Chapter 4: Discounted cash flow valuation

1. Chapter 4: Discounted cash flow valuation Corporate Finance Ross, Westerfield, and Jaffe

2. Outline 4.1 Future value 4.2 Present value 4.3 Other parameters 4.4 Multiple cash flows 4.5 Comparing rates 4.6 Loan types

3. What Time Value of Money Is? • In the most general sense, the phrase time value of money refers to the fact that a dollar in hand today is worth more than a dollar promised at some time in the future. • On a practical level, one reason for this is that you could earn interest while you waited; so a dollar today would grow to more than a dollar later. • The trade-off between money now and money later thus depends on, among other things, the rate you can earn by investing.

4. What Time Value of Money Is? • Our goal in this chapter is to explicitly evaluate this trade-off between dollars today and dollars at some future time.

5. Definitions • Present value (PV): earlier money on a time line. • Future value (FV): later money on a time line. • Interest rate (i), e.g., discount rate, required rate, cost of capital: exchange rate between earlier money and later money. • The number of time periods on a time line (N). • PV  FV: “time value of money” via the exchange rate, i.e., interest rate, i.

6. One equation; one solution • In general, we have one equation: 0 = f (PV, FV, i, N). • Since we have only one equation, we can only allow for one unknown parameter (variable). That is, if we’d like to calculate the value of a parameter, say FV, the values of the remaining parameters, i.e., PV, i, and N, need to be known.

7. Future Value and Compounding • Future value (FV) refers to the amount of money an investment will grow to over some period of time at some given interest rate.

8. FV example I • Suppose that we buy a 12-month CD at 12% annual interest rate for $10,000. • FV = PV  (1 + i)N = $10,000  (1 + 12%)1 = $11,200.

9. Do not compare apples with oranges • Why N = 1 while the CD matures in 12 months? The key is that: • The time frequency of i and N must be the same. • If we use annual interest rate, then we need to measure the investment period using the unit of year. In this case, 12 months equal a year; so N = 1. • What is the value of N if the example provided us monthly interest rate, say 0.96% per month?

10. Compounding • Of course, the previous formula, FV = PV  (1 + i)N, is based on the notion of compounding. • Compounding: the process of accumulating interest on an investment over time to earn more interest. • Earn interest on interest. • Reinvest the interest. • A popular method.

11. FV example II • Deposit $50,000 in a bank account paying 5%. How much will you have in 6 years? • Formula: FV = PV  (1 + i)N = $50,000  (1 + 5%)6 = $67,000. • Financial table (Table A.3): FV = $50,000  1.3401 = $67,000. • Financial calculator: 6 N; 5 I/Y; 50000 PV; CPT FV. The answer is FV = -67,004.7820. Ignore the negative sign.

12. Texas Instruments BAII Plus (keys) • FV: future value. • PV: present value. • I/Y: period interest rate. • Interest is entered as a percent. • N = number of time periods. • Clear the registers (CLR TVM, i.e., 2nd FV) after each calculation; otherwise, your next calculation may come up with a wrong answer.

13. FV example, III • Jacob invested $1,000 in the stock of IBM. IBM pays a current dividend of $2 per share, which is expected to grow by 20% per year for the next 2 years. What will the dividend of IBM be after 2 years? • Formula: FV = PV  (1 + i)N = $2  (1 + 20%)2 = $2.88. • Table A.3: FV = $2  1.4400 = $2.88. • Calculator: 2 PV; 20 I/Y; 2 N; CPT FV. The answer is -2.8800.

14. Simple VS Compound

15. Discounting • Discounting: the process of calculating the present value of future cash flows. • We call i the discount rate when we try to solve for present value. Depending on the question, this rate can be interest rate, cost of capital, or opportunity cost.

16. PV example, I • Suppose that you need $4,000 to pay your tuition. 1-year CD interest rate is 7%. How much do you need to put up today? • Formula: PV = FV / (1 + i)N = $4,000 / (1 + 7%)1 = $3,738.3. • Table A.1: PV = $4,000  0.9346 = $3,738.4. • Calculator: 4000 FV; 7 I/Y; 1 N; CPT PV. The answer is -3,738.3178.

17. PV example, II • Suppose that you are 21 years old. Your annual discount (return) rate is 10%. How much do you need to invest today in order to reach $1 million by the time you reach 65? • Formula: PV = FV / (1 + i)N = $1,000,000 / (1 + 10%)44 = $15,091. • Table A.1 does not have the present value factor for N = 44. This is the limitation of using a financial table. Thus, we will focus on the other 2 methods in the following discussions. • Calculator: 1000000 FV; 10 I/Y; 44 N; CPT PV. The answer is -15,091.1332.

18. PV relationship, I • Holding interest rate constant – the longer the time period, the lower the PV. • What is the present value of $5,000 to be received in 5 years? 10 years? The discount rate is 8% • 5 years: 5 N; 8 I/Y; 5000 FV; CPT PV. The answer is PV = -3,402.9160. • 10 years: 10 N; 8 I/Y; 5000 FV; CPT PV. The answer is PV = -2,315.9674.

19. PV relationship, II • Holding time period constant – the higher the interest rate, the smaller the PV. • What is the present value of $5,000 received in 5 years if the interest rate is 10%? 15%? • 10%: 10 I/Y; 5 N; 5000 FV; CPT PV. The answer is PV = -3,104.6066. • 15%: 15 I/Y; 5 N; 5000 FV; CPT PV. The answer is PV = -2,485.8837.

20. The other parameters • Recall that 0 = f (PV, FV, i, N). • We can find the value of i or N as long as we know about the values of the other parameters. • The easiest way is to use a financial calculator. • They are formulas, i.e., analytical solutions, for i and N as well. But these are not the focus of the course.

21. Interest rate example • Suppose that you deposit $5,000 today in a bank account paying interest rate i per year. If you reach $10,000 in 10 years, what rate of return are you being offered? • Calculator: 5000 PV; -10000 FV; 10 N; CPT I/Y. The answer is I/Y = 7.1773. • Note that for entering -10000 FV, this is the sequence: 10000 +/– FV.

22. Time period example • Suppose that you have $10,000 today. You want to retire as a millionaire. The annual rate of return that you can earn on the market is 10%. In how many years can you retire? • Calculator: 10000 PV; -1000000 FV; 10 I/Y; CPT N. The answer is: N = 48.3177.

23. Multiple cash flows • When there are multiple cash flows need to be discounted or compounded, the PV or FV of multiple cash flows are simply the sum of individual PV’s or FV’s, respectively.

24. Multiple cash flow example • Dennis has won the Kentucky State Lottery and will receive $2,000 (cash flow 1)in a year and $5,000 (cash flow 2) in 2 years. Dennis can earn 6% in his money market account, so the appropriate discount rate is 6%. • PV = PV1 + PV2 = $2,000 / (1 + 6%)1 + $5,000 / (1 + 6%)2 = $6,337. • That is, Dennis is equally inclined toward receiving $6,337 today and receiving $2,000 and $5,000 over the next 2 years.

25. Multiple cash flow example, Excel

26. Annuity & Its Types • An annuity is a stream of equal cash flows that occurs at equal interval over a given period. • Receiving $1,000 per year at the end of each of the next eight years is an example of an annuity. • There are two types of annuities: ordinary annuities and annuities due. • The ordinary annuity is the most common type of annuity. • It is characterized by cash flows that occur at the end of each compounding period. • This is a typical cash flow pattern for many investment and business finance applications.

27. Annuity PV example • The other type of annuity is called an annuity due, where payments or receipts occur at the beginning of each period (i.e., the first payment is today at t = 0). • FV = C { [ (1 + i)N – 1] / i}. PV = C { [ 1 – 1 / (1 + i)N ] / i}. • C is the fixed periodical payment.

28. Annuity PV example • Suppose that you want to buy a car. You can afford to pay $632 per month for the next 48 months. You borrow at 1% per month for 48 months. How much can you borrow? • Formula: PV = C { [ 1 – 1 / (1 + i)N ] / i } = $632  { [ 1 – 1 / (1 + 1%)48 ] / 1%} = $24,000. • Calculator: 632 PMT; 1 I/Y; 48 N; CPT PV. The answer is: PV = -23,999.5424. • In the solution manual (textbook), PVIFA(PVIA) stands for the PV of an annuity. • PVIFA(i,N) = [ 1 – 1 / (1 + i)N ] / i .

29. PV of an ordinary annuity

30. PV of an ordinary annuity

31. PV of an ordinary annuity

32. FV Of An Ordinary Annuity • What is the future value of an ordinary annuity that pays $150 per year at the end of each of the next 15 years, given the investment is expected to earn a 7% rate of return?

33. FV Of An Ordinary Annuity Formula: FV = C { [ (1 + i)N – 1] / i }

34. Annuity due • Annuity due: an annuity for which the cash flows occur at the beginning of the period. • For calculating PV and FV of an annuity due, we can use the following formula: Annuity due value = ordinary annuity value  (1 + i).

38. Perpetuity • Perpetuity: a constant stream of cash flows without end. • PV = C / i.

39. Perpetuity example • Preferred stock promises the buyer a fixed cash dividend every period (usually every quarter) forever. Suppose that VTinsurance Inc. wants to sell preferred stock. The quarterly dividend is $1 per share. The required rate of return for this issue is 2.5% per quarter. What is the fair value of this issue? • PV = C / i = $1 / 2.5% = $40 (per share).

40. Loan Payments and Amortization • Loan amortization is the process of paying off a loan with a series of periodic loan payments, whereby a portion of the outstanding loan amount is paid off, or amortized, with each payment. • When a company or individual enters into a long-term loan, the debt is usually paid off over time with a series of equal, periodic loan payments, and each payment includes the repayment of principal and an interest charge..

41. Loan Payments and Amortization • The payments may be made monthly, quarterly, or even annually. • Regardless of the payment frequency, the size of the payment remains fixed over the life of the loan. • The amount of the principal and interest component of the loan payment, however, does not remain fixed over the term of the loan. • Let's look at some examples to more fully develop the concept of amortization

43. Example: Principal and interest component of a specific loan payment Suppose you borrowed $10,000 at 10% interest to be paid semiannually over ten years. Calculate the amount of the outstanding balance for the loan after the second payment is made.

44. Solution First, the amount of the payment must be determined by entering the relevant information and computing the payment. PV = -$10,000; IN= 10 / 2 = 5; N = 10 x 2 = 20; CPT -+ PMT = $802.43

45. Comparing rates, I • Rates are quoted in many different ways. • Tradition. • Legislation. • Effective annual rate (EAR): the actual rate paid (or received) after accounting for compounding that occurs during the year. • When comparing two alternative investments with different compounding frequencies, one needs to compute the EARs and use them for reaching a decision.

46. Comparing rates, II • Annual percentage rate (APR) or stated annual interest rate (SAIR): the annual rate without consideration of compounding. • APR = period rate  the number of periods per year, m. • EAR = [1 + (APR / m)]m – 1.

47. Rate example, I • You went to a bank to borrow $10,000. You were told that the rate is quoted as “8% compounded semiannually.” What is the amount of debt after a year? • FV = PV  (1 + i)N = $10,000  (1 + 4%)2 = $10,816. • EAR = [1 + (APR / m)]m – 1= [1 + (8% / 2)]2 – 1 = 8.16%.

48. Rate example, II • What is the APR if the monthly rate is 1%? • APR = 1%  12 = 12%. • What is the monthly (period) rate if the APR is 6% with monthly compounding? • Period (monthly) rate = 6% / 12 = 0.5%.

49. Continuously compounding • FV = PV × eAPR×the number of years , where e has the value of 2.718. • Suppose that you invest $1,000 at a continuously compounded rate of 10% for a year. • FV = PV × eAPR×the number of years = $1,000 × e10%×1 = $1,105.20. So, EAR = 10.52%.

50. APR vs. EAR in real life • By Trust-in-saving law, banks need to disclose EAR ( or called annual percentage yield (APY), or effective annual yield (EAY)). So you get the correct rate when you save. • By Trust-in-lending law, banks need to disclose APR, the stated (quoted) rate. So you get a seemingly low rate when you borrow.