# Discounted Cash Flow Valuation

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1 Discounted Cash Flow Valuation Chapter 5 Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute loan payments Be able to find the interest rate on a loan Understand how loans are amortized or paid off Understand how interest rates are quoted Time Lines Cash Inflows Cash Outflows 1

2 Time Lines Time lines show timing of cash flows. Tick marks are at ends of each period, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period r% CF 0 CF 1 CF 2 CF 3 Time Lines Series of Uneven Cash Flows r% Multiple Cash Flows - FV What is the Future Value of the cash flow stream at the end of year % 7,000 4,000 4,000 4,000 2

3 Multiple Cash Flows - FV Find the value at the end of Year 3 of each cash flow and add them together. CF0 FV = 7,000(1.08) 3 = 8, CF1 FV = 4,000(1.08) 2 = 4, CF2 FV = 4,000(1.08) = 4,320 CF3 FV = PV = 4,000 Total value in 3 years 8, , , ,000 = 21, Multiple Cash Flows - FV % 7,000 4,000 4,000 4,000 4, , , FV = 21, Multiple Cash Flows - FV 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? 3

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

5 Multiple Cash Flows - FV Suppose you plan to deposit \$100 into an account in one year and \$300 into the account in three years. How much will be in the account in five years if the interest rate is 8%? FV = 100(1.08) (1.08) 2 FV = = Multiple Cash Flows - FV 0 1 8% FV = Multiple Cash Flows - PV What is the Present Value of the cash flow stream? %

6 Multiple Cash Flows - PV % Multiple Cash Flows - PV Find the PV of each cash flow and add them CF1: PV = 200/(1.12) 1 = CF2 PV = 400/(1.12) 2 = CF3 PV = 600/(1.12) 3 = CF4 PV = 800/(1.12) 4 = Total PV = = 1, Multiple Cash Flows - PV Financial Calculator Solution Enter the data in the following order: 0 [CFj] 200 [CFj] 400 [CFj] 600 [CFj] 800 [CFj] 12 [I/YR] [Gold] [NPV] Answer: = 1,

7 Multiple Cash Flows PV You are considering an investment that will pay you \$1,000 in one year, \$2,000 in two years and \$3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? Multiple Cash Flows - PV % 1,000 2,000 3,000 Multiple Cash Flows - PV Financial Calculator Solution Enter the data in the following order: 0 [CFj] 1,000 [CFj] 2,000 [CFj] 3,000[CFj] 10 [I/YR] [Gold] [NPV] Answer: 4,

8 Decisions, Decisions Your broker calls you and tells you that he has this great investment opportunity. If you invest \$100 today, you will receive \$40 in one year and \$75 in two years. If you require a 15% return on investments of this risk, should you take the investment? Decisions, Decisions Use the CF keys to compute the value of the investment 0 [CFj] 40 [CFj] 75 [CFj] 15 [I/YR] [Gold][NPV] Answer: The broker is charging more than you would be willing to pay. 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%? 8

9 Saving For Retirement Financial Calculator Solution Enter the data in the following order: 0 [CFj] Note: This is CF0 0 [CFj] 39 [Gold] [Nj] Note: These are CF1 CF39 25,000 [CFj] 5 [Gold] [Nj] Note: These are CF40 CF44 12 [I/YR] [Gold] [NPV] Answer: 1, Quick Quiz Suppose you are looking at the following possible cash flows: Year 1 CF = \$100 Years 2 and 3 CFs = \$200 Years 4 and 5 CFs = \$300 The required discount rate is 7%. Quick Quiz 0 1 7%

10 Quick Quiz What is the value of the cash flows today? What is the value of the cash flows at year 5? Quick Quiz What is the value of the cash flows today? 0 [CFj] 100 [CFj] 200 [CFj] 200 [CFj] 300 [CFj] 300 [CFj] 7 [I/YR] [Gold] [NPV] Answer: Quick Quiz What is the value of the cash flows at year 5? Using NPV to find FV [PV] 7 [I/YR] 5 [N] [FV] Answer: -1,

11 Annuities and Perpetuities Annuity Finite series of equal payments that occur at regular intervals Ordinary Annuity Annuity Due Perpetuity Infinite series of equal payments Also known as consols Annuities Ordinary Annuity r% PMT PMT PMT Annuity Due r% PMT PV PMT PMT FV Annuities and Perpetuities Perpetuity: PV = CF/r Annuities: PV FV = = CF CF 1 1 t (1 + r ) r t (1 + r ) 1 r 11

12 Annuities You can use the PMT key on the calculator for the equal payment Ordinary Annuity versus Annuity Due To switch from Ordinary Annuities to Annuities Due, you use [Gold] [BEG/END]. If you see BEGIN in the display of your calculator, you have it set for an annuity due Most problems are ordinary annuities Annuity Car Loan A car you are interested in purchasing costs \$25,000. The bank will loan you the money for 4 years at an interest rate of 6%. What are the annual payments? Annuity Car Loan Since you are borrowing the money today, the loan amount is the Present Value. Financial Calculator Solution 25,000 [PV] 6 [I/YR] 4 [N] [PMT] Answer: -7,

13 Annuity Car Loan What would be the monthly payments? Financial Calculator Solution 12 [Gold] [P/YR] 25,000 [PV] 6 [I/YR] 48 [N] Or use 4 [Gold] [x P/YR] [PMT] Answer: Note: ( x 12) 7, Why not? Annuity - Lottery Suppose you win the lottery for \$10 million. The money can be paid in 30 equal annual installments of \$333, beginning today, or in a lump sum of \$5,000,000. If the appropriate discount rate is 5%,which alternative should you take? Annuity - Lottery Since the payments begin today, this is an Annuity Due. [Gold][BEG/END] 333, [PMT] 30 [N] 5 [I/YR] [PV] Answer: -5,380,357 5,380,357 > 5,000,000 so take the payments 13

14 Annuity - 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 for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house? Annuity - Buying a House Bank loan Monthly Income \$36,000/12 = \$3,000 Maximum Payment.28(\$3,000) = \$ [Gold] [P/YR] 840 [PMT] 360 [N] 6 [I/YR] [PV] Answer: -140,105 Annuity - Buying a House Total Price Closing Costs.04(\$140,105) = \$5,604 Down Payment \$20,000 \$5,604 = \$14,396 Total Price Loan Amount + Down Payment \$140,105 + \$14,396 = \$154,501 14

15 Annuity - Retirement You want to receive \$5,000 per month in retirement. If you expect to earn 9% per year and you expect to need the income for 25 years, how much do you need to have in your account when you retire? Annuity - Retirement Financial Calculator Solution 12 [Gold] [P/YR] 5,000 [PMT] 300 [N] 9 [I/YR] [PV] Answer: -595,808 Annuity - Retirement Assume you have 35 years to save up for your retirement. How much will you need to save each month to achieve your goal if you make your first deposit in one month? Financial Calculator Solution 12 [Gold] [P/YR] 595,808 [FV] 420 [N] 9 [I/YR] [PMT] Answer:

16 Annuity Credit Card Payments You have a \$1,000 charge on your credit card, which charges an interest rate of 18%. If you only make a \$20 monthly payment, how long will it take you to pay off the credit card? Annuity Credit Card Payments Financial Calculator Solution 12 [Gold] [P/YR] 1,000 [PV] 18 [I/YR] -20 [PMT] [N] Answer: months. This is 7.78 years! And this is only if you don t charge anything else! Annuity Number of Payments Suppose you borrow \$2,000 at 5% and you are going to make annual payments of \$ How long before you pay off the loan? 1 [Gold] [P/YR] 2,000 [PV] 5 [I/YR] [PMT] [N] Answer: 3 years 16

17 Annuity - 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 annual interest rate? 12 [Gold] [P/YR] 10,000 [PV] [PMT] 60 [N] [I/YR] Answer: 9 Annuity - Travel After graduation, you want to travel around the world for 5 years. You estimate that you will need \$5,000 to live on while traveling. How much would you need to have on your graduation day if you can will make your first withdrawal on your graduation day and you can earn 7.5%? Annuity - Travel Financial Calculator Solution Put calculator on BEGIN mode [Gold] [BEG/END] 12 [Gold] [P/YR] 5,000 [PMT] 60 [N] 7.5 [I/YR] [PV] Answer: -251,086 17

18 Annuity - Travel Suppose you have only have \$200,000. How many months could you travel? Calculator should still be on BEGIN mode 12 [Gold] [P/YR] 5,000 [PMT] 7.5 [I/YR] -200,000 [PV] [N] Answer: months Annuities - Future Value Suppose you begin saving for your retirement by depositing \$2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years if you make your first deposit beginning in one year? Put calculator back on END mode: [Gold] [BEG/END] 1 [Gold] [P/YR] 2,000 [PMT] 7.5 [I/YR] 40 [N] [FV] Answer: -454,513 Annuities - Future Value You are saving for a new house and you put \$10,000 per year in an account paying 8%. The first deposit is made today. How much will you have at the end of 3 years? FV = 10,000[( ) /.08](1.08) = 35,

19 Annuities - Future Value Financial Calculator Solution Put calculator on BEGIN mode [Gold] [BEG/END] 1 [Gold] [P/YR] 10,000 [PMT] 3 [N] 8 [I/YR] [FV] Answer: -35, Annuities - Future Value You want to have \$1 million to use for retirement in 35 years. If you can earn 12% per year, how much do you need to deposit on a monthly basis if the first payment is made in one month? Put calculator back on END mode: [Gold] [BEG/END] 12 [Gold] [P/YR] 1,000,000 [FV] 420 [N] 12 [I/YR] [PMT] Answer: Annuities - Future Value What if the interest rate is only 6%? 12 [Gold] [P/YR] 1,000,000 [FV] 420 [N] 6 [I/YR] [PMT] Answer:

20 Annuities - Future Value What if the first payment is made today? Put calculator on BEGIN mode [Gold] [BEG/END] 12 [Gold] [P/YR] 1,000,000 [FV] 420 [N] 6 [I/YR] [PMT] Answer: Perpetuity Perpetuity Formula PV = CF/r You win a lottery where you are to receive \$10,000 a year forever. If current interest rates are 8%, what is the present value of your winnings? PV = \$10,000/.08 = \$125,000 Perpetuity Preferred Stock You are considering preferred stock that pays a quarterly dividend of \$1.50. If your desired return is 3% per quarter, how much would you be willing to pay? PV = \$1.50/.03 = \$50/share. 20

21 Effective Annual Rate (EAR) This is the actual rate paid (or received) after accounting for compounding that occurs during the year If you want to compare two alternative investments with different compounding periods you need to use the EAR. Annual Percentage Rate Also known as the Nominal Rate This is the annual rate that is quoted by law APR = Periodic rate times the number of periods per year Consequently, to get the periodic rate we rearrange the APR equation: Periodic rate = APR/number of periods per year You should NEVER divide the effective rate by the number of periods per year it will NOT give you the periodic rate Computing APRs What is the APR if the monthly rate is.5%?.5%*12 = 6% What is the APR if the semiannual rate is.5%?.5%*2 = 1% What is the monthly rate if the APR is 12% with monthly compounding? 12%/12 = 1% 21

22 Computing Effective Annual Rates APR EAR = 1 + m m 1 Remember that the APR is the quoted or nominal rate Computing Effective Annual Rates Suppose you can earn 1% per month on \$1 invested today. What is the APR? 1%(12) = 12% How much are you effectively earning? EAR = (1 +.12/12) 12 1 EAR = (1.01) 12 1 =.1268 = 12.68% Financial Calculator Solution 12 [Gold] [P/YR] 12 [Gold] [NOM%] [Gold] [EFF%] Answer: Computing EARs - Example Suppose if you put it in another account, you earn 3% per quarter. What is the APR? 3%(4) = 12% How much are you effectively earning? EAR = (1 +.12/4) 4 1 EAR = (1.03) 4 1 =.1255 = 12.55% Financial Calculator Solution 4 [Gold] [P/YR] 12 [Gold] [NOM%] [Gold] [EFF%] Answer:

23 Decisions, Decisions II You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? Decisions, Decisions II First Account: EAR = ( /365) = 5.39% 365 [Gold] [P/YR] 5.25 [Gold] [NOM%] [Gold] [EFF%] Answer: 5.39 Decisions, Decisions II Second account: EAR = ( /2) 2 1 = 5.37% [Gold] [P/YR] = [Gold] [NOM%] [Gold] [EFF%] Answer: 5.37 Which account should you choose? 23

24 Computing APRs from EARs If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: APR = m (1 + EAR) 1 m -1 APR - Example Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? APR - Example APR = (1 +.12) or 11.39% = [Gold] [P/YR] 12 [Gold] [EFF%] [Gold] [NOM%] Answer: =

25 Type of Loans Pure Discount Loans Interest-Only Loans Amortized Loans Pure Discount Loans A loan where the interest is deducted at the beginning of the loan. Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. Value of loan is PV of future principal amount. Interest Only Loan With interest only loans, interest is periodically paid and the entire principal is paid at the end of the loan. 25

26 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. Amortized Loan 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. Amortized Loan What is the annual payment? 1 [Gold] [P/YR] 5,000 [PV] 4 [N] 8 [I/YR] [PMT] Answer: -1,

27 Amortization Table Year 1 Beg. Balance 5, Payment Interest Paid Principal Paid End. Balance Totals Amortization Table #2 Construct an amortization table for a \$15,000 repaid in three equal payments at the end of the next three years at 8%. Be sure calculator is on END mode 3 [N] 8 [I/YR] 15,000 [PV] [PMT] Answer: -5, Amortization Table #2 Excel Formula =PMT(Periodic Rate,Number of Periods, Loan Amount) =-PMT(.08, 3, 15000) Note: The negative sign in the formula generate a positive payment. Year Beginning Amount Payment Interest Repayment of Principal Remaining Balance (1) (2) (3) (2) - (3) = (4) (1) - (4) = (5) 1 \$15, \$5, \$1, \$4, \$10, \$10, \$5, \$ \$4, \$5, \$5, \$5, \$ \$5, \$

28 Questions? 28

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