As the following loan table shows, the IRR of this investment is larger than 15 percent:Note that we have added an extra cell (B16) to this example. If the interest rate in cell B3 is indeed the IRR, then cellB16 should be 0. We can now use Excels Goal Seek (found on the Tools menu) to calculate the IRR:You can see the result in the following display:Of course, we could have simplified life by just using the IRR function:
1.8.2 Back to Finance—Continuous DiscountingIf the accretion factor for continuous compounding at interest r over t years is ert, then the discount factor for thesame period is e−rt. Thus a cash flow Ct occurring in year t and discounted at continuously compounded rate r will beworth Cte−rt today, as illustrated here.1.8.3 Calculating the Continuously Compounded Return from Price DataSuppose at time 0 you had $1,000 in the bank and suppose that one year later you had $1,200. What was yourpercentage return? Although the answer may appear obvious, it actually depends on the compounding method. If thebank paid interest only once a year, then the return would be 20 percent:However, if the bank paid interest twice a year, you would need to solve the following equation to calculate thereturn:The annual percentage return when interest is paid twice a year is therefore 2 * 9.5445 percent = 19.089 percent.In general, if there are n compounding periods per year, you have to solve −1 and thenmultiply the result appropriately. If n is very large, this solution converges to r = ln = 18.2322 percent:
1.8.4 Why Use Continuous Compounding?All of this may seem somewhat esoteric. However, continuous compounding and discounting are often used infinancial calculations. In this book, continuous compounding is used to calculate portfolio returns (Chapters 7–12)and in practically all of the options calculations (Chapters 13–19).Theres another reason to use continuous compounding—its ease of calculation. Suppose, for example, that your$1,000 grew to $1,500 in one year and nine months. Whats the annualized rate of return? The easiest —and mostconsistent—way to answer this question is to calculate the continuously compounded annual return. Since one yearand nine months equals 1.75 years, this return isExercises 1. You are offered an asset costing $600 that has cash flows of $100 at the end of each of the next 10 years. a. If the appropriate discount rate for the asset is 8 percent, should you purchase it? b. What is the IRR of the asset? 2. You just took a $10,000, five-year loan. Payments at the end of each year are flat (equal in every year) at an interest rate of 15 percent. Calculate the appropriate loan table, showing the breakdown in each year between principal and interest. 3. You are offered an investment with the following conditions: n The cost of the investment is 1,000. n The investment pays out a sum X at the end of the first year; this payout grows at the rate of 10 percent per year for 11 years. If your discount rate is 15 percent, calculate the smallest X that would entice you to purchase the asset. For example, as you can see in the following display, X = $100 is too small—the NPV is negative. 4. The following cash-flow pattern has two IRRs. Use Excel to draw a graph of the NPV of these cash flows as a function of the discount rate. Then use the IRR function to identify the two IRRs. Would you invest in this project if the opportunity cost were 20 percent?
5. In this exercise we solve iteratively for the internal rate of return. Consider an investment that costs 800 and has cash flows of 300, 200, 150, 122, 133 in years 1–5 (see cells A8:B13 in the following spreadsheet). Setting up the loan table shows that 10 percent is greater than the IRR (because the return of principal at the end of year 5 is less than the principal at the beginning of the year). Setting the IRR? cell equal to 3 percent shows that 3 percent is less than the IRR, since the return of principal at the end of year 5 is greater than the principal at the beginning of year 5. By changing the IRR? cell, find the internal rate of return of the investment.6. An alternative definition of the IRR is the rate that makes the principal at the beginning of year 6 equal to zero.  This is shown in the preceding printout, in which cell E14 gives the principal at the beginning of year 6. Using the Goal Seek function of Excel, find this rate (we illustrate how the screen should look). (Of course, you should check your calculations by using the Excel IRR function.)
7. Calculate the flat annual payment required to pay off a five-year loan of $100,000 bearing an interest rate of 13 percent. 8. You have just taken a car loan of $15,000. The loan is for 48 months at an annual interest rate of 15 percent (which the bank translates to a monthly rate of 15 percent/12 = 1.25 percent). The 48 payments (to be made at the end of each of the next 48 months) are all equal. a. Calculate the monthly payment on the loan. b. In a loan table, calculate, for each month, the principal remaining on the loan at the beginning of the month and the split of that months payment between interest and repayment of principal. c. Show that the principal at the beginning of each month is the present value of the remaining loan payments at the loan interest rate (use the PV function). 9. You are considering buying a car from a local auto dealer. The dealer offers you one of two payment options: n You can pay $30,000 cash. n The "deferred payment plan": You can pay the dealer $5,000 cash today and a payment of $1,050 at the end of each of the next 30 months. As an alternative to the dealer financing, you have approached a local bank, which is willing to give you a car loan of $25,000 at the rate of 1.25 percent per month. a. Assuming that 1.25 percent is the opportunity cost, calculate the present value of all the payments on the dealers deferred payment plan. b. What is the effective interest rate being charged by the dealer? Do this calculation by preparing a spreadsheet like this (only part of the spreadsheet is shown—you have to do this calculation for all 30 months): Now calculate the IRR of the numbers in column F; this is the monthly effective interest rate on the deferred payment plan.10. You are considering a savings plan that calls for a deposit of $15,000 at the end of each of the next five years. If the plan offers an interest rate of 10 percent, how much will you accumulate at the end of year 5? Do this calculation by completing the following spreadsheet. This spreadsheet does the calculation twice— once using the FV function and once using a simple table that shows the accumulation at the beginning of each year.11. Redo the calculation of exercise 10, this time assuming that you make five deposits at the beginning of this year and the following four years. How much will you accumulate by the end of year 5?12. A mutual fund has been advertising that, had you deposited $250 per month in the fund for the last 10 years, you would now have accumulated $85,000. Assuming that these deposits were made at the beginning of each month for a period of 240 months, calculate the effective annual return fund investors got. Hint Set up the following spreadsheet, and then use Goal Seek
for the infinite sum on the first line of the formula to have a finite solution, the growth rates of the dividends must beless than the discount rate. We can use this formula in our spreadsheet:So—if the dividend to be paid one year from now is anticipated to be $3, and if this dividend is expected to grow by10 percent per year, and if the correct discount rate is 15 percent, then the value of the share should be $60. We canfix the technical problem by redefining the formula, as in the following spreadsheet:2.2.1 "Supernormal Growth" and the Gordon ModelNotice that if the condition |g| < rE is violated, the formula P0 = D1/ (rE − g) gives a negative answer. However, thisdoes not mean that the value of the share is negative; rather it means that the basic condition has been violated. Infinance examples, violations of |g| < rE usually occur for very fast-growing firms, in which—at least for short periodsof time—we anticipate very high growth rates, so that g>rE. In this case the original dividend discount formula showsthat P0 will have an infinite value. Since this result is clearly unreasonable (remember that we are valuing a security),it probably means either (1) that the long-term growth rate is less than the discount rate rE, or (2) that the discountrate rE is too low.The following spreadsheet illustrates an initial, very high, growth rate that ultimately slows to a lower rate. Weconsider a firm whose current dividend is $8 per share. The firms dividend is expected to grow at 35 percent for thenext five years, after which the growth rate will slow down to 8 percent per year. The cost of equity, the discount ratefor all of the dividends, is 18 percent:To calculate the value of the firms share, we first discount the dividends for years 1–5. Cell E4 shows that these fivefuture dividends are worth $40. Now look at years 6–∞. Denote the long-term growth rate by g2 (in our example thisis 8 percent). At time 0, the discounted dividend stream from years 6–∞ looks like:This last expression is basically the Gordon model discounted over five years.
n A second alternative to finding Fords cost of capital is to predict its future dividends by doing a full-blown financial model for the company. Such models—illustrated in the succeeding two chapters—are often used by analysts. Though they are complicated and time-consuming to build, they take into account all of the firms productive and financial activities. Potentially they are, therefore, a more accurate predictor of the dividend.2.9.2 Problems with the CAPMIn the following spreadsheet fragment you will find the return of the S&P 500 and Big City Bagels (notice that thespreadsheet fragment skips from row 8 to row 35—some of the rows of the data have been hidden). Immediatelyafter the spreadsheet is a graph that shows the calculation of Big Citys β, which is computed to be −0.6408.Big City Bagels stock is clearly risky—the annualized standard deviation of its returns is 152 percent as compared toabout 17 percent for the S&P 500 over the same period. However, the β of Big City Bagels is −0.6408, indicating thatBig City has—in a portfolio context—negative risk. Were this conclusion true, it would mean that adding Big City to aportfolio would lower the portfolio variance enough to justify a below-risk-free return for Big City. While this statementmight be true for some stocks, it is hard to believe that—in the long run—the β of Big City is indeed negative.The R2 of the regression between Big Citys returns and the S&P 500 is extraordinarily low, 0.0052, meaning that theS&P 500 simply doesnt explain any of the variation in Big City returns. For statistics mavens: The situation isactually worse—the standard error of the slope estimate is 1.57, which is 2.5 times larger than the slope itself
Use the Gordon model to calculate Chryslers cost of equity in 1996.5. On the spreadsheet associated with this chapter you will find the following monthly data for IBMs stock price and the S&P 500 index during 1998: a. Use these data to calculate IBMs β. b. Suppose that at the end of 1997, the risk-free rate was 5.50 percent. Assuming that the market risk premium, E (rm)−rf = 8 percent and that the corporate tax rate TC = 40 percent, calculate IBMs cost of equity using both the classic CAPM security market line and Benninga-Sarigs tax-adjusted security market line. c. At the end of 1997, IBM had 969,015,351 shares outstanding and had $39.9 billion of debt. Assuming that IBMs cost of debt is 6.10 percent, use your calculations for the cost of equity in part b to arrive at two estimates of IBMs weighted average cost of capital.
6. A firm has a current stock price of $50 and has just paid a dividend of $5 per share. a. Assuming that investors in the firm anticipate a dividend growth rate of 10 percent, what is the firms cost of equity? b. Draw a graph showing the relation between the cost of equity and the anticipated dividend growth rate. 7. Exercise on supernormal growth: ABC Corporation has just paid a dividend of $3 per share. You—an experienced analyst—feel quite sure that the growth rate of the companys dividends over the next 10 years will be 15 percent per year. After 10 years you think that the companys dividend growth rate will slow to the industry average, which is about 5 percent per year. If the cost of equity for ABC is 12 percent, what is the value today of one share of the company? 8. Consider a company that has βequity = 1.5 and βdebt = 0.4. Suppose that the risk-free rate of interest is 6 percent, the expected return on the market E(rm) is 15 percent and the corporate tax rate is 40 percent. If the company has 40 percent equity and 60 percent debt in its capital structure, calculate its weighted average cost of capital using both the classic CAPM and the Benninga-Sarig tax-adjusted CAPM. 9. You are considering buying the bonds of a very risky company. A bond with a $100 face value, a one-year maturity, and a coupon rate of 22% is selling for $95. You consider the probability that the company will actually survive to pay off the bond 80%. With 20% probability, you think that the company will default, in which case you think that you will be able to recover $40. What is the expected return on the bond? 10. It is January 1, 1997. Normal America, Inc. (NA) has paid a year-end dividend in each of the last 10 years, as shown by the following table.Calculate NAs β with respect to the SP500.Appendix 1: A Rule of Thumb for Calculating Debt BetasVanguard is a large manager of mutual funds. Among its funds is the Vanguard Index 500 fund, which tracks theS&P 500 portfolio. The company also has numerous bond funds. The following table shows the β of these bondfunds derived by calculating thefor each fund. When we do this calculation for a number of Vanguard funds, we get the following results.Regressing the β on the bond fund average maturity gives:The rules of thumb for debt betas in the chapter are based on this regression.
The regression results can be seen in the following graph:Appendix 2: Why Is β Such a Good Measure of Risk? Portfolio β versusIndividual Stock βAlthough β may not be a very good measure of the riskiness of an individual stock, the average β is a very goodmeasure of the riskiness of a diversified portfolio. This point is illustrated in this appendix. However, before we fireinto the illustration, we want to stress the meaning of the first sentence:If portfolio β is a good measure of portfolio risk, then—for holders of diversified portfolios (and these include mostinvestors)—the individual-share β is a good measure of the risk of a share, when this share is ultimately held in adiversified portfolio.To illustrate, consider the following graph, which gives the βs of 23 shares. As you can see, the R2 for theindividual regressions are not high (the highest R2 is 35 percent and the lowest is close to zero). The average R2 forthe 23 stocks is 16.05 percent, and the average β is 0.944.When we combine the 23 shares into an equally weighted portfolio, the portfolio β is 0.944, which is equal to theaverage beta of the component securities.  However, the portfolios R2 = 61.44 percent, which is much larger thanaverage R2 of the component securities. For a large, well-diversified portfolio, the portfolio R2 approaches 1.The meaning of this number is that—when we invest in large diversified portfolios—almost all of the risk is due to the
individual assets βs.Appendix 3: Getting Data from the InternetAll of the data used in this chapter were retrieved from the Internet. This appendix provides a brief description of howthey were obtained. Keep in mind that, since the Internet is a very lively place, some of the technical details andaddresses may have changed by the time you read this book. 1. By going to the Yahoo business news page (http://dailynews.yahoo.com/headlines/business/) you can indicate in the Get Quotes window the ticker symbol of the company you want to look up. In the following picture, I have asked for information on abt, the symbol for Abbott Laboratories. 2. Clicking on the Get Quotes box gives you the companys latest stock quote. 3. Clicking on Profile (in the box labeled More Info) gives a page of basic financial information about the company, including its β. Note that the β for Abbott is not the same as the one calculated in the chapter. There are two reasons for this difference: First, the chapter uses return data that include dividends, whereas the calculation on Yahoo excludes dividends. Second, the time horizon is different—the Yahoo calculation uses five years of data, whereas the calculations in the chapter are based on 10 years. Note also that the page can direct you to much more information about Abbott. The companys Web site has all its financial statements in downloadable form. 4. To get downloadable price information on Abbott Labs in Yahoo, click on Chart in the Basic Info box. You will then see a page like the following:
5. Clicking on Table and choosing the appropriate time interval (here we chose monthly) gives you the following:Clicking on Download Spreadsheet Format gives the data (already adjusted for dividends and splits) in a csv filethat can be opened with Excel. Note that you can change the Start Date and the End Date for the data.Final Note Ticker symbols for two indexes commonly used: ^SPX (S&P 500), ^DJI (the Dow-Jones 30 Industrials).β is calculated against return data for the S&P 500 for monthly return data from July 1994 through June 1999.This equality will always hold: Suppose we take n securities whose βs are β1, β2,…βn. Now suppose we take aportfolio in which the weight of each security is x1, x2, …xn, where . Then the portfolio β will be equal to theweighted average of the individual security βs: βportfolio = .
addition, the firm anticipates the following financial-statement relations. Current assets: Assumed to be 15 percent of end-of-year sales Current liabilities: Assumed to be 8 percent of end-of-year sales Net fixed assets: 77 percent of end-of year sales Depreciation: 10 percent of the average value of assets on the books during the year Fixed assets at cost: Sum of net fixed assets plus accumulated depreciation Debt: The firm neither repays any existing debt nor borrows any more money over the five-year horizon of the pro formas. Cash and marketable This is the balance sheet plug (see explanation that follows). Average balances of securities: cash and marketable securities are assumed to earn 8 percent interest.3.2.1 The "Plug"Perhaps the most important financial policy variable in the financial statement modeling is the "plug": deciding whichbalance-sheet item will "close" the model. As an example, consider the balance sheet of our first pro forma model: Assets Liabilities and Equity Cash and marketable securities Current liabilities Current assets Debt Fixed assets Equity Fixed assets at cost - Accumulated depreciation Stock (paid in capital) Net fixed assets Accumulated retained earnings Total assets Total liabilities and equityIn this balance sheet we assume that that cash and marketable securities will be the plug. This assumption has twomeanings: 1. The mechanical meaning of the plug: Formally, we define By using this definition, we guarantee that assets and liabilities will always be equal. 2. The financial meaning of the plug: By defining the plug to be cash and marketable securities, we are also making a statement about how the firm finances itself. In our next model, for example, the firm sells no additional stock, does not pay back any of its existing debt, and does not raise any more debt. This definition means that all incremental financing (if needed) for the firm will come from the cash and marketable securities account; it also means that if the firm has additional cash, it will go into this account. 3.2.2 Projecting Next Years Balance Sheet and Income StatementWe have already given the financial statement for year zero. We now project the financial statement for year one:
The formulas are mostly obvious. (The dollar signs—indicating that when the formulas are copied, the cell referencesto the model parameters should not change—are very important! If you fail to put them in, the model will not copycorrectly when you project years 2 and beyond.) Model parameters are in bold-face in the following list:Income Statement Equations n Sales = Initial sales * (1 + Sales growth)year n Costs of goods sold = Sales * Costs of goods sold/Sales The assumption is that the only expenses related to sales are costs of goods sold. Most companies also book an expense item called selling, general, and administrative expenses (SG&A). The changes you would have to make to accommodate this item are obvious (see an exercise at the end of this chapter). n Interest payments on debt = Interest rate on debt * Average debt over the year This formula allows us to accommodate changes in the model for repayment of debt, as well as rollover of debt at different interest rates. Note that in the current version of the model, debt stays constant; but in other versions of the model to be discussed later debt will vary over time. n Interest earned on cash and marketable securities = Interest rate on cash * Average cash and marketable securities over the year n Depreciation = Depreciation rate * Average fixed assets at cost over the year This calculation assumes that all new fixed assets are purchased during the year. We also assume that there is no disposal of fixed assets. n Profit before taxes = Sales − Costs of goods sold − Interest payments on debt + Interest earned on cash and marketable securities − Depreciation n Taxes = Tax rate * Profit before taxes n Profit after taxes = Profit before taxes − Taxes n Dividends = Dividend payout ratio * Profit after taxes The firm is assumed to pay out a fixed percentage of its profits as dividends. An alternative would be to assume that the firm has a target for its dividends per share. n Retained earnings = Profit after taxes − DividendsBalance Sheet Equations
n Cash and marketable securities = Total liabilities − Current assets − Net fixed assets As explained earlier, this formula means that cash and marketable securities are the balance sheet plug. n Current assets = Current Assets/Sales * Sales n Net fixed assets = Net fixed assets/Sales * Sales n Accumulated depreciation = Previous years accumulated depreciation + Depreciation rate * Average fixed assets at cost over the year. n Fixed assets at cost = Net fixed assets + Accumulated depreciation Note that this model does not distinguish between plant property and equipment (PP&E) and other fixed assets such as land. n Current liabilities = Current liabilities/Sales * Sales n Debt is assumed to be unchanged. An alternative model, which we will explore later, assumes that debt is the balance-sheet plug. n Stock doesnt change (the company is assumed to issue no new stock). n Accumulated retained earnings = Previous years accumulated retained earnings + Current years additions to retained earnings Financial statement models in Excel always involve cells that are mutually dependent. As a result, the solution of the model depends on the ability of Excel to solve circular references. To make sure your spreadsheet recalculates, you have to go to the Tools|Options|Calculation box and click Iteration. If you open a spreadsheet that involves iteration, and if this box is not clicked, you will see the following Excel error message: Depending on where you are in Excel when you open the file with the circular references, you may get a slightly different version of this message. Whatever message you see, get out of it and go to Tools|Options|Calculation|Iteration.In this dialog box click the box labeled Iteration:3.2.3 Extending the Model to Years 2 and Beyond
flows from the firms operating, investing, and financing activities. Consolidated Statement of Cash Flows—Abbott Labs Year ended December 31 (dollars in thousands) 1998 1997 1996 Cash Flow from (Used in) Operating Activities: Net earnings $2,333,231 $2,094,462 $1,882,033 Adjustments to reconcile net earnings to net cash from operating 784,243 727,754 686,085 activities—Depreciation and amortization Exchange (gains) losses, net (14,176) 31,005 (3,419) Investing and financing (gains) losses, net 90,798 113,999 57,224 Trade receivables (143,470) (222,427) (163,621) Inventories (111,649) (98,964) (125,726) Prepaid expenses and other assets (239,533) (491,769) (303,766) Trade accounts payable and other liabilities 178,979 485,407 342,407 Income taxes payable (145,522) (10,700) 10,845 Net Cash from Operating Activities 2,732,901 2,628,767 2,382,062 Cash Flow from (Used in) Investing Activities: Acquisition of International Murex in 1998, Sanofis parenteral (249,177) (200,475) (830,559) products businesses in 1997, and MediSense in 1996, net of cash acquired Acquisitions of property, equipment and other businesses (990,619) (1,007,296) (949,028) Purchases of investment securities (278,002) (25,115) (312,535) Proceeds from sales of investment securities 78,898 43,424 117,783 Other 18,034 (8,209) 19,098 Net Cash Used in Investing Activities (1,420,866) (1,197,671) (1,955,241) Cash Flow from (Used in) Financing Activities: Proceeds from (repayments of) commercial paper, net 42,000 402,000 317,000 Proceeds from issuance of long-term debt 400,000 — 500,000 Other borrowing transactions, net (59,499) 16,085 18,037 Purchases of common shares (875,407) (1,054,512) (808,816) Proceeds from stock options exercised 150,881 137,482 109,638 Dividends paid (891,661) (809,554) (728,147) Net Cash Used in Financing Activities (1,233,686) (1,308,499) (592,288) Effect of exchange rate changes on cash and cash equivalents (143) (2,782) (5,521) Net Increase (Decrease) in Cash and Cash Equivalents 78,206 119,815 (170,988) Cash and Cash Equivalents, Beginning of Year 230,024 110,209 281,197 Cash and Cash Equivalents, End of Year $308,230 $230,024 $110,209 Supplemental Cash Flow Information: Income taxes paid $1,060,479 $922,242 $801,107 Interest paid 153,875 132,645 89,509In the next example, we treat the cash and marketable securities account as if it were solely a cash account; we thenderive the increase in this account through a consolidated statement of cash flows: