Microsoft Excel 2010: Data Analysis and Business Modeling phần 2 pot

57 Chapter 8 Evaluating Investments by Using Net Present Value Criteria Questions answered in this chapter: ■ What is net present value (NPV)? ■ How do I use the Excel NPV function? ■ How can I compute NPV when cash ows are received at the beginning of a year or in the middle of a year? ■ How can I compute NPV when cash ows are received at irregular intervals? Consider the following two investments, whose cash ows are listed in the le NPV.xlsx and shown in Figure 8-1. ■ Investment 1 requires a cash outow of $10,000 today and $14,000 two years from now. One year from now, this investment will yield $24,000. ■ Investment 2 requires a cash outow of $6,000 today and $1,000 two years from now. One year from now, this investment will yield $8,000. Which is the better investment? Investment 1 yields total cash ow of $0, whereas Investment 2 yields a total cash ow of $1,000. At rst glance, Investment 2 appears to be better. But wait a minute. Most of the cash outow for Investment 1 occurs two years from now, while most of the cash outow for Investment 2 occurs today. Spending $1 two years from now doesn’t seem as costly as spending $1 today, so maybe Investment 1 is better than rst appears. To determine which investment is better, you need to compare the values of cash ows received at different points in time. That’s where the concept of net present value proves useful. FIGURE 8-1 To determine which investment is better, you need to calculate net present value. 58 Microsoft Excel 2010: Data Analysis and Business Modeling Answers to This Chapter’s Questions What is net present value? The net present value (NPV) of a stream of cash ow received at different points in time is simply the value measured in today’s dollars. Suppose you have $1 today and you invest this dollar at an annual interest rate of r percent. This dollar will grow to 1+r dollars in the rst year, (1+r) 2 dollars in two years, and so on. You can say, in some sense, that $1 today equals $(1+r) a year from now and $(1+r) 2 two years from now. In general, you can say that $1 today is equal to $(1+r) n n years from now. As an equation, this calculation can be expressed as follows: $1 now=$(1+r) n received n years from now If you divide both sides of this equation by (1+r) n , you get the following important result: 1/(1+r) n now=$1 received n years from now This result tells you how to compute (in today’s dollars) the NPV of any sequence of cash ows. You can convert any cash ow to today’s dollars by multiplying the cash ow received n years from now (n can be a fraction) by 1/(1+r) n . You then add up the value of the cash ows (in today’s dollars) to nd the investment’s NPV. Let’s assume r is equal to 0.2. You could calculate the NPV for the two investments we’re considering as follows: 24,000 (1 + 0.20) 1 = $277.78 –14,000 (1 + 0.20) 2 = Investment1NPV = –10,000 + + 8,000 (1 + 0.20) 1 = $–27.78 –1,000 (1 + 0.20) 2 = Investment2NPV = –6,000 + + On the basis of NPV, Investment 1 is superior to Investment 2. Although total cash ow for Investment 2 exceeds total cash ow for Investment 1, Investment 1 has a better NPV because a greater proportion of Investment 1’s negative cash ow comes later, and the NPV criterion gives less weight to cash ows that come later. If you use a value of .02 for r, Investment 2 has a larger NPV because when r is very small, later cash ows are not dis- counted as much, and NPV returns results similar to those derived by ranking investments according to total cash ow. Chapter 8 Evaluating Investments by Using Net Present Value Criteria 59 Note I randomly chose the interest rate r=0.2, skirting the issue of how to determine an appropriate value of r. You need to study nance for at least a year to understand the issues involved in determining an appropriate value for r. The appropriate value of r used to compute NPV is often called the company’s cost of capital. Sufce it to say that most U.S. companies use an annual cost of capital between 0.1 (10 percent) and 0.2 (20 percent). If the annual interest rate is chosen according to accepted nance practices, projects with NPV>0 increase the value of a company, projects with NPV<0 decrease the value of a company, and projects with NPV=0 keep the value of a company unchanged. A company should (if it had unlimited investment capital) invest in every available investment having positive NPV. To determine the NPV of Investment 1 in Excel, I rst assigned the range name r_ to the in- terest rate (located in cell C3). I then copied the Time 0 cash ow from C5 to C7. I determined the NPV for Investment 1’s Year 1 and Year 2 cash ows by copying from D7 to E7 the for- mula D5/(1+r_)^D$4. The caret symbol (^), located over the number 6 on the keyboard, raises a number to a power. In cell A5, I computed the NPV of Investment 1 by adding the NPV of each year’s cash ow with the formula SUM(C7:E7). To determine the NPV for Investment 2, I copied the formulas from C7:E7 to C8:E8 and from A5 to A6. How do I use the Excel NPV function? The Excel NPV function uses the syntax NPV(rate,range of cells). This function determines the NPV for the given rate of the cash ows in the range of cells. The function’s calculation as- sumes that the rst cash ow is one period from now. In other words, entering the formula NPV(r_,C5:E5) will not determine the NPV for Investment 1. Instead, this formula (entered in cell C14) computes the NPV of the following sequence of cash ows: –$10,000 a year from now, $24,000 two years from now, and –$14,000 three years from now. Let’s call this Investment 1 (End of Year). The NPV of Investment 1 (End of Year) is $231.48. To compute the actual cash NPV of Investment 1, I entered the formula C7+NPV(r_,D5:E5) in cell C11. This formula does not discount the Time 0 cash ow at all (which is correct because Time 0 cash ow is already in today’s dollars), but rst multiplies the cash ow in D5 by 1/1.2 and then multiplies the cash ow in E5 by 1/1.2 2 . The formula in cell C11 yields the correct NPV of Investment 1, $277.78. How can I compute NPV when cash ows are received at the beginning of a year or in the middle of a year? To use the NPV function to compute the net present value of a project whose cash ows al- ways occur at the beginning of a year, you can use the approach I described to determine the NPV of Investment 1: separate out the Year 1 cash ow and apply the NPV function to the remaining cash ows. Alternatively, observe that for any year n, $1 received at the beginning 60 Microsoft Excel 2010: Data Analysis and Business Modeling of year n is equivalent to $(1+r) received at the end of year n. Remember that in one year, a dollar will grow by a factor (1+r). Thus, if you multiply the result obtained with the NPV func- tion by (1+r), you can convert the NPV of a sequence of year-end cash ows to the NPV of a sequence of cash ows received at the beginning of the year. You can also compute the NPV of Investment 1 in cell D11 with the formula (1+r_)*C14. Of course, you again obtain an NPV of $277.78. Now suppose the cash ows for an investment occur in the middle of each year. For an orga- nization that receives monthly subscription revenues, you can approximate the 12 monthly revenues received during a given year as a lump sum received in the middle of the year. How can you use the NPV function to determine the NPV of a sequence of mid-year cash ows? For any Year n, $ 1 + r received at the end of Year n is equivalent to $1 received at the middle of Year n because in half a year $1 will grow by a factor of 1 + r If you assume the cash ows for Investment 1 occur mid year, you can compute the NPV of the mid-year version of Investment 1 in cell C17 with the formula SQRT(1+r_)*C14. You obtain a value of $253.58. How can I compute NPV when cash ows are received at irregular intervals? Cash ows often occur at irregular intervals, which makes computing the NPV or internal rate of return (IRR) of these cash ows more difcult. Fortunately, the Excel XNPV function makes computing the NPV of irregularly timed cash ows a snap. The XNPV function uses the syntax XNPV(rate,values,dates). The rst date listed must be the earliest, but other dates need not be listed in chronological order. The XNPV function computes the NPV of the given cash ows assuming the current date is the rst date in the sequence. For example, if the rst listed date is 4/08/13, the NPV is computed in April 8, 2013 dollars. To illustrate the use of the XNPV function, look at the example on the NPV as of rst date worksheet in the le XNPV.xlsx, which is shown in Figure 8-2. Suppose that on April 8, 2013, you paid out $900. Later you receive the following amounts: ■ $300 on August 15, 2013 ■ $400 on January 15, 2014 ■ $200 on June 25, 2014 ■ $100 on July 3, 2015 Chapter 8 Evaluating Investments by Using Net Present Value Criteria 61 If the annual interest rate is 10 percent, what is the NPV of these cash ows? I entered the dates (in Excel date format) in D3:D7 and the cash ows in E3:E7. Entering the formula XNPV(A9,E3:E7,D3:D7) in cell D11 computes the project’s NPV in April 8, 2013 dollars because that is the rst date listed. This project would have an NPV, in April 8, 2013 dollars, of $20.63. FIGURE 8-2 Using the XNPV function. The computations performed by the XNPV function are as follows: 1. Compute the number of years after April 8, 2013, that each date occurred. (I did this in column F.) For example, August 15 is 0.3534 years after April 8. 2. Discount cash ows at the rate 1/(1+rate) years after . For example, the August 15, 2013 cash ow is discounted by 1 (1 + 0.1) 3534 = 0.967 3. Sum up in cell E11 overall cash ows: (cash ow value)*(discount factor). Suppose that today’s date is actually July 11, 2010. How would you compute the NPV of an investment in today’s dollars? Simply add a row with today’s date and 0 cash ow and include this row in the range for the XNPV function. (See Figure 8-3 and the Today worksheet.) The NPV of the project in today’s dollars is $15.88. FIGURE 8-3 NPV converted to today’s dollars. I’ll close by noting that if a cash ow is left blank, the NPV function ignores both the cash ow and the period. If a cash ow is left blank, the XNPV function returns a #NUM error. 62 Microsoft Excel 2010: Data Analysis and Business Modeling Problems 1. An NBA player is to receive a $1,000,000 signing bonus today and $2,000,000 one year, two years, and three years from now. Assuming r=0.10 and ignoring tax considerations, would he be better off receiving $6,000,000 today? 2. A project has the following cash ows: Now One year from now Two years from now Three years from now –$4 million $4 million $4 million –$3 million If the company’s cost of capital is 15 percent, should it proceed with the project? 3. Beginning one month from now, a customer will pay his Internet provider $25 per month for the next ve years. Assuming all revenue for a year is received at the middle of a year, estimate the NPV of these revenues. Use r=0.15. 4. Beginning one month from now, a customer will pay $25 per month to her Internet provider for the next ve years. Assuming all revenue for a year is received at the middle of a year, use the XNPV function to obtain the exact NPV of these revenues. Use r=0.15. 5. Consider the following set of cash ows over a four-year period. Determine the NPV of these cash ows if r=0.15 and cash ows occur at the end of the year. Year 1 2 3 4 -$600 $550 -$680 $1,000 6. Solve Problem 5 assuming cash ows occur at the beginning of each year. 7. Consider the following cash ows: Date Cash ow 12/15/01 –$1,000 1/11/02 $300 4/07/03 $600 7/15/04 $925 If today is November 1, 2001, and r=0.15, what is the NPV of these cash ows? 8. After earning an MBA, a student will begin working at an $80,000-per-year job on September 1, 2005. She expects to receive a 5 percent raise each year until she retires on September 1, 2035. If the cost of capital is 8 percent a year, determine the total present value of her before-tax earnings. 9. Consider a 30-year bond that pays $50 at the end of Years 1–29 and $1,050 at the end of Year 30. If the appropriate discount rate is 5 percent per year, what is a fair price for this bond? 63 Chapter 9 Internal Rate of Return Questions answered in this chapter: ■ How can I nd the IRR of cash ows? ■ Does a project always have a unique IRR? ■ Are there conditions that guarantee a project will have a unique IRR? ■ If two projects each have a single IRR, how do I use the projects’ IRRs? ■ How can I nd the IRR of irregularly spaced cash ows? ■ What is the MIRR and how do I compute it? The net present value (NPV) of a sequence of cash ows depends on the interest rate (r) used. For example, if you consider cash ows for Projects 1 and 2 (see the worksheet IRR in the le IRR.xlsx, shown in Figure 9-1), you nd that for r=0.2, Project 2 has a larger NPV, and for r=0.01, Project 1 has a larger NPV. When you use NPV to rank investments, the outcome can depend on the interest rate. It is the nature of human beings to want to boil everything in life down to a single number. The internal rate of return (IRR) of a project is simply the interest rate that makes the NPV of the project equal to 0. If a project has a unique IRR, the IRR has a nice interpretation. For example, if a project has an IRR of 15 percent, you receive an annual rate of return of 15 percent on the cash ow you invested. In this chapter’s exam- ples, you’ll nd that Project 1 has an IRR of 47.5 percent, which means that the $400 invested at Time 1 is yielding an annual rate of return of 47.5 percent. Sometimes, however, a project might have more than one IRR or even no IRR. In these cases, speaking about the project’s IRR is useless. FIGURE 9-1 Example of the IRR function. 64 Microsoft Excel 2010: Data Analysis and Business Modeling Answers to This Chapter’s Questions How can I nd the IRR of cash ows? The IRR function calculates internal rate of return. The function has the syntax IRR(range of cash ows,[guess]), where guess is an optional argument. If you do not enter a guess for a project’s IRR, Excel begins its calculations with a guess that the project’s IRR is 10 percent and then varies the estimate of the IRR until it nds an interest rate that makes the project’s NPV equal 0 (the project’s IRR). If Excel can’t nd an interest rate that makes the project’s NPV equal 0, Excel returns #NUM. In cell B5, I entered the formula IRR(C2:I2) to compute Project 1’s IRR. Excel returns 47.5 percent. Thus, if you use an annual interest rate of 47.5 percent, Project 1 will have an NPV of 0. Similarly, you can see that Project 2 has an IRR of 80.1 percent. Even if the IRR function nds an IRR, a project might have more than one IRR. To check whether a project has more than one IRR, you can vary the initial guess of the project’s IRR (for example, from –90 percent to 90 percent). I varied the guess for Project 1’s IRR by copy- ing from B8 to B9:B17 the formula IRR($C$2:$I$2,A8). Because all the guesses for Project 1’s IRR yield 47.5 percent, I can be fairly condent that Project 1 has a unique IRR of 47.5 percent. Similarly, I can be fairly condent that Project 2 has a unique IRR of 80.1 percent. Does a project always have a unique IRR? In the worksheet Multiple IRR in the le IRR.xlsx (see Figure 9-2), you can see that Project 3 (cash ows of –20, 82, –60, 2) has two IRRs. I varied the guess about Project 3’s IRR from –90 percent to 90 percent by copying from C8 to C9:C17 the formula IRR($B$4:$E$4,B8). FIGURE 9-2 Project with more than one IRR. Note that when a guess is 30 percent or less, the IRR is –9.6 percent. For other guesses, the IRR is 216.1 percent. For both these interest rates, Project 3 has an NPV of 0. Chapter 9 Internal Rate of Return 65 In the worksheet No IRR in the le IRR.xlsx (shown in Figure 9-3), you can see that no matter what guess you use for Project 4’s IRR, you receive the #NUM message. This message indicates that Project 4 has no IRR. When a project has multiple IRRs or no IRR, the concept of IRR loses virtually all meaning. Despite this problem, however, many companies still use IRR as their major tool for ranking investments. FIGURE 9-3 Project with no IRR. Are there conditions that guarantee a project will have a unique IRR? If a project’s sequence of cash ows contains exactly one change in sign, the project is guaranteed to have a unique IRR. For example, for Project 2 in the worksheet IRR, the sign of the cash ow sequence is – + + + + +. There is only one change in sign (between Time 1 and Time 2), so Project 2 must have a unique IRR. For Project 3 in the worksheet Multiple IRR, the signs of the cash ows are – + – +. Because the sign of the cash ows changes three times, a unique IRR is not guaranteed. For Project 4 in the worksheet No IRR, the signs of the cash ows are + – +. Because the signs of the cash ows change twice, a unique IRR is not guaran- teed in this case either. Most capital investment projects (such as building a plant) begin with a negative cash ow followed by a sequence of positive cash ows. Therefore, most capital investment projects do have a unique IRR. If two projects each have a single IRR, how do I use the projects’ IRRs? If a project has a unique IRR, you can state that the project increases the value of the company if and only if the project’s IRR exceeds the annual cost of capital. For example, if the cost of capital for a company is 15 percent, both Project 1 and Project 2 would increase the value of the company. 66 Microsoft Excel 2010: Data Analysis and Business Modeling Suppose two projects are under consideration (both having unique IRRs), but you can under- take at most one project. It’s tempting to believe that you should choose the project with the larger IRR. To illustrate that this belief can lead to incorrect decisions, look at Figure 9-4 and the Which Project worksheet in IRR.xlsx. Project 5 has an IRR of 40 percent, and Project 6 has an IRR of 50 percent. If you rank projects based on IRR and can choose only one project, you would choose Project 6. Remember, however, that a project’s NPV measures the amount of value the project adds to the company. Clearly, Project 5 will (for virtually any cost of capital) have a larger NPV than Project 6. Therefore, if only one project can be chosen, Project 5 is it. IRR is problematic because it ignores the scale of the project. Whereas Project 6 is better than Project 5 on a per-dollar-invested basis, the larger scale of Project 5 makes it more valu- able to the company than Project 6. IRR does not reect the scale of a project, whereas NPV does. FIGURE 9-4 IRR can lead to an incorrect choice of which project to pursue. How can I nd the IRR of irregularly spaced cash ows? Cash ows occur on actual dates, not just at the start or end of the year. The XIRR function has the syntax XIRR(cash ow, dates, [guess]). The XIRR function determines the IRR of a sequence of cash ows that occur on any set of irregularly spaced dates. As with the IRR function, guess is an optional argument. For an example of how to use the XIRR function, look at Figure 9-5 and worksheet XIRR of the le IRR.xlsx. FIGURE 9-5 Example of the XIRR function. The formula XIRR(F4:F7,E4:E7) in cell D9 shows that the IRR of Project 7 is -48.69 percent. What is the MIRR and how do I compute it? In many situations the rate at which a company borrows funds is different from the rate at which the company reinvests funds. IRR computations implicitly assume that the rate at which a company borrows and reinvests funds is equal to the IRR. If we know the actual rate at which we borrow money and the rate at which we can reinvest money, then the modied internal rate of return (MIRR) function computes a discount rate that makes the NPV of all [...]... Year 2 Year 3 –$4,000 $2, 000 $4,000 68 Microsoft Excel 20 10: Data Analysis and Business Modeling 3 Find all IRRs for the following project: Year 1 Year 3 $100 Year 2 –$300 $25 0 4 Find all IRRs for a project having the given cash flows on the listed dates 1/10 /20 03 7/18 /20 04 3 /20 /20 05 4/1 /20 05 1/10 /20 06 –$1,000 7/10 /20 03 5 /25 /20 04 $900 $700 $500 $350 $800 $500 5 Consider the following two projects, and. .. can compute a 15-month moving average is April, 18 72, so I begin my calculations in row 24  92 Microsoft Excel 20 10: Data Analysis and Business Modeling FIGURE 12- 4  Moving-average trading rule beats buy and hold Let’s assume I first owned the stock in April 18 72, so Yes is entered in cell C24 ■ By copying from D24 to D25:D1590 the formula AVERAGE(B9:B23), I compute the 15-month moving average for each... through 1 ,20 0, the cost is 2. 70*A9 ■ If A9 is from 1 ,20 1 through 2, 000, the cost is 2. 30*A9 ■ If A9 is more than 2, 000, the cost is 2* A9 Chapter 12 IF Statements 89 Begin by linking the range names in A2:A4 to cells B2:B4, and linking the range names in cells D2:D5 to cells C2:C5 Then you can implement this logic in cell B9 with the following formula: IF(A9 . period. 72 Microsoft Excel 20 10: Data Analysis and Business Modeling In worksheet FV of le Excel nfunctions.xlsx (see Figure 10 -2) I entered in cell B3 the formula =FV(0.08,40, 20 00) to nd. PV(rate,#per,[pmt],[fv],[type]), where pmt, fv, and type are optional arguments. 70 Microsoft Excel 20 10: Data Analysis and Business Modeling Note When working with Microsoft Excel nancial functions, I use. Year 2 Year 3 –$4,000 $2, 000 $4,000 68 Microsoft Excel 20 10: Data Analysis and Business Modeling 3. Find all IRRs for the following project: Year 1 Year 2 Year 3 $100 –$300 $25 0 4. Find all IRRs