Chapter Discounted Cash Flow Valuation 6-1 McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc All rights reserved Chapter Outline • • Multiple Cash Flows: Future and Present Values Multiple Equal Cash Flows: Annuities and Perpetuities • • • 6-2 Comparing Rates: the Effect of Compounding Loan Types Loan Amortization Chapter Outline • Multiple Cash Flows: Future and Present Values • Multiple Equal Cash Flows: Annuities and Perpetuities • Comparing Rates: the Effect of Compounding • Loan Types • Loan Amortization 6-3 Single Cash Flows In the previous chapter, we used single cash flows and moved them forward and backward in time Today 6-4 PV Compounding PV Today 2 Discounting FV FV Multiple Cash Flows What if we have more than one cash flow? The concept (and formula) are identical if we simply look at the problem as a series of single payments 6-5 Multiple Cash Flows Future Value Suppose you have $1,000 now in a savings account that is earning 6% You want to add $500 one year from now and $700 two years from now Today $1,000 Year $500 Years $700 ? How much will you have two years from now in your savings account (after you make your $700 deposit)? 6-6 Multiple Cash Flows Future Value Simply look at each payment separately and move them through time as we did in the earlier chapter Today $1,000 Year $ 500 Years $ 700 $1,124 Now just add them up $ 530 because they are all 6-7 adjusted to be in “year 3” value $2,354 TI BA II Plus years = N 1123.60 6% = I/Y -$1,000 = PV 1st CPT ? = FV 2nd 6-8 years = N HP 12-C 6% = i -$1,000 = PV ? = FV 1123.60 6-9 Multiple Cash Flows Future Value 1B Could we this problem another way? Bring each of the cash flows forward one year at a time and add them up each year Today $1,000 Year $ 500 Years $ 700 $1,060 $1,560 6-10 $1,654 $2,354 Work the Web There are web sites available that can easily prepare amortization tables Click on the web surfer to check out the Bankrate.com site and work the following example: You have a loan of $25,000 and will repay the loan over years at 8% interest What is your loan payment? What does the amortization schedule look like? 6-91 Ethics Issues Suppose you are in a hurry to get your income tax refund If you mail your tax return, you will receive your refund in weeks If you file the return electronically through a tax service, you can get the estimated refund tomorrow The service subtracts a $50 fee and pays you the remaining expected refund 6-92 Ethics Issues (Continued) The actual refund is then mailed to the preparation service Assume you expect to get a refund of $978 What is the APR with weekly compounding? What is the EAR? How large does the refund have to be for the APR to be 15%? What is your opinion of this practice? 6-93 Comprehensive Problem Part  An investment will provide you with $100 at the end of each year for the next 10 years What is the present value of that annuity if the discount rate is 8% annually?  What is the present value of the above if the payments are received at the beginning of each year? 6-94 Comprehensive Problem Part  If you deposit those payments into an account earning 8%, what will the future value be in 10 years?  What will the future value be if you open the account with $1,000 today, and then make the $100 deposits at the end of each year? 6-95 Terminology • • • • • • • • 6-96 Present value of multiple cash flows Future value of multiple cash flows Annuity Annuity Due Perpetuity Effective Annual Rate (EAR) Annual Percentage Rate (APR) Continuous Compounding Terminology (continued) • • • • 6-97 Pure Discount Loan Interest-only Loan Amortized Loan with Fixed Principle Amortized Loan with Fixed Payment Formulas Perpetuity: Annuity: 6-98 PV = C / r   1 − (1 + r ) t  PV = C   r        (1 + r ) t −  FV = C   r   Formulas (continued) t   (1 + g )   C   PV = 1 −  r − g   (1 + r )     C PV = r−g 6-99 Formulas (continued) m  APR  EAR = 1 + −  m    APR = m (1 + EAR)  6-100 m  -1  Key Concepts and Skills • Compute the future value of multiple cash flows • Compute the present value of multiple cash flows • Distinguish between an ordinary annuity and an annuity due 6-101 Key Concepts and Skills • Compute loan payments • Find the interest rate on a loan • Explain how interest rates are quoted • Give examples of different ways a loan can be paid off 6-102 What are the most important topics of this chapter? Multiple cash flows are moved both forward in time and into present value just like single payments Annuities are a special case of multiple cash flows in that they are a “fixed, equal, uninterrupted series of payments or receipts” 6-103 What are the most important topics of this chapter? Effective Annual Rate and Annual Percent Rate are two methods to determine the cost of a loan Loans can be pure discount, interest-only, or amortized – with fixed principal or fixed payments 6-104 Questions? 6-105 [...]... value of the cash flows at year 5?  What is the value of the cash flows today?  What is the value of the cash flows at year 3? 6-27 Chapter Outline • • Multiple Cash Flows: Future and Present Values Multiple Equal Cash Flows: Annuities and Perpetuities • • • 6-28 Comparing Rates: the Effect of Compounding Loan Types Loan Amortization Annuities and Perpetuities Definitions Annuity – finite series of. .. two years If you require a 15% return on investments of this risk, should you take the investment? Use the CF keys to compute the present value of the proposed investment’s cash flows CF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1 NPV; I = 15; CPT NPV = $91.49 No! – the broker is charging more than you would be willing to pay ($100 versus the PV of $91.49) 6-22 TI BA II Plus CF 0 = CF0 ; ↓ 91.49 40... $75 in two years If you require a 15% return on investments of this risk, should you take the investment? Use the CF keys to compute the present value of the proposed investment’s cash flows “g” CF0 = 0; “g” CFj = 40; “g” CFj = 75 i = 15; “f” NPV = 91.49 No! – the broker is charging more than you would be willing to pay ($100 versus the PV of $91.49) 6-25 0 = CF0 HP 12-C 40 = CFj 75 = CFj 15% = i 91.49... 200 2 400 3 600 4 800 Multiple Cash Flows Present Value - 1 To compute the present value of this future stream of cash, we just take each year to the present, one at a time: Today ? 178.57 318.88 427.07 508.41 6-17 1,432.93 1 200 2 400 3 600 4 800 Multiple Cash Flows Present Value -1 Using a calculator, find the PV of each cash flow and just add them up! Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV =... Definitions 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 6-29 Annuities and Perpetuities Basic Formulas Perpetuity: PV = C / r Annuity: 1  1 − (1 + r ) t PV... $2,354 6-12 3 4 5 ? $2,803 Multiple Cash Flows Future Value 1C We could do this two different ways: 2 Bring each of the three original dollars to year 5 and add them all up Today $1,000 1 500 2 700 3 4 5 $1,338 ? $ 631 $ 833 6-13 $2,803 Multiple Cash Flows Present Value To compute the present value of multiple cash flows, we again just bring the payments into the present value – one year at a time 6-14 Multiple... 508.41 Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 6-18 $1,432.93 Multiple Cash Flows Using a Spreadsheet  You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows  Setting the data up is half the battle – if it is set up properly, then you can just copy the formulas  6-19 Click on the Excel icon for an example Multiple Cash Flows Present Value... 1 −  (1.01) 48  PV = 632   = 23,999.54 01     Annuity: Sweepstakes Winner Suppose you win the Publishers Clearinghouse $10 million sweepstakes The money is paid in equal annual end -of- year installments of $333,333.33 over 30 years If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? 6-36 ... your calculator between the two types by using the 2   nd nd BGN 2 Set on the TI BA-II Plus HP 12C: “g” 7 sets the calculator for Beginning and “g” 8 for End If you see “BGN” or “Begin” in the display of your calculator, you have it set for an annuity due Most problems are ordinary annuities 6-33 Annuity: Saving for a Car After carefully going over your budget, you have determined you can afford to... flow keys 1 Press CF and enter the cash flows beginning with year 0 2 You have to press the “Enter” key for each cash flow 3 Use the down arrow key to move to the next cash flow 4 The “F” is the number of times a given cash flow occurs in consecutive periods 5 Use the NPV key to compute the present value by entering the interest rate for I, press “Enter”, then the down arrow, and then “CPT” computing ... required discount rate is 7%  What is the value of the cash flows at year 5?  What is the value of the cash flows today?  What is the value of the cash flows at year 3? 6-27 Chapter Outline... 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... Spreadsheet  You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows  Setting the data up is half the battle – if it is set up properly, then you