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: ∑TY–nTY b= ∑T2–nT2 a = Y – bT The parameters are calculated below: Calculation in the Least Squares Method T Y TY 1 2,000 2,000 2 2,200 4,400 3 2,100 6,300 4 2,300 9,200 5 2,500 12,500 6 3,200 19,200 7 3,600 25,200 8 4,000 32,000 9 3,900 35,100 10 4,000 40,000 11 4,200 46,200 12 4,300 51,600 13 4,900 63,700 14 5,300 74,200 ∑ T = 105 ∑ Y = 48,500 ∑ TY = 421,600 T = 7.5 Y = 3,464 ∑TY–nTY b= ∑T2–nT2 57,880 =
421,600 – 14 x 7.5 x 3,464 =
= 254
227.5 a = Y – bT = 3,464 – 254 (7.5) = 1,559
1,015 – 14 x 7.5 x 7.5
T2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 ∑ T 2 = 1,015
Thus linear regression is Y = 1,559 + 254 T 2. In general, in exponential smoothing the forecast for t + 1 is Ft + 1 = Ft + α et Where Ft + 1 = forecast for year ) α = smoothing parameter et = error in the forecast for year t = St = Ft F1 is given to be 2100 and α is given to be 0.3 The forecasts for periods 2 to 14 are calculated below: Period t Data (St) Forecast Error Forecast for t + 1 (Ft) (et St =Ft) (Ft + 1 = Ft + α et) 1 2 3 4 5 6 7 8 9 10 11 12 13
2,000 2,200 2,100 2,300 2,500 3,200 3,600 4,000 3,900 4,000 4,200 4,300 4,900
2100.0 2070 2109.0 2111.7 2168.19 2267.7 2547.4 2863.2 3204.24 3413 3589.1 3772.4 3930.7
-100 130 -9 188.3 331.81 932.3 1052.6 1136.8 695.76 587.0 610.9 527.6 969.3
F2 = 2100 + 0.3 (-100) = 2070 F3 = 2070 + 0.3(130) = 2109 F4 = 2109 + 0.3 (-9) = 2111.7 F5 = 2111.7 + 0.3(188.3) = 2168.19 F6 = 2168.19 + 0.3(331.81) = 2267.7 F7 = 2267.7 + 0.3(9332.3) = 2547.4 F8 = 2547.4 + 0.3(1052.6) = 2863.2 F9 = 2863.2 + 0.3(1136.8) = 3204.24 F10 = 33204.24 + 0.3(695.76) = 3413.0 F11 = 3413.0 + 0.3(587) = 3589.1 F12 = 3589.1 + 0.3(610.9) = 3773.4 F13 = 3772.4 + 0.3(527.6) = 3930.7 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)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
2,000 2,200 2,100 2,300 2,500 3,200 3,600 4,000 3,900 4,000 4,200 4,300 4,900 5,300
Forecast (Ft)
2100 2200 2300 2667 3100 3600 3833 3967 4033 4167 4467
Forecast for t + 1 Ft + 1 = (St+ S t – 1 + S t – 2)/ 3
F4 = (2000 + 2200 + 2100)/3 = 2100 F5 =(2200 + 2100 + 2300)/3= 2200 F6 = (2100 + 2300 + 2500)/3 = 2300 F7 = (2300 + 2500 + 3200)/3= 2667 F8 = (2500 + 3200 + 3600)/3 = 3100 F9 = (3200 + 3600 + 4000)/3 = 3600 F10 = (3600 + 4000 + 3900)/3 = 3833 F11 = (4000 + 3900 + 4000)/3 =3967 F12 =(3900 + 4000 + 4200)/3 = 4033 F13 = (4000 + 4200 + 4300)/3 = 4167 F14 = (4200 + 4300 + 4900) = 4467
4. Q1 = 60 Q2 = 70 I1 = 1000 I2 = 1200
Q1 – Q2
Income Elasticity of Demand E1 =
x I2 - I1
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 E1 =
1000 + 1200 x
1200 – 1000 22000
E1 =
= 0.846 26000
70 + 60
I1 + I2 Q2 – Q1
5. P1 = Rs.40 P2 = Rs.50 Q1 = 1,00,000 Q2 = 95,000
Q2 – Q1
Price Elasticity of Demand = Ep =
P1 + P2 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 Ep =
40 + 50 x
50 - 40 - 45 Ep =
= - 0.0231 1950
95000 + 100000
Chapter 6 FINANCIAL ESTIMATES AND PROJECTIONS 1. Projected Cash Flow Statement Sources of Funds Profit before interest and tax Depreciation provision for the year Secured term loan Total (A) Disposition of Funds Capital expenditure Increase in working capital Repayment of term loan Interest Tax Dividends Total (B) Opening cash balance Net surplus (deficit) (A – B) Closing cash balance
(Rs. in million)
4.5 1.5 1.0 7.0
1.50 0.35 0.50 1.20 1.80 1.00 6.35 1.00 0.65 1.65
Projected Balance Sheet
Liabilities Share capital Reserves & surplus Secured loans Unsecured loans Current liabilities & provisions
5.00 4.50 4.50 3.00 6.30 1.05 24.35
(Rs. in million) Assets Fixed assets 11.00 Investments .50 Current assets 12.85 * Cash 1.65 * Receivables 4.20 * Inventories 7.00 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 Sources Share capital Term loan Short-term bank borrowing Profit before interest and tax Depreciation Write off preliminary expenses
1st Operating year
1800 3000
4800 Uses Capital expenditure Current assets (other than cash) Interest Preliminary expenses Pre-operative expenses
3900 150 600 4650 Opening cash balance 0 Net surplus / deficit 150 Closing balance 150 Projected Balance Sheet Liabilities 31/3/n+1 31/3/n+2 Assets Share capital 1800 1800 Fixed assets (net) Reserves & surplus 122 Secured loans : Current assets - Term loan 3000 3600 - Cash - Short-term bank 1800 Other current assets borrowing Unsecured loans Miscellaneous expenditures & losses Current liabilities and - Preliminary provisions expenses 4800 7322
600 1800 1166 319 15 3900 2400 1044 3444 150 456 606 31/3/n+1 4500
31/3/n+2 4181
150
606 2400
150
135
4800
7322
Notes : i.
Allocation of Pre-operative Expenses : Rs. Type
Costs before allocation Land 120 Building 630 Plant & machinery 2700 Miscellaneous fixed assets 450 3900 ii.
19 97 415 69 600
Costs after allocation 139 727 3115 519 4500
Depreciation Schedule :
Opening balance Depreciation Closing balance iii.
Allocation
Land
Building
Plant & machinery
139 139
727 25 702
3115 252 2863
M.Fixed assets 519 42 477
Total (Rs.)
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
4500 319 4181
Chapter 7 THE TIME VALUE OF MONEY 1.
2.
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
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.
6.
Let A be the annual savings. A x FVIFA (12%, 10 years) = A x 17.549 =
1,000,000 1,000,000
So A = 1,000,000 / 17.549 =
Rs.56,983.
1,000 x FVIFA (r, 6 years)
=
10,000
FVIFA (r, 6 years)
=
10,000 / 1000 = 10
= =
9.930 10.980
From the tables we find that FVIFA (20%, 6 years) FVIFA (24%, 6 years)
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) FVIF (r,10 years)
= =
From the tables we find that FVIF (16%, 10 years) = FVIF (18%, 10 years) =
5,000 5,000 / 1000 = 5
4.411 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% 9.
PV
= 10,000 x PVIF (r = 15%, 8 years) = 10,000 x 0.327 = Rs.3,270 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) PVIFA (18%, 10 years)
= =
5.019 4.494
Using linear interpolation we get:
r = 15% +
5.019 – 5.00 ---------------5.019 – 4.494
x 3%
= 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 =
PVIF(12%, 8 years) 21.
= Rs.40,388 2.476
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
From the tables we find that 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
A
=
Rs.50,000 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.
of
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 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)
Rs.180 x
(1.01)180 - 1 ---------------- = 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 = P x 1.817 =
P
27.
=
Rs.14,998 Rs.14,998 Rs.14,998 ------------ = Rs.8254 1.817
Rs.300 x PVIFA(r, 24 months) = Rs.6,000 PVIFA (4%,24) =
Rs.6000 / Rs.300
From the tables we find that: PVIFA(1%,24) = PVIFA (2%, 24) =
= 20
21.244 18.914
Using a linear interpolation
r = 1% +
21.244 – 20.000 ---------------------21.244 – 18,914
x 1%
= 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 From the tables we find that 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
n=7+
30.
5.000 – 4.868 ----------------5.335 – 4.868
Equated annual installment
x 1 = 7.3 years
= 500000 / PVIFA(14%,4) = 500000 / 2.914 = Rs.171,585
Loan Amortisation Schedule
Year 1 2 3 4
Beginning amount
Annual installment
500000 398415 282608 150588
171585 171585 171585 171585
Interest
Principal repaid
Remaining balance
70000 55778 39565 21082
101585 115807 132020 150503
398415 282608 150588 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
= Rs.300 million x
1 – (1 + g)n / (1 + i)n -----------------------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
= (b)
- 44837
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) b) c)
Payback period NPV
= 5 years = 40000 x PVIFA (12%,10) – 200 000 = 26000 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) PVIFA (16%,10)
= =
5.019 4.883
Linear interporation in this range yields
d)
r = =
15 + 1 x (0.019 / 0.136) 15.14%
BCR
= = =
Benefit Cost Ratio PVB / I 226,000 / 200,000 = 1.13
Investment B a)
Payback period
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)
=
9 years
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%,5) + 60,000 x PVIF (12%,6) + 40,000 x PVIFA (12%,4) x PVIF (12%,6) - 210,000 111,371
c)
80.000 x PVIF (12%,1) + 60,000 x PVIF (12%,2) + 80,000 x PVIF (12%,3) + 60,000 x PVIF (12%,4)
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) Payback period (in years)
A
5
B
C
D
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.e., I 7.
8.
PV of benefits (PVB) = + + + + = Investment = Benefit cost ratio =
= =
I 141256
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 100,000 1.23 [= (A) / (B)]
(A) (B)
The NPV’s of the three projects are as follows: Project Discount rate 0% 5% 10% 15% 25% 30%
9. (a)
P
Q
400 223 69 - 66 - 291 - 386
500 251 40 - 142 - 435 - 555
R 600 312 70 - 135 - 461 - 591
NPV profiles for Projects P and Q for selected discount rates are as follows: Project Discount rate (%) 0 5 10 15 20
b)
(i)
P
Q
2950 1876 1075 471 11
500 208 - 28 - 222 - 382
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)
10. (a)
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% 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 PV of cash flows up to the end of year 2 PV of cash flows up to the end of year 3 PV of cash flows up to the end of year 4
= = = =
12% p.a 24.97 47.75 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 PV of cash flows up to the end of year 1 PV of cash flows up to the end of year 2
= = =
12% per annum 33.93 51.47
DPB lies between 1 and 2 years. Interpolating in this range we get an approximate DPB of 1.92 years.
(c)
Project M Cost of capital NPV
= =
= Project N Cost of capital NPV
12% per annum - 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
= 12% per annum = 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 Cost of capital NPV
= 10% per annum = Rs.25.02 million
Project N Cost of capital NPV
= 10% per annum = Rs.23.08 million
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 Cost of capital = NPV =
15% per annum 16.13 million
Project N Cost of capital: NPV =
15% per annum Rs.17.23 million
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 8000 =
1,000 -
10,000 +
2,000 +
(1+r) (1+r)2 (1+r)3 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
Year
(1+r)4
Unrecovered Investment Balance Unrecovered Interest for the Cash flow at the investment balance at year Ft-1 (1+r) end of the year CFt the beginning Ft-1
1 2 3 4
-8000 -7480 -9863.8 -1688.60
-1480 -1383.8 -1824.80 -312.39
2000 -1000 10000 2000
13.
Unrecovered investment balance at the end of the year Ft-1 (1+r) + CFt -7480 -9863.8 -1688.60 0
Rs. in lakhs Year Investment Depreciation Income before interest and tax Interest Income before tax Tax Income after tax
1 24.0 3.0 6.0
2 21.0 3.0 6.5
3 18.0 3.0 7.0
4 15.0 3.0 7.0
5 12.0 3.0 7.0
6 9.0 3.0 6.5
7 6.0 3.0 6.0
8 3.0 3.0 5.0
Sum 108 24.0 51.0
Average 13.500 3.000 6.375
2.5 3.5 3.5
2.5 4.0 1.0 3.0
2.5 4.5 2.5 2.0
2.5 4.5 2.5 2.0
2.5 4.5 2.5 2.0
2.5 4.0 2.2 1.8
2.5 3.5 1.9 1.6
2.5 2.5 1.4 1.1
20.0 31.0 14.0 17.0
2.500 3.875 1.750 2.125
Measures of Accounting Rate of Return A.
Average income after tax
2.125 =
Initial investment
= 8.9% 24
B.
Average income after tax
2.125 =
Average investment C.
= 15.7% 13.5
Average income after tax but before interest
2.125 + 2.5 =
= 19.3%
Initial investment D.
24
Average income after tax but before interest
2.125 + 2.5 =
= 34.3%
Average investment
E.
13.5
Average income before interest and taxes
6.375 =
Initial investment F.
= 26.6% 24
Average income before interest and taxes
6.375 =
Average investment G.
= 47.2% 13.5
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
Year
0
1. Plant & machinery
(150)
(Rs. in million)
1
2
3
4
5
6
7
3. Revenues
250
250
250
250
250
250
250
4. Costs (excluding depreciation & 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
2. Working capital
(50)
9. Net salvage value of plant & machinery
48
10. Recovery of working capital 11. Initial outlay (=1+2) 12. Operating CF (= 8 + 5) 107.00 13. Terminal CF ( = 9 +10)
50
(200) 116.25 113.44 111.33 109.75 108.56 107.67
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
Year
0
1
2
3
(Rs. in million) 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 (17+18-19+20)
(b)
(140) 10.20
20.55 31.46 62.80 49.25 35.94 55.00
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)
3. (a)
=
Rs.1.70 million
A.
Initial outlay (Time 0) i. ii. iii iv.
B.
Cost of new machine Salvage value of old machine Incremental working capital requirement Total net investment (=i – ii + iii)
Rs.
3,000,000 900,000 500,000 2,600,000
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. ii. iii. iv.
D. Year NCF
Salvage value of new machine Salvage value of old machine Recovery of incremental working capital Terminal cash flow ( i – ii + iii)
Rs.
1,500,000 200,000 500,000 1,800,000
Net cash flows associated with the replacement project (in Rs) 0 (2,600,000)
1
2
3
4
620000
578750
547813
524609
5 307207
(b)
NPV of the replacement project =
4.
- 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
Tax shield (savings) on depreciation (in Rs)
Year
Depreciation charge (DC)
Tax shield =0.4 x DC
PV of tax shield @ 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. ii. iii.
B.
Cost of new machine Salvage value of the old machine Net investment
Operating cash flow (years 1 through 5)
Rs.
400,000 90,000 310,000
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 depreciation ( ii – i)
82000
60600
44730
32972
24268
iv. Tax savings on incremental depreciation ( 0.35 x (iii)) 28700
21210
15656
11540
8494
v. Operating cash flow
21210
15656
11540
8494
C.
Terminal cash flow (year 5) i. ii. iii.
D. Year NCF
28700
Salvage value of new machine Salvage value of old machine Incremental salvage value of new machine = Terminal cash flow
Rs.
15000
Net cash flows associated with the replacement proposal. 0 (310000)
1 28700
2
3
21210
15656
25000 10000
4 11540
5 23494
6.
Net Cash Flows Relating to Equity (Rs. in million)
Particulars 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Equity funds Revenues Operating costs Depreciation Interest on working capital advance Interest on term loan Profit before tax Tax Profit after tax Preference dividend Net salvage value of fixed assets Net salvage value of current assets Repayment of term-loans Redemption of preference capital Repayment of short-term bank borrowings Retirement of trade creditors Initial investment (1) Operating cash flows (9-10+4) Liquidation and retirement cash flows (11+12-13-14-15-16) Net cash flows (17+18+19)
Year 0 (100)
1
2
3
500 320 83.33 18.00
500 320 55.56 18.00
500 320 37.04 18.00
30.00 48.67 24.335 24.335
28.50 77.94 38.97 38.97
22.50 102.46 51.23 51.23
-
40
40
7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Fixed assets Working capital margin Revenues Operating costs Depreciation Interest on working capital advance Interest on term loan Profit before tax Tax @ 50% Profit after tax Net salvage value of fixed assets Net recovery of working capital margin Initial investment (1+2) Operating cash inflow (9+5+7 (1-T) ) Terminal cash flow (11+12) Net cash flow (13+14+15)
500 320 24.69 18.00 16.50 120.81 60.405 60.405
40
5 500 320 16.46 18.00 10.50 135.04 67.52 67.52
40
6 500 320 10.97 18.00 4.50 146.53 73.265 73.265 200 40
100 50 (100)
(100)
107.665 107.665
94.53 54.53
88.27 48.27
85.095 45.095
83.98 43.98
84.235 90
107.665
54.53
48.27
45.095
43.98
174.235
Net Cash Flows Relating to Long-term Funds Particulars Year 1. 2. 3. 4. 5. 6.
4
0 (250) (50)
1
500 320 83.33 18.00
2
3
(Rs. in million) 4
500 320 24.69 18.00
5
6
500 320 16.46 18.00
500 320 10.97 18.00
500 320 55.56 18.00
500 320 37.04 18.00
30.00 48.67 24.335 24.335
28.50 77.94 38.97 38.97
22.50 102.46 51.23 51.23
16.50 120.81 60.405 60.405
10.50 135.04 67.52 67.52
4.50 146.53 73.265 73.265 80 50
122.665
108.78
99.52
93.345
89.23
86.845
122.665
108.78
99.52
93.345
89.23
130.00 216.485
(300)
(300)
Cash Flows Relating to Total Funds (Rs. in million) Year 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Total funds Revenues Operating costs Depreciation Interest on term loan Interest on working capital advance Profit before tax Tax Profit after tax Net salvalue of fixed assets Net salvage value of current assets Initial investment (1) Operating cash inflow 9+4+6 (1-t) + 5(1-t) Terminal cash flow (10+11) Net cash flow (12+13+14)
0 (450)
1
2
3
4
5
6
500 320 83.33 30.00 18.00
500 320 55.56 28.50 18.00
500 320 37.04 22.50 18.00
500 320 24.69 16.50 18.00
500 320 16.46 10.50 18.00
500 320 10.97 4.50 18.00
48.67 24.34 24.34
77.94 38.97 38.97
102.46 51.23 51.23
120.81 60.41 60.41
135.04 67.52 67.52
146.53 73.265 73.265 80 200
131.67
117.78
108.52
102.35
98.23
95.485
131.67
117.78
108.52
102.35
98.23
280 375.485
(450)
(450)
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:
rD
=
(b) After tax cost =
2.
WACC
=
9 + (100 – 92)/6 -------------------0.4 x100 + 0.6x92
=
0.1085 (or) 10.85%
=
0.4 x 13% x (1 – 0.35) + 0.6 x 18% 14.18%
= 4.
5.
12.60 x (1 – 0.35) = 8.19%
Define rp as the cost of preference capital. Using the approximate yield formula rp can be calculated as follows: rp
3.
14 + (100 – 108)/10 ------------------------ x 100 = 12.60% 0.4 x 100 + 0.6x108
Cost of equity = (using SML equation) Pre-tax cost of debt = After-tax cost of debt = Debt equity ratio = WACC = =
10% + 1.2 x 7% = 18.4% 14% 14% x (1 – 0.35) = 9.1% 2:3 2/5 x 9.1% + 3/5 x 18.4% 14.68%
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 β = 14.9% where β denotes the beta of Azeez’s equity. Solving this equation we get β = 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.
Source Equity Debentures – first series Debentures – second series Bank loan Total
Book value 800 (0.54) 300 (0.20) 200 (0.13) 200 (0.13) 1500 (1.00)
(Rs. in million) Market value 2400 (0.78) 270 (0.09) 204 (0.06) 200 (0.07) 3074 (1.00)
8.
9.
(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.
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
Equity capital Preference capital Retained earnings Debentures Term loans
Component Book value Book value cost Rs. in million proportion (1) (2) (3) 14.5% 100 0.28 15.9% 10 0.03 14.5% 120 0.33 9.6% 50 0.14 6.0% 80 0.22 360 Average cost capital
Product of (1) & (3) 4.06 0.48 4.79 1.34 1.32 11.99%
The average cost of capital using market value proportions is calculated below :
Source of capital
Equity capital and retained earnings Preference capital Debentures Term loans
Component cost (1)
Market value Market value Product of Rs. in million (2) (3) (1) & (3)
14.5% 15.9% 9.6% 6.0%
200 7.5 40 80 327.5
11. (a)
WACC
= =
0.62 0.02 0.12 0.24 Average cost capital
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
8.99 0.32 1.15 1.44 11.90%
Chapter 11 RISK ANALYSIS OF SINGLE INVESTMENTS 1. (a)
NPV of the project
(b)
NPVs under alternative scenarios:
= =
-250 + 50 x PVIFA (13%,10) Rs.21.31 million
Pessimistic
(Rs. in million) Expected Optimistic
Investment Sales Variable costs Fixed costs Depreciation Pretax profit Tax @ 28.57% Profit after tax Net cash flow Cost of capital
300 150 97.5 30 30 - 7.5 - 2.14 - 5.36 24.64 14%
250 200 120 20 25 35 10 25 50 13%
200 275 154 15 20 86 24.57 61.43 81.43 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)
2. (a)
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
Sensitivity of NPV with respect to quantity manufactured and sold: (in Rs) Pessimistic Expected Optimistic Initial investment Sale revenue Variable costs Fixed costs Depreciation Profit before tax Tax Profit after tax Net cash flow NPV at a cost of capital of 10% p.a and useful life of 5 years
(b)
= Rs.205.31 million
30000 24000 16000 3000 2000 3000 1500 1500 3500
30000 42000 28000 3000 2000 9000 4500 4500 6500
30000 54000 36000 3000 2000 13000 6500 6500 8500
-16732
- 5360
2222
Sensitivity of NPV with respect to variations in unit price.
Initial investment Sale revenue Variable costs Fixed costs
Pessimistic
Expected
Optimistic
30000 28000 28000 3000
30000 42000 28000 3000
30000 70000 28000 3000
Depreciation Profit before tax Tax Profit after tax Net cash flow NPV (c)
2000 9000 4500 4500 6500 (-) 5360
2000 37000 18500 18500 20500 47711
Sensitivity of NPV with respect to variations in unit variable cost.
Initial investment Sale revenue Variable costs Fixed costs Depreciation Profit before tax Tax Profit after tax Net cash flow NPV (d)
2000 -5000 -2500 -2500 - 500 - 31895
Pessimistic
Expected
Optimistic
30000 42000 56000 3000 2000 -11000 -5500 -5500 -3500 -43268
30000 42000 28000 3000 2000 9000 4500 4500 6500 - 5360
30000 42000 21000 3000 2000 16000 8000 8000 10000 7908
Accounting break-even point i. ii. iii.
Fixed costs + depreciation Contribution margin ratio Break-even level of sales
= Rs.5000 = 10 / 30 = 0.3333 = 5000 / 0.3333 = Rs.15000
Financial break-even point
2.
i. ii.
Annual cash flow PV of annual cash flow
iii. iv.
Initial investment Break-even level of sales
= 0.5 x (0.3333 Sales – 5000) = 2000 = (i) x PVIFA (10%,5) = 0.6318 sales – 1896 = 30000 = 31896 / 0.6318 = Rs.50484
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 12
= = =
4.7 / 1.1 +5.8 / (1.1)2 + 3.9 / (1.1)3 – 10 Rs.2.00 million 0.41
22 32
= =
0.56 0.49
NPV =
12
2
22 +
(1.1)2
32 +
(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 = = = 4.
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
Expected NPV 4 At = - 25,000 t=1 (1.06)t
…. (1)
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) = t=1 (1.06)2t 12
= =
22
= =
32
= =
42
= =
…….. (2)
[(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 [(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 [(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 [(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 Prob (NPV < 0) = Prob.
0 - NPV <
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
Value
Prob
Cumulative Prob.
800 1,000 1,200 1,400 1,600 1,800
0.10 0.10 0.20 0.30 0.20 0.10
0.10 0.20 0.40 0.70 0.90 1.00
PRICE Two digit random numbers 00 to 09 10 to 19 20 to 39 40 to 69 70 to 89 90 to 99
Value
Prob
Cumulative Prob.
20 30 40 50
0.40 0.40 0.10 0.10
0.40 0.80 0.90 1.00
VARIABLE COST Two digit random numbers 00 to 39 40 to 79 80 to 89 90 to 99
Value
Prob
15 20 40
0.30 0.50 0.20
Cumulative Prob. 0.30 0.80 1.00
Two digit random numbers 00 to 29 30 to 79 80 to 99
Exhibit 2 Simulation Results Run
1 2 3 4 5 6 7 8 9 Run
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Run
QUANTITY (Q) Random CorresNumber ponding Value 03 800 32 1,200 61 1,400 48 1,400 32 1,200 31 1,200 22 1,200 46 1,400 57 1,400 QUANTITY (Q) Random CorresNumber ponding Value 92 1,800 25 1,200 64 1,400 14 1,000 05 800 07 800 34 1,200 79 1,600 55 1,400 57 1,400 53 1,400 36 1,200 32 1,200 49 1,400 21 1,200 08 .800 85 1,600 61 1,400 25 1,200 51 1,400 32 1,200 52 1,400 76 1,600 43 1,400 70 1,600 67 1,400 26 1,200 QUANTITY (Q) Random Corres-
PRICE (P) Random CorresNumber ponding value 38 20 69 30 30 20 60 30 19 20 88 40 78 30 11 20 20 20 PRICE (P) Random CorresNumber ponding value 77 30 65 30 04 20 51 30 39 20 90 50 63 30 91 50 54 30 12 20 78 30 79 30 22 20 93 50 84 40 70 30 63 30 68 30 81 40 76 30 47 30 61 30 18 20 04 20 11 20 35 20 63 30 PRICE (P) Random Corres-
VARIABLE COST (V) Random CorresNumber ponding value 17 15 24 15 03 15 83 40 11 15 30 20 41 20 52 20 15 15 VARIABLE COST (V) Random CorresNumber ponding value 38 20 36 20 83 40 72 20 81 40 40 20 67 20 99 40 64 20 19 15 22 15 96 40 75 20 88 40 35 20 27 15 69 20 16 15 39 20 38 20 46 20 58 20 41 20 49 20 59 20 26 15 22 15 VARIABLE COST (V) Random Corres-
NPV 1.8955 Q(P-V)-31,895.5
-24,314 2,224 -18,627 -58,433 -20,523 13,597 -9,150 -31,896 -18,627 NPV 1.8955 Q(P-V)-31,895.5
2,224 -9,150 -84,970 -12,941 -62,224 13,597 -9,150 -1,568 -5,359 -18,627 7,910 -54,642 -31,896 -5,359 13,597 -9,150 -1,568 7,910 13,597 -5,359 -9,150 -5,359 -31,896 -31,896 -31,896 -18,627 2,224 NPV 1.8955 Q(P-V)-31,895.5
Number 37 38 39 40 41 42 43 44 45 46 47 48 49 50
ponding Value 1,600 1,800 .800 1,400 1,800 1,000 1,200 1,600 1,400 1,000 1,200 1,400 1,600 1,800
89 94 09 44 98 10 38 83 54 16 20 61 82 90
Expected NPV
Number 86 00 15 84 23 53 44 30 71 70 65 61 48 50
= = = =
Variance of NPV
Number 59 25 29 21 79 77 31 10 52 19 87 70 97 43
ponding value 20 15 15 15 20 20 20 15 20 15 40 20 40 20
28,761 -14,836 -24,314 34,447 -31,896 -12,941 -9,150 -16,732 -5,359 -3,463 -54,642 -5,359 -62,224 2,224
NPV 50 1/ 50 NPVi i=1 1/50 (-7,20,961) 14,419 50 NPVi – NPV)2 i=1
=
1/50
=
1/50 [27,474.047 x 106] = 549.481 x 106
Standard deviation of NPV
6.
ponding value 40 20 20 40 20 30 30 20 30 30 30 30 30 30
= =
549.481 x 106 23,441
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 I k F D T N
Range Rs.30,000 – Rs.30,000 10% - 10% Rs.3,000 – Rs.3,000 Rs.2,000 – Rs.2,000 0.5 – 0.5 5–5
Most likely value Rs.30,000 10% Rs.3,000 Rs.2,000 0.5 5
0–0 0 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 S Q
** 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 t=1 (1.1) (1.1)5
=
5 t=1
5Q - 500 - 30,000 t
(1.1)
The net present values for various values of Q are given in the following table: Q NPV
800 -16,732
1,000 -12,941
1,200 -9,150
1,400 -5,359
1,600 -1,568
1,800 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:
NPV
=
5
[1,400 (P-20) – 3,000 – 2,000] x 0.5 + 2,000
0 +
t=1
t
(1.1)
- 30,000 (1.1)5
700 P – 14,500
5 = t=1
- 30,000 (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 (Rs.in lakhs) PI 0.9
0
5
10
15
20
1.00
1.10
1.20
1.30
1.40
Prob.
0.03
0.10
0.40
0.30
0.15
0.02
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)
10.
= 11,000 x PVIFA (14%, 5 years) – 30,000 = Rs.7763
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 t=1 (1.12)
Project B:
NPV
=
5 t=1
11,000 - 30,000 = Rs.7,763 t
(1.14)
PI and IRR for the two projects are as follows: Project
A
B
PI IRR
1.23 20%
1.26 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 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
1.
= 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 =3x 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) p= 9 w A 2
2
2
A
wA wBAB
t 2x 2
1 2 1 1 2 = 9 x x A + 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 Return on Deviation of stock B, market return on RB (%) portfolio, stock B from RM (%) its mean (RB - RB)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
15 9 16 12 10 6 -15 4 -5 16 14 11 10 10 15 12 12 9 -4 8 -2 12 12 14 15 -6 12 2 10 8 9 7 12 9 9 10 22 37 13 10 180 200 Σ RB = 180 ΣRM = 200 RB = 9% RM = 10%
6 7 1 -24 -14 5 1 6 3 -13 -11 3 6 3 1 0 3 0 13 4
Deviation of return on market portfolio from its mean (RM – RM)
Product of the deviation (RB – RB) (RM – RM)
-1 2 -4 -6 6 1 0 2 -1 -2 2 4 -16 -8 -2 -3 -1 0 27 0
-6 14 -4 144 -84 5 0 12 -3 26 -22 12 -96 -24 -2 0 -3 0 351 0 Σ(RB – RB) (RM – RM) = 320
Square of the deviation of return on market portfolio, from its mean (RM – RM)2 1 4 16 36 36 1 0 4 1 4 4 16 256 64 4 9 1 0 729 0 Σ(RB – RB)2 = 1186
Beta of stock B is equal to: Cov (RB, RM) 2M Cov (RB, RM) =
2
Σ (RB - RB) (RM – RM) = n –1
=
= 16.84 19
Σ (RM – RM)2 M
320
1186 =
n –1
= 62.42 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. A =
E
[1 + D/E (1 –T)] E = 1.25 D/E = 1.6
T = 0.3
So, Pariman Company’s asset beta is: 1.25 = 0.59 [1 + 1.6 (0.7)]
7. (a) Asset beta for a petrochemicals project is: A =
E
1.30 =
[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 Cost of distemper painting Discount rate UAE of plastic emulsion painting UAE of distemper painting
= Rs.3,00,000 Life = 7 years = Rs. 1,80,000 Life = 3 years = 10% = Rs.3,00,000 / 4.868 = Rs.61,627 = 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 = + + 1.13 (1.13)2 (1.13)3
4,50,000 +
5,00,000 +
(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 Cost of less costly overhaul Cost of capital UAE of standard overhaul UAE of less costly overhaul
= = = = =
Rs.500,000 Rs.200,000 14% 500,000 / 3.889 = Rs.128,568 200,000 / 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 1. 2. 3. 4. 5. 6. 7. 8.
Initial outlay Economic life Annual operating and maintenance costs Present value of the stream of operating and maintenance costs at 12% discount rate Salvage value Present value of salvage value Present value of total costs (1+4-6) UAE of 7
Rs.2,500,000 12 years Rs.250,000 Rs.1,548,500
Counter plow Rs.1,500,000 9 years Rs.320,000 Rs.1,704,960
Rs.800,000 Rs.500,000 Rs.205,600 Rs.180,500 Rs.3,842,900 Rs.3,024,460 Rs.3,842,900 Rs.3,024,460 PVIFA (12%,12) PVIFA (12%,9) = 3,842,900 = 3,024,460 6.194 5.328 = Rs.620,423 = Rs.567,654
The Counter plow is a cheaper alternative 5.
The current value of different timing options is given below : Time 0 1 2 3 4
Net Future Value Rs. in million 10 15 19 23 26
Current Value Rs. in million 10 13.395 15.143 16.376 16.536
The optimal timing of the project is year 4. 6. Time (t)
(1) 1 2 3 4 5
Calculation of UAE (OM) for Various Replacement Periods Operating and maintenance costs (2) 20,000 25,000 35,000 50,000 70,000
Post-tax operating & maintenance costs (3) 12,000 15,000 21,000 30,000 42,000
PVIF (12%,t)
(4) 0.893 0.797 0.712 0.636 0.567
Present Cumulative value of present (3) value (5) 10,716 11,955 14,952 19,080 23,814
(6) 10,716 22,671 37,623 56,703 80,517
(Rupees) PVIFA UAE (12%,t) (OM)
(7) 0.893 1.690 2.402 3.037 3.605
(8) 12,000 13,415 15,663 18,671 22,335
Calculation of UAE (IO) for Various Replacement Periods Investment Outlay Rs. PVIFA (12%, t) UAE of investment outlay Rs. 80,000 0.893 89,586 80,000 1.690 47,337 80,000 2.402 33,306 80,000 3.037 26,342 80,000 3.605 22,191
Time (t) 1 2 3 4 5
Calculation of UAE (DTS) for Various Replacement Periods Time (t)
Depreciation charge R.s.
Depreciation tax shield
PVIF (12%, t)
(1) 1 2 3 4 5
(2) 20,000 15,000 11,250 8,438 6,328
(3) 8,000 6,000 4,500 3,375 2,531
(4) 0.893 0.797 0.712 0.636 0.567
Time (1) 1 2 3 4 5
PV of depreciation tax shield Rs.. (5) 7,144 4,782 3,204 2,147 1,435
Cumulative present value Rs.. (6) 7,144 11,926 15,130 17,277 18,712
PVIFA (12%, t) (7) 0.893 1.690 2.402 3.037 3.605
UAE of depreciation tax shield Rs.. (8) 8,000 7,057 6,299 5,689 5,191
Calculation of UAE (SV) for Various Replacement Periods Salvage PVIF Present value of PVIFA UAE of salvage value Rs. (12%, t) salvage value Rs. (12%, t) value Rs. (4) / (5) (2) (3) (4) (5) (6) 60,000 0.893 53,580 0.893 60,000 45,000 0.797 35,865 1.690 21,222 32,000 0.712 22,784 2.402 9,485 22,000 0.636 13,992 3.037 4,607 15,000 0.567 8,505 3.605 2,359
Summary of Information Required to Determine the Economic Life Replacement UAE UAE (IO) UAE UAE (SV) UAE UAE period (OM) Rs. Rs. (DTS) Rs. Rs. (CC) Rs. (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 IO DTS SV CC TC
-
Operating and Maintenance Costs Investment Outlay Depreciation Tax Shield Salvage Value Capital Cost 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.
(1) 1 2 3 4 5
(2) 800,000 1,000,000 1,300,000 1,900,000 2,800,000
Time 1 2 3 4 5
Post-tax O&M costs Rs. (3) 560,000 700,000 910,000 1,330,000 1,960,000
PVIF (12%,t) (4) 0.893 0.797 0.712 0.636 0.567
PV of posttax O&M costs Rs. (5) 500,080 557,900 647,920 845,880 1,111,320
Cumulative present value Rs. (6) 500,080 1,057,980 1,705,9000 2,551,780 3,663,100
PVIFA (12%, t) (7) 0.893 1.690 2.402 3.037 3.605
UAE of O&M costs Rs. (8) 560,000 626,024 710,200 840,230 1,016,117
Calculation of UAE (IO) for Various Replacement Periods Investment outlay Rs. PVIFA (12%, t) UAE of investment outlay Rs. 4,000,000 0.893 4,479,283 4,000,000 1.690 2,366,864 4,000,000 2.402 1,665,279 4,000,000 3.037 1,317,089 4,000,000 3.605 1,109,570 Calculation of UAE (DTS) for Various Replacement Periods
Time (t)
Depreciation charge Rs.
1 2 3 4 5
1,000,000 750,000 562,500 421,875 316,406
Time (1) 1 2 3 4 5
Depreciaton tax shield Rs. 300,000 225,000 168,750 126,563 94,922
PVIF (12%, t) 0.893 0.797 0.712 0.636 0.567
PV of depreciation tax shield Rs. 267,940 179,325 120,150 80,494 53,821
Cumulative present value Rs. 267,900 447,225 567,375 647,869 701,690
PVIFA (12%, t) 0.893 1.690 2.402 3.037 3.605
UAE of depreciation tax shield Rs. 300,000 264,630 236,209 213,325 194,643
Calculation of UAE (SV) for Various Replacement Peiods Salvage PVIF Present value of PVIFA UAE of salvage value Rs. (12%, t) salvage value Rs. (12%, t) value Rs. (4)/ (5) (2) (3) (4) (5) (6) 2,800,000 0.893 267,900 0.893 2,800,000 2,000,000 0.797 1,594,000 1.690 943,195 1,400,000 0.712 996,80 2.402 414,988 1,000,000 0.636 636,000 3.037 209,417 800,000 0.567 453,600 3.605 125,825
Summary of Information Required to Determine the Economic Life Replacement UAE UAE (IO) UAE UAE (SV) UAE (CC) UAE (TC) period (OM) Rs. (DTS) Rs. Rs. Rs. Rs. 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
Year (t)
1 2 3 4 5 6
Computation of Tax Shield Associated with Debt Finance Debt outstanding Interest Tax shield Present value of at the beginning tax shield Rs. Rs. Rs. Rs. 8,000,000 1,440,000 432,000 366,102 8,000,000 1,440,000 432,000 310,256 7,000,000 1,260,000 378,000 230,062 6,000,000 1,080,000 324,000 167,116 5,000,000 900,000 270,000 118,019 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.
b.
Year
1 2 3 4 5 6
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 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 Debt outstanding at the Interest Tax shield Present value of tax beginning shield Rs.5,000,000 5,000,000 4,000,000 3,00,000 2,000,000 1,000,000
Rs.750,000 750,000 600,000 450,000 300,000 150,000
Rs.300,000 300,000 240,000 180,000 120,000 60,000
Rs.260,869 226,843 157,804 102,916 59,66 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 1 2 3 4 5
Forecasted spot exchange rate Rs.46 (1.11 / 1.06)1 = Rs.48.17 Rs.46 (1.11 / 1.06)2 = Rs.50.44 Rs.46 (1.11 / 1.06)3 = Rs.52.82 Rs.46 (1.11 / 1.06)4 = Rs.55.31 Rs.46 (1.11 / 1.06)5 = Rs.57.92
The expected rupee cash flows for the project Year 0 1 2 3 4 5
Cash flow in dollars Expected exchange (million) rate -200 46 50 48.17 70 50.44 90 52.82 105 55.31 80 57.92
Cash flow in rupees (million) -9200 2408.5 3530.8 4753.8 5807.6 4633.6
Given a rupee discount rate of 20 per cent, the NPV in rupees is :
2408.5 NPV
=
-9200
+
+ (1.18)2
5807.6 +
3530.8 + (1.18)3
4633.6 +
(1.18)5
4753.8
(1.18)6
= Rs.3406.2 million The dollar NPV is : 3406.2 / 46 = 74.05 million dollars
(1.18)4
Chapter 14 SOCIAL COST BENEFIT ANALYSIS 1.
Social Costs and Benefits
Costs 1. Construction cost 2. Maintenance cost Benefits 3. Savings in the operation cost of existing ships 4. Increase in consumer satisfaction
Nature
Economic value (Rs in million)
Oneshot Annual
400
Annual
40
Explanation
3
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
400
=
50 t=1
40 + 3.6 – 3.0 = (1+r)t
50 t=1
40.6 (1+r)t
The solving value r is about 10.1% 2.
Social Costs and Benefits Costs Decrease in customer satisfaction as reflected 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) 2. Avoidance of annual cash loss Fare collection = 1000 x 250 x Rs.4 = Rs.1,000,000 Cash operating expenses = Rs.1,400,000
Rs.266,667
Rs.240,000 Rs.400,000
3.
The social costs and benefits of the project are estimated below: Costs & Benefits
0 0 1-40 0
Economic value 24 150 1 40
5. Labour cost
1-40
12
6. Decrease in the value of the timber output
2-40
4
1-40
0.5
1-5 6-40 1
10 50 20
1. 2. 3. 4.
Time
Construction cost Land development cost Maintenance cost Labour cost
Rs. in million Explanation
This includes the cost of transport and rehabilitation The shadow price of labour equals what others are willing to pay.
Benefits 7. Savings in the cost of shipping the agriculture produce 8. Income from cash crops 9. Income from the main crop 10. Increase in the value of timber output
Assuming that the life of the road is 40 years, the NPV of the stream of social costs and benefits at a discount rate of 10 percent is:
NPV
40 = - 24 - 150 - 40 - t=1 40 + t=1
0.5
(1.1)t
= - Rs.9.93 million
5 t=1
1 + 12 (1.1)t
40 t=2
10 40 (1.1)t t=6
50
4 (1.1)t 20
(1.1)t (1.1)1
4. Table 1 Social Costs Associated with the Initial Outlay Rs. in million Item
Financial cost 0.30 12.0
Imported equipment Indigeneous equipment Transport
15.0 80.0 2.0
Engineering and know-how fees Pre-operative expenses Bank charges Working capital requirement
6.0
Basis of conversion SCF = 1/1.5 T=0.50, L=0.25 R=0.25 CIF value CIF value T=0.65, L=0.25 R=0.10 SCF=1.5
6.0 3.7 25.0
SCF=1.0 SCF=0.02 SCF=0.8
Land Buildings
150.0
Tradeable value ab initio 0.20
6.0 0.074 20.0
T
L
R
6.0
3.0
3.0
1.3
0.5
0.2
7.3
3.5
3.2
9.0 60.0
9.0
104.274
Table 2 Conversion of Financial Costs into Social Costs Rs. in million Item Indigeneous raw material and stores Labour Salaries Repairs and maintenance Water, fuel, etc Electricity (Rate portion) Other overheads
Financial cost 85
Basis of conversion SCF=0.8
Tradeable value ab initio 68
7 5 1.2 6
SCF=0.5 SCF=0.8 SCF=1/1.5 T=0.5, L=0.25 R=0.25 T=0.71, L=0.13 R=0.16 SCF=1/1.5
3.5 4.0 0.8
5 10 119.2
6.667 82.967
T
L
R
3
1.5
1.5
3.55
0.65
0.8
6.55
2.15
2.3
As per table 1, the social cost of initial outlay is worked out as follows :
Rs. in million 104.274 4.867
Tradeable value ab initio Social cost of the tradeable component (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 : Tradeable value ab initio Social cost of the tradeable component ( 6.55 x 1/1.5 ) Social cost of labour component (2.15 x 0.5) Social cost of residual component (2.3 x 0.5) Total
82.967 4.367 1.075 1.150 89.559
The annual CIF value of the output is Rs.110 million. Hence the annual social net 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 project would be as follows : Year / ’s 0 1-19
Social flow (Rs. in million) -112.491 20.441
Chapter 15 MULTIPLE PROJECTS AND CONSTRAINTS 1. The ranking of the projects on the dimensions of NPV, IRR, and BCR is given below Project NPV (Rs.) Rank IRR (%) Rank BCR Rank M 60,610 3 34.1 2 2.21 1 N 58,500 4 34.9 1 1.59 3 O 40,050 5 18.6 4 1.33 5 P 162,960 1 26.2 3 2.09 2 Q 72,310 2 14.5 5 1.36 4 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 3 C 315,075 1 2.05 1 D 411,90 6 1.14 6 E 95,540 4 1.38 4 F 114,500 3 1.23 5 3. The two hypothetical projects are:
Initial outlay Cash inflows Year 1 Year 2 Year 3
A B
A 10000
B 1000
5000 5000 5000
600 600 600
NPV @ 10% 2435 492
Rank 1 2
IRR about 23% above 35%
Rank 2 1
4. The two hypothetical 4-year projects for which BCR and IRR criteria give different rankings are given below Project A B Investment outlay 20000 20000 Cash inflow Year 1 2000 8000 Year 2 2000 8000 Year 3 2000 8000 Year 4 31500 8000
Project A B
NPV 4822 4296
Rank 1 2
IRR 19% about 22%
Rank 2 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.5851 NPV (B) = 8000 x PVIFA(10%,8) – 50,000 = - Rs.840 NPV (C) = 15,000 x PVIFA(10%,8) – 75,000 = Rs.5025 NPV (D) = 15,000 x PVIFA(10%,12) – 100,000 = Rs.6,995 NPV (E) = 25,000 x PVIFA (10%,7) + 50,000 x PVIF(10%,7) – 150,000 = Rs.2,650 Since B and E have negative NPV, they are rejected. So we consider only A, C, and D. Further C and D are mutually exclusive. The feasible combinations, their outlays, and their NPVs are given below. Combination A C D A&C A&D
Outlay (Rs.) 20,000 75,000 100,000 95,000 120,000
NPV (Rs.) 5,851 5,025 6,995 10,876 12,846
The preferred combination is A & D. 6. The linear programming formulation of the capital budgeting problem under various constraints is as follows: Maximise 10 X1 + 15 X2 + 25 X3 + 40 X4 + 60 X5 + 100 X6 Subject to 15 X1 + 12 X2 + 8 X3 + 35 X4 + 100 X5 + 50 X6 + SF1 = 150
Funds constraint for year 1
5 X1 + 13 X2 + 40 X3 + 25 X4 + 10 X5 + 110 X6 ≤ 200 + 1.08 SF1
Funds constraint for year 2
5 X1 + 6 X2 + 5 X3 + 10 X4 + 12 X5 + 40 X6 ≤ 60
Power constraint
15 X1 + 20 X2 + 30 X3 + 35 X4 + 40 X5 + 60 X6 ≤ 120
Managerial constraint
0 ≤ Xj ≤ 1 (j = 1,….8) and SF1 ≥ 0
Rupees are expressed in ’000s. Power units are also expressed in ’000s. 7. Given the nature of the problem, in addition to the decision variables X1 through X10 for the original 10 projects, two more decision variables are required as follows: X11 X12
is the decision variable to represent the delay of projects 8 by one year 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
55 X1 + 75 X2 + 50 X3 + 60 X4 + 105 X5 + 12 X6 + 60 X7 + 120 X8 + 50 X9 + 40 X10 + 100 X11+ 178.2 X12
Subject to
75 X1 + 80 X2 + 75 X3 + 35 X4 + 80 X5 + 20 X6 + 70 X7 + 155 X8 + 55 X9 + 10 X10 + 109.3 X12 + SF1 = 400 40 X1 + 85 X2 + 8 X3 + 100 X4 + 160 X5 + 9 X6 + 5 X7 + 100 X8 + 20 X9 + 90 X10 + 155 X11+ 247 X12 + SF2 = 350 + SF1 (1 + r) X3 + X7 X5 + X8 + X9 + X10 X2 X8 X4 + X5 + X12 X8 + X11 Xj = {0,1} SFi ≥ 0
≥1 ≥2 ≤ X6 ≤ X9 ≤1 ≤1 j = 1, 2….12 i = 1, 2
It has been assumed that surplus funds can be shifted from one period to the next and they will earn a post-tax return of r percent. –
–
–
–
–
–
–
+
8. Minimise [P1(3d1+ 2 d 2 + d 3) + P 2 (4 d 4 + 2 d 5 + d 6) + P 3 (d 7 – d 7 )] Subject to: Economic Constraints 12 X1 + 14 X2 + 15 X3 + 16 X4 + 11 X5 + 23 X6 + 20 X7 ≤ 65 Goal Constraints 1.2 X1 + 1.6 X2 + 0.6 X3 + 1.5 X4 + 0.5 X5
–
+
+ 0.9 X6 + 1.8 X7 + d 1 – d 1 = 6
Net income for year 1
1.1 X1 + 1.2 X2 + 1.2 X3 + 1.6 X4 + 1.2 X5 –
+
+ 2.5 X6 + 2.0 X7 + d 2 – d 2 = 8
Net income for year 2
1.6 X1 + 1.5 X2 + 2.0 X3 + 1.8 X4 + 1.5 X5 –
+
+ 4.0 X6 + 2.2 X7 + d 3 – d 3 = 10
Net income for year 3
1.0 X1 + 1.2 X2 + 0.5 X3 + 1.8 X4 + 0.6 X5 –
+
+ 1.0 X6 + 2.0 X7 + d 4 – d 4 = 6
Sales growth for year 1
1.5 X1 + 1.0 X2 + 1.2 X3 + 2.0 X4 + 1.4 X5 –
+
+ 3.0 X6 + 3.0 X7 + d 5 – d 5 = 8
Sales growth for year 2
1.8 X1 + 1.2 X2 + 2.5 X3 + 2.2 X4 + 1.8 X5 –
+
+ 3.5 X6 + 3.5 X7 + d 6 – d 6 = 10
Sales growth for year 3
4 X1 + 5 X2 + 6 X3 + 8 X4 + 4 X5 –
+
+ 9 X6 + 7 X7 + d 7 – d 7 = 50 Xj 0
–
NPV
+
d i, d i 0
9. The BCRs of the projects are converted into NPVs as of now as follows Project 1 2 3 4 5 6 7 8 9
Outlay (Rs.) 800,000 200,000 400,000 300,000 200,000 500,000 400,000 600,000 300,000
BCR 1.08 1.35 1.20 1.03 0.98 1.03 1.21 1.17 1.01
NPV (Rs.) 64,000 70,000 80,000 9,000 - 4,000 15,000/1.10 = 13,636 84,000/1.10 = 76,364 102,000/1.10 = 92,727 3,000/1.10 = 2,727
The integer linear programming formulation of the problem is as follows :
Maximise
64,000 X1 + 70,000 X2 + 80,000 X3 + 9,000 X4 + 13,636 X6 + 76,364 X7 + 92,727 X8 + 2,727 X9
Subject to 800,000 X1 + 200,000 X2 + 400,000 X3 + 300,000 X4 + SF1 = 20,00,000 500,000 X6 + 400,000 X7 + 600,000 X8 + 300,000 X9 ≤ 500,000 + SF1 (1.032) Xj = {0,1}
j = 1, 2, 3, 4, 6, 7, 8, 9
Chapter 16 VALUATION OF REAL OPTIONS 1.
S = 100 , uS = 150, dS = 90 u = 1.5 , d = 0.9, r = 1.15 R = 1.15 E = 100 Cu = Max (uS – E, 0) = Max (150 – 100,0) = 50 Cd = Max (dS – E, 0) = Max (90 – 100,0) = 0 =
Cu – C d
50 =
= 0.833
(u-d)S
0.6 x 100
u Cd – d Cu
0 – 0.9 x 50
B =
= (u-d)R
= - 65.22 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 = 5 r = 0.16, R = 1.16, E = 60 Cu = Max (uS – E, 0) = Max (60u – E, 0) Cd = Max (dS – E, 0) = Max (45 – 60, 0) = 0 =
Cu – C d
60u – 60 =
u–1 =
(u – 0.75)60
(u-d)S u Cd – d Cu
u – 0.75
– 0.75 (60u – 60)
B =
= (u – 0.75) 1.16
(u-d)R
45 (1 – u) = 1.16 (u – 0.75)
C = S+B (u – 1) 60 5 =
45 (1 – u) +
u – 0.75 1.16 (u – 0.75) Multiplying both the sides by u – 0.75 we get 45 5(u – 0.75) = (u – 1) 60 + (1 – u) 1.16
Solving this equation for u we get u = 1.077 So Beta’s equity can rise to 60 x 1.077 = Rs.64.62 3.
E C0 = S0 N(d1) -
N (d2) ert S0 = 70, E = 72, r = 0.12, 0.3, t = 0.50 S0 ln
1 2
+ r+ E
d1 =
t
2 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
N (d1) = 0.6010 N (d2) = 0.5175 E = ert
0.50 = 0.0439
72 = 67.81 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.
E C0 = S0 N(d1) -
N (d2) ert E = 50, t = 0.25, S = 40, 0.40, r = 0.14
S0 ln
1 2
+ r+ E
t
2
d1 =
t
40 ln
+ (0.14 + 0.5 x 0.16) 0.25 50
d1 = 0.40 0.25 - 0.2231 + 0.055 =
= - 0.8405 0.20
d2 = d1 -
t = - 0.8405 – 0.40
N (d1) = 0.2003 N (d2) = 0.1491 E = rt e
0.25 = -1.0405
50 = 48.28 0.14 x 0.25
e
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 now will be Rs.189.54 million (Two times the present value of the cash inflows of Comp-I). So, we have S0 = present value of the asset = 189.54 x e–0.20 x 4 = Rs.85.17 million E = exercise price = $ 200 million = 0.30 t = 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 = -0.923
Step 2 : Find N(d1) and N(d2) N(d1) = 0.3733 N(d2) = 0.1780 Step 3 : Estimate the present value of the exercise price 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 unit building yields a profit of Rs.1.0 million (15 x 1.2 – 17). Hence a 9 unit building is the 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: Sluggish (Apartment price: Rs.1.5 million) 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 – unit building (payoff: Rs.5.5 million) and if the market turns out to be sluggish the best alternative is the 9 – unit building (payoff: Rs.0.9 million). Given the above information, we can apply the binomial method for valuing the vacant land: Step 1: Calculate the risk-neutral probabilities.
The binomial tree of apartment values is Rs.1.60 million (1.6 + 0.1) p Rs.1.2 million 1- p
Rs.1.20 million (1.1 + 0.1)
Given a risk free rate of 10 percent, the risk-neutral probabilities must satisfy the following conditions: p x 1.6 + (1 – p) x 120 1.2 million = 1.10 Solving this we get p = 0.3 Step 2: Calculate the expected cash flow next year The expected cash flow next year is: 0.3 x 5.5 + 0.7 x 0.9 = Rs.2.28 million Step 3: Compute the current value 2.28/ 1.10 = Rs.2.07 million Since Rs.2.07 million is greater than Rs.1.80 million, the profit from constructing a 9 unit building now, it is advisable to keep the vacant land. The value of the vacant land is Rs.2.07 million. 7. S0 = current value of the asset = value of the developed reserve discounted for 3 years (the development lag) at the dividend yield of 5% = $20 x 100/ (1.05)3 = $1727.6 million. E = exercise price = development cost = $600 million = standard deviation of ln (oil price) = 0.25 t = life of the option = 20 years r = risk-free rate = 8% y = dividend yield = net production revenue/ value of reserve = 5% Given these inputs, the call option is valued as follows: Step 1 : Calculate d1 and d2 S 2 ln + r–y t E 2 d1 = t
ln (1727.6/ 600) + [.08 - .05 + (.0625/ 2)] 20 = 0.25 d2 = d1 - t
20
= 2.0417 – 1.1180 = 0.9237
Step 2 : Find N(d1) and N(d2) N(d1) = N(2.0417) = 0.9794 N(d2) = N(0.9237) = 0.8221 Step 3 : Estimate the present value of the exercise price E / ert = 600 / e.08 x 20 = 600/ 4.9530 = $121.14 million Step 4 : Plug the numbers obtained in the previous steps in the Black-Scholes formula: C = $1727.6 million x 0.9794 - $121.14 million x 0.8221 = $1592.42 million
Chapter 21 PROJECT MANAGEMENT
1. a. Cost variance: BCWP – ACWP = 5,500,000 – 5,800,000 = – Rs.300,000 b. Schedule variance in cost terms: BCWP – BCWS = 5,500,000 – 6,00,000 = – Rs.500,000 5,500,000 c. Cost performance index: BCWP/ ACWP =
= 0.948 5,800,000 5,500,000
d. Schedule performance index: BCWP/ BCWS =
= 0.916 6,000,000
BCTW e. Estimated cost performance index:
10,000,000 =
(ACWP + ACC)
5,800,000 + 5,000,000 = 0.926
Chapter 22 NETWORK TECHNIQUES FOR PROJECT MANAGEMENT 2. The net work diagram with the earliest and latest occurrence times for each event is shown in Exhibit 1. Exhibit 1 Network for the Project
2
1
5 11 11
2
3
4 4
4
1 0 0
5
3
4 9 9
2
5
7 14 14
6
3 2 3 There are two critical paths: 1-2-4-5-7 and 1-2-4-7. The minimum time required for completing the project is 14 weeks.
3. The time estimates for various activities are shown in Exhibit 2. Exhibit 2 Time Estimates Activity
Optimistic to
Most likely tm
Pessimistic tp
Average to + 4 tm + tp te =
1-2 1-3 1-4 1-7 2-4 2-6 2-7 3-4 4-5 5-6 3-7 6-7
4 3 5 2 6 3 5 3 2 1 2 1
6 7 6 4 10 4 9 7 4 3 5 2
10 12 9 6 20 7 15 12 5 6 8 6
(a) The network diagram with average time estimates is shown in Exhibit 3.
4 6 1/3 7 1/6 6 1/3 4 11 4 1/3 9 1/3 7 1/6 3 5/6 3 1/6 5 2 1/2
Exhibit 3 2 6⅓ 6⅓
EOT LOT
4⅓ 9⅓
11
6⅓ 3⅚
3⅙
4
5
17 ⅓ 17⅓
21 ⅓ 21 ⅓
6 24 ⅓ 24 ⅓
7⅙ 6⅓ 3 7⅙
2½
10 ⅙ 5
7⅙
1 0
4 0
(b) The critical path for the project is 1-2-4-5-6-7 (c) Exhibit 3 shows the event slacks.
7 26 ⅚ 26 ⅚
Event 1 2 3 4 5 6 7
Exhibit 3 Event slacks LOT 0 6 1/3 10 1/6 17 1/3 21 1/6 24 1/3 26 5/6
EOT 0 6 1/3 7 1/6 17 1/3 21 1/6 24 1/3 26 5/6
Slack = LOT – EOT 0 0 3 0 0 0 0
Free Float EOT(j) – EOT(i) – dij 0 0 11 22 5/6 0 13 2/3 11 1/6 3 14 2/3 0 0 0
Independent Float EOT(j) – LOT (i) – dij 0 0 11 22 5/6 0 13 2/3 11 1/6 0 11 2/3 0 0 0
Exhibit 4 shows the activity floats Exhibit 4 Activity Floats Activity (i –j) 1-2 1-3 1-4 1-7 2-4 2-6 2-7 3-4 3-7 4-5 5-6 6-7
Duration dij 6 1/3 7 1/6 6 1/3 4 11 4 1/3 9 1/3 7 1/6 5 3 5/6 3 1/6 2 1/2
Total Float LOT(j) – EOT(i) – dij 0 3 11 22 5/6 0 13 2/3 11 1/6 3 14 2/3 0 0 0
(d) Standard deviation of the critical path duration = [Sum of the variances of activity durations on the critical path]1/2 The variances of the activity durations on the critical path are shown in Exhibit 5. Exhibit 5 Variances of Activity Durations on critical path Activity tp to tp – to 2 = 6 1-2 10 4 1.00 1.00 2-4 12 6 1.00 1.00 4-5 5 2 0.50 0.25 5-6 6 1 0.83 0.69 6-7 6 1 0.83 0.69 The standard deviation of the duration of critical path is:
= (1.00 + 1.00 + 0.25 + 0.69 + 0.69)1/2 = (3.63)1/2 = 1.91 weeks. (e) Let D = specified completion date T = mean of the critical path duration c = standard deviation of the critical path duration T = sum of the mean values of the activity durations on the critical path = 6 1/3 + 9 2/3 + 3 5/6 + 3 1/6 + 2 ½ = 25 ½ D–T Prob (D< 30) = Prob
30 – 25.5 <
c
= Prob [ Z < 2.356] 1.91
= 0.87 4.
(a) The net work diagram is given in Exhibit 6. Exhibit 6 Network Diagram 6
10
2
6
4
5
1
7
9
7
7
4
3
6
12
5
12
9
(b) The all-normal critical paths are 1-2-4-6-7-9 and 1-3-4-6-7-9. For all-normal network, the project duration is 34 weeks and the total direct cost is Rs.66,000. (c) The time-cost slope of the activities constituting the project is given in Exhibit 7.
(1) Activity (1-2) (2-4) (1-3) (3-4) (4-7) (3-5) (4-6) (6-7) (7-9) (5-9)
Time in weeks (2) Normal 5 6 4 7 9 12 10 7 6 12
Exhibit 7 Time-Cost Slope of Activities Cost (Rs.) (3) Crash 2 3 2 4 5 3 6 4 4 7
(4) Normal 6,000 7,000 1,000 4,000 6,000 16,000 15,000 4,000 3,000 4,000
(5) Crash 9,000 10,000 2,000 8,000 9,200 19,600 18,000 4,900 4,200 8,500
Cost to expedite per week (Rs.) (6) [(5)-(4) (2)-(3)] 1,000 1,000 500 1333.35 800 400 750 300 600 900
Examining the time-cost slope of activities on the critical path, we find that activity (6-7) has the lowest slope on both the critical paths. The project network after crashing this activity is shown below in Exhibit 8. Exhibit 8 6
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6
4
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1
4
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7
7
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3
6
12
5
12
9
As per Exhibit 8, the critical paths are (1-3-4-6-7-9) and (1-2-4-6-7-9) with a length of 31 weeks and the total cost is Rs.66,900. Looking at the time-cost slope of the activities on the critical paths (1-3-4-6-7-9) and (1-2-4-6-7-9), we find that activities (1-3) and (7-9) have the least time-cost slopes on the two critical paths respectively. The project net work after crashing these activities is shown in Exhibit 9.
Exhibit 9 6
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3
4
12
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12
9
As per Exhibit 9, the critical path is (1-2-4-6-7-9) with a length of 29 weeks and the total direct cost is Rs.(66,900 + 2,200) = Rs.69,100. Activity (4-6) has the least time-cost slope on the critical path. Hence this is crashed the net work after crashing (4-6) is shown in Exhibit 10. Exhibit 10 6
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2
6
4
5
1
4
9
7
7
2
3
4
12
5
12
9
As per Exhibit 10, the critical path is (1-3-5-9), with a length of 26 weeks, and total direct costs of Rs.72,100. Looking at the time-cost slope of the non-crashed activities on this path we find that activity (3-5) has the lowest slope. Hence it is crashed. The project net work after such crashing is shown in Exhibit 11.
Exhibit 11
6
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3
12
9
As per Exhibit 11, the critical path is (1-2-4-6-7-9), with a length of 25 weeks and a total direct cost of Rs.75,700. Looking at the time cost slope of the activities on this critical path, we find both activities (1-2) and (2-4) have the same slope. We crash activity (2-4). The resulting project network net work is given in Exhibit 12. Exhibit 12 6
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1
4
4
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3
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7
7
2
3
12
9
As per Exhibit 12, the critical path is (1-3-4-6-7-9), with a length of 23 weeks and a total direct cost of Rs.78,700. Crashing activity (3-4), the only uncrashed activity on this critical path, we get the net work shown in Exhibit 13.
Exhibit 13 6
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3
4
3
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12
9
As per Exhibit 13, the critical path is (1-2-4-6-7-9), with a length of 22 weeks and a total direct cost of Rs.82,700. The only uncrashed activity on this critical path is (1-2). Crashing this we get Exhibit 14. Exhibit 14 6
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9
As per Exhibit 14, the critical path is (1-3-4-6-7-9) with a duration of 20 weeks, and a total direct cost of Rs.85,700. Since all activities on this path are crashed, there is no possibility of further time reduction. Exhibit 15 shows the time-cost relationship.
Exhibit 15 Project Duration and Total Cost Exhibit
Activities Crashed
6 8 9 10 11 12 13 14
none (6-7) (6-7), (1-3) and (7-9) (6-7), (1-3), (7-9) and (4-6) (6-7), (1-3), (7-9), (4-6) and (3-5) (6-7), (1-3), (7-9), (4-6), (3-5) and (2-4) (6-7), (1-3), (7-9), (4-6), (3-5), (2-4) and (3-4) (6-7), (1-3), (7-9), (4-6), (3-5), (2-4), (3-4) and (1-2)
Project duration (in weeks) 34 31 29 26 25 23 22 20
Total direct cost (Rs.) 66,000 66,900 69,100 72,100 75,700 78,900 80,500 83,500
Total indirect cost (Rs.) 34,000 31,000 29,000 26,000 25,000 23,000 22,000 20,000
Total cost (Rs.) 1,00,000 97,900 98,100 98,100 1,00,700 1,01,700 1,02,500 1,03,500
If the objective is to minimise the total cost of the project, the pattern to crashing suggested by Exhibit 10 may appear as the best. However, it is possible to reduce the cost further without increasing the project duration beyond 26 weeks by decrashing some activities on the non-critical paths. To do so, begin with the activity which has the highest time-cost slope and proceed in the order of decreasing time-cost slope.
Chapter 23 PROJECT REVIEW AND ADMINISTRATIVE PROJECTS 1. 1. 2.
3.
4. 5. 6. 7.
Calculation of Economic Rate of Return Year 1 Cash flow 25 Present value at the 146.895 beginning of the year; 12 percent discount rate Present value at the end 139.518 of the year, 12 percent discount rate Change in value during -7.377 the year (3 – 2) Economic income 17.623 (1 + 4) Economic rate of return 0.12 (5/2) Economic depreciation 7.377
2
3
4
5
6
30 139.518
40 126.268
45 101.408
50 68.543
30 26.786
126.268
101.408
68.543
26.786
0
-13.250
-24.860
-32.865
-41.757
-26.786
16.750
15.140
12.135
8.243
3.214
0.12
0.12
0.12
0.12
0.12
13.250
24.860
32.865
41.757
26.786
2 30 122.412
3 40 97.929
4 45 73.446
5 50 48.963
6 30 24.480
97.929
73.446
48.963
24.480
-24.483
-24.483
-24.483
-24.483
-24.483
5.517 0.045
15.517 0.158
20.517 0.274
25.517 0.521
5.52 0.225
24.483
24.483
24.483
24.483
24.483
Calculation of Book Return on Investment 1. 2.
3.
4. 5. 6. 7.
Year 1 Cash flow 25 Book value at the 146.895 beginning of the year, straight line depreciation Book value at the end of 122.412 the year, straight line depreciation Change in book value -24.483 during the year (3 – 2 Book income (1 + 4) 0.517 Book return on .004 investment (5/2) Book depreciation 24.483
2. SV = Rs.120 million
DV = Rs.175 million
30 PVCF
=
35 +
(1.12)
45 +
(1.12)2
50 +
(1.12)3
(1.12)4
-
30
25
=
= Rs.148.21 million (1.12)5
(1.12)6
Since DV > PVCF > SV it is advisable to sell it to the third party at Rs.175 million
