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.