

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  


AD  M  (ADÅD)SQ  (MM)SQ  (ADÅD)(MM)  
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 (1Beta) = .06*(10.735) =  
=0.0159  
d. The variance would be a good measure if undiversified =  112.5  
Proportion that can be diversified = 1ßsq*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) 


Problem 1  
NPV = 100000 + 56000 (PVA,12%,5 yrs) =  101867.467  
Breakeven number of units =  5411  (100X20X400000((100X20X  
40000020000)*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  (XYZXYZ)SQ  (MM)SQ  (XYZXYZ)(MM)  
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 = 108*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 


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 aftertax 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 aftertax rental payments over five years is the opp. cost  
= 3000*0.6(PVA,10%,5 yrs) =  $6,823.42  
4. Aftertax cashflow = (400000160000)  (240000100000)*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 onethird 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. 


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)+ (X1,000,000)(PVA,10%,5)(PF,10%,5) =  1437006.43  
Solving for X, we get X= $ 610,510  
Problem 2  
a. Unlevered beta (NukNuk) =  1.3/(1+(10.6)0.5) =  1  
Unlevered beta (Gerber) =  1.5/(1+ (10.5)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  (160005000)*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  (10000060000)*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+(10.5)1) =  1  
(Remember regression betas are always levered betas)  
Unlevered beta for division A =  1.31/(1+(10.5)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+ (10.5) 2) =  2..38 


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 610=  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  : (50000001000000)*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)+(.6x1000000(.6x10000001000000)*.4)(PVA,10yrs,8%)+.1x/1.08^10=0  
(10000000.1x)+(.6x1000000(.6x10000001000000)*.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)+(.6x1000000(.6x10000001000000)*.4)(PVA,10yrs,8%)+1000000/1.08^10=0  
2.4156x =  11,000,000+200000*6.71100000*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=  (10050)/50= 


for NEW=  (8044)/30= 


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 fiveyear period = 6%  
(Whether this was annualized or monthly was not specified; I assumed that it was annual)  
Riskfree rate ( 1 beta) =  .5%(11.2) =  0.001  
Alpha (Intercept) =  0.002  
Alpha  Riskfree rate(1Beta) =  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) = 


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 15 at the end of yr 10=  17.239923  (FVA,5 yrs,8%)(FV,5yrs,8%)  
FV of $ 3 million outflow in years 610 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 aftertax 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 = 31842.6115163.15 =  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 = 1813.631500 = 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 = (5054+2)/54 =  0.03703704  
Expected Return on XYZ Stock = 5% + 1.2(25) = 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% 


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. Aftertax cashflow each year:  
Revenues 

Note:  The opporunity cost of the land could have been shown here  
 Op. costs 

and added on to the expenses.  
 Deprec'n 


Taxable Inc. 


Tax (40%) 


Net Income 


+ Deprecn 


ATCF 


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 322)  
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 (1Beta) = Jensen's Alpha  
Intercept  0.07% (11.2) = 0.01%  
Solving for the intercept,  
Intercept =  0.024  %  
4c. Not necessarily. It also depends upon the variance of the stock. 