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## Chapter 4 The Time Value of Money

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**Chapter Outline**4.1 The Timeline 4.2 The Three Rules of Time Travel 4.3 The Power of Compounding 4.4 Valuing a Stream of Cash Flows 4.5 The Net Present Value of a Stream of Cash Flows 4.6 Perpetuities, Annuities, and other Special Cases**Chapter Outline (cont’d)**4.7 Solving Problems with a Spreadsheet Program 4.8 Solving for Variables Other Than Present Value or Future Value**Learning Objectives**• Draw a timeline illustrating a given set of cash flows. • List and describe the three rules of time travel. • Calculate the future value of: • A single sum. • An uneven stream of cash flows, starting either now or sometime in the future. • An annuity, starting either now or sometime in the future. • Several cash flows occurring at regular intervals that grow at a constant rate each period.**Learning Objectives (cont'd)**• Calculate the present value of: • A single sum. • An uneven stream of cash flows, starting either now or sometime in the future. • An infinite stream of identical cash flows. • An annuity, starting either now or sometime in the future. • An infinite stream of cash flows that grow at a constant rate each period. • Several cash flows occurring at regular intervals that grow at a constant rate each period.**Learning Objectives (cont'd)**• Given four out of the following five inputs for an annuity, compute the fifth: (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate, (e) periodic payment. • Given three out of the following four inputs for a single sum, compute the fourth: (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate. • Given cash flows and present or future value, compute the internal rate of return for a series of cash flows.**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.**4.1 The Timeline (cont’d)**• Assume that you loan $10,000 to a friend. You will be repaid in two payments, one at the end of each year over the next two years.**4.1 The Timeline (cont’d)**• Differentiate between two types of cash flows • Inflows are positive cash flows. • Outflows are negative cash flows, which are indicated with a – (minus) sign.**4.1 The Timeline (cont’d)**• Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments. • The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow. • Timelines can represent cash flows that take place at the end of any time period.**4.2 Three Rules of Time Travel**• Financial decisions often require combining cash flows or comparing values. Three rules govern these processes.**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?”**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:**The 2nd Rule of Time Travel (cont’d)**• Future Value of a Cash Flow**Using a Financial Calculator: The Basics**• TI BA II Plus • Future Value • Present Value • I/Y • Interest Rate per Year • Interest is entered as a percent, not a decimal • For 10%, enter 10, NOT .10**Using a Financial Calculator: The Basics (cont'd)**• TI BA II Plus • Number of Periods • 2nd → CLR TVM • Clears out all TVM registers • Should do between all problems**Using a Financial Calculator: Setting the keys**• TI BA II Plus • 2ND → P/Y • Check P/Y • 2ND → P/Y → # → ENTER • Sets Periods per Year to # • 2ND → FORMAT → # → ENTER • Sets display to # decimal places**Using a Financial Calculator**• TI BA II Plus • Cash flows moving in opposite directions must have opposite signs.**Financial Calculator Solution**• Inputs: • N = 2 • I = 10 • PV = 1,000 • Output: • FV = −1,210**The 2nd Rule of Time Travel—Alternative Example**• To move a cash flow forward in time, you must compound it. • 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. The time line looks like this:**The 2nd Rule of Time Travel—Alternative Example (cont’d)**• 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.**Financial Calculator Solution**• Inputs: • N = 5 • I = 10 • PV = 5,000 • Output: • FV = –8,052.55**The 3rd Rule of Time Travel**• To move a cash flow backward in time, we must discount it. • Present Value of a Cash Flow**Example 4.2 Financial Calculator Solution**• Inputs: • N = 10 • I = 6 • FV = 15,000 • Output: • PV = –8,375.92**The 3rd Rule of Time Travel—Alternative Example**• 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?**The 3rd Rule of Time Travel—Alternative Example (cont’d)**• The $10,000 is worth: • $10,000 ÷ (1.10)5 = $6,209**Alternative Example: Financial Calculator Solution**• Inputs: • N = 5 • I = 10 • FV = 10,000 • Output: • PV = –6,209.21**Combining Values Using 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?**Combining Values Using the Rules of Time Travel (cont'd)**• The time line would look like this:**Combining Values Using the Rules of Time**Travel—Alternative Example • Assume that an investment will pay you $5,000 now and $10,000 in five years. • The time line would like this:**Combining Values Using the Rules of Time**Travel—Alternative Example (cont'd) • You can calculate the present value of the combined cash flows by adding their values today. The present value of both cash flows is $11,209.**Combining Values Using the Rules of Time**Travel—Alternative Example (cont'd) • You can calculate the future value of the combined cash flows by adding their values in Year 5. • The future value of both cash flows is $18,053.**Combining Values Using the Rules of Time**Travel—Alternative Example (cont'd)**4.3 The Power of Compounding: An Application**• Compounding • Interest on Interest • As the number of time periods increases, the future value increases, at an increasing rate since there is more interest on interest.**4.4 Valuing a Stream of Cash Flows**• Based on the first rule of time travel we can derive a general formula for valuing a stream of cash flows: if we want to find the present value of a stream of cash flows, we simply add up the present values of each.**4.4 Valuing a Stream of Cash Flows (cont’d)**• Present Value of a Cash Flow Stream