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## Chapter 4

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**Chapter 4**The Time Value of Money**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**Chapter Outline (cont’d)**4.6 Solving Problems with a Spreadsheet Program 4.7 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.**Learning Objectives**Several cash flows occurring at regular intervals that grow at a constant rate each period. 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.**Learning Objectives**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 made a loan 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 – a month, a week, a day, etc.**4.2 Three Rules of Time Travel**Financial decisions often require combining cash flows or comparing values. Three rules govern these processes. Table 4.1The Three Rules of Time Travel**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**Textbook Example 4.2 Financial Calculator Solution for n=7**years Inputs: N = 7 I = 10 PV = 1,000 Output: FV = –1,948.72**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.**Alternative Example 4.2 (cont’d)**Solution The time line looks like this: 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.**Alternative Example 4.2Financial 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**Textbook Example 4.3 Financial Calculator Solution**Inputs: N = 10 I = 6 FV = 15,000 Output: PV = –8,375.92**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?**Alternative Example 4.3 (cont’d)**Solution The $10,000 is worth: $10,000 ÷ (1.10)5 = $6,209**Alternative Example 4.3: Financial Calculator Solution**Inputs: N = 5 I = 10 FV = 10,000 Output: PV = –6,209.21**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?**Applying the Rules of Time Travel (cont'd)**The time line would look like this:**Applying the Rules of Time Travel**Table 4.1 The Three Rules of Time Travel**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:**Alternative Example 4.4 (cont'd)**Solution 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.**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. The future value of both cash flows is $18,053.**4.3 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.**4.3 Valuing a Stream of Cash Flows (cont’d)**Present Value of a Cash Flow Stream