# Solutions Manual to accompany Corporate Finance

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Solutions_Manual_to_accompany_Corporate_Finance/Ross_Jaffe_-_Instructor_resources/Excel Templates for Students/chap02.xlsMain MenuFinancial Analysis Spreadsheet TemplatesMAIN MENU -- CHAPTER 2Problem 2-3Problem 2-8Corporate Finance by Ross, Westerfield, and Jaffe -- Seventh EditionCopyright 2005 Irwin/McGraw-Hill and KMT Software, Inc. (www.kmt.com)

Problem 2-3`Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 2-3 ObjectivePrepare an Income StatementStudent Name:Course Name:Student ID:Course Number:Prepare an income statement from the following data.Sales\$500,000Cost of goods sold200,000Administrative expenses100,000Interest expense50,000The firm's tax rate is 34 percent.SolutionProblem 2-3InstructionsUse the template below to prepare an income statementIncome StatementSales\$500,000Less: Cost of goods soldAdministrative expenses0Earnings before interest and taxes\$500,000Less: Interest expenseEarnings before taxes\$500,000Taxes Rate =0Net income\$500,000

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 2-3|>>

Problem 5-10Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 5-10 ObjectiveCalculating the value of bondsStudent Name:Course Name:Student ID:Course Number:HexCorp Inc. has two different bonds currently outstanding. Bond A has a face value of \$40,000and matures in 20 years. The bond makes no payments for the first six years and then pays \$2,000semiannually for the subsequent eight years, and finally pays \$2,500 semiannually for the last six years.Bond B also has a face value of \$40,000 and a maturity of 20 years. However, it makes no couponpayments over the life of the bond. If the stated annual interest is 12 percent, compounded semiannually,a. What is the current price of Bond A?b. What is the current price of Bond B?SolutionProblem 5-10InstructionsUse formulas and the Excel PV function to calculate the current vales for Bond A and B.Bond A face value\$40,000Term of bond A20yearsInterest\$2,000semiannually years 7 - 14\$2,500semiannually years 15 -20Bond B face value\$40,000Term of bond B20yearsInterestNoneRequired rate of return12%Compounded semiannuallyCurrent value of Bond AFORMULACurrent value of Bond BFORMULA

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 5-10| 1 Then Sheets(1).Select End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE PREVIOUS SHEET'' ========================================================================Sub PrevSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index If iHere 1 Then iPrev = iHere - 1 While Worksheets(iPrev).Visible True And iPrev > 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 1 Then Sheets(1).Select End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE PREVIOUS SHEET'' ========================================================================Sub PrevSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index If iHere 1 Then iPrev = iHere - 1 While Worksheets(iPrev).Visible True And iPrev > 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext

Problem 11-3Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 11-3 ObjectiveDetermining systematic and unsystematic risk.Student Name:Course Name:Student ID:Course Number:Suppose a factor model is appropriate to describe the returns on a stock. Information about those factors ispresented in the following chart.Beta ofExpectedActualFactorFactorValue (%)Value (%)Growth in GNP2.043.5%4.8%Interest rates-1.9014.0%15.2%Stock return10.0%a. What is the systematic risk of the stock return?b. The firm announced that its market share had unexpectedly increased from 23 percent to 27 percent. Investorsknow from their past experience that the stock returns will increase by .36 percent per an increase of 1percentin its market share. What is the unsystematic risk of the stock?c. What is the total return of this stock?SolutionProblem 11-3InstructionsEnter formulas to solve the requirements of this problem. When possible, use cell references to the data in thestock table above.a. What is the systematic risk of the stock return?Systematic risk0.372%b. The firm announced that its market share had unexpectedly increased from 23 percent to 27 percent.Investors know from their past experience that the stock returns will increase by .36 percent per anincrease of 1percent in its market share. What is the unsystematic risk of the stock?Unsystematic riskFORMULAc. What is the total return of this stock?Total returnFORMULA

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 11-3| 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext >

Problem 17-8Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 17-8 ObjectiveEvaluate a potential capital investmentStudent Name:Course Name:Student ID:Course Number:Neon Corporation's stock returns have a covariance with the market of 0.031. The standard deviation of themarket returns is 0.16, and the historical market premium is 8.5 percent. Neon's bonds yield 11% perannum. The market value of the bonds is \$24 million. Neon has 4 million shares of common stock outstanding,each worth \$15. Neon's CEO considers the firm's current debt-to-equity ratio optimal. The tax rate is34 percent, and the Treasury bill rate is 7% per annum. Neon is considering the purchase of additional equipmentthat would cost \$27.5 million. The expected unlevered cash flows (UCF) from the new equipment are\$9 million a year for five years. Unlevered cash flows are defined as the after-tax cash flows the equipmentwould generate under all-equity financing.) Purchasing the equipment will not change the risk level of the firm.a. Use the weighted average cost of capital approach to determine whether or not Neon should purchasethe equipment.b. Suppose Neon decides to fund the purchase of the equipment entirely with debt. By how much will theweighted average cost of capital used in part a change? Explain your answer.SolutionProblem 17-8InstructionsEnter formulas to calculate the requirements of this problem. Use the Excel PV function to calculate theNet Present Value (NPV).AssumptionsStock returns covariance with market0.031Standard deviation of market returns0.16Historical market premium8.50%Bond coupon rate13.00%Bond yield11.00%Market value of bonds\$24,000,000Number of shares of stock outstanding4,000,000Share price of stock\$15Tax rate34%Treasury bill rate7%a. Use the weighted average cost of capital approach to determine whether or not Neon should purchasethe equipment.Weighted Average Cost of CapitalCost of Equity:BetaFORMULARequired rate of return on stockFORMULACost of DebtFORMULAValue of firm\$84,000,000Weighted average cost calculationWeightsCostsWACCStockFORMULA0.00%0.000%BondsFORMULA0.00%0.000%0.0%0.000%Net present value calculationAssumptionsCost of equipment\$27,500,000Annual cash flow\$9,000,000Term5yearsNPVFORMULAYesNoShould Neon buy the equipment? (enter an X)b. Suppose Neon decides to fund the purchase of the equipment entirely with debt. By how much will theweighted average cost of capital used in part a change? Explain your answer.

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 17-8| 1 Then Sheets(1).Select End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE PREVIOUS SHEET'' ========================================================================Sub PrevSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index If iHere 1 Then iPrev = iHere - 1 While Worksheets(iPrev).Visible True And iPrev > 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext

Problem 18-6Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 18-6 ObjectiveValue the stock of a liquidating corporationStudent Name:Course Name:Student ID:Course Number:Andahl Corporation stock, of which you own 500 shares, will pay a \$2-per-share dividend one year fromtoday. Two years from now Andahl will close its doors; stockholders will receive liquidating dividends of\$17.5375 per share. The required rate of return on Andahl stock is 15 percent.a. What is the current price of Andahl stock?b. You prefer to receive equal amounts of money in each of the next two years. How will you accomplish this?SolutionProblem 18-6InstructionsEnter formulas to complete the requirements of this problem. In part b., use the Excel PV function tocalculate the two year annuity amount (hint: divide the present value of the stock by the present value ofa \$1 annuity).a. What is the current price of Andahl stock?AssumptionsRequired rate of return15%Year from today dividend\$2.00Liquidating dividend\$17.5375Years till liquidation2yearsPresent value of stockFORMULAb. You prefer to receive equal amounts of money in each of the next two years. How will you accomplish this?AssumptionsNumber of shares500Current value of stock holdings\$0Annuity (2 years)FORMULAHow will you accomplish this? Complete the schedule below to show how you can generate the annuityfrom the stock.AssumptionEnd of year 1 price\$15.25\$ by SellingTotal Cash# of shares# of sharesYearDividendSharesFlowto sellRemaining1\$1,000(\$1,000.00)\$0.00FORMULA5002FORMULA\$0.000

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 18-6|>

Problem 20-7Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 20-7 ObjectiveEvaluating the effect of a coupon rate on bond pricingStudent Name:Course Name:Student ID:Course Number:Bowdeen Manufacturing intends to issue callable, perpetual bonds. The bonds are callable at \$1,250. One-year interest rates are 12 percent. There is a 60-percent probability that long-term interest rates one yearfrom today will be 15 percent. With a 40-percent probability, long-term interest rates will be 8 percent. Tosimplify the firm's accounting, Bowdeen would like to issue the bonds at par (\$1,000). What must thecoupon be for Bowdeen to be able to sell them at par?SolutionProblem 20-7InstructionsRead the information below before attempting the solution to this problem. This problem can be solvedusing the Excel Goal Seek tool found under the Tools menu. Below, there are more specific instructions onhow you can utilize Goal Seek in this problem.If interest rates rise to 15%, the price of the Bowdeen bonds will fall, which means the bonds would not becalled. In that case, the bond would be worth C/.15 where C is the coupon payment. The total bond holdingwould be worth C + C/.15.If interest rates fall to 8%, it is highly likely that the price of the bonds will rise above the call price. If thishappens, Bowdeen will call the bonds. In this case, the bondholders will receive the call price, \$1,250, plusthe coupon payment, C.The selling price of the bonds is the present value of the expected payoffs to the bondholders. The expectedpayoff is based on the probabilities mentioned in the problem. The expected payoff is reflected by thisformula:.6(C+C/.15) +.4(C+\$1,250)Where C is equal to the coupon.Since the company wants the bond to sell at par value, you must solve for C (coupon payment) so that theexpected payoff is \$1,000. Enter you answer for the coupon payment amount in the cell below, if theexpected payoff is equal to \$1,000 -- then you have correctly calculated the coupon. You can alsouse the Excel Goal Seek facility which you can find under Tools. You can use Goal Seek to adjust theCoupon value until the Expected payoff cell reaches the target value of \$1,000 (par). To use Goal Seekfirst take your best guess at the Coupon amount that will result in an expected payoff of \$1,000. Thenchoose Tools Goal Seek and enter the Set cell (expected payoff cell), To value (1,000), and the By changingcell (Coupon).CouponExpected payoff\$0.00

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 20-7|>

Problem 21-3Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 21-3 ObjectiveEvaluating a buy versus lease of equipment.Student Name:Course Name:Student ID:Course Number:Super Sonics Entertainment is considering borrowing money at 11 percent and purchasing a facility thatcosts \$350,000. The machine will be depreciated over five years by the straight-line method and will beworthless in five years. Super Sonic can lease the machine with the year-end payments of \$94,200. Thecorporate tax rate is 35 percent. Should Super Sonics buy or lease?SolutionProblem 21-3InstructionsComplete the lease/buy analysis below and then comment on whether the firm should buy or lease themachine.AssumptionsInterest rate on borrowing11%Facility cost\$350,000Useful life5yearsAnnual lease payments\$94,200Corporate tax rate35%Incremental cash flows from leasing as opposed to buyingLease minus buyYear 0Year 1-5LeaseLease payment(\$94,200)Tax benefit of leaseFORMULABuyCost of machine(\$350,000)Lost depreciation tax benefitFORMULATotal(\$350,000)(\$94,200)Present value of lease(\$384,692)Present value of buy(\$350,000)Conclusion (Buy or Lease)

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 21-3|

Problem 23-1Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 23-1 ObjectiveUse the Black-Scholes model to value an executive stock option.Student Name:Course Name:Student ID:Course Number:William Hurt is the Chief Executive Officer of the First Pacific Trading Company (FPTC). His annual salary is\$1 million. The current value of FPTC stock is \$50 per share. Mr. Hurt has just been granted options on\$1 million shares of FPTC stock at the money by FPTC's board of directors. The risk-free rate is 6 percent.The options have a maturity of four years. The volatility of FPTC stock has been about 25 percent on anannual basis. Determine the value of Mr. Hurt's stock options.SolutionProblem 23-1InstructionsGiven the assumptions below and the information presented in the problem, use Table 23.3 to determine thecumulative probabilities of the standard normal distribution function, enter those below and then enter theformula to calculate the call price. Where possible, use cell references to assumption data from theproblem. Use the Excel EXP(1) function (constant e) in the call price formula.AssumptionsStock price\$50Risk free rate6%Time interval4Annual volatility0.25d10.740d2FORMULAN(d1)0.7704N(d2)0.3974Value of stock optionFORMULA

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 23-1| 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext

Problem 28-5Corporate FinanceRoss, Westerfield, and Jaffe -- Seventh EditionProblem 28-5 ObjectiveUse the Excel Goal Seek feature to set a price for a product that will be sold on credit.Student Name:Course Name:Student ID:Course Number:Theodore Bruin Corporation, a manufacturer of high-quality stuffed animals, does not extend credit to itscustomers. A study has shown that, by offering credit, the company can increase its sales from the current750 units to 1,000 units. The cost per unit, however, will increase from \$43 to \$45, reflecting the expense ofmanaging accounts receivable. The current price of a toy is \$48. The probability of a customer making apayment on a credit sale is 92 percent, and the appropriate discount rate is 2.7 percent.How much should Theodore Bruin increase the price to make offering credit an attractive strategy?SolutionProblem 28-5InstructionsUse the Excel Goal Seek feature to solve this problem. Goal Seek is found under Tools. In Goal Seek, youwant the "Set cell" to be the Gross profit with credit plan amount, the "To value" to be equal to the currentgross profit amount, and the "By changing cells" to be the price that needs to be charged.AssumptionsCurrent price of a toy\$48Probability of credit sale92%Discount rate2.70%Current sales in units750Potential sales in units1,000Current cost per unit\$43Potential cost per unit\$45Current gross profit\$3,750Price that needs to be chargedGross profit with credit plan\$0

&L&8Copyright 2005 Irwin/McGraw-Hill&CFAST Workbooks by Ross, Westerfield, and Jaffe&R&8Problem: 28-5| 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext 1 Then Sheets(1).Select End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE PREVIOUS SHEET'' ========================================================================Sub PrevSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index If iHere 1 Then iPrev = iHere - 1 While Worksheets(iPrev).Visible True And iPrev > 1 iPrev = iPrev - 1 Wend If iPrev > 0 Then Worksheets(iPrev).Select End If End If Cells(1, 1).Select ActiveWindow.ScrollRow = 1 ActiveWindow.ScrollColumn = 1 ThisWorkbook.Saved = bSaveStateEnd Sub

' ========================================================================'' GO TO THE NEXT SHEET'' ========================================================================Sub NextSheet() Dim bSaveState As Boolean Dim iHere, iEnd, iNext As Integer On Error Resume Next bSaveState = ThisWorkbook.Saved Windows(ActiveWorkbook.Name).Activate iHere = ActiveSheet.Index iEnd = Worksheets.Count If iHere iEnd Then iNext = iHere + 1 While Worksheets(iNext).Visible True And iNext < iEnd iNext = iNext + 1 Wend If iNext YTM, price > par value (premium bond)

When coupon rate < YTM, price < par value (discount bond)

3.A bond with longer maturity has higher relative (percentage) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical.

4. A lower coupon bond has a higher relative price change than a higher coupon bond when interest rate (YTM) changes. All other features are identical.

Present Value of Common Stocks: Dividend Growth Models [PowerPoint slide 516 though 5-27]

The valuation principles are the same for stocks and bonds. The current price of a share of stock is the present value of expected future cash flows: i.e. the expected dividend plus expected price at the end of the holding period. For a single holding period,

But what determines P1? An investor valuing the stock in the next period (t = 1) will apply the same principles:

Expected P1 = Therefore, PV0 = Divl / (l+r)+ Div2 / (l+r)2 + P2 / (l+r)2Since common stock has no expiration date, applying the same principles to P2, P3, and so on eventually results in:

PV0=

The value of common stock depends only on the timing, size, and riskiness of expected future dividends. This is the essence of the dividend valuation model. How do we estimate future dividends?

Obviously, it is not feasible to estimate dividends for each period individually. To make the above valuation formula (equation 5.4 from the text) operational, we introduce three models for estimating future dividends: zero growth, constant dividend growth, and differential growth.

The three dividend growth models apply to firms in different stages of its life cycle. Young companies usually have a high growth rate. After a while they slow down and grow at a more normal rate. Finally, they may shrink or go out of business entirely.

We always emphasize a few caveats with the dividend growth models. First, growth is hard to forecast and the growth rate has a large impact on estimated firm value. Second, it is dividends, not earnings, that should be used. Using earnings would ignore the cost of reinvesting earnings in the firm and would over-estimate the firms value by double-counting cash flows retained in the firm to generate future dividends.

Case 1: Zero Growth [PowerPoint slide 5-15]

Assume that dividends will remain at the same level forever, i.e. D1 = D2 ==Dt. Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:

Example of a zero growth stock:

Suppose a firm's earnings and dividends are expected to remain constant at \$1 per share forever. The discount rate appropriate for the risk of the dividends is 10%. The value of the firm is then

Price = (\$1)/(.1) = \$10 per share.

The zero growth model fits many mature companies surprisingly well if cash flows and the discount rate are estimated in real terms. It fits exactly in real terms when nominal cash flows are expected to increase at the rate of inflation.

Case 2: Constant Dividend Growth [PowerPoint slide 5-16]

Assume that dividends will grow at a constant rate, g, forever, i.e.,

D1 = D0 (1 + g)

D2 = D1 (1 + g), etc., and

Dt = D0 (1 + g)tSince future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:

Example of a constant growth stock:

Suppose a firm just paid a dividend of \$10 per share. Future dividends are expected to increase at a 5% annual rate. The required return is 25% per year. The value of the firm is estimated as:

Divl = Div0 (1 + g) = (\$10)(1.05) = \$10.50

Price = Divl / (r g) = (\$10.50) / (.25 .05) = \$52.50.

Case 3: Differential Growth [PowerPoint slide 5-19 though 5-27]

Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter. This general type of model is especially useful for valuing firms in the growth stage of their life cycle.

To value a Differential Growth Stock, we need to:

1. Estimate future dividends in the foreseeable future.

2. Estimate the future stock price when the stock becomes a Constant Growth stock (case 2).

3. Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.

Example of a differential growth stock:

Problem 5.13

A common stock pays a current dividend of \$2. The dividend is expected to grow at an 8% annual rate for the next three years; then it will grow at 4% in perpetuity. The appropriate discount rate is 12%. What is the price of this stock today?

r = 12% (required return)

g1 = g2 = g3 = 8%

D0 = \$2

D1 = \$2 1.08 = \$2.16, D2 = \$2.33, D3 = \$2.52

g4 = gn = 4%

Constant growth rate applies to D4 > use Case 2 (constant growth) to compute P3D4 = \$2.52 1.04 = \$2.62

P3 = \$2.62 / (.12 .04) = \$32.75

Expected future cash flows of this stock:

0

1

2

3

|

|

|

|(r = 12%)

|

|

|

|

D1

D2

D3 + P3

2.16

2.33

2.52 + 32.75

P0 = 2.16/1.12 + 2.33/1.122 + 35.27/1.123 = \$28.89

Growth Opportunities and the Dividend Growth Model

Earnings (EPS) or Dividend (D)?Suppose a firm has no positive NPV opportunities and hence does not need retained earnings. The firm generates a constant stream of earnings per share which are paid out as dividends, so that g = 0 and EPS = D. The value of this zero growth firm is:

PV0 = EPS / r = D / r

Suppose our nogrowth firm takes its earnings in period 1 and, instead of paying a dividend, invests in a new project at time 1. The discount rate on this new project is r, the same as on the old assets. (Since we haven't discussed risk as yet, we'll assume certainty. It could just as well represent a riskadjusted discount rate.) The project will produce a change in EPS in period 2 and thereafter due to the new project, but otherwise the firm is the same. All earnings are still paid out as dividends, except in period 1.

PV0 =

= [PV of firm without new project] [Cost of new project] + [PV of new project]

= PV of firm without growth + NPV of growth opportunities

= EPS / r + NPVGO

Decomposing firm value into these two components help relate the valuation principles to the goal of the firm. Accepting projects with positive NPV will increase firm value and is consistent with the goal of the firm and the goal of stockholders. Negative NPV projects should be rejected because they reduce firm value.

The P/E ratio is a commonly cited statistic in the financial press. We point out to students that the P/E ratios reported in the financial press are computed using historic EPS and current stock price, rather than the no-growth EPS and the intrinsic value in the text. The economic intuition that applies to the P/E ratio in the text will apply to historic P/E ratio if 1) the historic EPS = the no-growth EPS and 2) current price = intrinsic value.

The NPVGO valuation model can be restated as:

PV0/EPS= 1/r + NPVGO/EPS

Therefore, P/E ratio is positively related to the net present value of growth opportunities (NPVGO) and negatively related to the discount rate r. High growth firms typically have high P/E ratios. According to this relationship, a risky firm with no growth opportunities and a high required return will have a low P/E. In fact, many high growth firms are riskier than average. For these firms, the P/E ratio will depend upon which effect, NPVGO or r, dominates.

CHAPTER 6 Some Alternative Investment Rules

Chapter Outline [PowerPoint slide 6-1]

6.1 Why Use Net Present Value?

6.2 The Payback Period Rule

6.3 The Discounted Payback Period Rule

6.4 The Average Accounting Return

6.5 The Internal Rate of Return

6.6 Problems with the IRR Approach

6.7 The Profitability Index

6.8 The Practice of Capital Budgeting

6.9 Summary and Conclusions

This chapter covers five methods for evaluating capital budgeting problems. For each method we discuss its advantages and disadvantages (compared to the NPV rule) and the decision rule for mutually exclusive and independent projects. Since the calculation for all the methods are rather straight forward (except for IRR which we dont require students to compute), we use one example at the end that utilizes the four methods.

Mutually Exclusive versus Independent Projects [PowerPoint slide 6-18]

Mutually Exclusive Projects: only ONE of several potential projects can be chosen, e.g. acquiring an accounting system.

RANK all alternatives and select the best one.

Independent Projects: accepting or rejecting one project does not affect the decision of the other projects.

Must exceed a MINIMUM acceptance criteria.

Although a simple concept, we stress the difference between mutually exclusive and independent capital budgeting decisions. In fact, all capital budgeting decisions can be viewed as mutually exclusive because investors can always invest in financial assets instead of real assets. Therefore, even for independent projects, they should be evaluated against what is available in the financial markets.

The Net Present Value (NPV) Rule [PowerPoint slide 6-2,3,4]

Net Present Value (NPV) = Total PV of future CFs + Initial Investment

Estimating NPV:

1. Estimate future cash flows: how much (?) and when?

2. Estimate discount rate

3. Estimate initial costs

Minimum Acceptance Criteria: Accept if NPV > 0

Ranking Criteria: Choose the highest NPVGood Attributes of the NPV Rule [PowerPoint slide 6-4]

1. Uses cash flows

2. Uses ALL cash flows of the project

3. Discounts ALL cash flows properly

Reinvestment assumption: the NPV rule assumes that all cash flows can be reinvested at the discount rate.

The Payback Period Rule [PowerPoint slide 65, 6, 7]

How long does it take the project to pay back its initial investment?

Payback Period = # of years to recover initial costs

Minimum Acceptance Criteria: set by management

Ranking Criteria: set by management

Ignores the time value of money

Ignores CF after payback period

Biased against long-term projects

Payback period may not exist or multiple payback periods

Requires an arbitrary acceptance criteria

A project accepted based on the payback criteria may not have a positive NPV

Easy to understand

Biased toward liquidity

The discounted payback period rule is identical to the payback rule except that it uses discounted cash flows. However, it does not fix the other problems with the payback period rule. Since the discounted payback period rule requires calculating the present values of cash flows, it no longer has the advantage of being easy to compute.

The Average Accounting Return (AAR) Rule [PowerPoint slide 68]

AAR= Average NI / Average Book Value of Investment

Minimum Acceptance Criteria: set by management

Ranking Criteria: set by management

Ignores the time value of money

Uses an arbitrary benchmark cutoff rate

Based on book values, not cash flows and market values

The accounting information is usually available

Easy to calculate

The Internal Rate of Return (IRR) Rule [PowerPoint slide 6-9 though 6-17]

IRR: the discount that sets the NPV to zero

When NPV = 0, initial cost = PV of future cash flows. Hence, the yield to maturity of a bond is simply the internal rate of return on the bond. Just like YTM, the only way to compute the IRR is by trial and error (often with the help of a financial calculator or computer). Minimum Acceptance Criteria: Accept if the IRR > required return

Ranking Criteria: Select alternative with the highest IRR

Reinvestment assumption: the IRR calculation assumes that all future cash flows are reinvested at the IRR

Does not distinguish between investing and financing

IRR may not exist or there may be multiple IRR

Problems with mutually exclusive investments

Easy to understand and communicate

The disadvantages of the IRR rule warrant more discussions. Problem # 1: Investing or Financing?

Cash flowstreams which are opposite in sign have the same IRR but NPV is opposite in sign. For simple cash flow streams, this is seldom a problem. Whether a project is an investment or a financing may not be apparent for more complex cash flow streams.

Problem #2: Multiple IRR and non-existing IRR

Unfortunately, there are as many solutions to an IRR problem as there are changes in sign in the cash flow stream. Consider the cash flow stream [ \$100,+\$230, \$132] for periods 0, 1, and 2.

NPV = \$100 + \$230 [1/(1 + r)] \$132 [1/(1 + r)]20 = \$100 + \$230 [1/(1 + IRR)] \$132 [1/(1 + IRR)]2This is a simple quadratic equation. It may be easier for students to recognize the problem by replacing [1/(1 + r)] with x.

0 = a + bx + cx20 = 100 + 230x 132x2The roots of a quadratic equation are: x = [ b +/ (b2 4ac)](1/2) / 2a

Hence x = 1.1 and 1.2 and IRR = 10% and 20%.

In fact, the IRR solution can contain imaginary (i = 1(1/2)) roots or no roots at all. For example, the cash flow stream [+\$1,+\$2,+\$3] has no internal rate of return.

In practice, these problems with IRR may not be detrimental because many project cash flows change sign only once (negative cash flow for initial investment and positive cash flows thereafter).

Problem #3: Problems with Mutually Exclusive Investments

1. The SCALE Problem

2. The TIMING Problem

The example at the end of the lecture notes demonstrates the conflicting decisions from the IRR rule versus the NPV rule when selecting mutually exclusive projects. The NPV rule always gives the correct answer.

Profitability Index [PowerPoint slide 619]

This index gives a measure of a project's bang for the buck; that is, the amount of NPV generated for each investment dollar. It is especially useful when the firm or a division faces a budget constraint.

PI = Total Present Value of future CFs / Initial Investment

Minimum Acceptance Criteria: Accept if PI > 1

Ranking Criteria: Select alternative with highest PI

Problems with mutually exclusive investments

May be useful when available investment funds are limited

Easy to understand and communicate

Correct decision when evaluating independent projects

Example of Investment Rules [PowerPoint slide 6-21 though 6-25]

Compute the IRR, NPV, PI, and payback period for the following two projects. Assume the required return is 10%.

YearProject AProject B

0 \$200 \$150

1\$200\$50

2\$800\$100

3 \$800\$150

Computing NPV, IRR, PI

Project AProject B

CF0 \$200 \$150

PV0 of CF1 3\$241.92\$240.80

NPV =\$41.92\$90.80

IRR =0%, 100%36.19%

PI =1.20961.6053

Payback Period

Project A

Project B

YearCFCum. CFCFCum. CF

0 200 200 150 150

1200050 100

28008001000

3 8000150150

Payback period for project B = 2 years

Payback period for project A = 1 or 3 years?

Relationship between NPV and IRR]

Discount rateNPV for ANPV for B

10% 87.52234.77

0%0.00150.00

20%59.2647.92

40%59.48 8.60

60%42.19 43.07

80%20.85 65.64

100%0.00 81.25

120% 18.93 92.52

NPV Profiles [T6-14]

CHAPTER 7 Net Present Value and Capital Budgeting

Chapter Outline [PowerPoint slide 7-1]

7.1 Incremental Cash Flows

7.2 The Baldwin Company: An Example

7.3 The Boeing 777: A Real-World Example

7.4 Inflation and Capital Budgeting

7.5 Investments of Unequal Lives: The Equivalent Annual Cost Method

7.6 Summary and Conclusions

Appendix 7A: Depreciation

To compute NPV, we must estimate future cash flows (size and timing) and determine the appropriate discount rate. Chapter 7 provides a detailed discussion on estimating future cash flows and introduces two factors (inflation and risk) that should be considered when determining the discount rate.

Incremental Cash Flows [PowerPoint slide 7-2 through 7-6]

We first go over the concept of incremental cash flows. The easiest way to determine whether a cash flow item is incremental is by asking two questions:

1. What will this cash flow be with the project?

2. What will this cash flow be without the project?

If the answers differ then this cash flow item is incremental, otherwise, it is irrelevant.

The following three types of cash flows often cause confusions. We give examples of each type and ask students to determine whether they are incremental.

1. Sunk costs

2. Opportunity costs

3. Side effects

In many cases, we use information from financial statements to estimate cash flows. We briefly review the income statement and balance sheet and assign Chapter 2 as supplemental readings.

Cash flows from operations = EBIT + Depreciation Taxes

Sometimes the income statement is not available and we can estimate cash flows from operations as follows:

Cash flows from operations = After-tax revenues After-tax costs + Tax Shield from Depreciation

Note: Tax Shield from Depreciation = depreciation expense tax rate

Project net after-tax cash flows = Cash flows from operations Addition to fixed assets Addition to net working capital

We use problem 7.2 as a simple example. We go over the Baldwin example at the second half of the lecture.

Appropriate Discount Rate

We discuss the effects of inflation on future cash flows in The relationship between nominal interest rates, real interest rates, and inflation is:

(1 + nominal interest rate) = (1 + real interest rate)(1 + inflation rate).

Some students may recognize this as the Fisher relation.

The important lesson in choosing the appropriate discount rate is to be consistent. Nominal cash flows must be discounted at the nominal interest rate and real cash flows must be discounted at the real interest rate. The relationship between risk and return will be discussed in Part III but we briefly introduce this concept here. Higher risk cash flows should provide higher returns. Cash flows with the same risk should be discounted at the same interest rate.

Since inflation may affect each cash flow item differently (e.g. depreciation expense does not change with inflation) and the risk of each cash flow may differ, we may want to discount each cash flow item with its appropriate discount rate. The NPV of the project will be the sum of the present value of each cash flow item. In practice, it is often infeasible to take into account varying risk within the same project. With regards to accounting for the effects of inflation, many companies estimate cash flows in nominal terms. Industry trade groups, consulting companies, and the government often provide inflation estimates for specific industries and products.

See slides 7-17 through 7-20 as an example of capital budgeting under inflation.

Investments of Unequal Lives [PowerPoint slide 733 through 7-41]

Investments of unequal lives must be put on a comparable basis, either through matching cycles or through the Equivalent Annual Cost (EAC) method. The EAC method can also be applied to replacement decisions.

Replacement Chain (comparing investments of unequal lives)

Assumption: Both projects can and will be repeated

Matching cycle

Repeat projects until they begin and end at the same time

Compute NPV for the repeated projects

The Equivalent Annual Cost Method

NPV = EAC { 1/r 1 / [r (1 + r)T]}

NPV is the net present value of the project

T is the life of the new machine

r is the discount rate used in computing NPV

Replacement projects

Replace if EAC < Cost of keeping the machine one more year

The Baldwin Example [PowerPoint slide 78 to 716]

We work through the Baldwin example from the text as an integrated problem for capital budgeting. Working through this example takes up much of the class period in an introductory course. We use this example to conclude our discussions on capital budget and demonstrate how all the pieces of the puzzles fit together.

Initial Facts

Expected life of bowling ball machine

5 years

Cost of test marketing (sunk cost already spent!)

\$250,000

Current market value of proposed factory site (opportunity cost)

\$150,000

Cost of bowling ball machine

\$100,000

Estimated market value at the end of five years

\$30,000

Production (in units) by year during the 5-year life of the machine (5000, 8000, 12000, 1000, 6000)

Price during first year

\$20

(increases 2% per year)

Production costs during first year (per unit)

\$10

(increase 10% per year)

Annual inflation rate

5%

Additional working capital requirements are initially \$10,000, increase in the first several years, and fall to zero by the end of year 5.

Depreciation for the Baldwin Company

We present the depreciation schedules based on a \$100,000 depreciable basis and the IRS half-year convention for the following recovery period classes:

Recovery Period Class

Year

3 years

5 years

7 years

1

33,340

20,000

14,280

2

44,440

32,000

24,490

3

14,810

19,200

17,490

4

7,410

11,520

12,500

5

11,520

8,920

6

5,760

8,920

7 8,920

8 4,480

Total

100,000

100,000

100,000

Finally, heres a problem thats not in the book or in the notes as an example:Your firm is considering developing an apartment complex. The firm owns land that could be used for the project; it was bought last year for \$500,000. Real estate has gone up sharply in the last year: the land could be sold today for \$625,000. Real estate values are expected to increase at 5 percent per year going forward from today. The permitting fees and sewage infrastructure development required for development cost \$250,000 (the bill is due now and the project could not have proceeded without this work). This expense will be depreciated over the 30-year life of the project.The apartment buildings will be four-plex townhouses (i.e. buildings that have four apartments each). Each apartment will rent for \$1,000 per month. Initially, your firm will build 20 townhouses; if the project is successful, there is space on the land for as many as twenty more townhouses.

Each townhouse will cost \$120,000 and will be depreciated straight-line to zero over the 30-year life of the project. In addition, the project will require a net working capital investment of \$20,000 todaythis amount will be recovered at the end of the project. At the end of the project, the townhouses will be completely trashedthe plan is to demolish them and sell the land. Demolition costs will be \$150,000 before tax. Your variable cost is \$1,250 per year per townhouse. Fixed costs are \$130,000 per year. Your firms tax rate is 34% and the cost of capital is 10% 1 What is the after-tax incremental cash flow for year 0?

\$3,002,500

20\$120,000 + \$20,000 + \$625,000 .34125,000

2 What is the after-tax incremental cash flow for year 3?

\$558,500

Revenue (\$1,000204)12\$960,000Variable cost (\$1,25020)\$25,000Depreciation (20120,000)/30\$80,000Fixed Cost \$130,000Earnings Before Interest and Taxes\$725,000Taxes\$246,500After-Tax Net Income\$478,500The incremental after-tax cash flow\$558,500

3 WITH THE EXCEPTION OF THE PROJECTS OPERATING CASH FLOW FROM OPERATIONS (THAT IS IGNORING YOUR ANSWER TO QUESTION 2), what is the after-tax incremental cash flow for year 30? DO NOT INCLUDE YOUR ANSWER TO QUESTION 2 IN YOUR CALCULATION.

\$1,873,801.23= \$20,000 + 1,952,801.23 \$150,000(1 .34)In year 30 the land will be unencumbered.

With increases of 5 percent per year, the land will be worth \$2,701,213.98 After tax, our after-tax opportunity benefit is

\$2,701,213.98 (2,701,213.98 -500,000) .34 = \$1,952,801.23

4 What is the NPV?

\$2,369,817

OCF0 = \$3,002,500

OCF1-29 = \$558,500OCF30 = \$2,432,301

I/Y= 10 percent

NPV = \$2,369,817

This is an interesting problem to return to in the next chapter, as the break-even quantity is a challenge.CHAPTER 8 Strategy and Analysis in Using Net Present Value

Chapter Outline [PowerPoint slide 8-1]

8.1 Decision Trees

8.2 Sensitivity Analysis, Scenario Analysis, and Break-Even Analysis

8.3 Monte Carlo Simulation

8.4 Options

8.5 Summary and Conclusions

Decision Trees [PowerPoint slides 82 and 8-3]

Decision trees are a convenient way of representing sequential decisions over time. Such decisions often arise when the uncertainty surrounding an investment can be reduced by some initial informationgathering such as test marketing a new product or preparing a feasibility study. We find the following example especially useful in intermediate and casemethod courses.

Example of a Decision Tree (Note: this example is not discussed in the text or the PowerPoint)Wild Kitty Drilling Company owns some land in Alaska, but is not sure if there is oil to be found. The land has no other commercial value. An exploratory well can be drilled today at a cost of \$20 million. There is an 80% chance the exploratory well will come up dry. If the exploratory well is dry, there is still a chance that there is oil to be found. Whether the exploratory well is successful or not, production capacity can be installed in one year for \$100 million. The discount rate for both phases of the project is estimated to be 10% (see Warning below).

Once production capacity is installed, the same amount of aftertax cash flow is expected to be generated in perpetuity. The actual amount will not be known until after production capacity is installed. If the exploratory well is successful, annual cash flow is expected to be \$30M. If the exploratory well is unsuccessful, annual cash flow is expected to be \$7.5M. Should Wild Kitty invest in the exploratory well?

The firm faces sequential decisions. Just as in solving a maze, it is easiest if we begin at the end and then move toward the start. The decision tree for Wild Kitty is shown below. The NPVs at the right-hand side represent the possible outcomes at t=2 given different production decisions. The solid lines represent the decision regarding the production capacity at t=1. The dashed lines represent the possible outcomes of the exploratory well at t=1. The dotted lines represent the decision regarding the exploratory well at t=0.

Decision Tree for Wild Kitty

Decision at t=1: Given the possible outcomes in year 2, should production begin in one year?

This decision is made at t=1. The NPVs provided on the far right side of the decision tree depend on the outcome of the exploratory well and the cost of the production capacity. The \$20M of the exploratory well is a sunk cost at this point in time and does not affect the decision at t=1.

Consider the \$181.82M expected NPV for production capacity given a successful exploratory well. The expected annual cash flow for years 2 and beyond given a successful exploratory well is \$30M. The NPV (discounted back to t=0) of this expected perpetual cash flow less the cost of the production capacity is:

E[NPVt=0 | successful exploratory well] = [\$30M/(.1) \$100M] / (1 + .1) = \$181.82M.

On the other hand, the expected annual cash flow for years 2 and beyond given an unsuccessful exploratory well is \$7.5M. The NPV (discounted back to t=0) of this expected perpetual cash flow less the cost of the production capacity is:

E[NPVt=0 | unsuccessful exploratory well] = [\$7.5M/(.1) \$100M)/(1+.1] = \$22.73M

Since the expected NPV when the exploratory well is unsuccessful will be less than \$0, we will not invest in the production capacity. The yellow lines in [T8-3] represent the decisions made at t=1. We point out that not all the possible outcomes (and NPV) are included in the analysis. Once a decision is made, only the NPVs from the chosen paths should be included. In this example, the expected NPV is \$181.82M if the exploratory well is successful and \$0 if the well is unsuccessful. The path that results in NPV = \$22.73M will not be chosen and is therefore irrelevant. (See Warning 2 from text.)

Decision at t=0: Should Wild Kitty dig an exploratory well today?

This decision (at t=0) depends on the possible outcomes at t=2 given the decisions made at t=1. At this time (today), the firm has not invested in the exploratory well and its cost must be included in the analysis.

E(NPVt=0) = (.8)(\$0) + (.2)(\$181.82M) \$20M = \$16.36M

Warning: Different Discount Rates for Different Alternatives

This analysis assumes that the risks of the exploratory well and production capacity are the same. In sequential decisions, this is often not the case. The riskiness of this project may warrant a 10% discount rate as of time 0. Once the outcome of the exploratory well is known, the appropriate discount rate may be less. Sensitivity Analysis

Sensitivity analysis looks at the sensitivity of project NPV to varying outcomes of a single variable. Sensitivity analysis serves many purposes. We discuss two common applications. First, sensitivity analysis helps us focus in on variables that have the greatest impact on the NPV of a project. For example, if a 1% change in revenue forecast results in a 50% change in project NPV, more time and effort should be spent in obtaining an accurate revenue forecast. On the other hand, if a 1% change in shipping costs results in a 0.01% change in project NPV, shipping costs should be a relatively low priority for the analyst. Second, sensitivity analysis can indicate whether NPV analysis should be trusted.

Scenario Analysis

Sensitivity analysis looks at the sensitivity of NPV to variation in a single variable. In contrast, scenario analysis allows several variables to change at once in an attempt to identify outcomes characteristic of, say, a most likely (or 'best guess'), an optimistic, and a pessimistic scenario. When we teach case courses, we invariably spend much of the time on this type of 'whatif' analysis.

Breakeven Analysis

Accounting Profit Breakeven Point

Many students have learnt break-even analysis in an accounting course. The variable of interest in accounting break-even analysis is usually sales quantity. We start with the basic net income equation:

Net Income = [(Price Variable cost per unit) (Quantity) Fixed costs Depreciation](l Tc)

The break-even quantity that sets net income to zero is:

Quantity = (Fixed costs + Depreciation) / (Price Variable cost)

Note: the above formula is different from the one on page 199 in the text. (1 Tc) appears in both the numerator and denominator in the formula in the text. The above formula cancels out (1 Tc).

Some students may remember the accounting break-even quantity as the sales quantity that sets EBIT to zero. If interest expense is zero, the EBIT break-even quantity is the same as the accounting profit break-even quantity discussed in the text.

Present Value Breakeven [Point PowerPoint slides 8-12 through 8-30]

Applying financial principles to traditional accounting profit breakeven analysis requires some refinements. First, financial analysis of a new project focuses on incremental cash flows, not net income. Second, accounting break-even quantity is computed for only one year. Present value break-even applies to the life of the entire project. The present value break-even example in the text assumes that all the variables (sales price, variable costs, fixed costs, etc.) are constant throughout the project. This assumption is necessary for the following formula to work:

Present value break-even quantity = [EAC + (Fixed costs)(1 Tc) (Depreciation)( Tc)]/(Price Variable cost) (l Tc)

where EAC = (initial investment) / (annuity factor)

In practice, the constant cash flow assumption is often violated. Fortunately, the principles of present value break-even can easily be applied using computers. Most spreadsheet applications and financial modeling software can numerically solve for the sales quantity (or any other variable) that sets the NPV to zero.

The following is a break-even price and break-even quantity problem that is not in the notes or the book. It follows up on an example in chapter 7 of the instructors manual. It has a unique challenge in the break-even quantity.

Your firm is considering developing an apartment complex. The firm owns land that could be used for the project; it was bought last year for \$500,000. Real estate has gone up sharply in the last year: the land could be sold today for \$625,000. Real estate values are expected to increase at 5 percent per year going forward from today. The permitting fees and sewage infrastructure development required for development cost \$250,000 (the bill is due now and the project could not have proceeded without this work). This expense will be depreciated over the 30-year life of the project.

The apartment buildings will be four-plex townhouses (i.e. buildings that have four apartments each). Each apartment will rent for \$1,000 per month. Initially, your firm will build 20 townhouses; if the project is successful, there is space on the land for as many as twenty more townhouses.

Each townhouse will cost \$120,000 and will be depreciated straight-line to zero over the 30-year life of the project. In addition, the project will require a net working capital investment of \$20,000 todaythis amount will be recovered at the end of the project. At the end of the project, the townhouses will be completely trashedthe plan is to demolish them and sell the land. Demolition costs will be \$150,000 before tax. Your variable cost is \$1,250 per year per townhouse. Fixed costs are \$130,000 per year. Your firms tax rate is 34% and the cost of capital is 10%

1 What is the after-tax incremental cash flow for year 0?

\$3,002,500

20\$120,000 + \$20,000 + \$625,000 .34125,000

2 What is the after-tax incremental cash flow for year 3?

\$558,500

Revenue (\$1,000204)12\$960,000Variable cost (\$1,25020)\$25,000Depreciation (20120,000)/30\$80,000Fixed Cost \$130,000Earnings Before Interest and Taxes\$725,000Taxes\$246,500After-Tax Net Income\$478,500The incremental after-tax cash flow\$558,500

3 WITH THE EXCEPTION OF THE PROJECTS OPERATING CASH FLOW FROM OPERATIONS (THAT IS IGNORING YOUR ANSWER TO QUESTION 2), what is the after-tax incremental cash flow for year 30? DO NOT INCLUDE YOUR ANSWER TO QUESTION 2 IN YOUR CALCULATION.

\$1,873,801.23

= \$20,000 + 1,952,801.23 \$150,000(1 .34)In year 30 the land will be unencumbered.

With increases of 5 percent per year, the land will be worth \$2,701,213.98 After tax, our after-tax opportunity benefit is

\$2,701,213.98 (2,701,213.98 -500,000) .34 = \$1,952,801.23

4 What is the NPV?

\$2,369,817

OCF0 = \$3,002,500

OCF1-29 = \$558,500OCF30 = \$2,432,301

I/Y= 10 percent

NPV = \$2,369,817

5 What is the break-even price?

The PV of the costs is \$307,111064 = Q1 + PV Q3

The 30-year EAC is \$307,111.64

Revenue (\$1,000204)12\$579,108.55Variable cost (\$1,25020)\$25,000Depreciation (20120,000)/30\$80,000Fixed Cost \$130,000Earnings Before Interest and Taxes344,108.55Taxes\$xxxxxxAfter-Tax Net Income\$227,111.64

The incremental after-tax cash flow\$307,111.64

PBE = 579,108.55/20*4*12 = \$603.24

6 What is the break-even quantity?

If we really get into it, QBE = 8

The PV of the costs is = QBE \$120,000 + \$20,000 + \$625,000 .34125,000 \$1,873,801.23/1.1^30

= QBE \$120,000 + \$20,000 + \$625,000 .34125,000 \$107,38484

= QBE \$120,000 + 495,115.16

The 30-year EAC is \$12,729.51 QBE + 52,521.44

The other tricky thing here is to realize that depreciation depends on QBERevenue (\$1,000 QBE 4)12Variable cost (\$1,250 QBE )Depreciation (QBE 120,000)/30Fixed Cost \$130,000EBIT[\$12,729.51 QBE + \$52,521.44 (QBE 120,000)/30]/0.66Taxes\$?After-Tax Net Income\$12,729.51 QBE + \$52,521.44 (QBE 120,000)/30The incremental after-tax cash flow\$12,729.51 QBE + 52,521.44

[\$12,729.51 QBE + \$52,521.44 (QBE 120,000)/30]/0.66 + \$130,000 =

(\$1,000 QBE 4)12 (\$1,250 QBE ) (QBE 120,000)/30

QBE = 7.09 townhouses; since we cant build a partial, we need at least 8 to break even.

Investment Options [PowerPoint slides 8-46 through 8-57]

When evaluating investment strategies, it is important to recognize the options to expand or to abandon a project as more information is gathered and if business conditions change. Such options are valuable. The Decision Tree example (Wild Kitty) has an implied option to abandon if future conditions turn unfavorable.

Suppose when Wild Kitty purchased the land from the Alaskan government they had agreed to install the production capacity if they do any digging, including an exploratory well. With this contract, Wild Kitty must invest the \$100M regardless of the outcome of the exploratory well. In other words, Wild Kitty no longer has the option to abandon the project. How does this contract affect the analysis?

Decision at t=1: Wild Kitty has no choice. It must install the production capacity.

E[NPVt=0 | successful exploratory well] = [\$30M/(.1) \$100M] / (1 + .1) = \$181.82M

E[NPVt=0 | unsuccessful exploratory well] = [\$7.5M/(.1) \$100M]/(1+.1) = \$22.73M

Decision at t=0: Should Wild Kitty dig an exploratory well today?

E(NPVt=0) = (.8)(\$-22.73) + (.2)(\$181.82M) \$20M = \$1.82M

The expected NPV of the exploratory well was \$16.36M in the original example. The expected NPV of the investment is higher when there is an option to abandon the project. We will abandon the project if the expected NPV is negative. The value of this option is:

NPVt=0(Abandonment Option) = (.8)(\$22.73M) = 18.18M

which is equal to the difference between the NPV of the project with and without the abandonment option (\$16.36M \$1.82M). The Option to Wait: [PowerPoint slide 8-57] (Note: this topic is not discussed in the text)

The textbook discusses the expansion and abandonment options in capital budgeting. Ingersol and Ross [1992] show that the ability to delay a project means that nearly every investment project must also compete with itself postponed. Moreover, the effect of interest rate uncertainty on the optimal investment delay is sizable. Therefore, the traditional textbook NPV rule that suggests accepting all positiveNPV projects may be incorrect given the option to wait.

Financial Analysis Tools and Financial Modeling (Note: this topic is not discussed in the text)

In practice, we usually construct financial models to analyze investment strategies. A financial model is an abstract of the real world. The correct way to construct a model should list all the assumptions explicitly. The assumptions of the Wild Kitty example are:

1. Cost of the exploratory well

2. Cost of the production capacity

3. Expected future cash flows given the outcome of the exploratory well

4. Probability of success of the exploratory well

5. Production strategy (including the option to abandon)

6. Discount rate

These assumptions determine the inputs (information on costs, CF, probabilities, and discount rate) and the structure (production strategy) of the financial model. The financial analysis tools discussed in this chapter can be used to fine-tune the assumptions and strategies. The law of GIGO also applies to financial models. Common sense (sometimes called intuition) is a powerful tool against GIGO. When in doubt, we should always check the assumptions: Is an important variable omitted? Have the relationships between the variables changed? Are the strategies consistent (especially in a sequential decision problem)?

Supplemental Bibliography

Ingersol, J.E. and S.A. Ross, "Waiting to Invest: Investment and Uncertainty", Journalof Business 65, 1992, pp. 129.

Part III Risk

To motivate students, it is useful to link the concept of return and risk to the overall goal of the financial manager, i.e. to maximize stockholder wealth. Managers can achieve this goal by accepting positive NPV projects. To compute NPV, we need future cash flows and a discount rate. So far, the discount rate is treated as a given. To estimate the appropriate discount rate, we need to define the relationship between return and risk. First, we will look at the history of returns on different assets in Chapter 9. Intuitively, and confirmed by history, investors demand higher returns for riskier investments. Chapters 10 and 11 introduce two theories of capital asset pricing, which provide a rigorous relationship between return and risk. Chapter 12 applies one of these theories, the Capital Asset Pricing Model (CAPM), to estimate the appropriate discount rate for making investment decisions.

Chapter 9 Capital Market Theory: An Overview

Chapter Outline [PowerPoint slide 9-1]

9.1 Returns

9.2 Holding-Period Returns

9.3 Return Statistics

9.4 Average Stock Returns and Risk-Free Returns

9.5 Risk Statistics

9.6 Summary and Conclusions

Appendix 9A: The Historical Market Risk Premium: The Very Long Run

This chapter covers the historic return statistics in the U.S. capital market. In addition to defining return and risk, this chapter also provides quantifiable measures of return and risk for a single asset. Lastly, the risk-free rate and risk premiums are introduced.

Historic Return and Risk Statistics for A Single Asset [PowerPoint slides 9-2 though 9-5]

It is important to emphasize that when we look to the past to learn about return and risk, we must keep in mind that history may not repeat itself. Historic, Ex Post, return and risk statistics are useful places to start but may not be sufficient for estimating discount rates to be used for evaluating future projects.

Dollar Returns versus Percentage Returns

Dollar Returns = Dividend + Change in Market Value

Percentage Returns = Dollar Returns / Beginning Market Value

Dividend Yield = Dividend / Beginning Market Value

Capital Gain = Change in Market Value / Beginning Market Value

Percentage Returns = Dividend + Capital Gain

Holding-Period Returns

Holding-Period Return = (1 + R1) (1 + R2) .. (1 + RT) 1

Ending Market value = Beginning Market Value (1 + Holding-Period Return)

Geometric Mean Return versus Arithmetic Mean Return [PowerPoint slide 9-9]

Geometric Mean Return:

Arithmetic Mean Return:

The geometric mean return takes into account the effects of compounding whereas the arithmetic mean return is a simple average. The geometric mean return is always smaller than the arithmetic mean return except if the actual return is identical in each time period, then both mean returns are the same. Conceptually the geometric mean is more appealing because using the geometric mean return to compute future value will give the actual ending value. Using the arithmetic mean return to compute future value will overstate the actual ending value because the effects of compounding is not included in the calculation of the arithmetic mean return. However, most financial publications, including Stocks, Bills and Inflations: 20xx Yearbook, by Roger G. Ibbotson and Rex A. Sinquefield, provide only arithmetic mean returns. Students should be alerted to the overstated ending value due to using the arithmetic mean return. Most importantly, students should not compare a geometric mean return against an arithmetic mean return.

The historic (ex post) average return is the arithmetic mean return. Through out Chapter 9 and in most places in the text, the term average return refers to the arithmetic mean return.

Return Variance and Return Standard DeviationThe historic (ex post) return variance:

where is the arithmetic mean return

The historic (ex post) return standard deviation:

Return Statistics: An Example

Year

Return on the NYSE Composite Index

1990 8.84%

199128.45%

19924.82%

19938.00%

1994 2.84%

199531.42%

199618.27%

199731.23%

Remind students that this is a sample and the statistics we will calculate are the sample statistics.

Historic average (arithmetic) mean return:

R = ( 8.84% + 28.45% + 4.82% + 8.00% 2.84% + 31.42% + 18.27% +31.23%) / 8 = 13.81%

Historic return variance:

(2 = [( .0884 .1381)2 + (.2845 .1381)2 + (.0482 .1381)2 + (.0800 .1381)2 +

( .0284 .1381)2 + (.3142 .1381)2 + (.1827 .1381)2 + (.3123 .1381)2] / (8 1)

= .0250

Historic standard deviation:

( = .1582 = 15.82%

You may also want to demonstrate the difference between geometric and arithmetic mean returns.

Geometric mean return:

R = (.9116 1.2845 1.0482 1.08 .9716 1.3142 1.1827 1.3123).125 1

= .1283 = 12.83%

For the NYSE composite index, which has a standard deviation of 15.82%, the difference between the arithmetic and geometric mean returns is quite large.

Understanding the Return Statistics and the Normal Distribution

It is useful to digress here to discuss how to interpret the return statistics. The variance measures the squared deviation from the average. Therefore, using the variance (and standard deviation) as a measure of risk puts equal emphasis on returns that are above average and returns that are below average. There are other measures of risk, such as the downside risk (semi-variance) and the range.

Variance (or standard deviation) is especially useful as a measure of risk for a normal distribution. Identify the percentage of the normal distribution that lies (1 (68%), (2 (95%), and (3 (99.7%) from the mean and the application of the normal distribution. For example, if returns on the NYSE Composite Index are normally distributed with mean = 13.81% and ( = 15.82%, there is a 68% chance that the actual return in a given year will be between 2.01% and 29.63%. Another way to put this is that there is only a 32% chance that actual return in a given year will be lower than 2.01% or higher than 29.63%. In fact, there is only a 0.15% chance that the actual return will be below 33.6%.

Capital Market History [PowerPoint slides 9-11 through 9-19]

Historic Average Returns on Different Assets

Figure 9.4 from the text illustrates the return history of five indexes:

Inflation

Treasury bills

Long-term government bonds

Large company stocks

Small company stocks

Point out that the smoothness of the line indicates the volatility (risk) of the index. The graphs make it easy for students to observe the positive relationship between return and risk.

Table 9.2 from the text presents the historic average returns and historic standard deviations on seven indexes. The numbers in Table 9.2 confirm the observation from Figure 9.4 that higher average returns are associated with higher risks, measured as standard deviations in this case.

The Treasury bill is usually used as the risk-free return. Treasury bills are U.S. federal government debt that will mature in one year or less.

Risk Premiums = Returns on Risky Securities Risk-free Return

We will use the Stock Market Risk Premium a lot in the next few chapters and throughout the text:

Market Risk Premium = Return on the Stock Market Risk-free Return

From Table 9.2, Average Return on the Stock Market = 12.2%, Treasury Bills = 3.7%, therefore, Market Risk Premium = 12.2% 3.8% = 8.4%

This is a good place to introduce the concept of a capital asset pricing theory. A capital asset pricing theory provides a formal and rigorous definition of the relationship between return and risk. Common sense suggests that a rational investor will demand a higher return for a riskier investment and history of the capital market confirms this intuition. A capital asset pricing theory explicitly predicts how much more in return is needed to compensate for an incremental unit of risk. Chapters 10 and 11 will discuss two theories of capital asset pricing.

Appendix 9A: The Historical Market Risk Premium: The Very Long Run

The market risk premium is an important input variable for estimating the discount rate in capital budgeting. The two studies cited in the text highlight the controversy over the stability and magnitude of the market risk premium. The challenge for the financial manager is to estimate the future market risk premium for new projects.

CHAPTER 10 Return And Risk: The CapitalAssetPricing Model

Chapter Outline [PowerPoint slide 10-1]

10.1 Individual Securities

10.2 Expected Return, Variance, and Covariance

10.3 The Return and Risk for Portfolios

10.4 The Efficient Set for Two Assets

10.5 The Efficient Set for Many Securities

10.6 Diversification: An Example

10.7 Riskless Borrowing and Lending

10.8 Market Equilibrium

10.9 Relationship between Risk and Expected Return (CAPM)

10.10 Summary and Conclusions

This chapter covers the basic elements of portfolio theory and asset pricing. In Chapter 9, we study historic returns and statistics for an individual security. Chapter 10 uses a singleperiod 'stateoftheworld' framework to study statistics for portfolios, specifically the effects of covariance on diversification and portfolio risk. To motivate the importance of asset pricing theories, we point out that capital budgeting analysis requires a discount rate that applies to future projects. Therefore, knowing the historic returns and historic (ex post) statistics is not sufficient. Asset pricing theories allow us to estimate expected (ex ante) discount rate given a projects risk.

Expected (ex ante) Return, Variance, and Covariance [PowerPoint slide 10-2 through 10-11]

Expected Return of a Security = E(R) =

Variance of a Single Security = (2 =

Standard Deviation of a Single Security = = s

Covariance Between Two Securities = (AB =

Correlation Coefficient Between Two Securities = (AB = (AB / ((A (B)

Note: ps is the probability of state s occurring;

Rs is the return on the security if state s occurs.

We use the following example to illustrate calculation of return and risk statistics depending on various states of the world:

Outcomes

Probability (ps)RARB

Boom

0.25

20%5%

Normal

0.50

10%10%

Bust

0.25

0%15%

= (.25)(.20)+(.5)(.10)+(.25)(.00) = .10

= (.25)(.05)+(.5)(.10)+(.25)(.15) = .10

(A2 = (.25)(.20 .10)2 +(.5)(.10 .10)2+(.25)(.00 .10)2 = .00500

(B2 = (.25)(.05 .10)2 +(.5)(.10 .10)2+(.25)(.15 .10)2 = .00125

(A = (.00500)(1/2) = .07071 = 7.071%

(B = (.00125) (1/2) = .03536 = 3.536%

(AB = (.25)[(.20 .10)(.05 .10)]+(.5)[(.10 .10)(.10 .10)]+(.25)[(.00 .10)(.15 .10)] = 0.0025

(AB = 0.0025 / (0.07071)(0.03536) = 1

Security B is a very unusual investment because it has its highest returns in economic downturns and its lowest returns during boom times. We use this extreme example to demonstrate the concept of diversification.

Diversification

Suppose in the previous example we invest \$100 in security A and \$200 in B. Dollar returns under each possible outcome are:

CF on

CF on

Total

%Return on

OutcomeProbability\$100 in A\$200 in BCF

\$300 in A&B

Boom

.25

\$120

\$210

\$330

10%

Normal.50

\$110

\$220

\$330

10%

Bust

.35

\$100

\$230

\$330

10%

Expected Return10%

Variance

0.00

Standard Deviation0%

In economic upturns, we would do well in security A but poorly in B. In downturns, higher returns to B would offset lower returns to A. Combining securities A and B together, we can form a portfolio with a certain return of \$330. Students are often surprised to find that a portfolio of risky assets can have no risk (variability of return) whatsoever. Since security B is an extreme example, it is important to point out that diversification works in many cases. The correlation between two securities falls into one of the following cases: Positively correlated

0 < AB < 1

Perfectly Positively correlated AB = 1

Negatively correlated

1 < AB < 0

Perfectly Negatively correlated AB = 1

Uncorrelated

AB = 0

As long as two securities are not perfectly positively correlated (AB = 1), diversification will work. We examine the effects of diversification in more details in the next section.Return and Risk for Portfolios [PowerPoint slides 1012 through 10-18]

Expected Return of a Portfolio = E(Rp) =

Variance of a Portfolio = (p2 =

where Xi is the percentage of the portfolio in security i and N is the number of securities in the portfolio.

The above formulas apply to portfolios with many securities. The text only provides the formulas for return and risk for a 2-asset portfolio:

E(Rp) = XAE(R)A + XB E(R)B(p2 = XA2(A2 + XB2(B2 + 2 XA XB(ABApplying the portfolio formulas to the previous example:

XA = \$100/\$300 = 1/3

XB = 1 XA = 2/3

E(RP) =(1/3)(0.10)+ (2/3)(0.10) = 0.10

(p2 = XA2(A2 + XB2(B2 + 2 XA XB(AB= (1/3)2(.00500) + (2/3)2(.00125) + (2)(1/3)(2/3)( .0025) = 0

An Example of Portfolio Return and Risk]

It is helpful to demonstrate portfolio return and risk with an equally weighted portfolio of two assets that have a zero correlation. The example in uses two stocks: IBM and Homestake Mining (HM). A \$10,000 portfolio will invest in IBM and in HM. Assume that (IBM,HM = 0 and each stock has the following characteristics:

Stock InvestmentXiE.(Ri)(i2

IBM\$500050%0.090.01

HM\$500050%0.130.04

Total

\$10000100%

Expected return on the portfolio = E[Rp] = (0.5)(0.09) + (0.5)(0.13) = 11%

Variance of the portfolio = (p2 = XIBM2(IBM2 + XHM2(HM2 + 2 XIBM XHM(IBM,HM= (.5)2(.01) + (.5)2(.04) + 2(.5)(.5)(0) = 0.0125

The standard deviation is p = (0.0125)(1/2) = 0.1118.

Portfolios with Many Securities

As the number of securities increase in a portfolio, the covariance terms outnumber the variance terms. For example, a portfolio with 4 securities has 12 covariance terms and 4 variance terms. In general, there are N2 N covariance terms and N variance terms in a portfolio. Consequently, the variance of return on a portfolio with many securities is more dependent on the covariances between the individual securities than on the variances of the individual securities. This concept is central to the Modern Portfolio Theory.

Efficient Sets And Diversification [PowerPoint slides 1019 through 10-37]

The first example in this manual includes 2 stocks (A and B) that are perfectly negatively correlated (( = 1) and the second example includes 2 stocks (IBM and HM) that are uncorrelated (( = 0). Most common stocks have a correlation coefficient between 0.2 and 0.5. We now examine the effect of the correlation between two securities on a portfolio. The standard deviation-expected return graph is a useful tool. It illustrates two general rules on the return-risk characteristics of portfolios:1. The expected return on a portfolio is the weighted average of the expected returns on the individual securities.

2. As long as ( < 1, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities. When ( = 1 the standard deviation of a portfolio of two securities is the weighted average of the standard deviations of the individual securities.

Using just two securities, we can create many different portfolios by varying the weights on each security. The investment opportunity set includes all such portfolios. The efficient set includes portfolios within the investment opportunity set that represents the best return-risk combinations. To be included in the efficient set, a portfolio must have

1) the highest expected return at a given standard deviation relative to all other portfolios in the investment opportunity set

OR

2) the lowest standard deviation at a given expected return relative to all other portfolios in the investment opportunity set.

In [PowerPoint slide 10-27], the efficient set is the portion of the curve northwest of the minimum variance (MV) portfolio. An important assumption of the Modern Portfolio Theory and CAPM is that investors are mean-variance optimizers, i.e. an investor prefers more return given the risk (greed) and an investor prefers less risk given the same return (risk aversion). Such investors will choose portfolios in the efficient set over any other portfolio.

The Efficient Set for Many Securities PowerPoint slides 10-25, 26, and 27]

The intuitions behind the 2-asset case apply to the case with many securities. The investment opportunity set for many securities is an area, rather than a curve, and the efficient set (frontier) is the upper boundary curve northwest of the minimum variance portfolio. Since most common stocks have a correlation coefficient between 0.2 and 0.5., the variance of a well-diversified portfolio is less than the weighted average of the variance of individual stocks. In fact, the efficient set is composed mostly of well-diversified portfolios.

Systematic versus Unsystematic Risks

Since a rational investor will only hold a well-diversified portfolio, the standard deviation of his portfolio is less than the weighted average standard deviation of the underlying securities. For this investor, diversification eliminates part of the risk of an individual security. Therefore, the only relevant risk is the securitys contribution towards the standard deviation of the well-diversified portfolio. In other words:

Total risk of individual security = portfolio (systematic) risk + unsystematic (diversifiable) risk

The Capital Asset Pricing Model

TWO additional assumptions are necessary to complete the development of the CAPM.

Riskless Borrowing And Lending: when there is a riskfree asset (F), we can construct the capital market line in [PowerPoint slides 10-29 through 10-37].

The riskfree asset is assumed to have zero standard deviation ( = 0) and hence zero covariance with any risky asset. The return and risk of combining the riskfree asset (F) and any risky asset (A) are:

RP = XA RA + (1 XA) RFP = XA A + 0

This is the equation of a straight line. An investor can combine the riskfree asset with any risky asset in the opportunity set. However, the line that is tangent to the efficient set of risky assets provide investors with the highest return at any given standard deviation. Under the assumption that investors are greedy and risk averse, investors will prefer combinations along the tangent line to any other lines. The portfolio of risky assets (M) that lies on the tangent line is the optimal risky portfolio and the tangent line is the capital market line. An investor who dislikes risk will choose a point lower on this line and a more aggressive investor will choose a point higher on this line. This insight leads to the separation principle of investment decision-making. The principle states that the decisio

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