Projects Solutions

Chapter 4

MARKET AND DEMAND ANALYSIS

1. We have to estimate the parameters a and b in the linear relationship

Yt = a + bT

Using the least squares method.

According to the least squares method the parameters are:

T Y n T Y b =

T 2 n T 2

a = Y bT The parameters are calculated below:

Calculation in the Least Squares Method

T Y TY T 2

1 2,000 2,000 1

2 2,200 4,400 4

3 2,100 6,300 9

4 2,300 9,200 16

5 2,500 12,500 25

6 3,200 19,200 36

7 3,600 25,200 49

8 4,000 32,000 64

9 3,900 35,100 81

10 4,000 40,000 100

11 4,200 46,200 121

12 4,300 51,600 144

13 4,900 63,700 169

14 5,300 74,200 196

T = 105 Y = 48,500 TY = 421,600 T 2 = 1,015

T = 7.5 Y = 3,464

T Y n T Y 421,600 14 x 7.5 x 3,464 b = =

T 2 n T 2 1,015 14 x 7.5 x 7.5

57,880

= = 254

227.5

a = Y bT = 3,464 254 (7.5) = 1,559

Thus linear regression is

Y = 1,559 + 254 T

2. In general, in exponential smoothing the forecast for t + 1 is

Ft + 1 = Ft + et

Where Ft + 1 = forecast for year ) = smoothing parameter et = error in the forecast for year t = St = Ft

F1 is 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) = 2863.2

8 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 method

St + S t 1 ++ S t n +1 Ft + 1 =

n

where Ft + 1 = forecast for the next period

St = sales for the current period

n = period over which averaging is done

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

Period t Data (St) Forecast

(Ft)

Forecast for t + 1

Ft + 1 = (St+ S t 1 + S t 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 = 3100

8 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 = 70

I1 = 1000

I2 = 1200

Q1 Q2 I1 + I2 Income Elasticity of Demand E1 = x

I2 - I1 Q2 Q1 E1 = Income Elasticity of Demand

Q1 = Quantity demanded in the base year

Q2 = Quantity demanded in the following year

I1 = Income level in base year

I2 = Income level in the following year

70 60 1000 + 1200 E1 = x

1200 1000 70 + 60

22000

E1 = = 0.846

26000

5.

P1 = Rs.40

P2 = Rs.50

Q1 = 1,00,000

Q2 = 95,000

Q2 Q1 P1 + P2 Price Elasticity of Demand = Ep = x

P2 P1 Q2 + Q1

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

P2, Q2 = Price per unit and quantity demanded in the following year

Ep = Price Elasticity of Demand

95000 - 100000 40 + 50

Ep = x

50 - 40 95000 + 100000

- 45

Ep = = - 0.0231

1950

Chapter 6

FINANCIAL ESTIMATES AND PROJECTIONS

1.

Projected Cash Flow Statement (Rs. in million)

Sources of Funds

Profit before interest and tax 4.5

Depreciation provision for the year 1.5

Secured term loan 1.0

Total (A) 7.0

Disposition of Funds

Capital expenditure 1.50

Increase in working capital 0.35

Repayment of term loan 0.50

Interest 1.20

Tax 1.80

Dividends 1.00

Total (B) 6.35

Opening cash balance 1.00

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

Projected Balance Sheet

(Rs. in million)

Liabilities Assets

Share capital 5.00 Fixed assets 11.00

Reserves & surplus 4.50 Investments .50

Secured loans 4.50 Current assets 12.85

Unsecured loans 3.00 * Cash 1.65

Current liabilities 6.30 * Receivables 4.20

& provisions 1.05 * Inventories 7.00

24.35 24.35

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

2. Projected Income Statement for the 1st Operating Year

Rs.

Sales 4,500

Cost of sales 3,000

Depreciation 319

Interest 1,044

Write off of Preliminary expenses 15

Net profit 122

Projected Cash Flow Statements

Construction period 1st Operating year

Sources

Share capital 1800 -

Term loan 3000 600

Short-term bank borrowing 1800

Profit before interest and tax 1166

Depreciation 319

Write off preliminary expenses 15

4800 3900

Uses

Capital expenditure 3900 -

Current assets (other than cash) - 2400

Interest - 1044

Preliminary expenses 150 -

Pre-operative expenses 600 -

4650 3444

Opening cash balance 0 150

Net surplus / deficit 150 456

Closing balance 150 606

Projected 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 - - Miscellaneous

expenditures & losses

Current liabilities and

provisions

- Preliminary

expenses

150 135

4800 7322 4800 7322

Notes :

i. Allocation of Pre-operative Expenses : Rs.

Type Costs before

allocation

Allocation Costs after

allocation

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 :

Land 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.720

Interest on short term bank borrowings of Rs,1800 @ 18% = Rs.324

= Rs.1044

Chapter 7

THE TIME VALUE OF MONEY

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

r = 8% FV5 = 1000 x FVIF (8%, 5 years)

= 1000 x 1.469 = Rs.1469

r = 10% FV5 = 1000 x FVIF (10%, 5 years)

= 1000 x 1.611 = Rs.1611

r = 12% FV5 = 1000 x FVIF (12%, 5 years)

= 1000 x 1.762 = Rs.1762

r = 15% FV5 = 1000 x FVIF (15%, 5 years)

= 1000 x 2.011 = Rs.2011

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

According to the Rule of 72 at 12 percent interest rate doubling takes place

approximately 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 initial

deposit. 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 is equivalent to saving Rs.2000 a year for 15 years and Rs.1000 a year for the

years 6 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.

5. Let A be the annual savings.

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

A 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) = 9.930

FVIFA (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) = 5,000

FVIF (r,10 years) = 5,000 / 1000 = 5


FVIF (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)

= 10,000 x 0.327 = Rs.3,270

9. 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 of

Rs.10,000.

11. The amount that can be withdrawn annually is:

100,000 100,000

A = ------------------ ------------ = ----------- = Rs.10,608

PVIFA (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 from now, 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 this

sum is:

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

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) = 5.019

PVIFA (18%, 10 years) = 4.494

Using linear interpolation we get:

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

5.019 4.494 = 15.1%

16. PV (Stream A) = Rs.100 x PVIF (12%, 1 year) + Rs.200 x PVIF (12%, 2 years) + Rs.300 x PVIF(12%, 3 years) + Rs.400 x

PVIF (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 (Stream B) = Rs.3,625.2

PV (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

Stated rate (%) 12 24 24

Frequency of compounding 6 times 4 times 12 times

Effective rate (%) (1 + 0.12/6)6- 1 (1+0.24/4)4 1 (1 + 0.24/12)12-1 = 12.6 = 26.2 = 26.8

Difference between the

effective rate and stated

rate (%) 0.6 2.2 2.8

20. Investment required at the end of 8th year to yield an income of Rs.12,000 per year 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 deposited

now is:

Rs.100,000 Rs.100,000

= = Rs.40,388

PVIF(12%, 8 years) 2.476

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

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

Rs.20,000

FVIF (r,10 years) = = 4.000

Rs.5,000


FVIF (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:

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.50,000

A = 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 from 2005 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

If A is the amount deposited at the end of each year from 1995 to 2000 then

A x FVIFA (12%, 6 years) = Rs.51,335

A x 8.115 = Rs.51,335

A = Rs.51,335 / 8.115 = Rs.6326

25. The discounted value of the annuity of Rs.2000 receivable for 30 years, evaluated as 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 is 0.30 x Rs.600 = Rs.180

Assuming that the monthly interest rate corresponding to an annual interest rate

of 12% is 1%, the discounted value of an annuity of Rs.180 receivable at the end of

each month for 180 months (15 years) is:

Rs.180 x PVIFA (1%, 180)

(1.01)180 - 1

Rs.180 x ---------------- = Rs.14,998

.01 (1.01)180

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

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

P x 1.817 = Rs.14,998

Rs.14,998

P = ------------ = 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) = 21.244

PVIFA (2%, 24) = 18.914

Using a linear interpolation

21.244 20.000 r = 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 years evaluated 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

If A is the annual deposit to be made in the sinking fund for the years 1 to 5, then

A x FVIFA (8%, 5 years) = Rs.2.21 million

A x 5.867 = Rs.2.21 million

A = 5.867 = Rs.2.21 million

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 withdrawn annually.

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

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


PVIFA (10%, 7 years) = 4.868

PVIFA (10%, 8 years) = 5.335

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

5.000 4.868 n = 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 amount installment Interest repaid balance

1 500000 171585 70000 101585 398415

2 398415 171585 55778 115807 282608

3 282608 171585 39565 132020 150588

4 150588 171585 21082 150503 85*

(*) rounding off error

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

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

PVIFA (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 million

Expected present value of the iron ore that can be mined over the next 15 years

assuming 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

= Rs.300 million x 1 (1.06)15 / (1.16)15 0.16 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

= - 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. Investment A

a) Payback period = 5 years

b) NPV = 40000 x PVIFA (12%,10) 200 000 = 26000

c) IRR (r ) can be obtained by solving the equation: 40000 x PVIFA (r, 10) = 200000

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

From the PVIFA tables we find that

PVIFA (15%,10) = 5.019

PVIFA (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 equation

40,000 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 that

r = 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 in this range provides an approximate payback period of 2.88 years.

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 equation

80,000 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 get

r = 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 in this 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 equation

200,000 x PVIF (r,1) + 200,000 x PVIF (r,2)

+ 200,000 x PVIF (r,9) + 50,000 x PVIF (r,10)

= 320000

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

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

Comparative Table

Investment A B C D

a) Payback period

(in years) 5 9 2.88 8.5

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 C because it has the a. Lowest payback period

b. Highest NPV

c. Highest IRR

d. Highest BCR

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

i.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 get

r = 1.61, -0.61; (or) 161%, (-)61%

Note : Given two changes in the signs of cashflow, we get two values for the

IRR of the cashflow stream. In such cases, the IRR rule breaks down.

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

equation

NCF x PVIFA (10%,8) = 500000

NCF = 500000 / 5.335

= 93271

6. Define I as 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 value

of I can be obtained from the following equation

25000 x PVIFA (12%,10) = I

i.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)

+ 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 R

Discount rate

0% 400 500 600

5% 223 251 312

10% 69 40 70

15% - 66 - 142 - 135

25% - 291 - 435 - 461

30% - 386 - 555 - 591

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

Project P Q

Discount rate (%)

0 2950 500

5 1876 208

10 1075 - 28

15 471 - 222

20 11 - 382

b) (i) The IRR (r ) of project P can be obtained by solving the following

equation 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 for r' -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%

NPV (P) = 1075

NPV (Q) = - 28

Given that NPV (P), NPV (Q) and NPV (P) > 0, I would choose project P.

From (a) we find that at a cost of capital of 20%

NPV (P) = 11

NPV (Q) = - 382

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

d) Project P

PV of investment-related costs

= 1000 x PVIF (12%,0)

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

+ 250 x PVIF (12%,3)

= 2728

TV of cash inflows = 2000 x (1.12) + 4000 = 6240

The MIRR of the project P is given by the equation:

2728 = 6240 x PVIF (MIRR,5)

(1 + MIRR)5 = 2.2874

MIRR = 18%

(c) Project Q PV of investment-related costs = 1600

TV of cash inflows @ 15% p.a. = 2772

The MIRR of project Q is given by the equation:

16000 (1 + MIRR)5 = 2772

MIRR = 11.62%

10.

(a) Project A NPV 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) = 100

i.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]

(b) Difference in capital outlays between projects A and B is Rs.50 million Difference in net annual cash flow between projects A and B 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) Project M

The pay back period of the project lies between 2 and 3 years. Interpolating in

this 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 we

get an approximate pay back period of 1.55 years.

(b) Project M

Cost of capital = 12% p.a

PV of cash flows up to the end of year 2 = 24.97



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 annum



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

approximate DPB of 1.92 years.

(c) Project M


NPV = - 50 + 11 x PVIFA (12%,1)

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

+ 37 x PVIF (12%,4)

= Rs.21.26 million

Project N


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 capital

constraint.

(d) Project M



Project N



Since the two projects are mutually exclusive, we need to choose the project

with the higher NPV i.e., choose project M.

Note : The MIRR can also be used as a criterion of merit for choosing between

the two projects because their initial outlays are equal.

(e) Project M


NPV = 16.13 million

Project N

Cost of capital: 15% per annum


Again the two projects are mutually exclusive. So we choose the project with the

higher NPV, i.e., choose project N.

(f) Project M Terminal value of the cash inflows: 114.47

MIRR 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: 115.41

MIRR of the project is given by the equation

50 ( 1+ MIRR)4 = 115.41

i.e., MIRR = 23.26%

12. The internal rate of return is the value of r in the equation

2,000 1,000 10,000 2,000

8000 = - + +

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

At r = 18%, the right hand side is equal to 8099

At r = 20%, the right hand side is equal to 7726

Thus the solving value of r is :

8,099 8,000 18% + x 2% = 18.5%

8,099 7,726

Unrecovered Investment Balance

Year Unrecovered

investment balance at

the beginning Ft-1

Interest for the

year Ft-1 (1+r)

Cash flow at the

end of the year CFt

Unrecovered

investment 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 lakhs Year 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 before

interest 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 1.750

Income 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%


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 before

Depreciation 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

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 50

capital

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

project is 55.17%.

2. Post-tax Incremental Cash Flows (Rs. in million)

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 80

4. Raw material cost 24 36 48 60 48 36 24

5. Variable mfg cost. 8 12 16 20 16 12 8

6. Fixed operating & maint. 10 10 10 10 10 10 10

cost

7. Variable selling expenses 8 12 16 20 16 12 8

8. Incremental overheads 4 6 8 10 8 6 4

9. Loss of contribution 10 10 10 10 10 10 10

10.Bad debt loss 4

11. Depreciation 30 22.5 16.88 12.66 9.49 7.12 5.34

12. Profit before tax -14 11.5 35.12 57.34 42.51 26.88 6.66

13. Tax - 4.2 3.45 10.54 17.20 12.75 8.06 2.00

14. Profit after tax - 9.8 8.05 24.58 40.14 29.76 18.82 4.66

15. Net salvage value of

capital equipments 25

16. Recovery of working 16

capital

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.

(a) A. Initial outlay (Time 0)

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

ii. Salvage value of old machine 900,000

iii Incremental working capital requirement 500,000

iv. 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 in

manufacturing costs 455,000 455,000 455,000 455,000 455,000

ii. Incremental

depreciation 550,000 412,500 309,375 232,031 174,023

iii. Tax shield on

incremental dep. 165,000 123,750 92,813 69,609 52,207

iv. Operating cash

flow ( 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 200,000

iii. Recovery of incremental working capital 500,000

iv. 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) 620000 578750 547813 524609 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)

= 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. 400,000

ii. Salvage value of the old machine 90,000

iii. Net investment 310,000

B. Operating cash flow (years 1 through 5)

Year 1 2 3 4 5

i. Depreciation

of old machine 18000 14400 11520 9216 7373

ii. Depreciation

of new machine 100000 75000 56250 42188 31641

iii. Incremental depre-

ciation ( ii i) 82000 60600 44730 32972 24268

iv. Tax savings on inc-

remental depreciation

( 0.35 x (iii)) 28700 21210 15656 11540 8494

v. Operating cash flow 28700 21210 15656 11540 8494

C. Terminal cash flow (year 5)

i. Salvage value of new machine Rs. 25000

ii. Salvage value of old machine 10000

iii. Incremental salvage value of new

machine = Terminal cash flow 15000

D. Net cash flows associated with the replacement proposal.

Year 0 1 2 3 4 5

NCF (310000) 28700 21210 15656 11540 23494

6. Net Cash Flows Relating to Equity

(Rs. in million)

Particulars Year

0 1 2 3 4 5 6

1. Equity funds (100) 2. Revenues 500 500 500 500 500 500 3. Operating costs 320 320 320 320 320 320 4. Depreciation 83.33 55.56 37.04 24.69 16.46 10.97 5. Interest on working capital

advance

18.00 18.00 18.00 18.00 18.00 18.00

6. Interest on term loan 30.00 28.50 22.50 16.50 10.50 4.50 7. 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 73.265 9. Profit after tax 24.335 38.97 51.23 60.405 67.52 73.265 10. Preference dividend 11. Net salvage value of fixed assets 200 12. Net salvage value of current

assets

- 40 40 40 40 40

13. Repayment of term-loans 14. Redemption of preference capital 15. Repayment of short-term bank

borrowings

100

16. Retirement of trade creditors 50 17. Initial investment (1) (100) 18. Operating cash flows (9-10+4) 107.665 94.53 88.27 85.095 83.98 84.235 19. 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 6

1. Fixed assets (250) 2. Working capital margin (50) 3. Revenues 500 500 500 500 500 500 4. Operating costs 320 320 320 320 320 320 5. Depreciation 83.33 55.56 37.04 24.69 16.46 10.97 6. Interest on working capital

advance

18.00 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 4.50 8. 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 73.265 10. Profit after tax 24.335 38.97 51.23 60.405 67.52 73.265 11. Net salvage value of fixed assets 80 12. Net recovery of working capital

margin

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)

Year

0 1 2 3 4 5 6

1. Total funds (450) 2. Revenues 500 500 500 500 500 500 3. Operating costs 320 320 320 320 320 320 4. 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 6. Interest on working capital

advance

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 73.265 9. Profit after tax 24.34 38.97 51.23 60.41 67.52 73.265 10. Net salvalue of fixed assets 80 11. Net salvage value of current assets 200 12. 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) 280 15. 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, rD

can be calculated as follows:

14 + (100 108)/10 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 formula rp can be calculated as follows:

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

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 : 3

WACC = 2/5 x 9.1% + 3/5 x 18.4%

= 14.68%

5. Given

0.5 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 debt

was originally issued. But we need to calculate the marginal cost of debt (cost

of raising new debt); and for this purpose we need to calculate the yield to

maturity of the debt as on the balance sheet date. The yield to maturity will not

be equal to 12% unless the book value of debt is equal to the market value of

debt on the balance sheet date.

(b) The cost of equity has been taken as D1/P0 ( = 6/100) whereas the cost of equity

is (D1/P0) + g where g represents the expected constant growth rate in dividend

per share.

7. The book value and market values of the different sources of finance are

provided in the following table. The book value weights and the market value

weights 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) Given

rD 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) Given

13% x (1 0.3) x 4/9 + rE x 5/9 = 15% rE = 19.72%, where rE represents 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 earnings costing 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 additional

equity costing 15 percent and Rs.2.5 million of debt costing 15 (1-.3) = 10.5

percent.

(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

10. The cost of equity and retained earnings rE = 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 is

19.1 (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 4.06

Preference capital 15.9% 10 0.03 0.48

Retained earnings 14.5% 120 0.33 4.79

Debentures 9.6% 50 0.14 1.34

Term 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 capital

and retained earnings 14.5% 200 0.62 8.99

Preference capital 15.9% 7.5 0.02 0.32

Debentures 9.6% 40 0.12 1.15

Term loans 6.0% 80 0.24 1.44

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)

Pessimistic Expected Optimistic

Investment 300 250 200

Sales 150 200 275

Variable costs 97.5 120 154

Fixed costs 30 20 15

Depreciation 30 25 20

Pretax profit - 7.5 35 86

Tax @ 28.57% - 2.14 10 24.57

Profit after tax - 5.36 25 61.43

Net 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 three

scenarios. It is also assumed that the salvage value of the

investment after ten years is zero.

(2) The investment is assumed to be depreciated at 10% per annum; and it is also assumed that this method and rate of

depreciation 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 offset against the taxable profit on other projects of the

company; and thus the company can claim a tax shield on

the loss in the same year.

(c) Accounting break even point (under expected scenario) Fixed costs + depreciation = Rs. 45 million

Contribution margin ratio = 60 / 200 = 0.3

Break even level of sales = 45 / 0.3 = Rs.150 million

Financial 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) = 1.1628 sales 38.74

iii. Initial investment = 200

iv. Financial break even level

of 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 30000

Sale revenue 24000 42000 54000

Variable costs 16000 28000 36000

Fixed costs 3000 3000 3000


Profit before tax 3000 9000 13000

Tax 1500 4500 6500

Profit after tax 1500 4500 6500

Net cash flow 3500 6500 8500

NPV at a cost of

capital of 10% p.a

and useful life of

5 years -16732 - 5360 2222

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



Sale revenue 28000 42000 70000


Fixed costs 3000 3000 3000


Profit before tax -5000 9000 37000

Tax -2500 4500 18500

Profit after tax -2500 4500 18500

Net cash flow - 500 6500 20500

NPV - 31895 (-) 5360 47711

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



Sale revenue 42000 42000 42000


Fixed costs 3000 3000 3000


Profit before tax -11000 9000 16000

Tax -5500 4500 8000

Profit after tax -5500 4500 8000

Net cash flow -3500 6500 10000

NPV -43268 - 5360 7908

(d) Accounting break-even point

i. Fixed costs + depreciation = Rs.5000

ii. Contribution margin ratio = 10 / 30 = 0.3333

iii. 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) = 2000 ii. PV of annual cash flow = (i) x PVIFA (10%,5)

= 0.6318 sales 1896 iii. Initial investment = 30000

iv. Break-even level of sales = 31896 / 0.6318 = Rs.50484

2. Define At as the random variable denoting net cash flow in year t.

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

12 22 32

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

= 1.00

(NPV) = Rs.1.00 million

3. Expected NPV 4 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 NPV 4 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

= 3,200

Substituting 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 2t

2 (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 get

490,000 / (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 - NPV Prob (NPV < 0) = Prob. <

NPV NPV

0 3044 = Prob Z <

1296

= Prob (Z < -2.35)

The required probability is given by the shaded area in the following normal

curve.

P (Z < - 2.35) = 0.5 P (-2.35 < Z < 0) = 0.5 P (0 < Z < 2.35) = 0.5 0.4906 = 0.0094

So the probability of NPV being negative is 0.0094

Prob (P1 > 1.2) Prob (PV / I > 1.2)

Prob (NPV / I > 0.2)

Prob. (NPV > 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 normal

curve:

P(Z > - 0.81) = 0.5 + P(-0.81 < Z < 0)

= 0.5 + P(0 < Z < 0.81)

= 0.5 + 0.2910

= 0.7910

So the probability of P1 > 1.2 as 0.7910

5. Given values of variables other than Q, P and V, the net present value model of Bidhan Corporation can be expressed as:

5

[Q(P V) 3,000 2,000] (0.5)+ 2,000 0 t=1

NPV = ---------------------------------------------------------- + ------- - 30,000

(1.1)t (1.1)5

5

0.5 Q (P V) 500 t=1

= ------------------------------------ - 30,000

(1.1)t

= [ 0.5Q (P V) 500] x PVIFA (10,5) 30,000 = [0.5Q (P V) 500] x 3.791 30,000 = 1.8955Q (P V) 31,895.5

Exhibit 1 presents the correspondence between the values of exogenous

variables and the two digit random number. Exhibit 2 shows the results of the

simulation.

Exhibit 1

Correspondence between values of exogenous variables and

two digit random numbers

QUANTITY PRICE VARIABLE COST

Value

Prob

Cumulative

Prob.

Two digit

random

numbers

Value

Prob

Cumulative

Prob.

Two digit

random

numbers

Value

Prob

Cumu-

lative

Prob.

Two digit

random

numbers

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 99

1,600 0.20 0.90 70 to 89

1,800 0.10 1.00 90 to 99

Exhibit 2

Simulation Results

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

Run Random

Number

Corres-

ponding

Value

Random

Number

Corres-

ponding

value

Random

Number

Corres-

ponding

value

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

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


Run Random

Number

Corres-

ponding

Value

Random

Number

Corres-

ponding

value

Random

Number

Corres-

ponding

value

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 -5,359

19 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 7,910

28 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 2,224


Run Random Corres- Random Corres- Random Corres- 1.8955 Q(P-V)-31,895.5

Number ponding

Value

Number ponding

value

Number ponding

value

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 -12,941

43 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

Expected NPV = NPV

50

= 1/ 50 NPVi i=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.30,000 k 10% - 10% 10%

F Rs.3,000 Rs.3,000 Rs.3,000 D Rs.2,000 Rs.2,000 Rs.2,000 T 0.5 0.5 0.5 N 5 5 5

S 0 0 0 Q Can assume any one of the values - 1,400*

800, 1,000, 1,200, 1,400, 1,600 and 1,800

P Can assume any of the values 20, 30, 30**

40 and 50

V Can assume any one of the values 20*

15,20 and 40

----------------------------------------------------------------------------------------

* The most likely values in the case of Q, P and V are the values that

have the highest probability associated with them

** In the case of price, 20 and 30 have the same probability of

occurrence viz., 0.4. We have chosen 30 as the most likely value

because the expected value of the distribution is closer to 30

Sensitivity Analysis with Reference to Q

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

variables is given by

5 [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 of Q are given in the following table:

Q 800 1,000 1,200 1,400 1,600 1,800

NPV -16,732 -12,941 -9,150 -5,359 -1,568 2,224

Sensitivity analysis with reference to P

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

variables 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

5 700 P 14,500

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

The net present values for various values of P are 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

Expected PI = PI = (PI)j P j j=1

= 1.24

6

Standard deviation = (PIj - PI) 2 P j o f P1 j=1

= .01156 = .1075

The standard deviation of P1 is .1075 for the given investment with an expected

PI 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 maximum

risk acceptable to the company for the given level of expected PI, the company

should accept the investment.

9. Investment A

Outlay : Rs.10,000

Net cash flow : Rs.3,000 for 6 years

Required 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 B

Outlay : Rs.30,000

Net cash flow : Rs.11,000 for 5 years

Required 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 are

as follows:

6 3,000

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

5 11,000

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

PI and IRR for the two projects are as follows:

Project A B

PI 1.23 1.26

IRR 20% 24.3%

B is superior to A in terms of NPV, PI, and IRR. Hence the company must

choose B.

Chapter 12

RISK ANALYSIS OF SINGLE INVESTMENTS

1. 2p = wi wj ij i j

2 p = w2121 + w2222 + w2323 + w2424 + w2525

+ 2 w1 w2 12 12 + 2 w1 w3 13 13 + 2 w1 w4 14 14 + 2 w1 w5 15

15 + 2 w2 w3 23 23 + 2 w2 w4 24 24 + 2 w2 w5 25 25 + 2 w3 w4

34 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 + 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

= 66.448

p = (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) all the 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 wBAB] 1 2 1 1

= 3 x x 2A + 6 x x x AB 3 3 3

1 2

= 2A + AB 3 3

(ii) Since there are 9 securities, there are 9 variance terms and 36 covariance

terms. Note that if the number of securities is n, the number of covariance

terms is n(n 1)/2.

In this case all the variance terms are the same (2A), all the covariance terms are 1

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

So,

n(n-1)

2p = 9 w2A 2A t 2 x wA wBAB 2

1 2 1 1

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

1 72

= 2A + AB 9 81

3. The beta for stock B is calculated below:

Period Return of

stock B,

RB (%)

Return on

market

portfolio,

RM (%)

Deviation of

return on

stock B from

its mean

(RB - RB)

Deviation

of return

on market

portfolio

from its

mean

(RM RM)

Product of

the

deviation

(RB RB)

(RM RM)

Square of

the

deviation

of return

on market

portfolio,

from its

mean

(RM RM)2

1 15 9 6 -1 -6 1

2 16 12 7 2 14 4

3 10 6 1 -4 -4 16

4 -15 4 -24 -6 144 36

5 -5 16 -14 6 -84 36

6 14 11 5 1 5 1

7 10 10 1 0 0 0

8 15 12 6 2 12 4

9 12 9 3 -1 -3 1

10 -4 8 -13 -2 26 4

11 -2 12 -11 2 -22 4

12 12 14 3 4 12 16

13 15 -6 6 -16 -96 256

14 12 2 3 -8 -24 64

15 10 8 1 -2 -2 4

16 9 7 0 -3 0 9

17 12 9 3 -1 -3 1

18 9 10 0 0 0 0

19 22 37 13 27 351 729

20 13 10 4 0 0 0

180 200 (RB RB) (RB RB)2 RB = 180 RM = 200 (RM RM) = 1186 RB = 9% RM = 10% = 320

Beta of stock B is equal to:

Cov (RB, RM)

2M (RB - RB) (RM RM) 320 Cov (RB, RM) = = = 16.84

n 1 19

(RM RM)2 1186

2M = = = 62.42 n 1 19

So the beta for stock B is:

16.84

= 0.270

62.42

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

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

Given a risk-free rate (Rf ) of 11 percent and the expected market risk premium

(E(RM Rf ) of 6 percent we get the following:

Project Beta Required rate(%) Expected rate (%)

A 0.5 11 + 0.5 x 6 = 14 15

B 0.8 11 + 0.8 x 6 = 15.8 16

C 1.2 11 + 1.2 x 6 = 18.2 21

D 1.6 11 + 1.6 x 6 = 20.6 22

E 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:

E A = [1 + D/E (1 T)]

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

Firm Asset Beta

1.25

A = 0.49

[1 + (2.25) x 0.7]

1.25

B = 0.48

[1 + (2.00) x 0.7]

1.10

C = 0.45

[1 + (2.1) x 0.7]

0.49 + 0.48 + 0.45

Average of the asset betas of sample firms = = 0.47

3

The 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. E A =

[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 1.30

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

= 0.68

The equity beta (systematic risk) for the petrochemicals project of Growmore,

when D/E = 1.25 and T = 0.4, is

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

(b) The cost of equity for the petrochemicals project is

12% + 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 is

1 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 years

Cost of distemper painting = Rs. 1,80,000 Life = 3 years

Discount rate = 10%

UAE of plastic emulsion painting = Rs.3,00,000 / 4.868 = Rs.61,627

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

Since plastic emulsion painting has a lower UAE, 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.1,372,013

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

Present value of costs of internal transportation = 1,500,000 1,372,013 system 162,828 = Rs.27,09,185

UAE of the internal transportation system = 27,09,185 / 3.517 = Rs.7,70,311

3. Cost of standard overhaul = Rs.500,000

Cost of less costly overhaul = Rs.200,000

Cost of capital = 14%

UAE of standard overhaul = 500,000 / 3.889 = Rs.128,568

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

Since the less costly overhaul has a lower UAE, 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.1,500,000 2. Economic life 12 years 9 years 3. Annual operating and maintenance costs Rs.250,000 Rs.320,000 4. Present value of the stream of operating

and maintenance costs at 12% discount rate

Rs.1,548,500 Rs.1,704,960

5. Salvage value Rs.800,000 Rs.500,000 6. Present value of salvage value Rs.205,600 Rs.180,500 7. Present value of total costs (1+4-6) Rs.3,842,900 Rs.3,024,460 8. UAE of 7 Rs.3,842,900

PVIFA (12%,12)

= 3,842,900

6.194

= Rs.620,423

Rs.3,024,460

PVIFA (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 million Rs. in million

0 10 10

1 15 13.395

2 19 15.143

3 23 16.376

4 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 12,000

2 25,000 15,000 0.797 11,955 22,671 1.690 13,415

3 35,000 21,000 0.712 14,952 37,623 2.402 15,663

4 50,000 30,000 0.636 19,080 56,703 3.037 18,671

5 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 47,337

3 80,000 2.402 33,306

4 80,000 3.037 26,342

5 80,000 3.605 22,191

Calculation of UAE (DTS) for Various Replacement Periods Time

(t)

Depreciation

charge R.s.

Depreciation

tax shield

PVIF

(12%, t)

PV of

depreciation

tax shield Rs..

Cumulative

present

value Rs..

PVIFA

(12%, t)

UAE of

depreciation

tax shield Rs..

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

1 20,000 8,000 0.893 7,144 7,144 0.893 8,000

2 15,000 6,000 0.797 4,782 11,926 1.690 7,057

3 11,250 4,500 0.712 3,204 15,130 2.402 6,299

4 8,438 3,375 0.636 2,147 17,277 3.037 5,689

5 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 21,222

3 32,000 0.712 22,784 2.402 9,485

4 22,000 0.636 13,992 3.037 4,607

5 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 33,586

2 13,415 47,337 7,057 21,222 19,058 32,473

3 15,663 33,306 6,299 9,485 17,522 33,185

4 18,671 26,342 5,689 4,607 16,046 34,717

5 22,335 22,191 5,190 2,359 14,642 36,977

OM - Operating and Maintenance Costs

IO - Investment Outlay

DTS - Depreciation Tax Shield

SV - Salvage Value

CC - Capital Cost

TC - Total Cost

UAE (CC) = UAE (IO) [UAE (DTS) + UAE (SV)] UAE (TC) = UAE (OM) + UAE (CC)

7. Calculation of UAE (OM) for Various Replacement periods Time O&M costs

Rs.

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 0.893 560,000

2 1,000,000 700,000 0.797 557,900 1,057,980 1.690 626,024

3 1,300,000 910,000 0.712 647,920 1,705,9000 2.402 710,200

4 1,900,000 1,330,000 0.636 845,880 2,551,780 3.037 840,230

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 4,479,283

2 4,000,000 1.690 2,366,864

3 4,000,000 2.402 1,665,279

4 4,000,000 3.037 1,317,089

5 4,000,000 3.605 1,109,570

Calculation of UAE (DTS) for Various Replacement Periods Time

(t)

Depreciation

charge Rs.

Depreciaton

tax shield

Rs.

PVIF

(12%, t)

PV of

depreciation

tax shield Rs.

Cumulative

present

value Rs.

PVIFA

(12%, t)

UAE of

depreciation

tax shield Rs.

1 1,000,000 300,000 0.893 267,940 267,900 0.893 300,000

2 750,000 225,000 0.797 179,325 447,225 1.690 264,630

3 562,500 168,750 0.712 120,150 567,375 2.402 236,209

4 421,875 126,563 0.636 80,494 647,869 3.037 213,325

5 316,406 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 2,800,000

2 2,000,000 0.797 1,594,000 1.690 943,195

3 1,400,000 0.712 996,80 2.402 414,988

4 1,000,000 0.636 636,000 3.037 209,417

5 800,000 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 -819,283

2 626,024 2,366,864 264,630 943,195 1,159,039 1,785,063

3 710,200 1,665,279 236,209 414,988 1,014,082 1,724,282

4 840,230 1,317,089 213,325 209,417 894,347 1,734,577

5 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

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 = 0.112

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

r* = r LrDT 1 + rD

1 + 0.16

= 0.16 0.6 x 0.15 x 0.5 x 1 + 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 million

Issue cost of equity = 12%

Rs.4,000,000

Equity to be issued = = Rs.4,545,455

0.88

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 1,440,000 432,000 366,102

2 8,000,000 1,440,000 432,000 310,256

3 7,000,000 1,260,000 378,000 230,062

4 6,000,000 1,080,000 324,000 167,116

5 5,000,000 900,000 270,000 118,019

6 4,000,000 720,000 216,000 80,013

1,271,568

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.2,567,455

Present 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 million

Equity finance = Rs.3 million

Issue cost of equity = 10%

Rs.3,000,000

Hence, 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.260,869

2 5,000,000 750,000 300,000 226,843

3 4,000,000 600,000 240,000 157,804

4 3,00,000 450,000 180,000 102,916

5 2,000,000 300,000 120,000 59,66

6 1,000,000 150,000 60,000 25,940

843,033

Adjusted NPV = - 1004000 333333 + 834033 = - Rs.503,300 Adjusted NPV if issue cost of external

equity 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%

Adjusted cost of capital as per Miles and Ezzell formula:

1 + r

r* = r LrDT 1 + rD

1 + 0.19

= 0.19 0.5 x 0.16 x 0.5 x 1 + 0.16

= 0.149 = 14.9%

12. S0 = Rs.46 , rh = 11 per cent , rf = 6 per cent

Hence the forecasted spot rates are :

Year Forecasted spot exchange rate

1 Rs.46 (1.11 / 1.06)1 = Rs.48.17

2 Rs.46 (1.11 / 1.06)2 = Rs.50.44

3 Rs.46 (1.11 / 1.06)3 = Rs.52.82

4 Rs.46 (1.11 / 1.06)4 = Rs.55.31

5 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 -9200

1 50 48.17 2408.5

2 70 50.44 3530.8

3 90 52.82 4753.8

4 105 55.31 5807.6

5 80 57.92 4633.6

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

2408.5 3530.8 4753.8

NPV = -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 :

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

2. Maintenance cost Annual 3 Benefits

3. Savings in the operation cost of existing ships

Annual 40

4. Increase in consumer satisfaction

Annual 3.6 The number of passenger hours

saved will be : (75,000 x 2 +

50,000 + 50,000 x 2) = 600000.

Multiplying this by Rs.6 gives

Rs.3.6 million

The IRR of the stream of social costs and benefits is the value of r in 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 r is about 10.1%

2. Social Costs and Benefits

Costs

Decrease in customer satisfaction as reflected Rs.266,667

in the opportunity cost of the extra time taken

by bus journey

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

Benefits

1. Resale value of the diesel train (one time) Rs.240,000 2. Avoidance of annual cash loss Rs.400,000

Fare collection = 1000 x 250 x Rs.4

= Rs.1,000,000

Cash 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 24 2. Land development cost 0 150 3. Maintenance cost 1-40 1 4. Labour cost 0 40 This includes the cost of

transport and rehabilitation

5. Labour c

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