MIDTERM; OCTOBER 1985 PROBLEM 1 PV of annual expenses from age 65 to 85 = 170271.274 Annuity needed each year for 40 years to have FV of 170271= 384.712752 PROBLEM 2 Initial Investment = 50000 Annual Cashflow NPV= -50000 +51200 (PVA, 10, 15%)= 206960.954 Revenues 250000 Rent 48000 Salary Exp 120000 Deprec'n 5000 Taxable Income 77000 Tax 30800 Net Income 46200 +Depreciation 5000 ATCF 51200 B. BREAKEVEN Breakeven ATCF = 9962.60313 ! NPV = -50000 + X (PVA, 10, 15%) = 0 ! Solve for X Breakeven net income= 4962.60313 ! Y + 5000 = 9962 ! Solve for Y Breakeven taxable income= 8271.00521 ! Taxable income = 4962.60 / (1 - 0.4) Breakeven revenue= 181271.005 ! 8271 + 5000 + 120000 + 48000 ! All your costs are fixed. Breakeven # members= 362.54201 ! 181271 / 500 PROBLEM 3 PV of tax savings using straight line depreciation= 3032.62942 PV of tax savings using DDB = 3243.99103 (I switched to straightline in year 4 because it was higher) B. OPPORTUNITY COST PV of rental revenues = 100000* 0.6 *(PVA,15%,10) = 301126.118 (The depreciation is not a cost because you will get it anyway) PROBLEM 4 See Risk & Return problem set

 MIDTERM: SPRING 1988 AD M (AD-ÅD)SQ (M-M)SQ (AD-ÅD)(M-M) 10 5 25 4 -10 5 15 0 64 0 -5 8 100 1 -10 20 12 225 25 75 -5 -5 100 144 120 5 7 450 238 175 BETA = 175/238 = 0.73529412 ALPHA = 5 - 0.74 (7) = -0.145 b. EXPECTED RETURN = 6 +0.74 (8.5) = 12.29 c. AD did worse than expected. Compare alpha to Riskfree rate (1-Beta) = .06*(1-0.735) = =0.0159 d. The variance would be a good measure if undiversified = 112.5 Proportion that can be diversified = 1-ßsq*Var(m)/Var(j) 71.41% We know the Beta of AD before divestment = 0.735. Then the beta of divested division = 1.47 Let the beta after divestment be X Then, 0.735 = 0.8 X + 0.2 (1.47) Solving for X, 0.55125 2. Initial Investment = - 500000 - 50000 + 50000 = -500000 ATCF per year = 1000000 - 500000 - 200000 - .5 (300000 -100000) = 200000 NPV of this project = -500000 + 200000*(AF,10%,5 years)= 258157.354 NPV of investment banking job = 75000*.5*(Af,10%,5 years) = 142154.504 TAKE THE INVESTMENT! ALTERNATIVELY, YOU CAN SHOW THE INVESTMENT BANKING JOB AS AN OPPORTUNITY COST IN THE ANALYSIS. Remember that the interest you could have made on the CD should not be considered as an explicit opp. cost. It is already taken into account through discounting. 3. Annual payment on loan (at 8%)= 50091.2909 NPV of loan at 12% = +200000 - 50091 (AF, 12%,5 years) = 19433.1552 Adding on the loan will make your NPV positive. To be precise, this payment should have been broken down into interest and principal since interest will provide added tax savings. 4. (1) True (2) False (Different lives) (3) False (Think of the two IRRs case)

 CORPORATE FINANCE: SPRING 1989 Problem 1 NPV = -100000 + 56000 (PVA,12%,5 yrs) = 101867.467 Breakeven number of units = 5411 (100X-20X-400000-((100X-20X -400000-20000)*0.4)= 27740 ) Problem 2 Year Potential sales Lost sales Lost profits PV lost profits 1 27500 0 \$0 \$0 2 30250 250 \$9,000 \$7,438 3 33275 3275 \$117,900 \$88,580 4 36603 6603 \$237,690 \$162,345 5 40263 10263 \$369,459 \$229,405 6 44289 14289 \$514,405 \$290,368 7 48718 18718 \$673,845 \$345,789 8 50000 20000 \$720,000 \$335,885 9 50000 20000 \$720,000 \$305,350 10 50000 20000 \$720,000 \$277,591 OPPORTUNITY COST 2042752.6 Problem 3 XYZ M (XYZ-XYZ)SQ (M-M)SQ (XYZ-XYZ)(M-M) 20 15 100 49 70 -10 -5 400 169 260 30 25 400 289 340 10 15 0 49 0 0 -10 100 324 180 10 8 1000 880 850 BETA = 850/880 = 0.96590909 ALPHA = 10-8*0.97= 2.27272727 ! IT DID BETTER THAN EXPECTED EXPECTED RETURN = 9 +0.97 *8.3 (OR 8.5) = 17.0170455 Problem 4 PV of obligations = 1 (PVA,5 yrs, 10%) + 2 (PVA,5 yrs,10%)(PF,5 yrs,10%) + 5 (PVA,10 yrs,10%)(PF,10 yrs,10%) = 20.343331

 SPRING 1990 MIDTERM PROBLEM 1 Present Value of Liabilities = 100000(PVA,8%,5 yrs) (1/1.08^5) + 250000(PVA,8%,10yrs)(1/1.08^10) +100000(PVA,8%,5 yrs) (1/1.08^20) =271737+777016+85662 =1134416 Current Assets = 500000 Remaining liabilities = 1134416 - 500000 = 634416 Annual cashflow required over next five years = 634416 (APV,8%,5 yrs) = 158894 PROBLEM 2 1. There is no cost the first three years. The after-tax salary paid in last two years is an opp. cost = 80,000*0.6/1.1^4 + 80000*0.6/1.1^5 = 62589 2. The opportunity cost is the difference in PV of investing in year 4 instead of year 8 = 250000/1.1^4 - 250000/1.1^8 = \$54,126 3. The present value of after-tax rental payments over five years is the opp. cost = 3000*0.6(PVA,10%,5 yrs) = \$6,823.42 4. After-tax cashflow = (400000-160000) - (240000-100000)*0.4 = 184000 5. NPV = -500000 -62589 - 54126 - 6823 + 184000(1-(.1.1)^-5)/.1= 73966.7656 PROBLEM 3 NPV(I) = -12000 - 500/0.1 = -17000 EAC(I) = -17000*0.1 = -1700 ! Remember this is a perpetuity: PV = A/i; A = PV*i; NPV(II) = -5000 - 1000(1-(1.1)^(-20))/.1 = -13514 EAC(II) = -1587 NPV(III) = -3500 -1200(1-(1.1)^(-15))/0.1 = -12627 EAC(III) = -1660 CHOOSE OPTION II (GAS HEATING SYSTEM) PROBLEM 4 a. (R) = 6 + 1.5*8.3 = 18.45% b. 1 - R squared = 60% is diversifiable c. First unlever the firm's beta = 1.5/(1+(0.6)(1)) = 0.9375 ! 0.9375 = (0.33) (1) + (0.67) (Beta of remaining company)! Solve for this beta. Estimate the beta of the firm after divestment = (0.9375 - 0.33)/0.67 = 0.90671642 (The divested division has a beta of one and a market value of \$20 million. This is one-third of the market value of the firm (\$60 million)) Estimate the unlevered beta of the firm after new acquizition = 0.91 * (4/9) + 2 (5/9) = 1.51555556 ! Equity : Existing= 40; Equity: New =50 (The new division has a market value of \$ 50 million, and the value of the total firm is \$90 million) Estimate the levered beta after acquizition =1.52(1+(0.6)(2)) = 3.344 (The new debt equity ratio is 2. The new debt (\$30 million) plus old debt (\$30 million) equals \$60 million. The equity stays at \$ 30 million.

 FALL 1990 MIDTERM EXAM Problem 1 a. PV of Strawberry's offer = 4,000,000 (PVA, 10%, 5 years) = \$15163147 PV of counter offer = 3,000,000 (PVA,10%,5)+1,000,000(PVA,10%,5)(PF,10%,5) = \$1376,2141 Difference in PV = 1437006.43 b. 3,000,000 (PVA,10%,5)+ (X-1,000,000)(PVA,10%,5)(PF,10%,5) = 1437006.43 Solving for X, we get X= \$ 610,510 Problem 2 a. Unlevered beta (Nuk-Nuk) = 1.3/(1+(1-0.6)0.5) = 1 Unlevered beta (Gerber) = 1.5/(1+ (1-0.5)1.00) = 1 This project has no debt. So the appropriate beta = 1.00 Appropriate discount rate = 8.5 + 1.0 (8.5) = 0.17 (If you use 8.3% the discount rate = 16.8%) b. Revenues 30000 Expenses 12000 Garage cost 2000 BTCF 16000 Taxes 4400 (16000-5000)*0.4 ATCF 11600 Alternatively, you could consider the garaging cost separately as an opportunity cost, in which case ATCF=13600 If you considered working capital increase in year 1, the ATCF in year 1 alone=4600. (Note that since working capital stays at 7500, there are no working capital changes after the inital year.) c. NPV = -57500 +11600 (PVA,17%,10 years)+6000(PF,17%,10 years) = -2211.97362 Problem 3 Cost of the new facility = 100000 - Capital gains from sale of facility = 10000 (100000-60000)*0.25 -Cost of new facility= 40000 -Depreciation lost on old facility= 14746.9611 (6000*0.4*(PVA,10%,10)) +Depreciation gained on new facility= 9831.30737 (4000*0.4*(PVA,10%,10)) OPPORTUNITY COST= 45084.3463 Problem 4 Unlevered beta of the firm = 1.5/(1+(1-0.5)1) = 1 (Remember regression betas are always levered betas) Unlevered beta for division A = 1.31/(1+(1-0.5)0.2)= 1.19090909 (Divisional betas are asset betas; hence the unleveraged beta will do) Setting the unlevered beta of the firm to the weighted averages of the divisonal betas, 1.00 = 0.6 ( 1.19) + 0.4 X Solving for X, X = 0.715 b. If the company divests itself of B, it is left with division A (and its unlevered beta of 1.19) New levered beta = 1.19 (1+ (1-0.5) 2) = 2..38
 SOLUTIONS TO SPRING 1991 FINAL EXAM 1a. Annuity needed to get \$10 million in 10 years at 8% = 690294.887 1b. Amount that you will have in the bank at the end of yr 5 = 4049684.65 Future value of \$4049684 in year 10 at 6%= 5419391.58 Shortfall that will have to be covered by annuity 6-10= 4580608.42 Annuity needed to get 4580608 in 5 years @ 6%= 812583.446 1014269.66 Increase in annuity needed because of rate drop= 122288.559 2a. Initial investment = 10 million (Distribution system) + 1 million (WC) = 11 million 2b. Incremental Revenues = 10000000 Variable costs (40%)= 4000000 Advertising Costs 1000000 BTCF 5000000 Taxes 1600000 : (5000000-1000000)*0.4 ATCF 3400000 2c. NPV = -11,000,000 + 3,400,000 (PVA,10 years,8%) + 1,000,000 (PF, 10 years, 8%) = 12277470.2 2d. Precise Breakeven : (-10000000 -.1x)+(.6x-1000000-(.6x-1000000-1000000)*.4)(PVA,10yrs,8%)+.1x/1.08^10=0 (-10000000-.1x)+(.6x-1000000-(.6x-1000000-1000000)*.4)(6.71)+.1x*0.4632=0 -.1x+2.4156x+.04632x = 10000000 +200000*6.71 2.36192x = 11342000 x = 4802025.47 or INCREASE 4.80% from initial level of 10% Approximate Breakeven (-11,000,000)+(.6x-1000000-(.6x-1000000-1000000)*.4)(PVA,10yrs,8%)+1000000/1.08^10=0 2.4156x = 11,000,000+200000*6.71-100000*0.4632 2.4156x = 12295680 x = 5090114.26 OF INCREASE 5.09% from initial level of 10% 3a. Year Old Product New Product Excess/Shortfall 1 50 30 20 2 52.5 33 14.5 3 55.125 36.3 8.575 4 57.88125 39.93 2.18875 5 60.7753125 43.923 -4.6983125 OUT OF CAPACITY 6 63.8140781 48.3153 -12.1293781 7 67.004782 53.14683 -20.151612 8 70.3550211 58.461513 -28.8165341 9 73.8727722 64.3076643 -38.1804365 10 77.5664108 70.7384307 -48.3048415 3b. Contribution margin for 1% of capacity : for OLD= (100-50)/50= 1 for NEW= (80-44)/30= 1.2 YOU WILL LOSE LESS CUTTING BACK ON OLD PRODUCT Year Lost Capacity \$ BT loss (m) \$AT loss (m) PV (loss) 5 -4.7 -4.7 -2.82 -1.75099813 6 -12.13 -12.13 -7.278 -4.10824126 7 -20.15 -20.15 -12.09 -6.20408165 8 -28.82 -28.82 -17.292 -8.06684562 9 -38.18 -38.18 -22.908 -9.71522824 10 -48.3 -48.3 -28.98 -11.1730445 TOTAL OPPORTUNITY COST= -41.0184394 3c. PV of Building facility in year 5 = 31.0460662 PV of depreciation benefits on this building = 2 million * 0.4 *(PVa, 10%, 25) * (PF, 10%, 5) = 4.50890216 Year in which you would have run out of capacity without new product = 14.2066991 ! YEAR 14 (Remember that growth rate on old product is 5%) PV of building facility in year 14 = 13.1665627 PV of depreciation benefits on this building = 2 million * 0.4 *(PVa, 10%, 25) * (PF, 10%, 14) = 1.91221467 NET OPPORTUNITY COST = (PV of Building in year 5 - PV of Depreciation on this building) - (PV of Building in year 14 - PV of Depreciation on this building) = = (31.05 - 4.51) - (13.17 - 1.91) = 15.2828159 4a. Riskfree rate during the five-year period = 6% (Whether this was annualized or monthly was not specified; I assumed that it was annual) Riskfree rate ( 1- beta) = .5%(1-1.2) = -0.001 Alpha (Intercept) = 0.002 Alpha - Riskfree rate(1-Beta) = 0.20 - (-0.10) = 0.30% better than expected 4b. R squared = (1.2^2) (20^2)/(40^2) = 0.36 Hence 64% of this firm's risk is diversifiable (If stated in terms of %, Unsystematic risk = 40^2 - 1.2^2 (20^2) = 1024%) 4c. Expected Return = Current riskfree rate + beta * 8.5% = 7% + 1.2*8.5 = 17.2% Dividend Yield = 4% Expected price appreciation = 17.2% - 4% = 13.2 % Expected price = 50*(1.132) = 56.6 4d. Current levered beta = 1.2 Current Debt = 5 million Current equity = 5 million Current D/E ratio= 1 Unlevered beta = 1.2/(1+0.6*1) = 0.75 0.75 = 0.5 X + 0.5 (0.5) ! X is the unlevered beta of what's left of the firm X = 1.00 New Debt = 3 million New Equity = 2 million New D/E ratio = 1.5 New levered beta = 1.00 *(1+0.6*1.5) = 1.9
 MIDTERM - FALL 1992 Problem 1 a. FV of \$ 5 million at the end of year 10 = 10.794625 (FV,10yrs,8%) FV of \$ 2 million in years 1-5 at the end of yr 10= 17.239923 (FVA,5 yrs,8%)(FV,5yrs,8%) FV of \$ 3 million outflow in years 6-10 in year 10= -17.5998029 (FVA,5 yrs,8%) FV in year 10 = 10.7946 + 17.2399 - 17.5998 = 10.4357 b. Annuity per year = 10.4357 * .08 = 0.834856 Problem 2 a. PV of lost rent (in after-tax terms) = 14000*0.6*(PVA,5 yrs,10%) = 31842.6089 PV of tax savings from depreciation = 10000*0.4*(PVA,5 yrs,10%) = 15163.1471 Opportunity Cost = 31842.61-15163.15 = 16679.46 b. Opportunity Cost of Salary = 50000*0.6*(PVA,5 yrs, 10%) = 113723.603 c. Revenues 500000 BTCF 250000 Fixed Costs 50000 Deprec'n 50000 Var. Costs 200000 Taxable Inc. 200000 BTCF 250000 Tax 80000 Tax 80000 Net income 120000 ATCF 170000 d. NPV = -350000 - 16679 - 113724 + 170000 (PVA,5,10%) + 100000(PV,5,10%) = 226122.883 ! Accept the project Problem 3 NPV of Wood Siding = -5000 - 1000 (PVA.10,10%) = -11144.5671 EAC of Wood Siding = -11144*(APV,10,10%) = -1813.63468 EAC of Aluminium Siding investment = -15000*.1 = -1500 Maintenance Cost for Aluminium Siding = 1813.63-1500 = 313.63 Problem 4 a. Expected Return = 3% + 1.2(8.5%) = 13.2% b. Expected Price one year from today = 50*(1.132)-2.50 = 54.10 c. Actual Return on XYZ Stock = (50-54+2)/54 = -0.03703704 Expected Return on XYZ Stock = 5% + 1.2(-2-5) = -3.4% Return on mkt= -5% + 3% = -2% Excess Return = -3.70% - (-3.4%) = -0.3% d. R squared = 1.44 * 20 / 50 = 56.60%
 SOLUTION TO FALL 1993 EXAM 1a. Present Value of this contract = \$9 m (PVA,8%,3 years) + \$12 m(PVA,8%,4 years) (PF,8%, 3 years) = 54.7451497 millions 1b. Present Value of alternative contract = \$8m(PVA,8%,7 years) + \$25m(PF,8%,7 years) = 56.2382204 millions 1c. Breakeven Payment needed at end of seventh year = \$25 million - FV of Difference in PV between contracts 22.4411394 millions 2a. Initial Investment = \$10 mil + \$5 mil (PF,15%,1 year) +\$5 mil(PF,15%,2 years) = 18.1285444 millions Opportunity Cost of the land = \$200,000*0.6*(PVA,15%,12 years) = 650474.28 (The land cannot be leased out even in the first two years, since you are building a park on top of it.) 2b. After-tax cashflow each year: Revenues 10 Note: The opporunity cost of the land could have been shown here - Op. costs 3 and added on to the expenses. - Deprec'n 1 Taxable Inc. 6 Tax (40%) 2.4 Net Income 3.6 + Deprecn 1 ATCF 4.6 2c. NPV of the project = -18,128,540 - 650,474 + 4,600,000 (PVA,15%,10) (PF,15%,2) + 10,000,000 (PF,15%,12) = 546644.027 I have assumed the following: (1) The cashflows start in year 3 and go through year 12. (2) The book value of the park = Initial Investment (\$20 mil) - Depreciation (\$1 mil * 10 years) 2d. If the project runs an extra ten years: Initial Investment = -18128540 Opportunity cost = -763039.529 (PV of 120,000 for 22 years) PV of operating CF = 21771587.7 (4.6 mill in years 3-22) NPV = 2880008.21 The book value of the park is zero. (The salvage value of land is irrelevant, since it would apply even if the land were leased.) 3. EAC of using outside printer = -2000 (It is already an annual cost. What is the point of calculating the NPV, and then recalculating the EAC?) NPV of buying printer = -10000 - 500 (PVA,10%,10 years) = -13072.2836 EAC of buying printer = -2127.45395 4a. R squared = (Beta^2)(Variance of the market)/Variance of the stock) = 0.48 Beta =Ã((.48*.6)/.2) = 1.2 4b. Intercept - Riskfree rate (1-Beta) = Jensen's Alpha Intercept - 0.07% (1-1.2) = -0.01% Solving for the intercept, Intercept = -0.024 % 4c. Not necessarily. It also depends upon the variance of the stock.