Projects Solutions Pfa

7/31/2019 Projects Solutions Pfa

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Chapter 4

MARKET AND DEMAND ANALYSIS

1. We have to estimate the parameters a and b in the linear relationshipYt= a + bT

Using the least squares method.According to the least squares method the parameters are:

T Yn T Yb =

T2 n T2

a = YbTThe parameters are calculated below:

Calculation in the Least Squares Method

T Y TY T 2

1 2,000 2,000 12 2,200 4,400 43 2,100 6,300 94 2,300 9,200 165 2,500 12,500 256 3,200 19,200 367 3,600 25,200 498 4,000 32,000 649 3,900 35,100 8110 4,000 40,000 100

11 4,200 46,200 12112 4,300 51,600 14413 4,900 63,700 16914 5,300 74,200 196

T = 105 Y = 48,500 TY = 421,600 T2 = 1,015

T= 7.5 Y= 3,464

T Yn T Y 421,600 14 x 7.5 x 3,464

b = = T2 n T2 1,015 14 x 7.5 x 7.5

57,880= = 254

227.5a = YbT

= 3,464 254 (7.5)

= 1,559


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Thus linear regression isY= 1,559 + 254 T

2. In general, in exponential smoothing the forecast fort+ 1 isFt+ 1 =Ft+ et

WhereFt+ 1 = forecast for year ) = smoothing parameteret = error in the forecast for yeart= St =Ft

F1is given to be 2100 and is given to be 0.3

The forecasts for periods 2 to 14 are calculated below:

Period t Data (St) Forecast

(Ft)

Error

(et St=Ft)

Forecast for t + 1

(Ft + 1 = Ft + et)

1 2,000 2100.0 -100 F2 = 2100 + 0.3 (-100) = 2070

2 2,200 2070 130 F3 = 2070 + 0.3(130) = 2109

3 2,100 2109.0 -9 F4 = 2109 + 0.3 (-9) = 2111.7

4 2,300 2111.7 188.3 F5 = 2111.7 + 0.3(188.3) = 2168.19

5 2,500 2168.19 331.81 F6 = 2168.19 + 0.3(331.81) = 2267.7

6 3,200 2267.7 932.3 F7 = 2267.7 + 0.3(9332.3) = 2547.4

7 3,600 2547.4 1052.6 F8 = 2547.4 + 0.3(1052.6) = 2863.28 4,000 2863.2 1136.8 F9 = 2863.2 + 0.3(1136.8) = 3204.24

9 3,900 3204.24 695.76 F10 = 33204.24 + 0.3(695.76) = 3413.0

10 4,000 3413 587.0 F11 = 3413.0 + 0.3(587) = 3589.1

11 4,200 3589.1 610.9 F12 = 3589.1 + 0.3(610.9) = 3773.4

12 4,300 3772.4 527.6 F13 = 3772.4 + 0.3(527.6) = 3930.7

13 4,900 3930.7 969.3 F14 = 3930.7 + 0.3(969.3) = 4221.5

3. According to the moving average methodSt+ St 1++ St n +1

Ft+ 1 = n

whereFt+ 1 = forecast for the next periodSt = sales for the current periodn = period over which averaging is done

Given n = 3, the forecasts for the period 4 to 14 are given below:


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Period t Data (St) Forecast

(Ft)

Forecast for t + 1

Ft + 1 = (St+ St 1+

St 2)/ 3

1 2,000

2 2,200

3 2,100 F4 = (2000 + 2200 + 2100)/3 = 2100

4 2,300 2100 F5 =(2200 + 2100 + 2300)/3= 2200

5 2,500 2200 F6 = (2100 + 2300 + 2500)/3 = 2300

6 3,200 2300 F7 = (2300 + 2500 + 3200)/3= 2667

7 3,600 2667 F8 = (2500 + 3200 + 3600)/3 = 31008 4,000 3100 F9 = (3200 + 3600 + 4000)/3 = 3600

9 3,900 3600 F10 = (3600 + 4000 + 3900)/3 = 3833

10 4,000 3833 F11 = (4000 + 3900 + 4000)/3 =3967

11 4,200 3967 F12 =(3900 + 4000 + 4200)/3 = 4033

12 4,300 4033 F13 = (4000 + 4200 + 4300)/3 = 4167

13 4,900 4167 F14 = (4200 + 4300 + 4900) = 4467

14 5,300 4467

4.Q1 = 60

Q2 = 70I1 = 1000I2 = 1200

Q1Q2 I1 +I2

Income Elasticity of DemandE1 = xI2 - I1 Q2Q1 E1 = Income Elasticity of Demand

Q1 = Quantity demanded in the base yearQ2 = Quantity demanded in the following yearI1 = Income level in base yearI2 = Income level in the following year

70 60 1000 + 1200E1 = x1200 1000 70 + 60

22000E1 = = 0.846

26000


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5.P1 = Rs.40

P2 = Rs.50Q1 = 1,00,000Q2 = 95,000

Q2Q1 P1 +P2Price Elasticity of Demand = Ep = x

P2P1Q2 + Q1 P1 , Q1 = Price per unit and quantity demanded in the base year

P2, Q2 = Price per unit and quantity demanded in the following yearEp= Price Elasticity of Demand

95000 - 100000 40 + 50Ep= x

50 - 40 95000 + 100000

- 45Ep= = - 0.0231

1950


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Chapter 6

FINANCIAL ESTIMATES AND PROJECTIONS

1. Projected Cash Flow Statement (Rs. in million)

Sources of Funds

Profit before interest and tax 4.5Depreciation provision for the year 1.5Secured term loan 1.0

Total (A) 7.0

Disposition of Funds

Capital expenditure 1.50

Increase in working capital

0.35Repayment of term loan 0.50Interest 1.20Tax 1.80Dividends 1.00

Total (B) 6.35

Opening cash balance 1.00Net surplus (deficit) (A B) 0.65Closing cash balance 1.65

Projected Balance Sheet

(Rs. in million)

Liabilities Assets

Share capital 5.00 Fixed assets 11.00Reserves & surplus 4.50 Investments .50Secured loans 4.50 Current assets 12.85Unsecured loans 3.00 * Cash 1.65Current liabilities 6.30 * Receivables 4.20

& provisions 1.05 * Inventories 7.0024.35 24.35

2. Projected Income Statement for the 1st Operating Year

Working capital here is defined as :(Current assets other than cash) (Current liabilities other than bank borrowings)In this case inventories increase by 0.5 million, receivables increase by 0.2 million and current

liabilities and provisions increase by 0.35 million. So working capital increases by 0.35 million


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Rs.Sales 4,500

Cost of sales 3,000Depreciation 319Interest 1,044Write off of Preliminary expenses 15Net profit 122

Projected Cash Flow Statements

Construction period 1stOperating yearSources

Share capital 1800 -Term loan 3000 600Short-term bank borrowing 1800Profit before interest and tax 1166Depreciation 319Write off preliminary expenses 15

4800 3900Uses

Capital expenditure 3900 -Current assets (other than cash) - 2400Interest - 1044Preliminary expenses 150 -Pre-operative expenses 600 -

4650 3444Opening cash balance 0 150Net surplus / deficit 150 456

Closing balance 150 606Projected Balance Sheet

Liabilities 31/3/n+1 31/3/n+2 Assets 31/3/n+1 31/3/n+2

Share capital 1800 1800 Fixed assets (net) 4500 4181

Reserves & surplus - 122

Secured loans : Current assets

- Term loan 3000 3600 - Cash 150 606

- Short-term bank

borrowing

1800 Other current assets 2400

Unsecured loans - - Miscellaneousexpenditures & losses

Current liabilities andprovisions

- Preliminaryexpenses

150 135

4800 7322 4800 7322

Notes :


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i. Allocation of Pre-operative Expenses : Rs.

Type Costs beforeallocation Allocation Costs afterallocation

Land 120 19 139

Building 630 97 727

Plant & machinery 2700 415 3115

Miscellaneous fixed assets 450 69 519

3900 600 4500

ii. Depreciation Schedule :

Lan

d

Building Plant & machinery M.Fixed

assets

Total (Rs.)

Opening balance 139 727 3115 519 4500

Depreciation - 25 252 42 319

Closing balance 139 702 2863 477 4181

iii. Interest Schedule :Interest on term loan of Rs.3600 @20% = Rs.720Interest on short term bank borrowings of Rs,1800 @ 18% = Rs.324

= Rs.1044

Chapter 7


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THE TIME VALUE OF MONEY

1. Value five years hence of a deposit of Rs.1,000 at various interest rates is asfollows:

r = 8% FV5 = 1000 x FVIF (8%, 5 years)= 1000 x 1.469 = Rs.1469




2. Rs.160,000 / Rs. 5,000 = 32 = 25

According to the Rule of 72 at 12 percent interest rate doubling takes placeapproximately in 72 / 12 = 6 years

So Rs.5000 will grow to Rs.160,000 in approximately 5 x 6 years = 30 years

3. In 12 years Rs.1000 grows to Rs.8000 or 8 times. This is 23 times the initialdeposit. Hence doubling takes place in 12 / 3 = 4 years.

According to the Rule of 69, the doubling period is:

0.35 + 69 / Interest rate

Equating this to 4 and solving for interest rate, we get

Interest rate = 18.9%.

4. Saving Rs.2000 a year for 5 years and Rs.3000 a year for 10 years thereafter isequivalent to saving Rs.2000 a year for 15 years and Rs.1000 a year for the years6 through 15.

Hence the savings will cumulate to:

2000 x FVIFA (10%, 15 years) + 1000 x FVIFA (10%, 10 years)= 2000 x 31.772 + 1000 x 15.937 = Rs.79481.


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5. LetA be the annual savings.

A x FVIFA (12%, 10 years) = 1,000,000A x 17.549 = 1,000,000

So A = 1,000,000 / 17.549 = Rs.56,983.

6. 1,000 x FVIFA (r, 6 years) = 10,000

FVIFA (r, 6 years) = 10,000 / 1000 = 10

From the tables we find that

FVIFA (20%, 6 years) = 9.930FVIFA (24%, 6 years) = 10.980

Using linear interpolation in the interval, we get:

20% + (10.000 9.930)

r= x 4% = 20.3%(10.980 9.930)

7. 1,000 x FVIF (r, 10 years) = 5,000FVIF (r,10 years) = 5,000 / 1000 = 5

From the tables we find thatFVIF (16%, 10 years) = 4.411

FVIF (18%, 10 years) = 5.234

Using linear interpolation in the interval, we get:

(5.000 4.411) x 2% r= 16% + = 17.4%

(5.234 4.411)

8. The present value of Rs.10,000 receivable after 8 years for various discount rates(r) are:r= 10% PV = 10,000 x PVIF(r= 10%, 8 years)

= 10,000 x 0.467 = Rs.4,670

r= 12% PV = 10,000 x PVIF (r= 12%, 8 years)= 10,000 x 0.404 = Rs.4,040

r= 15% PV = 10,000 x PVIF (r= 15%, 8 years)


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= 10,000 x 0.327 = Rs.3,2709. Assuming that it is an ordinary annuity, the present value is:

2,000 x PVIFA (10%, 5years)= 2,000 x 3.791 = Rs.7,582

10. The present value of an annual pension of Rs.10,000 for 15 years when r= 15%is:

10,000 x PVIFA (15%, 15 years)= 10,000 x 5.847 = Rs.58,470

The alternative is to receive a lumpsum of Rs.50,000.

Obviously, Mr. Jingo will be better off with the annual pension amount ofRs.10,000.

11. The amount that can be withdrawn annually is:100,000 100,000

A = ------------------ ------------ = ----------- = Rs.10,608PVIFA (10%, 30 years) 9.427

12. The present value of the income stream is:

1,000 x PVIF (12%, 1 year) + 2,500 x PVIF (12%, 2 years)+ 5,000 x PVIFA (12%, 8 years) x PVIF(12%, 2 years)

= 1,000 x 0.893 + 2,500 x 0.797 + 5,000 x 4.968 x 0.797 = Rs.22,683.

13. The present value of the income stream is:

2,000 x PVIFA (10%, 5 years) + 3000/0.10 x PVIF (10%, 5 years)= 2,000 x 3.791 + 3000/0.10 x 0.621= Rs.26,212

14. To earn an annual income of Rs.5,000 beginning from the end of 15 years fromnow, if the deposit earns 10% per year a sum of

Rs.5,000 / 0.10 = Rs.50,000

is required at the end of 14 years. The amount that must be deposited to get thissum is:

Rs.50,000 / PVIF (10%, 14 years) = Rs.50,000 / 3.797 = Rs.13,165


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15. Rs.20,000 =- Rs.4,000 x PVIFA (r, 10 years)PVIFA (r,10 years) = Rs.20,000 / Rs.4,000 = 5.00

From the tables we find that:

PVIFA (15%, 10 years) = 5.019PVIFA (18%, 10 years) = 4.494

Using linear interpolation we get:

5.019 5.00r= 15% + ---------------- x 3%

5.019 4.494= 15.1%

16. PV (StreamA) = Rs.100 x PVIF (12%, 1 year) + Rs.200 xPVIF (12%, 2 years) + Rs.300 x PVIF(12%, 3 years) + Rs.400 xPVIF (12%, 4 years) + Rs.500 x PVIF (12%, 5 years) +Rs.600 x PVIF (12%, 6 years) + Rs.700 x PVIF (12%, 7 years) +

Rs.800 x PVIF (12%, 8 years) + Rs.900 x PVIF (12%, 9 years) +Rs.1,000 x PVIF (12%, 10 years)

= Rs.100 x 0.893 + Rs.200 x 0.797 + Rs.300 x 0.712+ Rs.400 x 0.636 + Rs.500 x 0.567 + Rs.600 x 0.507+ Rs.700 x 0.452 + Rs.800 x 0.404 + Rs.900 x 0.361+ Rs.1,000 x 0.322

= Rs.2590.9

Similarly,PV (StreamB) = Rs.3,625.2PV (Stream C) = Rs.2,851.1

17. FV5 = Rs.10,000 [1 + (0.16 / 4)]5x4

= Rs.10,000 (1.04)20

= Rs.10,000 x 2.191= Rs.21,910

18. FV5 = Rs.5,000 [1+( 0.12/4)]5x4

= Rs.5,000 (1.03)20

= Rs.5,000 x 1.806= Rs.9,030

19. A B C


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Stated rate (%) 12 24 24Frequency of compounding 6 times 4 times 12 timesEffective rate (%) (1 + 0.12/6)6- 1 (1+0.24/4)4 1 (1 + 0.24/12)12-1

= 12.6 = 26.2 = 26.8Difference between theeffective rate and statedrate (%) 0.6 2.2 2.8

20. Investment required at the end of 8th year to yield an income of Rs.12,000 peryear from the end of 9th year (beginning of 10th year) for ever:

Rs.12,000 x PVIFA(12%, )= Rs.12,000 / 0.12 = Rs.100,000

To have a sum of Rs.100,000 at the end of 8th year , the amount to be depositednow is:

Rs.100,000 Rs.100,000

= = Rs.40,388PVIF(12%, 8 years) 2.476

21. The interest rate implicit in the offer of Rs.20,000 after 10 years in lieu ofRs.5,000 now is:

Rs.5,000 x FVIF (r,10 years) = Rs.20,000

Rs.20,000FVIF (r,10 years) = = 4.000

Rs.5,000

From the tables we find thatFVIF (15%, 10 years) = 4.046

This means that the implied interest rate is nearly 15%.I would choose Rs.20,000 for 10 years from now because I find a return of 15%

quite acceptable.

22. FV10 = Rs.10,000 [1 + (0.10 / 2)]10x2

= Rs.10,000 (1.05)20

= Rs.10,000 x 2.653= Rs.26,530

If the inflation rate is 8% per year, the value of Rs.26,530 10 years from now, in

terms of the current rupees is:


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Rs.26,530 x PVIF (8%,10 years)= Rs.26,530 x 0.463 = Rs.12,283

23. A constant deposit at the beginning of each year represents an annuity due.

PVIFA of an annuity due is equal to : PVIFA of an ordinary annuity x (1 + r)

To provide a sum of Rs.50,000 at the end of 10 years the annual deposit should be

Rs.50,000A = FVIFA(12%, 10 years) x (1.12)

Rs.50,000= = Rs.2544

17.549 x 1.12

24. The discounted value of Rs.20,000 receivable at the beginning of each year from2005 to 2009, evaluated as at the beginning of 2004 (or end of 2003) is:

Rs.20,000 x PVIFA (12%, 5 years)= Rs.20,000 x 3.605 = Rs.72,100.

The discounted value of Rs.72,100 evaluated at the end of 2000 is

Rs.72,100 x PVIF (12%, 3 years)= Rs.72,100 x 0.712 = Rs.51,335

IfA is the amount deposited at the end of each year from 1995 to 2000 thenA x FVIFA (12%, 6 years) = Rs.51,335A x 8.115 = Rs.51,335A = Rs.51,335 / 8.115 = Rs.6326

25. The discounted value of the annuity of Rs.2000 receivable for 30 years, evaluatedas at the end of 9th year is:

Rs.2,000 x PVIFA (10%, 30 years) = Rs.2,000 x 9.427 = Rs.18,854

The present value of Rs.18,854 is:Rs.18,854 x PVIF (10%, 9 years)

= Rs.18,854 x 0.424= Rs.7,994

26. 30 percent of the pension amount is0.30 x Rs.600 = Rs.180

Assuming that the monthly interest rate corresponding to an annual interest rate of12% is 1%, the discounted value of an annuity of Rs.180 receivable at the end of

each month for 180 months (15 years) is:


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Rs.180 x PVIFA (1%, 180)

(1.01)180 - 1Rs.180 x ---------------- = Rs.14,998

.01 (1.01)180

If Mr. Ramesh borrows Rs.Ptoday on which the monthly interest rate is 1%

P x (1.01)60 = Rs.14,998P x 1.817 = Rs.14,998

Rs.14,998P = ------------ = Rs.8254

1.817

27. Rs.300 x PVIFA(r, 24 months) = Rs.6,000

PVIFA (4%,24) = Rs.6000 / Rs.300 = 20

From the tables we find that:

PVIFA(1%,24) = 21.244PVIFA (2%, 24) = 18.914

Using a linear interpolation

21.244 20.000r = 1% + ---------------------- x 1%

21.244 18,914

= 1.53%Thus, the bank charges an interest rate of 1.53% per month.The corresponding effective rate of interest per annum is

[ (1.0153)12 1 ] x 100 = 20%

28. The discounted value of the debentures to be redeemed between 8 to 10 yearsevaluated at the end of the 5th year is:

Rs.10 million x PVIF (8%, 3 years)+ Rs.10 million x PVIF (8%, 4 years)+ Rs.10 million x PVIF (8%, 5 years)= Rs.10 million (0.794 + 0.735 + 0.681)= Rs.2.21 million

IfA is the annual deposit to be made in the sinking fund for the years 1 to 5, thenA x FVIFA (8%, 5 years) = Rs.2.21 millionA x 5.867 = Rs.2.21 million

A = 5.867 = Rs.2.21 million


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A = Rs.2.21 million / 5.867 = Rs.0.377 million

29. Let n be the number of years for which a sum of Rs.20,000 can be withdrawnannually.

Rs.20,000 x PVIFA (10%, n) = Rs.100,000PVIFA (15%, n) = Rs.100,000 / Rs.20,000 = 5.000

From the tables we find thatPVIFA (10%, 7 years) = 4.868PVIFA (10%, 8 years) = 5.335

Thus n is between 7 and 8. Using a linear interpolation we get

5.000 4.868n = 7 + ----------------- x 1 = 7.3 years

5.335 4.868

30. Equated annual installment = 500000 / PVIFA(14%,4)= 500000 / 2.914= Rs.171,585

Loan Amortisation Schedule

Beginning Annual Principal Remaining

Year amountinstallment Interestrepaid balance

1 50000017158570000 1015853984152 39841517158555778 1158072826083 28260817158539565 1320201505884 15058817158521082 150503 85*

(*) rounding off error

31. Define n as the maturity period of the loan. The value of n can be obtained fromthe equation.

200,000 x PVIFA(13%, n) = 1,500,000PVIFA (13%, n) = 7.500

From the tables or otherwise it can be verified that PVIFA(13,30) = 7.500

Hence the maturity period of the loan is 30 years.

32. Expected value of iron ore mined during year 1 = Rs.300 millionExpected present value of the iron ore that can be mined over the next 15 yearsassuming a price escalation of 6% per annum in the price per tonne of iron

1 (1 +g)n / (1 + i)n

= Rs.300 million x ------------------------

i -g


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= Rs.300 million x 1 (1.06)15 / (1.16)150.16 0.06

= Rs.300 million x (0.74135 / 0.10)= Rs.2224 million

Chapter 8

INVESTMENT CRITERIA

1.(a) NPV of the project at a discount rate of 14%.

100,000 200,000


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= - 1,000,000 + ---------- + ------------(1.14) (1.14)2

300,000 600,000 300,000+ ----------- + ---------- + ----------

(1.14)3 (1.14)4 (1.14)5

= - 44837

(b) NPV of the project at time varying discount rates

= - 1,000,000

100,000+

(1.12)

200,000+

(1.12) (1.13)

300,000+

(1.12) (1.13) (1.14)

600,000+

(1.12) (1.13) (1.14) (1.15)

300,000+

(1.12) (1.13) (1.14)(1.15)(1.16)

= - 1,000,000 + 89286 + 158028 + 207931 + 361620 + 155871= - 27264

2. InvestmentA

a) Payback period = 5 yearsb) NPV = 40000 x PVIFA (12%,10) 200 000

= 26000c) IRR (r) can be obtained by solving the equation:

40000 x PVIFA (r, 10) = 200000


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i.e., PVIFA (r, 10) = 5.000

From the PVIFA tables we find that

PVIFA (15%,10) = 5.019PVIFA (16%,10) = 4.883

Linear interporation in this range yields

r = 15 + 1 x (0.019 / 0.136)= 15.14%

d) BCR = Benefit Cost Ratio= PVB / I= 226,000 / 200,000 = 1.13

Investment B

a) Payback period = 9 years

b) NP V = 40,000 x PVIFA (12%,5)+ 30,000 x PVIFA (12%,2) x PVIF (12%,5)+ 20,000 x PVIFA (12%,3) x PVIF (12%,7)

- 300,000

= (40,000 x 3.605) + (30,000 x 1.690 x 0.567)+ (20,000 x 2.402 x 0.452) 300,000

= - 105339

c) IRR (r) can be obtained by solving the equation40,000 x PVIFA (r, 5) + 30,000 x PVIFA (r, 2) x PVIF (r,5) +20,000 x PVIFA (r, 3) x PVIF (r, 7) = 300,000

Through the process of trial and error we find thatr= 1.37%

d) BCR = PVB /I= 194,661 / 300,000 = 0.65

Investment C

a) Payback period lies between 2 years and 3 years. Linear interpolation inthis range provides an approximate payback period of 2.88 years.


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b) NPV = 80.000 x PVIF (12%,1) + 60,000 x PVIF (12%,2)+ 80,000 x PVIF (12%,3) + 60,000 x PVIF (12%,4)

+ 80,000 x PVIF (12%,5) + 60,000 x PVIF (12%,6)+ 40,000 x PVIFA (12%,4) x PVIF (12%,6)- 210,000

= 111,371

c) IRR (r) is obtained by solving the equation80,000 x PVIF (r,1) + 60,000 x PVIF (r,2) + 80,000 x PVIF (r,3)+ 60,000 x PVIF (r,4) + 80,000 x PVIF (r,5) + 60,000 x PVIF (r,6)

+ 40000 x PVIFA (r,4) x PVIF (r,6) = 210000

Through the process of trial and error we getr= 29.29%

d) BCR = PVB /I = 321,371 / 210,000 = 1.53

Investment D

a) Payback period lies between 8 years and 9 years. A linear interpolation inthis range provides an approximate payback period of 8.5 years.8 + (1 x 100,000 / 200,000)

b) NPV = 200,000 x PVIF (12%,1)+ 20,000 x PVIF (12%,2) + 200,000 x PVIF (12%,9)+ 50,000 x PVIF (12%,10)

- 320,000= - 37,160

c) IRR (r) can be obtained by solving the equation200,000 x PVIF (r,1) + 200,000 x PVIF (r,2)+ 200,000 x PVIF (r,9) + 50,000 x PVIF (r,10)

= 320000Through the process of trial and error we get r= 8.45%

d) BCR = PVB / I= 282,840 / 320,000 = 0.88Comparative Table

Investment A B C D

a) Payback period(in years) 5 9 2.88 8.5


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b) NPV @ 12% 26000 -105339 111371 -37160

c) IRR (%) 15.14 1.37 29.29 8.45

d) BCR 1.13 0.65 1.53 0.88

Among the four alternative investments, the investment to be chosen is Cbecause it has the a. Lowest payback period

b. Highest NPVc. Highest IRRd. Highest BCR

3. IRR (r) can be calculated by solving the following equations for the value of r.60000 x PVIFA (r,7) = 300,000i.e., PVIFA (r,7) = 5.000

Through a process of trial and error it can be verified that r= 9.20% p.a.

4. The IRR (r) for the given cashflow stream can be obtained by solving the

following equation for the value of r.-3000 + 9000 / (1+r) 3000 / (1+r) = 0

Simplifying the above equation we getr = 1.61, -0.61; (or) 161%, (-)61%

Note : Given two changes in the signs of cashflow, we get two values for theIRR of the cashflow stream. In such cases, the IRR rule breaks down.

5. Define NCF as the minimum constant annual net cashflow that justifies thepurchase of the given equipment. The value of NCF can be obtained from theequation

NCF x PVIFA (10%,8) = 500000NCF = 500000 / 5.335

= 93271

6. DefineIas the initial investment that is justified in relation to a net annual cash

inflow of 25000 for 10 years at a discount rate of 12% per annum. The valueofIcan be obtained from the following equation

25000 x PVIFA (12%,10) = Ii.e.,I = 141256

7. PV of benefits (PVB) = 25000 x PVIF (15%,1)+ 40000 x PVIF (15%,2)+ 50000 x PVIF (15%,3)

+ 40000 x PVIF (15%,4)


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+ 30000 x PVIF (15%,5)= 122646 (A)

Investment = 100,000 (B)Benefit cost ratio = 1.23 [= (A) / (B)]

8. The NPVs of the three projects are as follows:

Project

P Q RDiscount rate

0% 400 500 600

5% 223 251 31210% 69 40 7015% - 66 - 142 - 13525% - 291 - 435 - 46130% - 386 - 555 - 591

9. NPV profiles for ProjectsPand Q for selected discount rates are as follows:(a)

ProjectP Q

Discount rate (%)

0 2950 5005 1876 20810 1075 - 2815 471 - 22220 11 - 382

b) (i) The IRR (r) of projectPcan be obtained by solving the followingequation for `r.

-1000 -1200 x PVIF (r,1) 600 x PVIF (r,2) 250 x PVIF (r,3)+ 2000 x PVIF (r,4) + 4000 x PVIF (r,5) = 0

Through a process of trial and error we find that r= 20.13%

(ii) The IRR (r') of project Q can be obtained by solving the following

equation forr'

-1600 + 200 x PVIF (r',1) + 400 x PVIF (r',2) + 600 x PVIF (r',3)

+ 800 x PVIF (r',4) + 100 x PVIF (r',5) = 0

Through a process of trial and error we find that r' = 9.34%.

c) From (a) we find that at a cost of capital of 10%


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NPV (P) = 1075NPV (Q) = - 28

Given that NPV (P), NPV (Q) and NPV (P) > 0, I would choose projectP.From (a) we find that at a cost of capital of 20%

NPV (P) = 11NPV (Q) = - 382

Again NPV (P) > NPV (Q); and NPV (P) > 0. I would choose projectP.

d) Project PPV of investment-related costs

= 1000 x PVIF (12%,0)

+ 1200 x PVIF (12%,1) + 600 x PVIF (12%,2)+ 250 x PVIF (12%,3)

= 2728TV of cash inflows = 2000 x (1.12) + 4000 = 6240The MIRR of the projectPis given by the equation:

2728 = 6240 x PVIF (MIRR,5)(1 + MIRR)5 = 2.2874MIRR = 18%

(c) ProjectQPV of investment-related costs = 1600TV of cash inflows @ 15% p.a. = 2772The MIRR of project Q is given by the equation:

16000 (1 + MIRR)5 = 2772MIRR = 11.62%

10.

(a) ProjectANPV at a cost of capital of 12%

= - 100 + 25 x PVIFA (12%,6)= Rs.2.79 million

IRR (r) can be obtained by solving the following equation for r.25 x PVIFA (r,6) = 100i.e., r= 12,98%

Project B

NPV at a cost of capital of 12%= - 50 + 13 x PVIFA (12%,6)= Rs.3.45 million

IRR (r') can be obtained by solving the equation

13 x PVIFA (r',6) = 50

i.e., r' = 14.40% [determined through a process of trial and error]


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(b) Difference in capital outlays between projectsA andB is Rs.50 millionDifference in net annual cash flow between projectsA andB is Rs.12 million.NPV of the differential project at 12%

= -50 + 12 x PVIFA (12%,6)= Rs.3.15 million

IRR (r'') of the differential project can be obtained from the equation

12 x PVIFA (r'', 6) = 50

i.e., r'' = 11.53%

11.(a) ProjectM

The pay back period of the project lies between 2 and 3 years. Interpolating inthis range we get an approximate pay back period of 2.63 years.

Project N

The pay back period lies between 1 and 2 years. Interpolating in this range weget an approximate pay back period of 1.55 years.

(b) ProjectMCost of capital = 12% p.aPV of cash flows up to the end of year 2 = 24.97PV of cash flows up to the end of year 3 = 47.75PV of cash flows up to the end of year 4 = 71.26

Discounted pay back period (DPB) lies between 3 and 4 years. Interpolating in

this range we get an approximate DPB of 3.1 years.

Project N

Cost of capital = 12% per annumPV of cash flows up to the end of year 1 = 33.93PV of cash flows up to the end of year 2 = 51.47

DPB lies between 1 and 2 years. Interpolating in this range we get an

approximate DPB of 1.92 years.(c) ProjectM

Cost of capital = 12% per annumNPV = - 50 + 11 x PVIFA (12%,1)

+ 19 x PVIF (12%,2) + 32 x PVIF (12%,3)+ 37 x PVIF (12%,4)

= Rs.21.26 millionProject N

Cost of capital = 12% per annum


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NPV = Rs.20.63 million

Since the two projects are independent and the NPV of each project is (+) ve,both the projects can be accepted. This assumes that there is no capitalconstraint.

(d) ProjectMCost of capital = 10% per annumNPV = Rs.25.02 million

Project N

Cost of capital = 10% per annumNPV = Rs.23.08 million

Since the two projects are mutually exclusive, we need to choose the projectwith the higher NPV i.e., choose projectM.

Note : The MIRR can also be used as a criterion of merit for choosing betweenthe two projects because their initial outlays are equal.

(e) Project MCost of capital = 15% per annumNPV = 16.13 million

Project N

Cost of capital:15% per annumNPV = Rs.17.23 million

Again the two projects are mutually exclusive. So we choose the project with thehigher NPV, i.e., choose projectN.

(f) ProjectMTerminal value of the cash inflows: 114.47MIRR of the project is given by the equation

50 (1 + MIRR)4 = 114.47

i.e., MIRR = 23.01%Project N

Terminal value of the cash inflows: 115.41MIRR of the project is given by the equation

50 ( 1+ MIRR)4 = 115.41i.e., MIRR = 23.26%

12. The internal rate of return is the value ofrin the equation


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2,000 1,000 10,000 2,0008000 = - + +

(1+r) (1+r)2 (1+r)3 (1+r)4

At r= 18%, the right hand side is equal to 8099At r= 20%, the right hand side is equal to 7726Thus the solving value ofris :

8,099 8,00018% + x 2% = 18.5%

8,099 7,726

Unrecovered Investment Balance

Year Unrecovered investment balance at

the beginning Ft-1

Interest for theyear Ft-1 (1+r)

Cash flow at theend of the year CFt

Unrecoveredinvestment balance at

the end of the year Ft-1(1+r) + CFt

1 -8000 -1480 2000 -7480

2 -7480 -1383.8 -1000 -9863.8

3 -9863.8 -1824.80 10000 -1688.60

4 -1688.60 -312.39 2000 0

13. Rs. in lakhsYear 1 2 3 4 5 6 7 8 Sum Average

Investment 24.0 21.0 18.0 15.0 12.0 9.0 6.0 3.0 108 13.500

Depreciation 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 24.0 3.000

Income beforeinterest and tax

6.0 6.5 7.0 7.0 7.0 6.5 6.0 5.0 51.0 6.375

Interest 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 20.0 2.500

Income before tax 3.5 4.0 4.5 4.5 4.5 4.0 3.5 2.5 31.0 3.875

Tax - 1.0 2.5 2.5 2.5 2.2 1.9 1.4 14.0 1.750Income after tax 3.5 3.0 2.0 2.0 2.0 1.8 1.6 1.1 17.0 2.125

Measures of Accounting Rate of Return

A. Average income after tax 2.125= = 8.9%

Initial investment 24

B. Average income after tax 2.125= = 15.7%

Average investment 13.5

C. Average income after tax but before interest 2.125 + 2.5= = 19.3%



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D. Average income after tax but before interest 2.125 + 2.5= = 34.3%


E. Average income before interest and taxes 6.375= = 26.6%


F. Average income before interest and taxes 6.375= = 47.2%


G. Total income after tax but beforeDepreciation Initial investment 17.0 + 24.0 24.0

=(Initial investment / 2) x Years (24 / 2) x 8

= 17.0 / 96.0 = 17.7%

Chapter 9

PROJECT CASH FLOWS

1.(a) Project Cash Flows (Rs. in million)

Year 0 1 2 3 4 5 6 7


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1. Plant & machinery (150)

2. Working capital (50)

3. Revenues 250 250 250 250 250 250 250

4. Costs (excluding de-preciation & interest) 100 100 100 100 100 100 100

5. Depreciation 37.5 28.13 21.09 15.82 11.87 8.90 6.67

6. Profit before tax 112.5 121.87 128.91 134.18 138.13 141.1 143.33

7. Tax 33.75 36.56 38.67 40.25 41.44 42.33 43.0

8. Profit after tax 78.75 85.31 90.24 93.93 96.69 98.77 100.33

9. Net salvage value of

plant & machinery 48

10. Recovery of working 50capital

11. Initial outlay (=1+2) (200)

12. Operating CF (= 8 + 5) 116.25 113.44 111.33 109.75 108.56 107.67

107.00

13. Terminal CF ( = 9 +10) 98

14. NCF (200) 116.25 113.44 111.33 109.75 108.56 107.67 205

(c) IRR (r) of the project can be obtained by solving the following equation for r

-200 + 116.25 x PVIF (r,1) + 113.44 x PVIF (r,2)+ 111.33 x PVIF (r,3) + 109.75 x PVIF (r,4) + 108.56 x PVIF (r,5)+107.67 x PVIF (r,6) + 205 x PVIF (r,7) = 0

Through a process of trial and error, we get r = 55.17%. The IRR of the projectis 55.17%.

2. Post-taxIncremental Cash Flows (Rs. in million)


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Year 0 1 2 3 4 5 6 7

1. Capital equipment (120)2. Level of working capital 20 30 40 50 40 30 20

(ending)3. Revenues 80 120 160 200 160 120 804. Raw material cost 24 36 48 60 48 36 245. Variable mfg cost. 8 12 16 20 16 12 86. Fixed operating & maint. 10 10 10 10 10 10 10

cost

7. Variable selling expenses 8 12 16 20 16 12 88. Incremental overheads 4 6 8 10 8 6 49. Loss of contribution 10 10 10 10 10 10 1010.Bad debt loss 411. Depreciation 30 22.5 16.88 12.66 9.49 7.12 5.3412. Profit before tax -14 11.5 35.12 57.34 42.51 26.88 6.6613. Tax - 4.2 3.45 10.54 17.20 12.75 8.06 2.0014. Profit after tax - 9.8 8.05 24.58 40.14 29.76 18.82 4.66

15. Net salvage value ofcapital equipments 25

16. Recovery of working 16capital

17. Initial investment (120)18. Operating cash flow 20.2 30.55 41.46 52.80 39.25 25.94 14.00

(14 + 10+ 11)

19. Working capital 20 10 10 10 (10) (10) (10)

20. Terminal cash flow 41

21. Net cash flow (140) 10.20 20.55 31.46 62.80 49.25 35.94 55.00(17+18-19+20)

(b) NPV of the net cash flow stream @ 15% per discount rate

= -140 + 10.20 x PVIF(15%,1) + 20.55 x PVIF (15%,2)+ 31.46 x PVIF (15%,3) + 62.80 x PVIF (15%,4) + 49.25 x PVIF(15%,5)+ 35.94 x PVIF (15%,6) + 55 x PVIF (15%,7)

= Rs.1.70 million

3.


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(a) A. Initial outlay (Time 0)

i. Cost of new machine Rs. 3,000,000

ii. Salvage value of old machine 900,000iii Incremental working capital requirement 500,000iv. Total net investment (=i ii + iii) 2,600,000

B. Operating cash flow (years 1 through 5)

Year 1 2 3 4 5

i. Post-tax savings inmanufacturing costs 455,000 455,000 455,000 455,000 455,000

ii. Incrementaldepreciation 550,000 412,500 309,375 232,031 174,023

iii. Tax shield onincremental dep. 165,000 123,750 92,813 69,609 52,207

iv. Operating cashflow ( i + iii) 620,000 578,750 547,813 524,609 507,207

C. Terminal cash flow (year 5)

i. Salvage value of new machine Rs. 1,500,000

ii. Salvage value of old machine 200,000iii. Recovery of incremental working capital 500,000iv. Terminal cash flow ( i ii + iii) 1,800,000

D. Net cash flows associated with the replacement project (in Rs)

Year 0 1 2 3 4 5

NCF (2,600,000) 620000578750547813524609 307207

(b) NPV of the replacement project

= - 2600000 + 620000 x PVIF (14%,1)+ 578750 x PVIF (14%,2)+ 547813 x PVIF (14%,3)+ 524609 x PVIF (14%,4)+ 2307207 x PVIF (14%,5)


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= Rs.267849

4. Tax shield (savings) on depreciation (in Rs)

Depreciation Tax shield PV of tax shield

Year charge (DC) =0.4 x DC @ 15% p.a.

1 25000 10000 8696

2 18750 7500 5671

3 14063 5625 3699

4 10547 4219 2412

5 7910 3164 1573--------22051--------

Present value of the tax savings on account of depreciation = Rs.22051

5. A. Initial outlay (at time 0)

i. Cost of new machine Rs. 400,000ii. Salvage value of the old machine 90,000iii. Net investment 310,000

B. Operating cash flow (years 1 through 5)

Year 1 2 3 4 5

i. Depreciationof old machine 18000 14400 11520 9216 7373

ii. Depreciationof new machine 100000 75000 56250 42188 31641


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advance6. Interest on term loan 30.00 28.50 22.50 16.50 10.50 4.507. Profit before tax 48.67 77.94 102.46 120.81 135.04 146.53

8. Tax 24.335 38.97 51.23 60.405 67.52 73.2659. Profit after tax 24.335 38.97 51.23 60.405 67.52 73.26510. Preference dividend11. Net salvage value of fixed assets 20012. Net salvage value of current

assets- 40 40 40 40 40

13. Repayment of term-loans14. Redemption of preference capital15. Repayment of short-term bank

borrowings100

16. Retirement of trade creditors 5017. Initial investment (1) (100)18. Operating cash flows (9-10+4) 107.665 94.53 88.27 85.095 83.98 84.23519. Liquidation and retirement cash

flows (11+12-13-14-15-16)107.665 54.53 48.27 45.095 43.98 90

20. Net cash flows (17+18+19) (100) 107.665 54.53 48.27 45.095 43.98 174.235

Net Cash Flows Relating to Long-term Funds (Rs. in million)

Particulars Year

0 1 2 3 4 5 61. Fixed assets (250)2. Working capital margin (50)3. Revenues 500 500 500 500 500 5004. Operating costs 320 320 320 320 320 3205. Depreciation 83.33 55.56 37.04 24.69 16.46 10.976. Interest on working capital

advance18.00 18.00 18.00 18.00 18.00 18.00

7. Interest on term loan 30.00 28.50 22.50 16.50 10.50 4.508. Profit before tax 48.67 77.94 102.46 120.81 135.04 146.53

9. Tax @ 50% 24.335 38.97 51.23 60.405 67.52 73.26510. Profit after tax 24.335 38.97 51.23 60.405 67.52 73.26511. Net salvage value of fixed assets 80

12.Net recovery of working capitalmargin

50

13. Initial investment (1+2) (300)

14. Operating cash inflow (9+5+7(1-T) )

122.665 108.78 99.52 93.345 89.23 86.845

15. Terminal cash flow (11+12) 130.00

16. Net cash flow (13+14+15) (300) 122.665 108.78 99.52 93.345 89.23 216.485 Cash Flows Relating to Total Funds

(Rs. in million)

Year0 1 2 3 4 5 6

1. Total funds (450)2. Revenues 500 500 500 500 500 5003. Operating costs 320 320 320 320 320 3204. Depreciation 83.33 55.56 37.04 24.69 16.46 10.97

5. Interest on term loan 30.00 28.50 22.50 16.50 10.50 4.50


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6. Interest on working capitaladvance

18.00 18.00 18.00 18.00 18.00 18.00

7. Profit before tax 48.67 77.94 102.46 120.81 135.04 146.53

8. Tax 24.34 38.97 51.23 60.41 67.52 73.2659. Profit after tax 24.34 38.97 51.23 60.41 67.52 73.26510. Net salvalue of fixed assets 8011. Net salvage value of current assets 20012. Initial investment (1) (450)

13. Operating cash inflow 9+4+6 (1-t)+ 5(1-t)

131.67 117.78 108.52 102.35 98.23 95.485

14. Terminal cash flow (10+11) 28015. Net cash flow (12+13+14) (450) 131.67 117.78 108.52 102.35 98.23 375.485

Chapter 10

THE COST OF CAPITAL

1(a) Define rD as the pre-tax cost of debt. Using the approximate yield formula, rDcan be calculated as follows:

14 + (100 108)/10


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rD = ------------------------ x 100 = 12.60%0.4 x 100 + 0.6x108

(b) After tax cost = 12.60 x (1 0.35) = 8.19%

2. Define rp as the cost of preference capital. Using the approximate yield formularp can be calculated as follows:

9 + (100 92)/6rp = --------------------

0.4 x100 + 0.6x92

= 0.1085 (or) 10.85%

3. WACC= 0.4 x 13% x (1 0.35)+ 0.6 x 18%

= 14.18%

4. Cost of equity = 10% + 1.2 x 7% = 18.4%(using SML equation)Pre-tax cost of debt = 14%After-tax cost of debt = 14% x (1 0.35) = 9.1%Debt equity ratio = 2 : 3WACC = 2/5 x 9.1% + 3/5 x 18.4%

= 14.68%

5. Given0.5 x 14% x (1 0.35) + 0.5 x rE = 12%

where rE is the cost of equity capital.Therefore rE 14.9%Using the SML equation we get

11% + 8% x = 14.9%where denotes the beta of Azeezs equity.

Solving this equation we get = 0.4875.

6 (a) The cost of debt of 12% represents the historical interest rate at the time the debtwas originally issued. But we need to calculate the marginal cost of debt (costof raising new debt); and for this purpose we need to calculate the yield tomaturity of the debt as on the balance sheet date. The yield to maturity will notbe equal to 12% unless the book value of debt is equal to the market value ofdebt on the balance sheet date.


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(b) The cost of equity has been taken asD1/P0 ( = 6/100) whereas the cost of equityis (D1/P0) + gwhere grepresents the expected constant growth rate in dividendper share.

7. The book value and market values of the different sources of finance areprovided in the following table. The book value weights and the market valueweights are provided within parenthesis in the table.

(Rs. in million)

Source Book value Market value

Equity 800 (0.54) 2400 (0.78)

Debentures first series 300 (0.20) 270 (0.09)Debentures second series 200 (0.13) 204 (0.06)Bank loan 200 (0.13) 200 (0.07)Total 1500 (1.00) 3074 (1.00)

8.

(a) GivenrD x (1 0.3) x 4/9 + 20% x 5/9 = 15%rD = 12.5%,where rD represents the pre-tax cost of debt.

(b) Given13% x (1 0.3) x 4/9 + rE x 5/9 = 15%rE = 19.72%, where rErepresents the cost of equity.

9. Cost of equity = D1/P0 +g

= 3.00 / 30.00 + 0.05= 15%

(a) The first chunk of financing will comprise of Rs.5 million of retained earningscosting 15 percent and Rs.25 million of debt costing 14 (1-.3) = 9.8 percent.The second chunk of financing will comprise of Rs.5 million of additionalequity costing 15 percent and Rs.2.5 million of debt costing 15 (1-.3) = 10.5percent.

(b) The marginal cost of capital in the first chunk will be :5/7.5 x 15% + 2.5/7.5 x 9.8% = 13.27%The marginal cost of capital in the second chunk will be :

5/7.5 x 15% + 2.5/7.5 x 10.5% = 13.50%

Note : We have assumed that(i) The net realisation per share will be Rs.25, after floatation costs, and(ii) The planned investment of Rs.15 million is inclusive of floatation costs


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10. The cost of equity and retained earningsrE = D1/PO +g

= 1.50 / 20.00 + 0.07 = 14.5%

The cost of preference capital, using the approximate formula, is :11 + (100-75)/10

rE = = 15.9%0.6x75 + 0.4x100

The pre-tax cost of debentures, using the approximate formula, is :13.5 + (100-80)/6

rD = = 19.1%0.6x80 + 0.4x100

The post-tax cost of debentures is19.1 (1-tax rate) = 19.1 (1 0.5)

= 9.6%The post-tax cost of term loans is

12 (1-tax rate) = 12 (1 0.5)= 6.0%

The average cost of capital using book value proportions is calculated below:

Source of capital Component Book value Book value Product of

cost Rs. in million proportion (1) & (3)(1) (2) (3)

Equity capital 14.5% 100 0.28 4.06Preference capital 15.9% 10 0.03 0.48Retained earnings 14.5% 120 0.33 4.79

Debentures 9.6% 50 0.14 1.34Term loans 6.0% 80 0.22 1.32

360 Average cost 11.99%capital

The average cost of capital using market value proportions is calculated below :

Source of capital Component Market value Market value Product of

cost Rs. in million(1) (2) (3) (1) & (3)

Equity capitaland retained earnings 14.5% 200 0.62 8.99Preference capital 15.9% 7.5 0.02 0.32Debentures 9.6% 40 0.12 1.15Term loans 6.0% 80 0.24 1.44


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327.5 Average cost 11.90%capital

11.(a) WACC= 1/3 x 13% x (1 0.3)

+ 2/3 x 20%= 16.37%

(b) Weighted average floatation cost= 1/3 x 3% + 2/3 x 12%

= 9%

(c) NPV of the proposal after taking into account the floatation costs= 130 x PVIFA (16.37%, 8) 500 / (1 - 0.09)= Rs.8.51 million

Chapter 11

RISK ANALYSIS OF SINGLE INVESTMENTS

1.(a) NPV of the project = -250 + 50 x PVIFA (13%,10)

= Rs.21.31 million

(b) NPVs under alternative scenarios:(Rs. in million)


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Pessimistic Expected Optimistic

Investment 300 250 200

Sales 150 200 275Variable costs 97.5 120 154Fixed costs 30 20 15Depreciation 30 25 20Pretax profit - 7.5 35 86Tax @ 28.57% - 2.14 10 24.57Profit after tax - 5.36 25 61.43Net cash flow 24.64 50 81.43

Cost of capital 14% 13% 12%

NPV - 171.47 21.31 260.10

Assumptions: (1) The useful life is assumed to be 10 years under all threescenarios. It is also assumed that the salvage value of theinvestment after ten years is zero.

(2) The investment is assumed to be depreciated at 10% perannum; and it is also assumed that this method and rate ofdepreciation are acceptable to the IT (income tax)authorities.

(3) The tax rate has been calculated from the given table i.e. 10/ 35 x 100 = 28.57%.

(4) It is assumed that only loss on this project can be offsetagainst the taxable profit on other projects of the company;and thus the company can claim a tax shield on the loss inthe same year.

(c) Accounting break even point (under expected scenario)

Fixed costs + depreciation = Rs. 45 millionContribution margin ratio = 60 / 200 = 0.3Break even level of sales = 45 / 0.3 = Rs.150 millionFinancial break even point (under expected scenario)

i. Annual net cash flow = 0.7143 [ 0.3 x sales 45 ] + 25= 0.2143 sales 7.14

ii. PV (net cash flows) = [0.2143 sales 7.14 ] x PVIFA (13%,10)


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= 1.1628 sales 38.74

iii. Initial investment = 200

iv. Financial break even levelof sales = 238.74 / 1.1628 = Rs.205.31 million

2.(a) Sensitivity of NPV with respect to quantity manufactured and sold:

(in Rs)


Initial investment 30000 30000 30000Sale revenue 24000 42000 54000Variable costs 16000 28000 36000Fixed costs 3000 3000 3000Depreciation 2000 2000 2000Profit before tax 3000 9000 13000Tax 1500 4500 6500

Profit after tax 1500 4500 6500Net cash flow 3500 6500 8500NPV at a cost ofcapital of 10% p.aand useful life of5 years -16732 - 5360 2222

(b) Sensitivity of NPV with respect to variations in unit price.


Initial investment 30000 30000 30000Sale revenue 28000 42000 70000Variable costs 28000 28000 28000Fixed costs 3000 3000 3000Depreciation 2000 2000 2000

Profit before tax -5000 9000 37000Tax -2500 4500 18500Profit after tax -2500 4500 18500Net cash flow - 500 6500 20500NPV - 31895 (-) 5360 47711

(c) Sensitivity of NPV with respect to variations in unit variable cost.



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Initial investment 30000 30000 30000Sale revenue 42000 42000 42000

Variable costs 56000 28000 21000Fixed costs 3000 3000 3000Depreciation 2000 2000 2000Profit before tax -11000 9000 16000Tax -5500 4500 8000Profit after tax -5500 4500 8000Net cash flow -3500 6500 10000NPV -43268 - 5360 7908

(d) Accounting break-even point

i. Fixed costs + depreciation = Rs.5000ii. Contribution margin ratio = 10 / 30 = 0.3333iii. Break-even level of sales = 5000 / 0.3333

= Rs.15000

Financial break-even point

i. Annual cash flow = 0.5 x (0.3333 Sales 5000) = 2000ii. PV of annual cash flow = (i) x PVIFA (10%,5)

= 0.6318 sales 1896iii. Initial investment = 30000iv. Break-even level of sales = 31896 / 0.6318 = Rs.50484

2. DefineAtas the random variable denoting net cash flow in yeart.

A1 = 4 x 0.4 + 5 x 0.5 + 6 x 0.1= 4.7

A2 = 5 x 0.4 + 6 x 0.4 + 7 x 0.2= 5.8

A3 = 3 x 0.3 + 4 x 0.5 + 5 x 0.2= 3.9

NPV = 4.7 / 1.1 +5.8 / (1.1)2 + 3.9 / (1.1)3 10= Rs.2.00 million

12 = 0.41

22 = 0.56

32

= 0.49


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12 22 32

2NPV = + +(1.1)2 (1.1)4 (1.1)6

= 1.00

(NPV) = Rs.1.00 million

3. Expected NPV4 At

= - 25,000 t=1 (1.08)t

= 12,000/(1.08) + 10,000 / (1.08)2 + 9,000 / (1.08)3

+ 8,000 / (1.08)4 25,000

= [ 12,000 x .926 + 10,000 x .857 + 9,000 x .794 + 8,000 x .735]- 25,000

= 7,708

Standard deviation of NPV

4 t t=1 (1.08)t

= 5,000 / (1.08) + 6,000 / (1.08)2 + 5,000 / (1,08)3 + 6,000 / (1.08)4

= 5,000 x .926 + 6,000 x .857 + 5000 x .794 + 6,000 x .735

= 18,152

4. Expected NPV4 At

= - 25,000 . (1) t=1 (1.06)t

A1 = 2,000 x 0.2 + 3,000 x 0.5 + 4,000 x 0.3= 3,100

A2 = 3,000 x 0.4 + 4,000 x 0.3 + 5,000 x 0.3= 3,900

A3 = 4,000 x 0.3 + 5,000 x 0.5 + 6,000 x 0.2= 4,900

A4 = 2,000 x 0.2 + 3,000 x 0.4 + 4,000 x 0.4


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= 3,200Substituting these values in (1) we get

Expected NPV = NPV

= 3,100 / (1.06)+ 3,900 / (1.06)2 + 4,900 / (1.06)3 + 3,200 / (1,06)4

- 10,000 = Rs.3,044

The variance of NPV is given by the expression

4 2t2 (NPV) = .. (2)

t=1 (1.06)2t

12 = [(2,000 3,100)2 x 0.2 + (3,000 3,100)2 x 0.5+ (4,000 3,100)2 x 0.3]

= 490,000

22 = [(3,000 3,900)2 x 0.4 + (4,000 3,900)2 x 0.3+ (5,000 3900)2 x 0.3]

= 690,000

32 = [(4,000 4,900)2 x 0.3 + (5,000 4,900)2 x 0.5+ (6,000 4,900)2 x 0.2]

= 490,000

42 = [(2,000 3,200)2 x 0.2 + (3,000 3,200)2 x 0.4+ (4,000 3200)2 x 0.4]

= 560,000

Substituting these values in (2) we get490,000 / (1.06)2 + 690,000 / (1.06)4

+ 490,000 / (1.06)6 + 560,000 / (1.08)8

[ 490,000 x 0.890 + 690,000 x 0.792+ 490,000 x 0.705 + 560,000 x 0.627 ]= 1,679,150

NPV = 1,679,150 = Rs.1,296

NPV NPV 0 -NPVProb (NPV < 0) = Prob. 0.2 x 10,000)Prob (NPV > 2,000)

Prob (NPV >2,000)= Prob (Z> 2,000- 3,044 / 1,296)Prob (Z> - 0.81)

The required probability is given by the shaded area of the following normalcurve:P(Z> - 0.81) = 0.5 +P(-0.81


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= [ 0.5Q (PV) 500] x PVIFA (10,5) 30,000= [0.5Q (PV) 500] x 3.791 30,000

= 1.8955Q (PV) 31,895.5

Exhibit 1 presents the correspondence between the values of exogenous variablesand the two digit random number. Exhibit 2 shows the results of the simulation.

Exhibit 1Correspondence between values of exogenous variables and

two digit random numbers

QUANTITY PRICE VARIABLE COST

Value ProbCumulative

Prob.

Two digit

randomnumbers Value Prob

CumulativeProb.

Two digit

randomnumbers Value Prob

Cumu-lative

Prob.

Two digit

randomnumbers

800 0.10 0.10 00 to 09 20 0.40 0.40 00 to 39 15 0.30 0.30 00 to 29

1,000 0.10 0.20 10 to 19 30 0.40 0.80 40 to 79 20 0.50 0.80 30 to 79

1,200 0.20 0.40 20 to 39 40 0.10 0.90 80 to 89 40 0.20 1.00 80 to 99

1,400 0.30 0.70 40 to 69 50 0.10 1.00 90 to 991,600 0.20 0.90 70 to 89

1,800 0.10 1.00 90 to 99

Exhibit 2Simulation Results

QUANTITY (Q) PRICE (P) VARIABLE COST (V) NPV

Run Random

Number

Corres-

pondingValue

Random

Number

Corres-

pondingvalue

Random

Number

Corres-

pondingvalue

1.8955 Q(P-V)-31,895.5

1 03 800 38 20 17 15 -24,314

2 32 1,200 69 30 24 15 2,224

3 61 1,400 30 20 03 15 -18,627

4 48 1,400 60 30 83 40 -58,433

5 32 1,200 19 20 11 15 -20,523

6 31 1,200 88 40 30 20 13,597

7 22 1,200 78 30 41 20 -9,150

8 46 1,400 11 20 52 20 -31,896


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9 57 1,400 20 20 15 15 -18,627

QUANTITY (Q) PRICE (P) VARIABLE COST (V) NPV

Run Random

Number

Corres-

pondingValue

Random

Number

Corres-

pondingvalue

Random

Number

Corres-

pondingvalue

1.8955 Q(P-V)-31,895.5

10 92 1,800 77 30 38 20 2,224

11 25 1,200 65 30 36 20 -9,150

12 64 1,400 04 20 83 40 -84,970

13 14 1,000 51 30 72 20 -12,941

14 05 800 39 20 81 40 -62,224

15 07 800 90 50 40 20 13,597

16 34 1,200 63 30 67 20 -9,150

17 79 1,600 91 50 99 40 -1,568

18 55 1,400 54 30 64 20 -5,35919 57 1,400 12 20 19 15 -18,627

20 53 1,400 78 30 22 15 7,910

21 36 1,200 79 30 96 40 -54,642

22 32 1,200 22 20 75 20 -31,896

23 49 1,400 93 50 88 40 -5,359

24 21 1,200 84 40 35 20 13,597

25 08 .800 70 30 27 15 -9,150

26 85 1,600 63 30 69 20 -1,568

27 61 1,400 68 30 16 15 7,91028 25 1,200 81 40 39 20 13,597

29 51 1,400 76 30 38 20 -5,359

30 32 1,200 47 30 46 20 -9,150

31 52 1,400 61 30 58 20 -5,359

32 76 1,600 18 20 41 20 -31,896

33 43 1,400 04 20 49 20 -31,896

34 70 1,600 11 20 59 20 -31,896

35 67 1,400 35 20 26 15 -18,627

36 26 1,200 63 30 22 15 2,224QUANTITY (Q) PRICE (P) VARIABLE COST (V) NPV

Run Random

Number

Corres-

pondingValue

Random

Number

Corres-

pondingvalue

Random

Number

Corres-

pondingvalue

1.8955 Q(P-V)-31,895.5

37 89 1,600 86 40 59 20 28,761

38 94 1,800 00 20 25 15 -14,836

39 09 .800 15 20 29 15 -24,314

40 44 1,400 84 40 21 15 34,447

41 98 1,800 23 20 79 20 -31,896

42 10 1,000 53 30 77 20 -12,94143 38 1,200 44 30 31 20 -9,150

44 83 1,600 30 20 10 15 -16,732

45 54 1,400 71 30 52 20 -5,359

46 16 1,000 70 30 19 15 -3,463

47 20 1,200 65 30 87 40 -54,642

48 61 1,400 61 30 70 20 -5,359

49 82 1,600 48 30 97 40 -62,224

50 90 1,800 50 30 43 20 2,224


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ExpectedNPV = NPV50

= 1/ 50 NPVii=1

= 1/50 (-7,20,961)= 14,419

50

Variance of NPV = 1/50 (NPVi NPV)2

i=1

= 1/50 [27,474.047 x 106]= 549.481 x 106

Standard deviation of NPV = 549.481 x 106

= 23,441

6. To carry out a sensitivity analysis, we have to define the range and the most likely

values of the variables in the NPV Model. These values are defined below

Variable Range Most likely value

I Rs.30,000 Rs.30,000 Rs.30,000k 10% - 10% 10%F Rs.3,000 Rs.3,000 Rs.3,000D Rs.2,000 Rs.2,000 Rs.2,000

T 0.5 0.5 0.5N 5 5 5S 0 0 0Q Can assume any one of the values - 1,400*

800, 1,000, 1,200, 1,400, 1,600 and 1,800P Can assume any of the values 20, 30, 30**

40 and 50V Can assume any one of the values 20*

15,20 and 40----------------------------------------------------------------------------------------* The most likely values in the case ofQ,Pand Vare the values that

have the highest probability associated with them

** In the case of price, 20 and 30 have the same probability ofoccurrence viz., 0.4. We have chosen 30 as the most likely valuebecause the expected value of the distribution is closer to 30


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Sensitivity Analysis with Reference to Q

The relationship between Q and NPV given the most likely values of other

variables is given by5 [Q (30-20) 3,000 2,000] x 0.5 + 2,000 0

NPV = + - 30,000 t=1 (1.1)t (1.1)5

5 5Q - 500

= - 30,000 t=1 (1.1)t

The net present values for various values ofQ are given in the following table:

Q 800 1,000 1,200 1,400 1,600 1,800NPV -16,732 -12,941 -9,150 -5,359 -1,568 2,224

Sensitivity analysis with reference to P

The relationship betweenPand NPV, given the most likely values of othervariables is defined as follows:

5 [1,400 (P-20) 3,000 2,000] x 0.5 + 2,000 0

NPV = + - 30,000 t=1 (1.1)t (1.1)5

5 700P 14,500

= - 30,000 t=1 (1.1)t

The net present values for various values ofPare given below :P(Rs) 20 30 40 50

NPV(Rs) -31,896 -5,359 21,179 47,716

8. NPV - 5 0 5 10 15 20(Rs.in lakhs)PI 0.9 1.00 1.10 1.20 1.30 1.40

Prob. 0.02 0.03 0.10 0.40 0.30 0.15

6


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Expected PI = PI= (PI)jPj j=1= 1.24

6

Standard deviation = (PIj - PI)2Pjo fP1 j=1

= .01156= .1075

The standard deviation ofP1 is .1075 for the given investment with an expectedPI of 1.24. The maximum standard deviation of PI acceptable to the company

for an investment with an expected PI of 1.25 is 0.30.

Since the risk associated with the investment is much less than the maximumrisk acceptable to the company for the given level of expected PI, the companyshould accept the investment.

9. Investment AOutlay : Rs.10,000

Net cash flow : Rs.3,000 for 6 yearsRequired rate of return: 12%

NPV(A) = 3,000 x PVIFA (12%, 6 years) 10,000= 3,000 x 4.11 10,000 = Rs.2,333

Investment BOutlay : Rs.30,000

Net cash flow : Rs.11,000 for 5 yearsRequired rate of return: 14%NPV(B) = 11,000 x PVIFA (14%, 5 years) 30,000

= Rs.7763

10. The NPVs of the two projects calculated at their risk adjusted discount rates areas follows:

6 3,000

ProjectA: NPV = - 10,000 = Rs.2,333t=1 (1.12)t

5 11,000

ProjectB: NPV = - 30,000 = Rs.7,763t=1 (1.14)t

PIand IRR for the two projects are as follows:


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Project A B

PI 1.23 1.26IRR 20% 24.3%

B is superior toA in terms of NPV, PI, and IRR. Hence the company mustchooseB.

Chapter 12

RISK ANALYSIS OF SINGLE INVESTMENTS

1. 2p = wiwjijij

2

p = w

2

12

1 + w

2

22

2 + w

2

32

3 + w

2

42

4 + w

2

52

5

+ 2 w1 w2 12 12 + 2 w1 w3 13 13 + 2 w1 w4 14 14 + 2 w1 w5 1515 + 2 w2 w3 23 23 + 2 w2 w4 24 24 + 2 w2 w5 25 25 + 2 w3 w434 34 + 2 w3 w5 35 35 + 2 w4 w5 45 45

= 0.12 x 82 + 0.22 x 92 + 0.32 x 102 + 0.32 x 162 + 0.12 x 122 + 2 x 0.1 x 0.2 x 0.1 x 8 x 9 + 2 x 0.1 x 0.3 x 0.5 x 8 x 10

+ 2 x 0.1 x 0.3 x 0.2 x 8 x 16 + 2 x 0.1 x 0.1 x 0.3 x 8 x 12


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+ 2 x 0.2 x 0.3 x 0.4 x 9 x 10 + 2 x 0.2 x 0.3 x 0.8 x 9 x 16+ 2 x 0.2 x 0.1 x 0.2 x 9 x 12 + 2 x 0.3 x 0.3 x0.1 x 10 x 16+ 2 x 0.3 x 0.1 x 0.6 x 10 x 12 + 2 x 0.3 x 0.1 x 0.1 x 16 x 12

= 66.448p = (66.448)1/2 = 8.152

2. (i) Since there are 3 securities, there are 3 variance terms and 3 covariance terms.Note that if there are n securities the number of covariance terms are: 1 + 2 ++

(n + 1) = n (n 1)/2. In this problem all the variance terms are the same (2A) allthe covariance terms are the same (AB) and all the securities are equally weighted( wA = )

So,

2p = [3 w2A 2A + 2 x 3 AB]2p = [3 w2A 2A + 6 wA wB AB]

1 2 1 1

= 3 x x 2A + 6 x x x AB3 3 3

1 2

= 2A + AB3 3

(ii) Since there are 9 securities, there are 9 variance terms and 36 covarianceterms. Note that if the number of securities is n, the number of covarianceterms is n(n 1)/2.

In this case all the variance terms are the same (2A), all the covariance terms are1

the same (AB) and all the securities are equally weighted wA =9

So,n(n-1)

2p = 9 w2A 2A t 2 x wA wB AB2

1 2 1 1

= 9 x x 2A + 9(8) x x AB9 9 9

1 72

= 2A + AB9 81

3. The beta for stock B is calculated below:

Period Return of Return on Deviation of Deviation Product of Square of


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stock B,

RB (%)

market

portfolio,

RM(%)

return on

stock B from

its mean

(RB - RB)

of return

on market

portfolio

from itsmean

(RM RM)

the

deviation

(RB RB)

(RM RM)

the

deviation

of return

on marketportfolio,

from its

mean

(RM RM)2

1 15 9 6 -1 -6 12 16 12 7 2 14 43 10 6 1 -4 -4 16

4 -15 4 -24 -6 144 365 -5 16 -14 6 -84 366 14 11 5 1 5 17 10 10 1 0 0 08 15 12 6 2 12 49 12 9 3 -1 -3 110 -4 8 -13 -2 26 411 -2 12 -11 2 -22 4

12 12 14 3 4 12 1613 15 -6 6 -16 -96 25614 12 2 3 -8 -24 6415 10 8 1 -2 -2 416 9 7 0 -3 0 917 12 9 3 -1 -3 118 9 10 0 0 0 019 22 37 13 27 351 729

20 13 10 4 0 0 0180 200 (RB RB) (RB RB)2

RB = 180 RM= 200 (RM RM) = 1186RB = 9% RM= 10% = 320

Beta of stock B is equal to:

Cov (RB,RM)

2M (RB -RB) (RMRM) 320

Cov (RB,RM) = = = 16.84n 1 19

(RMRM)2 1186

2M = = = 62.42

n 1 19


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So the beta for stockB is:16.84

= 0.27062.42

4. According to the CAPM, the required rate of return is:

E(Ri) =Rf+ (E(RMRf)i

Given a risk-free rate (Rf) of 11 percent and the expected market risk premium(E(RMRf) of 6 percent we get the following:

Project Beta Required rate(%) Expected rate (%)A 0.5 11 + 0.5 x 6 = 14 15B 0.8 11 + 0.8 x 6 = 15.8 16C 1.2 11 + 1.2 x 6 = 18.2 21D 1.6 11 + 1.6 x 6 = 20.6 22E 1.7 11 + 1.7 x 6 = 21.2 23

a. The expected return of all the 5 projects exceeds the required rate as per the CAPM.

So all of them should be accepted.b. If the cost of capital of firm which is 16 percent is used as the hurdle rate, project A

will be rejected incorrectly.

5. The asset beta is linked to equity beta, debt-equity ratio, and tax rate as follows:

EA =

[1 +D/E(1 T)]

The asset beta of A, B, and C is calculated below:

Firm Asset Beta

1.25A = 0.49

[1 + (2.25) x 0.7]

1.25

B = 0.48[1 + (2.00) x 0.7]

1.10C = 0.45

[1 + (2.1) x 0.7]


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0.49 + 0.48 + 0.45Average of the asset betas of sample firms = = 0.47

3The equity beta of the cement project is

E= A [ 1 +D/E(1 T)]= 0.47 [1 + 2 (1-0.3)] = 1.128

As per the CAPM model, the cost of equity of the proposed project is:12% + (17% - 12%) x 1.128 = 17.64%

The post-tax cost of debt is:16% (1 0.3) = 11.2%

The required rate of return for the project given a debt-equity ratio of 2:1 is:1/3 x 17.64% + 2/3 x 11.2% = 13.35%

6. EA =

[1 +D/E(1 T)]E = 1.25 D/E= 1.6 T= 0.3

So, Pariman Companys asset beta is:1.25

= 0.59[1 + 1.6 (0.7)]

7. (a) Asset beta for a petrochemicals project is:

E 1.30A = =

[1 +D/E( 1 T)] [1 + 1.5 (1 .4)]

= 0.68

The equity beta (systematic risk) for the petrochemicals project of Growmore,whenD/E= 1.25 and T= 0.4, is

0.68 [1 + 1.25 (1 .4)] = 1.19


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(b) The cost of equity for the petrochemicals project is12% + 1.19 (18% - 12%) = 19.14%

The cost of debt is

12% (1 0.4) = 7.2%Given, a debt equity ratio of 1.25 the required return for the petrochemicals

project is1 1.25

19.14% x + 7% x = 12.4%2.25 2.25

Chapter 13

SPECIAL DECISION SITUATIONS

1. PV Cost UAE =

PVIFAr,n

Cost of plastic emulsion painting = Rs.3,00,000 Life = 7 yearsCost of distemper painting = Rs. 1,80,000 Life = 3 yearsDiscount rate = 10%UAEof plastic emulsion painting = Rs.3,00,000 / 4.868 = Rs.61,627


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UAEof distemper painting = Rs.1,80,000 / 2.487 = Rs.72,376

Since plastic emulsion painting has a lowerUAE, it is preferable.

2. Present value of the operating costs :3,00,000 3,60,000 4,00,000 4,50,000 5,00,000

= + + + +1.13 (1.13)2 (1.13)3 (1.13)4 (1.13)5

= Rs.1,372,013Present value of salvage value = 3,00,000 / (1.13)5 = Rs.162,828

Present value of costs of internal transportation = 1,500,000 1,372,013system 162,828 = Rs.27,09,185UAEof the internal transportation system = 27,09,185 / 3.517 = Rs.7,70,311

3. Cost of standard overhaul = Rs.500,000Cost of less costly overhaul = Rs.200,000Cost of capital = 14%UAEof standard overhaul = 500,000 / 3.889 = Rs.128,568

UAEof less costly overhaul = 200,000 / 1.647 = Rs.121,433

Since the less costly overhaul has a lowerUAE, it is the preferred alternative

4. The details for the two alternatives are shown below :

Gunning plow Counter plow

1. Initial outlay Rs.2,500,000 Rs.1,500,0002. Economic life 12 years 9 years3. Annual operating and maintenance costs Rs.250,000 Rs.320,0004. Present value of the stream of operating and

maintenance costs at 12% discount rateRs.1,548,500 Rs.1,704,960

5. Salvage value Rs.800,000 Rs.500,0006. Present value of salvage value Rs.205,600 Rs.180,5007. Present value of total costs (1+4-6) Rs.3,842,900 Rs.3,024,460


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8. UAEof 7 Rs.3,842,900PVIFA (12%,12)= 3,842,900

6.194= Rs.620,423

Rs.3,024,460PVIFA (12%,9)= 3,024,460

5.328= Rs.567,654

The Counter plow is a cheaper alternative

5. The current value of different timing options is given below :

Time Net Future Value Current Value

Rs. in millionRs. in million0 10 101 15 13.3952 19 15.1433 23 16.3764 26 16.536

The optimal timing of the project is year 4.

6. Calculation of UAE (OM) for Various Replacement Periods (Rupees)

Time

(t)

Operating

and

maintenance

costs

Post-tax

operating &

maintenance

costs

PVIF

(12%,t)Present

value

of(3)

Cumulative

present

value

PVIFA

(12%,t)UAE

(OM)

(1) (2) (3) (4) (5) (6) (7) (8)

1 20,000 12,000 0.893 10,716 10,716 0.893 12,0002 25,000 15,000 0.797 11,955 22,671 1.690 13,4153 35,000 21,000 0.712 14,952 37,623 2.402 15,6634 50,000 30,000 0.636 19,080 56,703 3.037 18,6715 70,000 42,000 0.567 23,814 80,517 3.605 22,335

Calculation of UAE (IO) for Various Replacement Periods

Time (t) Investment Outlay Rs. PVIFA ( 12%, t) UAE of investment outlay Rs.1 80,000 0.893 89,586

2 80,000 1.690 47,3373 80,000 2.402 33,3064 80,000 3.037 26,3425 80,000 3.605 22,191

Calculation of UAE (DTS) for Various Replacement PeriodsTime

(t)Depreciation

charge R.s.Depreciation

tax shieldPVIF

(12%, t)PV of

depreciation

tax shield Rs..

Cumulativepresent

value Rs..

PVIFA(12%, t)

UAE ofdepreciation

tax shield Rs..


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(1) (2) (3) (4) (5) (6) (7) (8)1 20,000 8,000 0.893 7,144 7,144 0.893 8,0002 15,000 6,000 0.797 4,782 11,926 1.690 7,0573 11,250 4,500 0.712 3,204 15,130 2.402 6,2994 8,438 3,375 0.636 2,147 17,277 3.037 5,6895 6,328 2,531 0.567 1,435 18,712 3.605 5,191

Calculation of UAE (SV) for Various Replacement Periods

Time Salvage

value Rs.

PVIF

(12%, t)Present value of

salvage value Rs.

PVIFA

(12%, t)UAE of salvage

value Rs. (4) / (5)(1) (2) (3) (4) (5) (6)1 60,000 0.893 53,580 0.893 60,000

2 45,000 0.797 35,865 1.690 21,2223 32,000 0.712 22,784 2.402 9,4854 22,000 0.636 13,992 3.037 4,6075 15,000 0.567 8,505 3.605 2,359

Summary of Information Required to Determine the Economic Life

Replacement

period

UAE

(OM) Rs.

UAE (IO)

Rs.

UAE

(DTS) Rs.

UAE (SV)

Rs.

UAE

(CC) Rs.

UAE (TC)

Rs.

(1) (2) (3) (4) (5) (6) (7)1 12,000 89,586 8,000 60,000 21,586 33,5862 13,415 47,337 7,057 21,222 19,058 32,4733 15,663 33,306 6,299 9,485 17,522 33,1854 18,671 26,342 5,689 4,607 16,046 34,7175 22,335 22,191 5,190 2,359 14,642 36,977

OM - Operating and Maintenance Costs

IO - Investment OutlayDTS - Depreciation Tax ShieldSV - Salvage ValueCC - Capital CostTC - Total CostUAE (CC) = UAE (IO) [UAE (DTS) + UAE (SV)]UAE (TC) = UAE (OM) + UAE (CC)

7. Calculation of UAE (OM) for Various Replacement periodsTime O&M costsRs.

Post-tax

O&M costs

Rs.

PVIF

(12%,t)PV of post-

tax O&M

costs Rs.

Cumulative

present

value Rs.

PVIFA

(12%, t)UAE of

O&M

costs Rs.

(1) (2) (3) (4) (5) (6) (7) (8)1 800,000 560,000 0.893 500,080 500,080 0.893 560,0002 1,000,000 700,000 0.797 557,900 1,057,980 1.690 626,0243 1,300,000 910,000 0.712 647,920 1,705,9000 2.402 710,2004 1,900,000 1,330,000 0.636 845,880 2,551,780 3.037 840,230


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5 2,800,000 1,960,000 0.567 1,111,320 3,663,100 3.605 1,016,117

Calculation of UAE (IO) for Various Replacement Periods

Time Investment outlay Rs. PVIFA ( 12%, t) UAE of investment outlay Rs.1 4,000,000 0.893 4,479,2832 4,000,000 1.690 2,366,8643 4,000,000 2.402 1,665,2794 4,000,000 3.037 1,317,0895 4,000,000 3.605 1,109,570

Calculation of UAE (DTS) for Various Replacement Periods

Time(t)

Depreciationcharge Rs.

Depreciatontax shield

Rs.

PVIF(12%, t)

PV ofdepreciationtax shield Rs.

Cumulativepresentvalue Rs.

PVIFA(12%, t)

UAE ofdepreciationtax shield Rs.

1 1,000,000 300,000 0.893 267,940 267,900 0.893 300,0002 750,000 225,000 0.797 179,325 447,225 1.690 264,6303 562,500 168,750 0.712 120,150 567,375 2.402 236,2094 421,875 126,563 0.636 80,494 647,869 3.037 213,3255 316,406 94,922 0.567 53,821 701,690 3.605 194,643

Calculation of UAE (SV) for Various Replacement Peiods

Time Salvage

value Rs.

PVIF

(12%, t)Present value of

salvage value Rs.

PVIFA

(12%, t)UAE of salvage

value Rs. (4)/ (5)(1) (2) (3) (4) (5) (6)1 2,800,000 0.893 267,900 0.893 2,800,0002 2,000,000 0.797 1,594,000 1.690 943,1953 1,400,000 0.712 996,80 2.402 414,9884 1,000,000 0.636 636,000 3.037 209,4175 800,000 0.567 453,600 3.605 125,825

Summary of Information Required to Determine the Economic Life

Replacement

period

UAE

(OM)

Rs.

UAE (IO)

Rs.

UAE

(DTS)

Rs.

UAE (SV)

Rs.

UAE (CC)

Rs.

UAE (TC)

Rs.1 560,000 4,479,283 300,000 2,800,000 (-)1,379,283 -819,2832 626,024 2,366,864 264,630 943,195 1,159,039 1,785,0633 710,200 1,665,279 236,209 414,988 1,014,082 1,724,2824 840,230 1,317,089 213,325 209,417 894,347 1,734,5775 1,016,117 1,109,570 194,643 125,825 789,102 1,805,219

The economic life of the well-drilling machine is 3 years


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8. Adjusted cost of capital as per Modigliani Miller formula:r* = r(1 TL)r* = 0.16 (1 0.5 x 0.6)

= 0.16 x 0.7 = 0.112Adjusted cost of capital as per Miles Ezzell formula:

1 + rr* = rLrDT

1 + rD1 + 0.16

= 0.16 0.6 x 0.15 x 0.5 x1 + 0.15

= 0.115

9.a. Base case NPV = -12,000,000 + 3,000,000 x PVIFA (20%, b)

= -12,000,000 + 3,000,000 x 3,326= - Rs.2,022,000

b. Adjusted NPV = Base case NPV Issue cost + Present value of tax shield.

Term loan = Rs.8 million Equity finance = Rs.4 millionIssue cost of equity = 12%Rs.4,000,000

Equity to be issued = = Rs.4,545,4550.88

Cost of equity issue = Rs.545,455

Computation of Tax Shield Associated with Debt Finance

Year (t) Debt outstanding

at the beginning

Rs.

Interest

Rs.

Tax shield

Rs.

Present value of

tax shield

Rs.

1 8,000,000 1,440,000 432,000 366,1022 8,000,000 1,440,000 432,000 310,2563 7,000,000 1,260,000 378,000 230,0624 6,000,000 1,080,000 324,000 167,1165 5,000,000 900,000 270,000 118,0196 4,000,000 720,000 216,000 80,013

1,271,568


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Adjusted NPV = - Rs.2,022,000 Rs.545,455 + Rs.1,271,568= - Rs.1,295,887

Adjusted NPV if issue cost alone is considered = Rs.2,567,455Present Value of tax shield of debt finance = Rs.1,271,568

10.a. Base Case NPV = - 8,000,000 + 2,000,000 x PVIFA (18%, 6)

= - 8,000,000 + 2,000,000 x 3,498= - Rs.1,004,000

b. Adjusted NPV = Base case NPV Issue cost + Present value of tax shield.Term loan = Rs.5 millionEquity finance = Rs.3 millionIssue cost of equity = 10%

Rs.3,000,000Hence, Equity to be issued = = Rs.3,333,333

0.90

Cost of equity issue = Rs.333,333

Computation of Tax Shield Associated with Debt Finance

Year Debt outstanding at the

beginning

Interest Tax shield Present value of tax

shield

1 Rs.5,000,000 Rs.750,000 Rs.300,000 Rs.260,8692 5,000,000 750,000 300,000 226,843

3 4,000,000 600,000 240,000 157,8044 3,00,000 450,000 180,000 102,9165 2,000,000 300,000 120,000 59,666 1,000,000 150,000 60,000 25,940

843,033

Adjusted NPV = - 1004000 333333 + 834033 = - Rs.503,300Adjusted NPV if issue cost of externalequity alone is adjusted for = - Rs.1,004000 Rs.333333

= Rs.1337333

c. Present value of tax shield of debt finance = Rs.834,033

11. Adjusted cost of capital as per Modigliani Miller formula:r* = r(1 TL)r* = 0.19 x (1 0.5 x 0.5) = 0.1425 = 14.25%


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Adjusted cost of capital as per Miles and Ezzell formula:1 + r

r* = rLrDT

1 + rD1 + 0.19

= 0.19 0.5 x 0.16 x 0.5 x1 + 0.16

= 0.149 = 14.9%

12. S0 = Rs.46 , rh = 11 per cent , rf = 6 per centHence the forecasted spot rates are :

Year Forecasted spot exchange rate

1 Rs.46 (1.11 / 1.06)1 = Rs.48.172 Rs.46 (1.11 / 1.06)2 = Rs.50.443 Rs.46 (1.11 / 1.06)3 = Rs.52.824 Rs.46 (1.11 / 1.06)4 = Rs.55.315 Rs.46 (1.11 / 1.06)5 = Rs.57.92

The expected rupee cash flows for the project

Year Cash flow in dollars Expected exchange Cash flow in rupees

(million) rate (million)

0 -200 46 -92001 50 48.17 2408.52 70 50.44 3530.83 90 52.82 4753.84 105 55.31 5807.65 80 57.92 4633.6

Given a rupee discount rate of 20 per cent, the NPV in rupees is :

2408.5 3530.8 4753.8NPV = -9200 + + +

(1.18)2 (1.18)3 (1.18)4

5807.6 4633.6+ +

(1.18)5 (1.18)6

= Rs.3406.2 million

The dollar NPV is :


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3406.2 / 46 = 74.05 million dollars

Chapter 14

SOCIAL COST BENEFIT ANALYSIS

1. Social Costs and Benefits

Nature Economic

value (Rs

in million)

Explanation

Costs

1. Construction cost One-shot 400


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2. Maintenance cost Annual 3Benefits

3. Savings in the operation

cost of existing ships

Annual 40

4. Increase in consumersatisfaction

Annual 3.6 The number of passenger hourssaved will be : (75,000 x 2 +50,000 + 50,000 x 2) = 600000.Multiplying this by Rs.6 givesRs.3.6 million

The IRR of the stream of social costs and benefits is the value ofrin the

equation

50 40 + 3.6 3.0 50 40.6

400 = = t=1 (1+r)t t=1 (1+r)t

The solving value ris about 10.1%

2. Social Costs and BenefitsCosts

Decrease in customer satisfaction as reflected Rs.266,667in the opportunity cost of the extra time takenby bus journey

800 x (2/3) x 250 x Rs.2

Benefits

1. Resale value of the diesel train (one time) Rs.240,0002. Avoidance of annual cash loss Rs.400,000Fare collection = 1000 x 250 x Rs.4

= Rs.1,000,000Cash operating expenses = Rs.1,400,000

3. The social costs and benefits of the project are estimated below:Rs. in million

Costs & Benefits Time Economic

value

Explanation

1. Construction cost 0 242. Land development cost 0 1503. Maintenance cost 1-40 14. Labour cost 0 40 This includes the cost of

transport and rehabilitation

5. Labour cost 1-40 12 The shadow price of labourequals what others are willingto pay.


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6. Decrease in the value of the timberoutput

2-40 4

Benefits

7. Savings in the cost of shipping theagriculture produce 1-40 0.5

8. Income from cash crops 1-5 109. Income from the main crop 6-40 5010. Increase in the value of timber output 1 20

Assuming that the life of the road is 40 years, the NPV of the stream of social costs and benefits ata discount rate of 10 percent is:

40 1 + 12 40 4

NPV = - 24 - 150 - 40 - - t=1 (1.1)t t=2 (1.1)t

40 0.5 5 10 40 50 20

+ + + +t=1 (1.1)t t=1 (1.1)t t=6 (1.1)t (1.1)1

= - Rs.9.93 million

4.Table 1

Social Costs Associated with the Initial Outlay

Rs. in millionItem Financial

cost

Basis of

conversion

Tradeable value

ab initio

T L R

Land 0.30 SCF = 1/1.5 0.20Buildings 12.0 T=0.50,L=0.25

R=0.256.0 3.0 3.0

Imported equipment 15.0 CIF value 9.0Indigeneous equipment 80.0 CIF value 60.0


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Transport 2.0 T=0.65,L=0.25R=0.10

1.3 0.5 0.2

Engineering and know-how

fees

6.0 SCF=1.5 9.0

Pre-operative expenses 6.0 SCF=1.0 6.0Bank charges 3.7 SCF=0.02 0.074Working capitalrequirement

25.0 SCF=0.8 20.0

150.0 104.274 7.3 3.5 3.2

Table 2

Conversion of Financial Costs into Social Costs

Rs. in millionItem Financial

cost

Basis of

conversion

Tradeable value

ab initio

T L R

Indigeneous raw materialand stores

85 SCF=0.8 68

Labour 7 SCF=0.5 3.5Salaries 5 SCF=0.8 4.0Repairs and maintenance 1.2 SCF=1/1.5 0.8Water, fuel, etc 6 T=0.5,L=0.25

R=0.25

3 1.5 1.5

Electricity (Rate portion) 5 T=0.71,L=0.13R=0.16

3.55 0.65 0.8

Other overheads 10 SCF=1/1.5 6.667119.2 82.967 6.55 2.15 2.3

As per table 1, the social cost of initial outlay is worked out as follows :

Rs. in millionTradeable value ab initio 104.274Social cost of the tradeable component 4.867

(7.3 / 1.5)Social cost of labour component 1.75(3.5 x 0.5)

Social cost of residual component 1.60(3.2 x 0.5)

Total 112.491

As per Table 2, the annual social cost of operation is worked out as follows :


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Tradeable value ab initio 82.967Social cost of the tradeable component 4.367

( 6.55 x 1/1.5 )

Social cost of labour component 1.075(2.15 x 0.5)

Social cost of residual component 1.150(2.3 x 0.5)

Total 89.559

The annual CIF value of the output is Rs.110 million. Hence the annual socialnet benefit will be : 110 89.559 = Rs.20.441 million

Working capital recovery will be Rs.20 million at the end of the 20th

year.

Putting the above figures together the social flows associated with the projectwould be as follows :

Year / s Social flow (Rs. in million)

0 -112.4911-19 20.441

Chapter 15

MULTIPLE PROJECTS AND CONSTRAINTS

1. The ranking of the projects on the dimensions of NPV, IRR, and BCR is given belowProject NPV (Rs.) Rank IRR (%) Rank BCR Rank

M 60,610 3 34.1 2 2.21 1N 58,500 4 34.9 1 1.59 3O 40,050 5 18.6 4 1.33 5P 162,960 1 26.2 3 2.09 2Q 72,310 2 14.5 5 1.36 4


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2. The ranking of the projects on the dimensions of NPV and BCR is given below

Project NPV (Rs.) Rank BCR Rank

A 61,780 5 1.83 2

B 208,480 2 1.52 3C 315,075 1 2.05 1D 411,90 6 1.14 6E 95,540 4 1.38 4F 114,500 3 1.23 5

3. The two hypothetical projects are:

A BInitial outlay 10000 1000Cash inflows

Year 1 5000 600Year 2 5000 600Year 3 5000 600

NPV @ 10% Rank IRR Rank

A 2435 1 about 23% 2B 492 2 above 35% 1

4. The two hypothetical 4-year projects for which BCR and IRR criteria give differentrankings are given below

Project A B

Investment outlay 20000 20000Cash inflow

Year 1 2000 8000Year 2 2000 8000Year 3 2000 8000Year 4 31500 8000

Project NPV Rank IRR Rank

A 4822 1 19% 2B 4296 2 about 22% 1

5. The NPVs of the projects are as follows:NPV (A) = 6000 x PVIFA(10%,5) + 5000 x PVIF(10%,5) 20,000 = Rs.5851NPV (B) = 8000 x PVIFA(10%,8) 50,000 = - Rs.840NPV (C) = 15,000 x PVIFA(10%,8) 75,000 = Rs.5025NPV (D) = 15,000 x PVIFA(10%,12) 100,000 = Rs.6,995NPV (E) = 25,000 x PVIFA (10%,7) + 50,000 x PVIF(10%,7)

150,000 = Rs.2,650


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SinceB andEhave negative NPV, they are rejected. So we consider onlyA, C,and D. Further CandD are mutually exclusive. The feasible combinations, theiroutlays, and their NPVs are given below.

Combination Outlay

(Rs.)

NPV

(Rs.)

A 20,000 5,851C 75,000 5,025D 100,000 6,995

A & C 95,000 10,876A & D 120,000 12,846

The preferred combination isA &D.

6. The linear programming formulation of the capital budgeting problem under variousconstraints is as follows:

Maximise 10X1 + 15X2 + 25X3 + 40X4 + 60X5 + 100X6Subject to

15X1 + 12X2 + 8X3 + 35X4 + 100X5

+ 50X6 + SF1 = 150 Funds constraint for year 1

5X1 + 13X2 + 40X3 + 25X4 + 10X5+ 110X6 200 + 1.08 SF1 Funds constraint for year 2

5X1 + 6X2 + 5X3 + 10X4 + 12X5+ 40X6 60 Power constraint

15X1 + 20X2 + 30X3 + 35X4 + 40X5+ 60X6 120 Managerial constraint

0 Xj 1 (j = 1,.8) and SF1 0Rupees are expressed in 000s. Power units are also expressed in 000s.

7. Given the nature of the problem, in addition to the decision variablesX1 throughX10for the original 10 projects, two more decision variables are required as follows:

X11 is the decision variable to represent the delay of projects 8 by one yearX12 is the decision variable for the composite project which represents the

combination of projects 4 and 5.The integer linear programming formulation is as follows:

Maximise 55X1 + 75X2 + 50X3 + 60X4 + 105X5 + 12X6 + 60X7 + 120X8+ 50X9 + 40X10 + 100X11+ 178.2X12


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Subject to 75X1 + 80X2 + 75X3 + 35X4 + 80X5 + 20X6 + 70X7 + 155 X8 +55X9 + 10X10 + 109.3X12 + SF1 = 400

40X1 + 85X2 + 8X3 + 100X4 + 160X5 + 9X6 + 5X7 + 100 X8 + 20X9 + 90X10 + 155X11+ 247X12 +SF2 = 350 + SF1 (1 + r)

X3 +X7 1X5 +X8 + X9 +X10 2X2 X6

X8 X9 X4 +X5 + X12 1

X8 +X11 1

Xj = {0,1} j = 1, 2.12SFi 0 i = 1, 2

It has been assumed that surplus funds can be shifted from one period to the nextand they will earn a post-tax return ofrpercent.

+

8. Minimise [P1(3d1+ 2 d2 + d3) +P2 (4 d4 + 2 d5 + d6) +P3 (d7 d7

)]

Subject to:Economic Constraints

12X1 + 14X2 + 15X3 + 16X4 + 11X5 + 23X6 + 20X7 65

Goal Constraints

1.2X1 + 1.6X2 + 0.6X3 + 1.5X4 + 0.5X5 +

+ 0.9X6 + 1.8X7 + d1d1 = 6 Net income for year 1

1.1X1 + 1.2X2 + 1.2X3 + 1.6X4 + 1.2X5 +


1.6X1 + 1.5X2 + 2.0X3 + 1.8X4 + 1.5X5 +


1.0X1 + 1.2X2 + 0.5X3 + 1.8X4 + 0.6X5 +

+ 1.0X6 + 2.0X7 + d4d4 = 6 Sales growth for year 1

1.5X1 + 1.0X2 + 1.2X3 + 2.0X4 + 1.4X5


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+


1.8X1 + 1.2X2 + 2.5X3 + 2.2X4 + 1.8X5 +


4X1 + 5X2 + 6X3 + 8X4 + 4X5 +

+ 9X6 + 7X7 + d7d7 = 50 NPV +

Xj 0 di, di 0

9. The BCRs of the projects are converted into NPVs as of now as follows

Project Outlay (Rs.) BCR NPV (Rs.)

1 800,000 1.08 64,0002 200,000 1.35 70,0003 400,000 1.20 80,000

4 300,000 1.03 9,0005 200,000 0.98 - 4,0006 500,000 1.03 15,000/1.10 = 13,6367 400,000 1.21 84,000/1.10 = 76,3648 600,000 1.17 102,000/1.10 = 92,7279 300,000 1.01 3,000/1.10 = 2,727

The integer linear programming formulation of the problem is as follows :

Maximise 64,000X1 + 70,000X2 + 80,000X3 + 9,000X4 + 13,636X6+ 76,364X7 + 92,727X8 + 2,727X9

Subject to800,000X1 + 200,000X2 + 400,000X3 + 300,000X4 + SF1 = 20,00,000500,000X6 + 400,000X7 + 600,000X8 + 300,000X9 500,000 + SF1 (1.032)

Xj = {0,1} j = 1, 2, 3, 4, 6, 7, 8, 9


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Chapter 16

VALUATION OF REAL OPTIONS

1. S= 100 , uS= 150, dS= 90u = 1.5 , d= 0.9, r= 1.15 R = 1.15E= 100

Cu= Max (uSE, 0) = Max (150 100,0) = 50Cd= Max (dSE, 0) = Max (90 100,0) = 0

CuCd 50

= = = 0.833(u-d)S 0.6 x 100

uCddCu 0 0.9 x 50


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B = = = - 65.22(u-d)R 0.6 x 1.15

C = S+B = 0.833 x 100 65.22 = 18.08

2. S= 60 , dS= 45, d= 0.75, C= 5r= 0.16, R = 1.16, E= 60

Cu= Max (uSE, 0) = Max (60u E, 0)Cd= Max (dSE, 0) = Max (45 60, 0) = 0

CuCd 60u 60 u 1 = = =

(u-d)S (u0.75)60 u 0.75

uCddCu 0.75 (60u 60) 45 (1 u) B = = =

(u-d)R (u 0.75) 1.16 1.16 (u 0.75)

C = S+B

(u 1) 60 45 (1 u)5 = +

u 0.75 1.16 (u 0.75)Multiplying both the sides by u 0.75 we get

455(u 0.75) = (u 1) 60 + (1 u)

1.16

Solving this equation foru we getu = 1.077

So Betas equity can rise to60 x 1.077 = Rs.64.62

3. EC0 = S0 N(d1) - N(d2)

ert

S0 = 70,E= 72, r= 0.12, = 0.3, t= 0.50

S0 1

ln + r+ 2 t E 2


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d1=

t 70

ln + (0.12 + 0.5 x .09) x 0.50 72

=0.30 0.50

- 0.0282 + 0.0825= = 0.2560

0.2121

d2 = d1 - t = 0.2560 0.30 0.50 = 0.0439

N(d1) = 0.6010N(d2) = 0.5175

E 72= = 67.81

ert e0.12x 0.50

C0 = S0 x 0.6010 67.81 x 0.5175= 70 x 0.6010 67.81 x 0.5175 = Rs.6.98

4. EC0 = S0 N(d1) - N(d2)

ert

E= 50, t= 0.25, S= 40, = 0.40, r= 0.14

S0 1

ln + r+ 2 t E 2

d1=

t

40

ln + (0.14 + 0.5 x 0.16) 0.25 50d1=

0.40 0.25

- 0.2231 + 0.055= = - 0.8405

0.20


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d2 = d1 - t = - 0.8405 0.40 0.25 = -1.0405

N(d1) = 0.2003N(d2) = 0.1491

E 50= = 48.28

ert e0.14 x 0.25

C0 = S0 x 0.2003 48.28 x 0.1491= 40 x 0.2003 48.28 x 0.1491 = 0.8135

5. The NPV of the proposal to make Comp-I is:20 50 50 20 + 10

-100 + + + +1.20 (1.20)2 (1.20)3 (1.20)4

= -100 + 16.66 + 34.70 + 28.95 + 14.46= - Rs.5.23 million

The present value of the cash inflows of Comp II proposal, four years from nowwill be Rs.189.54 million (Two times the present value of the cash inflows of Comp-I).

So, we haveS0 = present value of the asset = 189.54 x e

0.20 x 4 = Rs.85.17 millionE = exercise price = $ 200 million

= 0.30t = 4 years

r = 12

Step 1 : Calculate d1 and d2

S0 2

ln + r+ t E1 2 -0.854 + (0.12 + (.09/2)) 4 -0.194

d1 = = = = -0.323

t 0.3 4 0.6

d2 = d1 - t = -0.323 0.60 = -0.923Step 2 : FindN(d1) andN(d2)

N(d1) = 0.3733N(d2) = 0.1780

Step 3 : Estimate the present value of the exercise price


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E. e-rt = 200 / 1.6161 = Rs.123.76 million

Step 4 : Plug the numbers obtained in the previous steps in the Black-Scholes formula:

C0 = 85.17 x 0.3733 123.76 x 0.1780= Rs.9.76

6. Presently a 9 unit building yields a profit of Rs.1.8 million (9 x 1.2 9) and a 15 unitbuilding yields a profit of Rs.1.0 million (15 x 1.2 17). Hence a 9 unit building isthe best alternative if the builder has to construct now.

However, if the builder waits for a year, his payoffs will be as follows:

Market Condition

Alternative Buoyant (Apartment price:Rs.1.5 million)

Sluggish (Apartment price:Rs.1.1million)

9 unit building 1.5 x 9 9 = 4.5 1.1 x 9 9 = 0.9

15 unit building 1.5 x 15 17 = 5.5 1.1 x 15 17 = -0.5

Thus, if the market turns out to be buoyant the best alternative is the 15 unitbuild

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