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4-47 Present Value of an Annuity Chapter 4 The Time Value of Money Copyright © 2011 Pearson Prentice Hall. All rights reserved. Chapter Outline 4.1 The Timeline 4.2 The Three Rules of Time Travel 4.3 Valuing a Stream of Cash Flows 4.4 Calculating the Net Present Value 4.5 Perpetuities, Annuities, and Other Special Cases Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-2 Chapter Outline (cont’d) 4.6 Solving Problems with a Spreadsheet Program 4.7 Solving for Variables Other Than Present Value or Future Value Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-3 4.1 The Timeline • A timeline is a linear representation of the timing of potential cash flows. • Drawing a timeline of the cash flows will help you visualize the financial problem. Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-4 Textbook Example 4.1 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-5 Textbook Example 4.1 (cont’d) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-6 4.2 Three Rules of Time Travel • Financial decisions often require combining cash flows or comparing values. Three rules govern these processes. Table 4.1 The Three Rules of Time Travel Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-7 The 1st Rule of Time Travel • A dollar today and a dollar in one year are not equivalent. • It is only possible to compare or combine values at the same point in time. – Which would you prefer: A gift of $1,000 today or $1,210 at a later date? – To answer this, you will have to compare the alternatives to decide which is worth more. One factor to consider: How long is “later?” Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-8 The 2nd Rule of Time Travel • To move a cash flow forward in time, you must compound it. – Suppose you have a choice between receiving $1,000 today or $1,210 in two years. You believe you can earn 10% on the $1,000 today, but want to know what the $1,000 will be worth in two years. The time line looks like this: Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-9 The 2nd Rule of Time Travel (cont’d) • Future Value of a Cash Flow =× +×+××+ =×+n FVn C (1 r) (1 r) (1 r) C (1 r) n times Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-10 Textbook Example 4.2 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-11 Textbook Example 4.2 (cont’d) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-12 Alternative Example 4.2 • Problem – Suppose you have a choice between receiving $5,000 today or $10,000 in five years. You believe you can earn 10% on the $5,000 today, but want to know what the $5,000 will be worth in five years. Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-13 Alternative Example 4.2 (cont’d) • Solution – The time line looks like this: 0 1 2 3 4 5 $5,000 x 1.10 $5, 500 x 1.10 $6,050 x 1.10 $6,655 x 1.10 $7,321 x 1.10 $8,053 – In five years, the $5,000 will grow to: $5,000 × (1.10)5 = $8,053 – The future value of $5,000 at 10% for five years is $8,053. – You would be better off forgoing the gift of $5,000 today and taking the $10,000 in five years. Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-14 The 3rd Rule of Time Travel • To move a cash flow backward in time, we must discount it. • Present Value of a Cash Flow C PV =÷+ C (1 r )n = (1 + r )n Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-15 Textbook Example 4.3 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-16 Textbook Example 4.3 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-17 Alternative Example 4.3 • Problem – Suppose you are offered an investment that pays $10,000 in five years. If you expect to earn a 10% return, what is the value of this investment today? Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-18 Alternative Example 4.3 (cont’d) • Solution – The $10,000 is worth: • $10,000 ÷ (1.10)5 = $6,209 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-19 Applying the Rules of Time Travel • Recall the 1st rule: It is only possible to compare or combine values at the same point in time. So far we’ve only looked at comparing. – Suppose we plan to save $1000 today, and $1000 at the end of each of the next two years. If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today? Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-20 Applying the Rules of Time Travel (cont'd) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-21 Textbook Example 4.4 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-22 Textbook Example 4.4 (cont’d) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-23 Alternative Example 4.4 • Problem – Assume that an investment will pay you $5,000 now and $10,000 in five years. – The time line would like this: 0 1 2 3 4 5 $5,000 $10,000 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-24 Alternative Example 4.4 (cont'd) • Solution – You can calculate the present value of the combined cash flows by adding their values today. 0 1 2 3 4 5 $5,000 $6,209 5 $10,000 $11,209 ÷ 1.10 – The present value of both cash flows is $11,209. Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-25 Alternative Example 4.4 (cont'd) • Solution – You can calculate the future value of the combined cash flows by adding their values in Year 5. 0 1 2 3 4 5 $10,000 $5,000 x 1.105 $8,053 $18,053 – The future value of both cash flows is $18,053. Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-26 Alternative Example 4.4 (cont'd) Present Value 0 1 2 3 4 5 $11,209 $18,053 ÷ 1.105 Future Value 0 1 2 3 4 5 $11,209 $18,053 x 1.105 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-27 4.3 Valuing a Stream of Cash Flows (cont’d) • Present Value of a Cash Flow Stream NNC = = n PV ∑∑ PV( Cn ) n nn = 0 = 0(1 + r) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-28 Textbook Example 4.5 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-29 Textbook Example 4.5 (cont’d) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-30 Future Value of Cash Flow Stream • Future Value of a Cash Flow Stream with a Present Value of PV n FVn = PV ×+ (1 r) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-31 Alternative Example 4.5 • Problem – What is the future value in three years of the following cash flows if the compounding rate is 5%? 0 1 2 3 $2,000 $2,000 $2,000 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-32 Alternative Example 4.5 (cont'd) • Solution 0 1 2 3 $2,000 $2,315 x 1.05 x 1.05 x 1.05 $2,000 $2,205 x 1.05 x 1.05 $2,000 $2,100 • Or x 1.05 $6,620 0 1 2 3 $2,000 $2,000 $2,000 $2,100 x 1.05 $4,100 x 1.05 $4,305 $6,305 Copyright © 2011 Pearson Prentice Hall. All rights reserved. $6,620 4-33 x 1.05 4.4 Calculating the Net Present Value • Calculating the NPV of future cash flows allows us to evaluate an investment decision. • Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs). Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-34 Textbook Example 4.6 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-35 Textbook Example 4.6 (cont'd) Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-36 Alternative Example 4.6 • Problem – Would you be willing to pay $5,000 for the following stream of cash flows if the discount rate is 7%? 0 1 2 3 $3,000 $2,000 $1,000 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-37 Alternative Example 4.6 (cont’d) • Solution – The present value of the benefits is: 3000 / (1.05) + 2000 / (1.05)2 + 1000 / (1.05)3 = 5366.91 – The present value of the cost is $5,000, because it occurs now. – The NPV = PV(benefits) – PV(cost) = 5366.91 – 5000 = 366.91 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-38 4.5 Perpetuities, Annuities, and Other Special Cases • When a constant cash flow will occur at regular intervals forever it is called a perpetuity. Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-39 4.5 Perpetuities, Annuities, and Other Special Cases (cont’d) • The value of a perpetuity is simply the cash flow divided by the interest rate. • Present Value of a Perpetuity C PV( C in perpetuity) = r Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-40 Textbook Example 4.7 Copyright © 2011 Pearson Prentice Hall. All rights reserved. 4-41 Textbook Example 4.7 (cont’d) Copyright © 2011 Pearson Prentice Hall.
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