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**Multiple Cash Flows 6.1 – FV Example 1**

You currently have $7,000 in a bank account earning 8% interest. You think you will be able to deposit an additional $4,000 at the end of each of the next three years. How much will you have in three years? © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows FV Example 1 continued**

Find the value at year 3 of each cash flow and add them together. Formula Approach Today (year 0): FV = 7000(1.08)3 = 8,817.98 Year 1: FV = 4,000(1.08)2 = 4,665.60 Year 2: FV = 4,000(1.08) = 4,320 Year 3: value = 4,000 Total value in 3 years = = 21,803.58 © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows FV Example 1 continued**

Calculator Approach Today (year 0 CF): 3 N; 8 I/Y; PV; CPT FV = Year 1 CF: 2 N; 8 I/Y; PV; CPT FV = Year 2 CF: 1 N; 8 I/Y; PV; CPT FV = 4320 Year 3 CF: value = 4,000 Total value in 3 years = = 21,803.58 © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows – FV Example 2**

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? Formula Approach FV = 500(1.09) (1.09) = Calculator Approach Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV = Year 1 CF: 1 N; -600 PV; 9 I/Y; CPT FV = Total FV = = © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows – FV Example 2 Continued**

How much will you have in 5 years if you make no further deposits? Formula Approach First way: FV = 500(1.09) (1.09)4 = Second way – use value at year 2: FV = (1.09)3 = Calculator Approach Year 0 CF: 5 N; -500 PV; 9 I/Y; CPT FV = Year 1 CF: 4 N; -600 PV; 9 I/Y; CPT FV = Total FV = = 3 N; PV; 9 I/Y; CPT FV = © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows – FV Example 3**

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%? Formula Approach FV = 100(1.08) (1.08)2 = = Calculator Approach Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV = Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV = Total FV = = © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows – PV Example 1**

You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the year after, and $800 at the end of the following year. You can earn 12% on similar investments. How much is this investment worth today? © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows - PV Example 1 - Timeline**

1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows - PV Example 1 continued**

Find the PV of each cash flow and add them Formula Approach Year 1 CF: 200 / (1.12)1 = Year 2 CF: 400 / (1.12)2 = Year 3 CF: 600 / (1.12)3 = Year 4 CF: 800 / (1.12)4 = Total PV = = © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows - PV Example 1 continued**

Calculator Approach Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = Total PV = = Remember the sign convention. The negative numbers imply that we would have to pay today to receive the cash flows in the future. © 2013 McGraw-Hill Ryerson Limited

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**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 Click on the Excel icon for an example Click on the tabs at the bottom of the worksheet to move from a future value example to a present value example. © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows – PV Example 2**

You are considering an investment that will pay you $1000 in one year, $2000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? © 2013 McGraw-Hill Ryerson Limited

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**Multiple Cash Flows – PV Example 2 continued**

Formula Approach PV = 1000 / (1.1)1 = PV = 2000 / (1.1)2 = PV = 3000 / (1.1)3 = PV = = Calculator Approach N = 1; I/Y = 10; FV = 1000; CPT PV = N = 2; I/Y = 10; FV = 2000; CPT PV = N = 3; I/Y = 10; FV = 3000; CPT PV = © 2013 McGraw-Hill Ryerson Limited

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**Multiple Uneven Cash Flows – Using the Calculator**

Another way to use the financial calculator for uneven cash flows is to use the cash flow keys Texas Instruments BA-II Plus Press CF and enter the cash flows beginning with year 0 You have to press the “Enter” key for each cash flow Use the down arrow key to move to the next cash flow The “F” is the number of times a given cash flow occurs in consecutive years Use the NPV key to compute the present value by entering the interest rate for I, pressing the down arrow and then compute Clear the cash flow keys by pressing CF and then CLR Work The next example will be worked using the cash flow keys. Note that with the BA-II Plus, the students can double check the numbers they have entered by pressing the up and down arrows. It is similar to entering the cash flows into spreadsheet cells. Other calculators also have cash flow keys. You enter the information by putting in the cash flow and then pressing CF. You have to always start with the year 0 cash flow, even if it is zero. Remind the students that the cash flows have to occur at even intervals, so if you skip a year, you still have to enter a 0 cash flow for that year. © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Decisions, Decisions LO1 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? Use the CF keys to compute the value of the investment 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. You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get – You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV. © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Saving For Retirement LO1 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%? © 2013 McGraw-Hill Ryerson Limited

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**Saving For Retirement Timeline**

LO1 … … K 25K 25K 25K 25K Notice that the year 0 cash flow = 0 (CF0 = 0) The cash flows years 1 – 39 are 0 (C01 = 0; F01 = 39) The cash flows years 40 – 44 are 25,000 (C02 = 25,000; F02 = 5) © 2013 McGraw-Hill Ryerson Limited

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**Saving For Retirement continued**

LO1 Calculator Approach Use cash flow keys: CF; CF0 = 0; C01 = 0; F01 = 39; C02 = 25000; F02 = 5; NPV; I = 12; CPT NPV = © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Quick Quiz – Part I LO1 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% 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? The easiest way to work this problem is to use the uneven cash flow keys and find the present value first and then compute the others based on that. CF0 = 0; C01 = 100; F01 = 1; C02 = 200; F02 = 2; C03 = 300; F03 = 2; I = 7; CPT NPV = Value in year 5: PV = ; N = 5; I/Y = 7; CPT FV = Value in year 3: PV = ; N = 3; I/Y = 7; CPT FV = Using formulas and one CF at a time: Year 1 CF: FV5 = 100(1.07)4 = ; PV0 = 100 / 1.07 = 93.46; FV3 = 100(1.07)2 = Year 2 CF: FV5 = 200(1.07)3 = ; PV0 = 200 / (1.07)2 = ; FV3 = 200(1.07) = 214 Year 3 CF: FV5 = 200(1.07)2 = ; PV0 = 200 / (1.07)3 = ; FV3 = 200 Year 4 CF: FV5 = 300(1.07) = 321; PV0 = 300 / (1.07)4 = ; PV3 = 300 / 1.07 = Year 5 CF: FV5 = 300; PV0 = 300 / (1.07)5 = ; PV3 = 300 / (1.07)2 = Value at year 5 = = Present value today = = (difference due to rounding) Value at year 3 = = © 2013 McGraw-Hill Ryerson Limited

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**Annuities and Perpetuities 6.2**

LO1 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 © 2013 McGraw-Hill Ryerson Limited

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**Annuities and Perpetuities – Basic Formulas**

LO1 Perpetuity: PV = C / r Annuities: © 2013 McGraw-Hill Ryerson Limited

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**Annuities and the Calculator**

LO1 You can use the PMT key on the calculator for the equal payment The sign convention still holds Ordinary annuity versus annuity due You can switch your calculator between the two types by using the 2nd BGN 2nd Set on the TI BA-II Plus 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 Other calculators also have a key that allows you to switch between Beg/End. Warn students to be careful with this function. Every term, it seems that there is always one student who, during the midterm exam, switches their calculator to beginning of period payments and forgets to switch it back for the next question. Another way of handling the annuity due calculation is to calculate it as if it were a regular annuity, and then simply multiply the answer given by (1+R). © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Annuity – Example 1 LO1 After carefully going over your budget, you have determined that you can afford to pay $632 per month towards a new sports car. Your bank will lend to you at 1% per month for 48 months. How much can you borrow? © 2013 McGraw-Hill Ryerson Limited

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**Annuity – Example 1 continued**

LO1 You borrow money TODAY so you need to compute the present value. Formula Approach Calculator Approach 48 N; 1 I/Y; -632 PMT; CPT PV = 23, ($24,000) © 2013 McGraw-Hill Ryerson Limited

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**Annuity – Sweepstakes Example**

LO1 Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333, over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? Formula Approach PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29 Calculator Approach 30 N; 5 I/Y; 333, PMT; CPT PV = -5,124,150.29 © 2013 McGraw-Hill Ryerson Limited

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**Annuities on the Spreadsheet - Example**

LO1 The present value and future value formulas in a spreadsheet include a place for annuity payments Click on the Excel icon to see an example © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Quick Quiz – Part II LO1 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? You want to receive $5,000 per month in retirement. If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement? Calculator PMT = 5000; N = 25*12 = 300; I/Y = .75; CPT PV = 595,808 Formula PV = 5000[1 – 1 / ] / = 595,808 © 2013 McGraw-Hill Ryerson Limited

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**Future Values for Annuities – Example 1**

LO1 Suppose you begin saving for your retirement by depositing $2000 per year in an RRSP. If the interest rate is 7.5%, how much will you have in 40 years? Formula Approach FV = 2000( – 1)/.075 = 454,513.04 Calculator Approach Remember the sign convention!!! 40 N 7.5 I/Y -2000 PMT CPT FV = 454,513.04 © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Annuity Due – Example 1 LO1 You are saving for a new house and you put $10,000 per year in an account paying 8% compounded annually. The first payment is made today. How much will you have at the end of 3 years? © 2013 McGraw-Hill Ryerson Limited

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**Annuity Due – Example 1 Timeline**

LO1 32,464 If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period. 35,061.12 © 2013 McGraw-Hill Ryerson Limited

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**Annuity Due – Example 1 continued**

LO1 Formula Approach FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12 Calculator Approach 2nd BGN 2nd Set (you should see BGN in the display) 3 N -10,000 PMT 8 I/Y CPT FV = 35,061.12 2nd BGN 2nd Set (be sure to change it back to an ordinary annuity) Note that the procedure for changing the calculator to an annuity due is similar on other calculators. What if it were an ordinary annuity? FV = 32,464 (so receive an additional by starting to save today.) Remind students to change their calculator back to the regular annuity setting before proceeding with other questions! © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Perpetuity – Example 1 LO1 The Home Bank of Canada want to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend would the Home Bank have to offer if its preferred stock is going to sell? © 2013 McGraw-Hill Ryerson Limited

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**Perpetuity – Example 1 continued**

LO1 Perpetuity formula: PV = C / r First, find the required return for the comparable issue: 40 = 1 / r r = .025 or 2.5% per quarter Then, using the required return found above, find the dividend for new preferred issue: 100 = C / .025 C = 2.50 per quarter This is a good preview to the valuation issues discussed in future chapters. The price of an investment such as a stock is just the present value of expected future cash flows. © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Growing Perpetuity LO1 The perpetuities discussed so far have constant payments Growing perpetuities have cash flows that grow at a constant rate and continue forever Growing perpetuity formula: © 2013 McGraw-Hill Ryerson Limited

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**Growing Perpetuity – Example 1**

LO1 Hoffstein Corporation is expected to pay a dividend of $3 per share next year. Investors anticipate that the annual dividend will rise by 6% per year forever. The required rate of return is 11%. What is the price of the stock today? © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Growing Annuity LO1 Growing annuities have a finite number of growing cash flows Growing annuity formula: © 2013 McGraw-Hill Ryerson Limited

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**Growing Annuity – Example 1**

LO1 Gilles Lebouder has just been offered a job at $50,000 a year. He anticipates his salary will increase by 5% a year until his retirement in 40 years. Given an interest rate of 8%, what is the present value of his lifetime salary? © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Quick Quiz – Part IV LO1 You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month? What if the first payment is made today? 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? Q1: 35(12) = 420 N; 1,000,000 FV; 1 I/Y; CPT PMT = 1,000,000 = C ( – 1) / .01 C = Q2: Set calculator to annuity due and use the same inputs as above. CPT PMT = 1,000,000 = C[( – 1) / .01] ( 1.01) C = Q3: PV = 1.50 / .03 = $50 © 2013 McGraw-Hill Ryerson Limited

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**© 2013 McGraw-Hill Ryerson Limited**

Work the Web Example LO1 Another online financial calculator can be found at MoneyChimp Click on the web surfer and work the following example Choose calculator and then annuity You just inherited $5 million. If you can earn 6% on your money, how much can you withdraw each year for the next 40 years? Payment = $332,307.68 Using MoneyChimp, select “Calculator” at the top of the screen, then select “Annuity” from the menu on the top, right-hand side of the screen. You can choose between a regular annuity or an annuity due by clicking on the appropriate button in the box. If you chose “annuity due”, then the amount will change to $313, © 2013 McGraw-Hill Ryerson Limited

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**Table 6.2 – Summary of Annuity and Perpetuity Calculations**

LO1 © 2013 McGraw-Hill Ryerson Limited

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© 2003 The McGraw-Hill Companies, Inc. All rights reserved. Discounted Cash Flow Valuation Chapter Six Prepared by Anne Inglis, Ryerson University.

© 2003 The McGraw-Hill Companies, Inc. All rights reserved. Discounted Cash Flow Valuation Chapter Six Prepared by Anne Inglis, Ryerson University.

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