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Because learning changes everything.® Chapter 05 Discounted Cash Flow Valuation © McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.  Key Co...

Because learning changes everything.® Chapter 05 Discounted Cash Flow Valuation © McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.  Key Concepts and Skills After studying this chapter, you should be able to: Determine the future value and present value of investments with multiple cash flows. Calculate loan payments, and find the interest rate on a loan. Describe how loans are amortized or paid off. Explain how interest rates are quoted (and misquoted). © McGraw Hill, LLC 2  Chapter Outline 5.1 Future and Present Values of Multiple Cash Flows. 5.2 Valuing Level Cash Flows: Annuities and Perpetuities. 5.3 Comparing Rates: The Effect of Compounding Periods. 5.4 Loan Types and Loan Amortization. © McGraw Hill, LLC 3  Future Value: Multiple Cash Flows Example 5.1 You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in three years? How much in four years? © McGraw Hill, LLC 4  Future Value: Multiple Cash Flows Example 5.1 - Formulas Find the value at Year 3 of each cash flow and add them together. Year 0: FV = $7,000(1.08)3 = $ 8,817.98 Year 1: FV = $4,000(1.08)2 = $ 4,665.60 Year 2: FV = $4,000(1.08)1 = $ 4,320.00 In the following table, read ‘$7,000(1.08)3’ as 7,000 dollars times 1.08 to the power of 3; read ‘$4,000(1.08)2’ as 4,000 dollars times 1.08 to the power of 2; read ‘$4,000(1.08)1’ as 4,000 dollars times 1.08 to the power of 1; Year 3: value = $ 4,000.00 Total value in 3 years = $21,803.58 Value at Year 4 = $21,803.58(1.08) = $23,547.87. © McGraw Hill, LLC 5  Future Value: Multiple Cash Flows Example 5.2 1 If you deposit $100 in one year, $200 in two years, and $300 in three years: How much will you have in three years at 7 percent interest? How much in five years at 7 percent interest if you do not add additional amounts? Year 1 FV100 =100×(1+0.07)^2 = $114.49 Year 2 FV1=200 =200×(1+0.07)^1 = $214.00 Year 3 FV300 =300×(1+0.07)^0 = $300.00 Total FV3 = $628.49 Total FV5 = $628.49(1.07)2 = $719.56 © McGraw Hill, LLC 6  Future Value: Multiple Uneven Cash Flows Example 5.2 – Formulas & Time Line If you deposit $100 in one year, $200 in two years, and $300 in three years: How much will you have in three years at 7 percent interest? How much in five years at 7 percent interest if you do not add additional amounts? Access the text alternative for slide images. © McGraw Hill, LLC 7  Future Value: Multiple Cash Flows Example 1 Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9 percent annually, how much will you have in two years? FV = $500(1.09)2 = $ 594.05 In the following table, read ‘$500(1.09)2’ + $600(1.09) = 654.00 as 500 dollars times 1.09 to the power of 2; = $1,248.05 © McGraw Hill, LLC 8  Future Value: Multiple Cash Flows Example 2 How much will you have in 5 years if you make no further deposits? First way: Second way – use value at Year 2: © McGraw Hill, LLC 9  Future Value: Multiple Cash Flows Example 3 - Formula 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 percent? © McGraw Hill, LLC 10  Example 3 Time Line Access the text alternative for slide images. © McGraw Hill, LLC 11  Present Value: Multiple Cash Flows Example 5.3 You are offered an investment that will pay. $200 in one year, $400 the next year, $600 the following year, and $800 at the end of the fourth year. You can earn 12 percent on similar investments. What is the most you should pay for this one? © McGraw Hill, LLC 12  Present Value: Multiple Cash Flows Example 5.3 - Formula Find the PV of each cash flow and add them: Year 1 CF: $200/1.121 = $ 178.57 Year 2 CF: $400/1.122 = In the following table, read ‘$200/1.121’ as 200 dollars divided by 1.12 to the power of 1; read ‘$400/1.122’ as 200 dollars divided by 1.12 to the power of 2; read ‘$600/1.123’ as 200 dollars divided by $ 318.88 1.12 to the power of 3; read ‘$800/1.124’ as 200 dollars divided by Year 3 CF: $600/1.123 1.12 to the power of 4; = $ 427.07 Year 4 CF: $800/1.124 = $ 508.41 Total PV = $1,432.93 Calculator and Excel Solution. © McGraw Hill, LLC 13  Example 5.3 Time Line Access the text alternative for slide images. © McGraw Hill, LLC 14  Present Value: Multiple Cash Flows Another Example – Formula Solution 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 percent on your money, how much would you be willing to pay? PV = $1,000/1.101 = $ 909.09 PV = $2,000/1.102 = In the following table, read ‘$1,000/1.101’ as 1,000 dollars divided by 1.10 to the power of 1; read ‘$2,000/1.102’ as 2,000 dollars divided by 1.10 to the power of 2; read $1,652.89 ‘$3,000/1.103’ as 3,000 dollars divided by 1.10 to the power PV = $3,000/1.103 = $2,253.94 of 3; PV = $4,815.92 Calculator and Excel Solution © McGraw Hill, LLC 15  Decisions, Decisions 1 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 percent return on investments of this risk, should you take the investment? No – the broker is charging more than you would be willing to pay. PV = $40/(1.15)1 = $ 34.78 PV = $75/(1.15)2 = $ 56.71 PV = $91.49 © McGraw Hill, LLC 16  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 percent? PV = $25,000/(1.12)40 = $ 268.67 PV = $25,000/(1.12)41 = $239.88 PV = $25,000/(1.12)42 = $214.18 PV = $25,000/(1.12)43 = $191.23 PV = $25,000/(1.12)44 = $170.74 Total PV = 1,084.70 © McGraw Hill, LLC 17  Saving For Retirement Time Line Notice that Year 0 cash flow = $0 (CF0 = 0). Cash flows Years 1 to 39 = $0 (C01 = 0; F01 = 39). Cash flows Years 40 to 44 = $25,000 (C02 = 25000; F02 = 5). © McGraw Hill, LLC 18  Quick Quiz: Part 1 1 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 percent. What is the value of the CFs at Year 5? What is the value of the CFs today? Calculator Solution. © McGraw Hill, LLC 19  Quick Quiz: Part 1 – Excel Solution Value of the cash flow today: Rate 7% Year Nper CF PV Formula 1 1 100 $93.46 = 100/(1+0.07)^1 2 2 200 $174.69 =100/(1+0.07)^2 3 3 200 $163.26 =100/(1+0.07)^3 4 4 300 $228.87 =100/(1+0.07)^4 5 5 300 $213.90 =100/(1+0.07)^5 Value of the cash flow atPV Total Year 5: $874.17 =SUM Year Nper CF PV Year 1 4 100 $131.08 =100×(1+0.07)^(5−1) 2 3 200 $245.01 =200×(1+0.07)^(5−2) N= Year 5 3 2 200 $228.98 =200×(1+0.07)^(5−3) T = The year 4 1 300 $321.00 =300×(1+0.07)^(5−4) of the cash flow 5 0 300 $300.00 =300×(1+0.07)^(5−5) © McGraw Hill, LLC Total FV $1,226.07 =SUM 20  Chapter 5 – Quick Quiz: Part 1 $ 874.12 PV $ 213.90 $ 228.87 $ 163.26 $ 174.69 $ 93.46 7% Period 0 1 2 3 4 5 CFs 0 100 200 200 300 300 $ 300.00 $ 321.00 $ 228.98 $ 245.01 $ 131.08 FV = $ 1,226.07 Access the text alternative for slide images. © McGraw Hill, LLC 21  Annuities and Perpetuities Annuity – finite series of equal payments that occur at regular intervals. If the first payment occurs at the end of the period, it is called an ordinary annuity. If the first payment occurs at the beginning of the period, it is called an annuity due. Perpetuity – infinite series of equal payments. © McGraw Hill, LLC 22  Annuities and Perpetuities Basic Formulas Perpetuity: Annuities: © McGraw Hill, LLC 23  Important Points to Remember Interest rate and time period must match! Annual periods → annual rate. Monthly periods → monthly rate. The Sign Convention. Cash inflows are positive. Cash outflows are negative. © McGraw Hill, LLC 24  Annuity Example 5.5 You can afford $632 per month. Going rate = 1 percent per month for 48 months. How much can you borrow? You borrow money TODAY so you need to compute the present value. © McGraw Hill, LLC 25  Annuity: Sweepstakes Example Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5 percent, how much is the sweepstakes actually worth today? Calculator and Excel Solution © McGraw Hill, LLC 26  Buying a House 1 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 percent of the loan value. You have an annual salary of $36,000. The bank is willing to allow your monthly mortgage payment to be equal to 28 percent of your monthly income. The interest rate on the loan is 6 percent per year with monthly compounding (.5 percent per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house? © McGraw Hill, LLC 27  Buying a House 2 1. Calculate Monthly Mortgage Payment Annual Salary = $36,000, Monthly income = Allowed Monthly Mortgage Payment =.28($3,000) = $840. 2. Determine the Amount bank will give you (Loan Amount/Principal) PV= $840/ 0.005×[1−(1/(1+0.005)360] =140,105 3. Calculate Maximum Offer you can make for the house (Total Price) Closing costs =.04($140,105) = $5,604. Remaining Down payment = $20,000 – 5,604 = $14,396. Total price = $140,105 + 14,396 = $154,501. © McGraw Hill, LLC 28  Quick Quiz: Part 2 1 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? © McGraw Hill, LLC 29  Quick Quiz: Part 2 2 You want to receive $5,000 per month in retirement. If you can earn.75 percent per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement? n = Tot. no. of payments (25 years ×12 months/year = 300 months) PV= 5000×[1−(1+0.0075) ^−300 ] 0.0075 = $595,808.11 © McGraw Hill, LLC 30  Finding the Number of Payments Example 5.6 $1,000 due on credit card. (PV) Payment = $20 month minimum. (PMT) Rate = 1.5 percent per month. Calculate Number of Payments = 93.11 months = 7.76 years © McGraw Hill, LLC 32  Finding the Number of Payments Another Example Suppose you borrow $2,000 at 5 percent and you are going to make annual payments of $734.42. How long before you pay off the loan? 5 I/Y 2000 PV −734.42 PMT 0 FV = 3 years © McGraw Hill, LLC 33  Finding the Rate ***Unfortunately, solving for r directly in this equation analytically is quite complex. The equation is generally solved using financial calculator Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? 60 I/Y 10000 PV −207.58 PMT = RATE(60,−207.58,10000,0) 0 FV CPT I/Y =.75% per month © McGraw Hill, LLC 34  Quick Quiz: Part 3 3 Suppose you have $200,000 to deposit and can earn.75 percent per month. How many months could you receive the $5,000 payment? N = 47.73 months ≈ 4 years © McGraw Hill, LLC 36  Future Values for Annuities Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5 percent, how much will you have in 40 years? = FV(.075,40,−2000,0) 40 N 7.5 I/Y 0 PV −2000 PMT CPT FV = $454,513.04 © McGraw Hill, LLC 37  Annuity Due You are saving for a new house and you put $10,000 per year in an account paying 8 percent. The first payment is made today. How much will you have at the end of 3 years? © McGraw Hill, LLC 38  Table 5.2 1 Table 5.2 Summary of annuity and perpetuity calculations. I. Symbols. PV = Present value, what future cash flows are worth today. FVt = Future value, what cash flows are worth in the future at Time t. r = Interest rate, rate of return, or discount rate per period—typically, but not always, one year. t = Number of periods—typically, but not always, the number of years. C = Cash amount. II. Future value of C invested per period for t periods at r percent per period A series of identical cash flows paid for a set number of periods is called an annuity, and the term is called the annuity future value factor. © McGraw Hill, LLC 39  Table 5.2 2 III. Present value of C per period for t periods at r percent per period The term is called the annuity present value factor. IV. Present value of a perpetuity of C per period A perpetuity has the same cash flow every period forever. © McGraw Hill, LLC 40  Perpetuity Example 5.7 Perpetuity formula: Current required return: r =.025, or 2.5% per quarter. $100 = PMT = $2.50 per quarter. © McGraw Hill, LLC 41  Quick Quiz: Part 4 3 You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3 percent per quarter, how much would you be willing to pay? © McGraw Hill, LLC 42  End of Main Content Because learning changes everything.® www.mheducation.com © McGraw Hill LLC. All rights reserved. No reproduction or distribution without the prior written consent of McGraw Hill LLC.