key Concepts and Skills Be able to compute the future value and/or present value of a single cash flow or series of cash flows Be able to compute the return on an investment Be able to use a financial calculator and/or spreadsheet to solve time value problems Understand perpetuities and annuities Dr. Xiao Ming USTB 2

3 Chapter Outline 4.1 Valuation: The One-Period Case 4.2 The Multiperiod Case 4.3 Compounding Periods 4.4 Simplifications 4.5 Loan Amortization 4.6 What Is a Firm Worth? Dr. Xiao Ming USTB 3

4 4.1 The One-Period Case If you were to invest \$10,000 at 5-percent interest for one year, your investment would grow to \$10,500. \$500 would be interest (\$10,000.05) \$10,000 is the principal repayment (\$10,000 1) \$10,500 is the total due. It can be calculated as: \$10,500 \$10,000 (1.05) The total amount due at the end of the investment is call the Future Value (FV). Dr. Xiao Ming USTB 4

5 Future Value In the one-period case, the formula for FV can be written as: FV C 0 (1 + r) Where C 0 is cash flow today (time zero), and r is the appropriate interest rate. Dr. Xiao Ming USTB 5

6 Present Value If you were to be promised \$10,000 due in one year when interest rates are 5-percent, your investment would be worth \$9, in today s dollars. \$ 9, \$10, The amount that a borrower would need to set aside today to be able to meet the promised payment of \$10,000 in one year is called the Present Value (PV). Note that \$10,000 \$9, (1.05). Dr. Xiao Ming USTB 6

7 Present Value In the one-period case, the formula for PV can be written as: PV C1 1 + r Where C 1 is cash flow at date 1, and r is the appropriate interest rate. Dr. Xiao Ming USTB 7

8 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. Suppose an investment that promises to pay \$10,000 in one year is offered for sale for \$9,500. Your interest rate is 5%. Should you buy? Dr. Xiao Ming USTB 8

9 Net Present Value NPV NPV \$9,500+ \$9,500+ \$10, \$9, NPV \$23.81 The present value of the cash inflow is greater than the cost. In other words, the Net Present Value is positive, so the investment should be purchased. Dr. Xiao Ming USTB 9

10 Net Present Value In the one-period case, the formula for NPV can be written as: NPV Cost + PV If we had not undertaken the positive NPV project considered on the last slide, and instead invested our \$9,500 elsewhere at 5 percent, our FV would be less than the \$10,000 the investment promised, and we would be worse off in FV terms : \$9,500 (1.05) \$9,975 < \$10,000 Dr. Xiao Ming USTB 10

11 4.2 The Multiperiod Case The general formula for the future value of an investment over many periods can be written as: FV C 0 (1 + r) T Where C 0 is cash flow at date 0, r is the appropriate interest rate, and T is the number of periods over which the cash is invested. Dr. Xiao Ming USTB 11

12 Future Value Suppose a stock currently pays a dividend of \$1.10, which is expected to grow at 40% per year for the next five years. What will the dividend be in five years? FV C 0 (1 + r) T \$5.92 \$1.10 (1.40) 5 Dr. Xiao Ming USTB 12

13 Future Value and Compounding Notice that the dividend in year five, \$5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original \$1.10 dividend: \$5.92 > \$ [\$ ] \$3.30 This is due to compounding. Dr. Xiao Ming USTB 13

14 Future Value and Compounding \$ 1.10 \$ \$ 1.10 (1.40) 2 (1.40) (1.40) \$ 1.10 \$ (1.40) 5 (1.40) \$1.10 \$1.54 \$2.16 \$3.02 \$4.23 \$ Dr. Xiao Ming USTB 14

15 Present Value and Discounting How much would an investor have to set aside today in order to have \$20,000 five years from now if the current rate is 15%? PV \$20, \$ 9, \$20,000 5 (1.15) Dr. Xiao Ming USTB 15

16 4.5 Finding the Number of Periods If we deposit \$5,000 today in an account paying 10%, how long does it take to grow to \$10,000? FV C (1 r) 0 + T T \$10,000 ( 1.10) \$5,000 \$ 10,000 \$5,000 2 T (1.10) T ln( 1.10) ln(2) T ln(2) ln(1.10) years Dr. Xiao Ming USTB 16

17 What Rate Is Enough? Assume the total cost of a college education will be \$50,000 when your child enters college in 12 years. You have \$5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child s education? About 21.15%. T 12 C0 (1 + r \$ 50,000 \$5,000 (1 + r) FV ) (1 + r) 12 \$50,000 \$5, ( 1+ r) r Dr. Xiao Ming USTB 17

18 Calculator Keys Texas Instruments BA-II Plus FV future value PV present value I/Y periodic interest rate P/Y must equal 1 for the I/Y to be the periodic rate Interest is entered as a percent, not a decimal N number of periods Remember to clear the registers (CLR TVM) after each problem Other calculators are similar in format Dr. Xiao Ming USTB 18

19 Multiple Cash Flows Consider an investment that pays \$200 one year from now, with cash flows increasing by \$200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows? If the issuer offers this investment for \$1,500, should you purchase it? Dr. Xiao Ming USTB 19

20 Multiple Cash Flows , Present Value < Cost Do Not Purchase Dr. Xiao Ming USTB 20

21 Valuing Lumpy Cash Flows First, set your calculator to 1 payment per year. Then, use the cash flow menu: CF0 0 CF3 600 I 12 CF1 200 F3 1 NPV 1, F1 1 CF4 800 CF2 400 F4 1 F2 1 Dr. Xiao Ming USTB 21

22 4.3 Compounding Periods Compounding an investment m times a year for T years provides for future value of wealth: FV C0 1+ r m m T Dr. Xiao Ming USTB 22

23 Compounding Periods For example, if you invest \$50 for 3 years at 12% compounded semi-annually, your investment will grow to.12 FV \$ \$50 (1.06) 6 \$70.93 Dr. Xiao Ming USTB 23

24 Effective Annual Rates of Interest A reasonable question to ask in the above example is what is the effective annual rate of interest on that investment? FV \$50 ( \$50 (1.06) The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-ofinvestment wealth after 3 years: ) \$70.93 \$50 (1 + EAR) 3 \$70.93 Dr. Xiao Ming USTB 24

25 Effective Annual Rates of Interest FV \$50 (1 + EAR) 3 \$70.93 EAR (1 + EAR) \$70.93 \$ \$70.93 \$ So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually. Dr. Xiao Ming USTB 25

26 Effective Annual Rates of Interest Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly. What we have is a loan with a monthly interest rate rate of 1½%. This is equivalent to a loan with an annual interest rate of 19.56%. 1 + r m m (1.015) Dr. Xiao Ming USTB 26

27 EAR on a Financial Calculator Texas Instruments BAII Plus keys: description: [2nd] [ICONV] Opens interest rate conversion menu [ ] [C/Y] 12 [ENTER] Sets 12 payments per year [ ][NOM] 18 [ENTER] Sets 18 APR. [ ] [EFF] [CPT] Dr. Xiao Ming USTB 27

28 Continuous Compounding The general formula for the future value of an investment compounded continuously over many periods can be written as: FV C 0 e rt Where C 0 is cash flow at date 0, r is the stated annual interest rate, T is the number of years, and e is a transcendental number approximately equal to e x is a key on your calculator. Dr. Xiao Ming USTB 28

29 4.4 Simplifications Perpetuity A constant stream of cash flows that lasts forever Growing perpetuity A stream of cash flows that grows at a constant rate forever Annuity A stream of constant cash flows that lasts for a fixed number of periods Growing annuity A stream of cash flows that grows at a constant rate for a fixed number of periods Dr. Xiao Ming USTB 29

30 Perpetuity A constant stream of cash flows that lasts forever C C C PV C C C + + ( 1+ r) (1 + r) 2 (1 + r) 3 C PV r +L Dr. Xiao Ming USTB 30

31 Perpetuity: Example What is the value of a British consol that promises to pay 15 every year for ever? The interest rate is 10-percent PV?5.10?50 Dr. Xiao Ming USTB 31

32 Growing Perpetuity A growing stream of cash flows that lasts forever C C (1+g) C (1+g) 2 0 PV C C (1 + g) C (1 + g) + + ( 1+ r) (1 + r) 2 (1 + r) 3 2 +L PV r C g Dr. Xiao Ming USTB 32

33 Growing Perpetuity: Example The expected dividend next year is \$1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream? \$1.30 \$1.30 (1.05) \$1.30 (1.05) PV \$ \$26.00 Dr. Xiao Ming USTB 33

34 Annuity A constant stream of cash flows with a fixed maturity C C C C L T PV C C C (1 + r) (1 + r) 2 (1 + r) 3 L C (1 + r) T PV C 1 1 r (1 + r) T Dr. Xiao Ming USTB 34

35 Annuity: Example If you can afford a \$400 monthly car payment, how much car can you afford if interest rates are 7% on 36- month loans? \$400 \$400 \$400 \$400 L PV \$ /12 1 ( ) 36 \$12, Dr. Xiao Ming USTB 35

36 What is the present value of a four-year annuity of \$100 per year that makes its first payment two years from today if the discount rate is 9%? PV \$100 (1.09) \$100 (1.09) \$100 (1.09) \$100 (1.09) \$100 (1.09) t t 1 \$ \$ \$ \$100 \$100 \$100 \$ \$ PV \$ Dr. Xiao Ming USTB 36

37 Growing Annuity A growing stream of cash flows with a fixed maturity C C (1+g) C (1+g) 2 C (1+g) T-1 L T PV C (1 + r) + C (1 + g) (1 + r) 2 + L+ C (1 + (1 + g) r) T T 1 PV r C g 1 1+ (1 + g r) T Dr. Xiao Ming USTB 37

38 Growing Annuity: Example A defined-benefit retirement plan offers to pay \$20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%? \$20, \$20,000 (1.03) 2 \$20,000 (1.03) 39 L 40 PV \$20, \$265, Dr. Xiao Ming USTB 38

39 Growing Annuity: Example You are evaluating an income generating property. Net rent is received at the end of each year. The first year's rent is expected to be \$8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%? 2 4 \$8,500 (1.07) \$8,500 (1.07) \$8,500 (1.07) 3 \$8,500 (1.07) \$8,500 \$ 9,095 \$9, \$10, \$11, \$34, Dr. Xiao Ming USTB 39

40 4.5 Loan Amortization Pure Discount Loans are the simplest form of loan. The borrower receives money today and repays a single lump sum (principal and interest) at a future time. Interest-Only Loans require an interest payment each period, with full principal due at maturity. Amortized Loans require repayment of principal over time, in addition to required interest. Dr. Xiao Ming USTB 40

41 Pure Discount Loans Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. If a T-bill promises to repay \$10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market? PV 10,000 / , Dr. Xiao Ming USTB 41

42 Interest-Only Loan Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is \$10,000. Interest is paid annually. What would the stream of cash flows be? Years 1 4: Interest payments of.07(10,000) 700 Year 5: Interest + principal 10,700 This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about them in greater detail later. Dr. Xiao Ming USTB 42

43 Amortized Loan with Fixed Principal Payment 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. Click on the Excel icon to see the amortization table Dr. Xiao Ming USTB 43

44 Amortized Loan with Fixed Payment Each payment covers the interest expense plus reduces principal Consider a 4 year loan with annual payments. The interest rate is 8%,and the principal amount is \$5,000. What is the annual payment? 4 N 8 I/Y 5,000 PV CPT PMT -1, Click on the Excel icon to see the amortization table Dr. Xiao Ming USTB 44

45 4.6 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. Dr. Xiao Ming USTB 45

46 Quick Quiz How is the future value of a single cash flow computed? How is the present value of a series of cash flows computed. What is the Net Present Value of an investment? What is an EAR, and how is it computed? What is a perpetuity? An annuity? Dr. Xiao Ming USTB 46

47 Dr. Xiao Ming USTB 47

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