Chapter 7
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 = Rs.1469
r = 10% FV5 = Rs.1611
r = 12% FV5 = Rs.1762
r = 15% FV5 = Rs.2011
2. 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
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 From the tables we find that
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)
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 The amount that can be withdrawn annually is: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 / 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
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 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 per cent 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 (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 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 5.000 – 4.868
n = 7 + --- x 1 = 7.3 years
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
MINICASE Solution:
1. How much money would Ramesh need 15 years from now? 500,000 x PVIFA (10%, 15years)
+ 1,000,000 x PVIF (10%, 15years) = 500,000 x 7.606 + 1,000,000 x 0.239 = 3,803,000 x 239,000
= Rs.4,042,000
2. How much money should Ramesh save each year for the next 15 years to be able to meet his investment objective?
Ramesh’s current capital of Rs.600,000 will grow to : 600,000 (1.10)15 = 600,000 x 4.177 = Rs 2,506,200 This means that his savings in the next 15 years must grow to :
4,042,000 – 2,506,200 = Rs 1,535,800 So, the annual savings must be :
1,535,800 1,535,800
= = Rs.48,338 FVIFA (10%, 15 years) 31.772
3. How much money would Ramesh need when he reaches the age of 60 to meet his donation objective?
200,000 x PVIFA (10% , 3yrs) x PVIF (10%, 11yrs) = 200,000 x 2.487 x 0.317 = 157,676
4. What is the present value of Ramesh’s life time earnings?
400,000 400,000(1.12) 400,000(1.12)14 46
1.12 15 1 – 1.08 = 400,000 0.08 – 0.12 = Rs.7,254,962
Chapter 8
VALUATION OF BONDS AND STOCKS 1. 5 11 100
P = +
t=1 (1.15) (1.15)5
= Rs.11 x PVIFA(15%, 5 years) + Rs.100 x PVIF (15%, 5 years) = Rs.11 x 3.352 + Rs.100 x 0.497
= Rs.86.7
2.(i) When the discount rate is 14% 7 12 100
P = +
t=1 (1.14) t (1.14)7
= Rs.12 x PVIFA (14%, 7 years) + Rs.100 x PVIF (14%, 7 years) = Rs.12 x 4.288 + Rs.100 x 0.4
= Rs.91.46
(ii) When the discount rate is 12% 7 12 100
P = + = Rs.100
t=1 (1.12) t (1.12)7
Note that when the discount rate and the coupon rate are the same the value is equal to par value.
3. The yield to maturity is the value of r that satisfies the following equality. 7 120 1,000
Rs.750 = + = Rs.100 t=1 (1+r) t (1+r)7
Try r = 18%. The right hand side (RHS) of the above equation is: Rs.120 x PVIFA (18%, 7 years) + Rs.1,000 x PVIF (18%, 7 years) = Rs.120 x 3.812 + Rs.1,000 x 0.314
= Rs.771.44
Try r = 20%. The right hand side (RHS) of the above equation is: Rs.120 x PVIFA (20%, 7 years) + Rs.1,000 x PVIF (20%, 7 years) = Rs.120 x 3.605 + Rs.1,000 x 0.279
= Rs.711.60
Using linear interpolation in this range, we get 771.44 – 750.00 Yield to maturity = 18% + 771.44 – 711.60 x 2% = 18.7% 4. 10 14 100 80 = + t=1 (1+r) t (1+r)10
Try r = 18%. The RHS of the above equation is
Rs.14 x PVIFA (18%, 10 years) + Rs.100 x PVIF (18%, 10 years) = Rs.14 x 4.494 + Rs.100 x 0.191 = Rs.82
Try r = 20%. The RHS of the above equation is
Rs.14 x PVIFA(20%, 10 years) + Rs.100 x PVIF (20%, 10 years) = Rs.14 x 4.193 + Rs.100 x 0.162
= Rs.74.9
Using interpolation in the range 18% and 20% we get: 82 - 80 Yield to maturity = 18% + --- x 2% 82 – 74.9 = 18.56% 5. 12 6 100 P = + t=1 (1.08) t (1.08)12
= Rs.6 x PVIFA (8%, 12 years) + Rs.100 x PVIF (8%, 12 years) = Rs.6 x 7.536 + Rs.100 x 0.397
= Rs.84.92
6. The post-tax interest and maturity value are calculated below:
* Post-tax interest (C ) 12(1 – 0.3) 10 (1 – 0.3)
=Rs.8.4 =Rs.7
* Post-tax maturity value (M) 100 - 100 -
[ (100-70)x 0.1] [ (100 – 60)x 0.1]
=Rs.97 =Rs.96
The post-tax YTM, using the approximate YTM formula is calculated below 8.4 + (97-70)/10
Bond A : Post-tax YTM = --- 0.6 x 70 + 0.4 x 97 = 13.73%
7 + (96 – 60)/6 Bond B : Post-tax YTM = ---
0.6x 60 + 0.4 x 96 = 17. 47% 7. 14 6 100 P = + t=1 (1.08) t (1.08)14 = Rs.6 x PVIFA(8%, 14) + Rs.100 x PVIF (8%, 14) = Rs.6 x 8.244 + Rs.100 x 0.341 = Rs.83.56 8. Do = Rs.2.00, g = 0.06, r = 0.12 Po = D1 / (r – g) = Do (1 + g) / (r – g) = Rs.2.00 (1.06) / (0.12 - 0.06) = Rs.35.33
Since the growth rate of 6% applies to dividends as well as market price, the market price at the end of the 2nd year will be:
P2 = Po x (1 + g)2 = Rs.35.33 (1.06)2
9. Po = D1 / (r – g) = Do (1 + g) / (r – g) = Rs.12.00 (1.10) / (0.15 – 0.10) = Rs.264 10. Po = D1 / (r – g) Rs.32 = Rs.2 / 0.12 – g g = 0.0575 or 5.75% 11. Po = D1/ (r – g) = Do(1+g) / (r – g) Do = Rs.1.50, g = -0.04, Po = Rs.8 So 8 = 1.50 (1- .04) / (r-(-.04)) = 1.44 / (r + .04) Hence r = 0.14 or 14 per cent
12. The market price per share of Commonwealth Corporation will be the sum of three components:
A: Present value of the dividend stream for the first 4 years B: Present value of the dividend stream for the next 4 years
C: Present value of the market price expected at the end of 8 years. A = 1.50 (1.12) / (1.14) + 1.50 (1.12)2 / (1.14)2 + 1.50(1.12)3 / (1.14)3 + + 1.50 (1.12)4 / (1.14)4 = 1.68/(1.14) + 1.88 / (1.14)2 + 2.11 / (1.14)3 + 2.36 / (1.14)4 = Rs.5.74 B = 2.36(1.08) / (1.14)5 + 2.36 (1.08)2 / (1.14)6 + 2.36 (1.08)3 / (1.14)7 + + 2.36 (1.08)4 / (1.14)8 = 2.55 / (1.14)5 + 2.75 / (1.14)6 + 2.97 / (1.14)7 + 3.21 / (1.14)8 = Rs.4.89 C = P8 / (1.14)8 P8 = D9 / (r – g) = 3.21 (1.05)/ (0.14 – 0.05) = Rs.37.45 So C = Rs.37.45 / (1.14)8 = Rs.13.14 Thus, Po = A + B + C = 5.74 + 4.89 + 13.14 = Rs.23.77
13. The intrinsic value of the equity share will be the sum of three components: A: Present value of the dividend stream for the first 5 years when the
growth rate expected is 15%.
B: Present value of the dividend stream for the next 5 years when the growth rate is expected to be 10%.
C: Present value of the market price expected at the end of 10 years. 2.00 (1.15) 2.00 (1.15)2 2.00 (1.15)3 2.00(1.15)4 2.00 (1.15)5 A = --- + --- +--- + --- + --- (1.12) (1.12)2 (1.1.2)3 (1.1.2)4 (1.12)5 = 2.30 / (1.12) + 2.65 / (1.12)2 + 3.04 / (1.12)3 + 3.50 / (1.12)4 + 4.02/(1.12)5 = Rs.10.84 4.02(1.10) 4.02 (1.10)2 4.02(1.10)3 4.02(1.10)4 4.02 (1.10)5 B = --- + --- + --- + --- + --- (1.12)6 (1.12)7 (1.12)8 (1..12)9(1.12)10 4.42 4.86 5.35 5.89 6.48 = --- + --- + --- + --- + --- (1.12)6 (1.12)7 (1.12)8 (1.1.2)9(1.12)10 = Rs.10.81 D11 1 6.48 (1.05) C = --- x --- = --- x 1/(1.12)10 r – g (1 +r)10 0.12 – 0.05 = Rs.97.20
The intrinsic value of the share = A + B + C = 10.84 + 10.81 + 97.20 = Rs.118.85 14. Terminal value of the interest proceeds
= 140 x FVIFA (16%,4) = 140 x 5.066
= 709.24
Terminal value of the proceeds from the bond = 1709.24
Define r as the yield to maturity. The value of r can be obtained from the equation 900 (1 + r)4 = 1709.24
r = 0.1739 or 17.39%
15. Intrinsic value of the equity share (using the 2-stage growth model) (1.18)6 2.36 x 1 - --- 2.36 x (1.18)5 x (1.12) (1.16)6 = --- + --- 0.16 – 0.18 (0.16 – 0.12) x (1.16)6 - 0.10801 = 2.36 x --- + 62.05 - 0.02 = Rs.74.80
16. Intrinsic value of the equity share (using the H model) 4.00 (1.20) 4.00 x 4 x (0.10)
= --- + --- 0.18 – 0.10 0.18 – 0.10 = 60 + 20
Chapter 9 RISK AND RETURN 1 (a) Expected price per share a year hence will be:
= 0.4 x Rs.10 + 0.4 x Rs.11 + 0.2 x Rs.12 = Rs.10.80 (b) Probability distribution of the rate of return is
Rate of return (Ri) 10% 20% 30%
Probability (pi) 0.4 0.4 0.2
Note that the rate of return is defined as: Dividend + Terminal price --- - 1
Initial price
(c ) The standard deviation of rate of return is : σ = pi (Ri – R)2
The σ of the rate of return on MVM’s stock is calculated below:
--- Ri pi pI ri (Ri-R) (R i- R)2 pi (Ri-R)2 --- 10 0.4 4 -8 64 25.6 20 0.4 8 2 4 1.6 30 0.2 6 12 144 28.8 --- R = pi Ri pi (Ri-R)2 = 56 σ = 56 = 7.48%
2 (a) For Rs.1,000, 20 shares of Alpha’s stock can be acquired. The probability distribution of the return on 20 shares is
Economic Condition Return (Rs) Probability
High Growth 20 x 55 = 1,100 0.3 Low Growth 20 x 50 = 1,000 0.3 Stagnation 20 x 60 = 1,200 0.2 Recession 20 x 70 = 1,400 0.2
= 330 + 300 + 240 + 280 = Rs.1,150
Standard deviation of the return = [(1,100 – 1,150)2 x 0.3 + (1,000 – 1,150)2 x 0.3 + (1,200 – 1,150)2 x 0.2 + (1,400 – 1,150)2 x 0.2]1/2
= Rs.143.18
(b) For Rs.1,000, 20 shares of Beta’s stock can be acquired. The probability distribution of the return on 20 shares is:
Economic condition Return (Rs) Probability
High growth 20 x 75 = 1,500 0.3 Low growth 20 x 65 = 1,300 0.3 Stagnation 20 x 50 = 1,000 0.2 Recession 20 x 40 = 800 0.2 Expected return = (1,500 x 0.3) + (1,300 x 0.3) + (1,000 x 0.2) + (800 x 0.2) = Rs.1,200
Standard deviation of the return = [(1,500 – 1,200)2 x .3 + (1,300 – 1,200)2 x .3 + (1,000 – 1,200)2 x .2 + (800 – 1,200)2 x .2]1/2 = Rs.264.58
(c ) For Rs.500, 10 shares of Alpha’s stock can be acquired; likewise for Rs.500, 10 shares of Beta’s stock can be acquired. The probability distribution of this option is:
Return (Rs) Probability (10 x 55) + (10 x 75) = 1,300 0.3 (10 x 50) + (10 x 65) = 1,150 0.3 (10 x 60) + (10 x 50) = 1,100 0.2 (10 x 70) + (10 x 40) = 1,100 0.2 Expected return = (1,300 x 0.3) + (1,150 x 0.3) + (1,100 x 0.2) + (1,100 x 0.2) = Rs.1,175 Standard deviation = [(1,300 –1,175)2 x 0.3 + (1,150 – 1,175)2 x 0.3 + (1,100 – 1,175)2 x 0.2 + (1,100 – 1,175)2 x 0.2 ]1/2 = Rs.84.41
d. For Rs.700, 14 shares of Alpha’s stock can be acquired; likewise for Rs.300, 6 shares of Beta’s stock can be acquired. The probability distribution of this
Return (Rs) Probability (14 x 55) + (6 x 75) = 1,220 0.3 (14 x 50) + (6 x 65) = 1,090 0.3 (14 x 60) + (6 x 50) = 1,140 0.2 (14 x 70) + (6 x 40) = 1,220 0.2 Expected return = (1,220 x 0.3) + (1,090 x 0.3) + (1,140 x 0.2) + (1,220 x 0.2) = Rs.1,165 Standard deviation = [(1,220 – 1,165)2 x 0.3 + (1,090 – 1,165)2 x 0.3 + (1,140 – 1,165)2 x 0.2 + (1,220 – 1,165)2 x 0.2]1/2 = Rs.57.66
The expected return to standard deviation of various options are as follows :
Option Expected return (Rs) Standard deviation (Rs) Expected / Standard return deviation a 1,150 143 8.04 b 1,200 265 4.53 c 1,175 84 13.99 d 1,165 58 20.09
Option `d’ is the most preferred option because it has the highest return to risk ratio. 3. Expected rates of returns on equity stock A, B, C and D can be computed as follows:
A: 0.10 + 0.12 + (-0.08) + 0.15 + (-0.02) + 0.20 = 0.0783 = 7.83% 6 B: 0.08 + 0.04 + 0.15 +.12 + 0.10 + 0.06 = 0.0917 = 9.17% 6 C: 0.07 + 0.08 + 0.12 + 0.09 + 0.06 + 0.12 = 0.0900 = 9.00% 6 D: 0.09 + 0.09 + 0.11 + 0.04 + 0.08 + 0.16 = 0.095 = 9.50% 6
(a) Return on portfolio consisting of stock A = 7.83% (b) Return on portfolio consisting of stock A and B in equal
proportions = 0.5 (0.0783) + 0.5 (0.0917) = 0.085 = 8.5%
(c ) Return on portfolio consisting of stocks A, B and C in equal proportions = 1/3(0.0783 ) + 1/3(0.0917) + 1/3 (0.090)
= 0.0867 = 8.67%
(d) Return on portfolio consisting of stocks A, B, C and D in equal proportions = 0.25(0.0783) + 0.25(0.0917) + 0.25(0.0900) +
0.25(0.095)
= 0.08875 = 8.88%
4. Define RA and RM as the returns on the equity stock of Auto Electricals Limited a and Market
portfolio respectively. The calculations relevant for calculating the beta of the stock are shown below: Year RA RM RA-RA RM-RM (RA-RA) (RM-RM) RA-RA/RM-RM 1 15 12 -0.09 -3.18 0.01 10.11 0.29 2 -6 1 -21.09 -14.18 444.79 201.07 299.06 3 18 14 2.91 -1.18 8.47 1.39 -3.43 4 30 24 14.91 8.82 222.31 77.79 131.51 5 12 16 0-3.09 0.82 9.55 0.67 -2.53 6 25 30 9.91 14.82 98.21 219.63 146.87 7 2 -3 -13.09 -18.18 171.35 330.51 237.98 8 20 24 4.91 8.82 24.11 77.79 43.31 9 18 15 2.91 -0.18 8.47 0.03 -0.52 10 24 22 8.91 6.82 79.39 46.51 60.77 11 8. 12 -7.09 -3.18 50.27 10.11 22.55 RA = 15.09 RM = 15.18 (RA – RA)2 = 1116.93 (RM – RM) 2 = 975.61 (RA – RA) (RM – RM) = 935.86
Beta of the equity stock of Auto Electricals (RA – RA) (RM – RM) (RM – RM) 2 = 935.86 = 0.96 975.61 Alpha = RA – βA RM = 15.09 – (0.96 x 15.18) = 0.52
Equation of the characteristic line is
RA = 0.52 + 0.96 RM
5. The required rate of return on stock A is:
RA = RF + βA (RM – RF)
= 0.10 + 1.5 (0.15 – 0.10) = 0.175
Intrinsic value of share = D1 / (r- g) = Do (1+g) / ( r – g)
Given Do = Rs.2.00, g = 0.08, r = 0.175
2.00 (1.08) Intrinsic value per share of stock A =
0.175 – 0.08 = Rs.22.74 6. The SML equation is RA = RF + βA (RM – RF) Given RA = 15%. RF = 8%, RM = 12%, we have 0.15 = .08 + βA (0.12 – 0.08) 0.07 i.e.βA = = 1.75 0.04 Beta of stock A = 1.75
7. The SML equation is: RX = RF + βX (RM – RF)
We are given 0.15 = 0.09 + 1.5 (RM – 0.09) i.e., 1.5 RM = 0.195
or RM = 0.13%
Therefore return on market portfolio = 13%
8. RM = 12% βX = 2.0 RX =18% g = 5% Po = Rs.30 Po = D1 / (r - g)
So D1 = Rs.39 and Do = D1 / (1+g) = 3.9 /(1.05) = Rs.3.71 Rx = Rf + βx (RM – Rf) 0.18 = Rf + 2.0 (0.12 – Rf) So Rf = 0.06 or 6%. Original Revised Rf 6% 8% RM – Rf 6% 4% g 5% 4% βx 2.0 1.8 Revised Rx = 8% + 1.8 (4%) = 15.2%
Price per share of stock X, given the above changes is 3.71 (1.04)
= Rs.34.45 0.152 – 0.04
Chapter 10
OPTIONS AND THEIR VALUATION
E = 105 r = 0.12 R = 1.12
The values of ∆ (hedge ratio) and B (amount borrowed) can be obtained as follows:
Cu – Cd ∆ = (u – d) S Cu = Max (150 – 105, 0) = 45 Cd = Max (80 – 105, 0) = 0 45 – 0 45 9 ∆ = = = = 0.6429 0.7 x 100 70 14 u.Cd – d.Cu B = (u-d) R (1.5 x 0) – (0.8 x 45) = 0.7 x 1.12 -36 = = - 45.92 0.784 C = ∆ S + B = 0.6429 x 100 – 45.92 = Rs.18.37
Value of the call option = Rs.18.37
2. S = 40 u = ? d = 0.8
R = 1.10 E = 45 C = 8
We will assume that the current market price of the call is equal to the pair value of the call as per the Binomial model.
Given the above data
∆ Cu – Cd R = x B u Cd – d Cu S ∆ Cu – 0 1.10 = x B -0.8Cu 40 = (-) 0.034375 ∆ = - 0.34375 B (1) C = ∆ S + B 8 = ∆ x 40 + B (2) Substituting (1) in (2) we get 8 = (-0.034365 x 40) B + B 8 = -0.375 B or B = - 21.33 ∆ = - 0.034375 (-21.33) = 0.7332
The portfolio consists of 0.7332 of a share plus a borrowing of Rs.21.33 (entailing a repayment of Rs.21.33 (1.10) = Rs.23.46 after one year). It follows that when u occurs either u x 40 x 0.7332 – 23.46 = u x 40 – 45 -10.672 u = -21.54 u = 2.02 or u x 40 x 0.7332 – 23.46 = 0 u = 0.8
Since u > d, it follows that u = 2.02.
Put differently the stock price is expected to rise by 1.02 x 100 = 102%.
3. Using the standard notations of the Black-Scholes model we get the following results:
ln (S/E) + rt + σ2 t/2
d1 =
= ln (120 / 110) + 0.14 + 0.42/2 0.4 = 0.08701 + 0.14 + 0.08 0.4 = 0.7675 d2 = d1 - t = 0.7675 – 0.4 = 0.3675 N(d1) = N (0.7675) ~ N (0.77) = 0.80785 N (d2) = N (0.3675) ~ N (0.37) = 0.64431 C = So N(d1) – E. e-rt. N(d2) = 120 x 0.80785 – 110 x e-0.14 x 0.64431 = (120 x 0.80785) – (110 x 0.86936 x 0.64431) = 35.33
Value of the call as per the Black and Scholes model is Rs.35.33.
4. t = 0.2 x 1 = 0.2
Ratio of the stock price to the present value of the exercise price 80 = --- 82 x PVIF (15.03,1) 80 = --- 82 x 0.8693 = 1.122
From table A6 we find the percentage relationship between the value of the call option and stock price to be 14.1 per cent. Hence the value of the call option is
0.141 x 80 = Rs.11,28. 5. Value of put option
= Value of the call option
+ Present value of the exercise price
The value of the call option gives an exercise price of Rs.85 can be obtained as follows: t = 0.2 1 = 0.2
Ratio of the stock price to the present value of the exercise price 80
= --- 85 x PVIF (15.03,1)
= 80 / 73.89 = 1.083
From Table A.6, we find the percentage relationship between the value of the call option and the stock price to be 11.9%
Hence the value of the call option = 0.119 x 80 = Rs.9.52
Plugging in this value and the other relevant values in (A), we get Value of put option = 9.52 + 85 x (1.1503)-1 – 80
= Rs.3.41 6. So = Vo N(d1) – B1 e –rt N (d2) = 6000 N (d1) – 5000 e – 0.1 N(d2) ln (6000 / 5000) + (0.1 x 1) + (0.18/2) d1 = --- 0.18 x 1 ln (1.2) + 0.19 = 0.4243 = 0.8775 = 0.88 N(d1) = N (0.88) = 0.81057 d2 = d1 - t = 0.8775 - 0.18 = 0.4532 = 0.45
N (d2) = N (0.45) = 0.67364 So = 6000 x 0.81057 – (5000 x 0.9048 x 0.67364) = 1816 B0 = V0 – S0 = 60000 – 1816 = 4184 Chapter 11
TECHNIQUES OF CAPITAL BUDGETING 1.(a) NPV of the project at a discount rate of 14%.
= - 1,000,000 + 100,000 + 200,000 --- --- (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% pa 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 Lowest payback period
Highest NPV Highest IRR 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% pa.
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 NPV’s 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)
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.
Cost of capital = 12% p.a PV of cash flows up to the end of year 2 = 24.97 PV of cash flows up to the end of year 3 = 47.75 PV of cash flows up to the end of year 4 = 71.26
Discounted pay back period (DPB) lies between 3 and 4 years. Interpolating in this range we get an approximate DPB of 3.1 years.
Project N
Cost of capital = 12% per annum
PV of cash flows up to the end of year 1 = 33.93 PV of cash flows up to the end of year 2 = 51.47
DPB lies between 1 and 2 years. Interpolating in this range we get an approximate DPB of 1.92 years.
(c ) Project M
Cost of capital = 12% per annum
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
Cost of capital = 12% per annum 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
Cost of capital = 10% per annum NPV = Rs.25.02 million Project N
Cost of capital = 10% per annum NPV = 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 = 15% per annum NPV = 16.13 million
Project N
Cost of capital: 15% per annum NPV = 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%
Chapter 12
ESTIMATION OF 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 112.5 121.87 128.91 134.18 138.13 141.1143.33 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 98.77100.33 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) 116.25 113.44 111.33 109.75 108.56 107.6 107.00 13. Terminal CF ( = 9 +10) 98 14. N C F (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.
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 2307207 (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 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
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
MINICASE Solution:
a. Cash flows from the point of all investors (which is also called the explicit cost funds point of view) Rs.in million Item 0 1 2 3 4 5 1. Fixed assets (15) 2. Net working capital (8) 3. Revenues 30 30 30 30 30 4. Costs (other than
depreciation and
interest) 20 20 20 20 20 5. Loss of rental 1 1 1 1 1 6. Depreciation 3.750 2.813 2.109 1.582 1.187 7. Profit before tax 5.250 6.187 6.891 7.418 7.813 8. Tax 1.575 1.856 2.067 2.225 2.344 9. Profit after tax 3.675 4.331 4.824 5.193 5.469 10. Salvage value of fixed assets 5.000 11. Net recovery of working capital 8.000 12. Initial outlay (23) 13. Operating cash inflow 7.425 7.144 6.933 6.775 6.656
14. Terminal cash
flow 13.000
15. Net cash flow (23) 7.425 7.144 6.933 6.775 19.656 b. Cash flows form the point of equity investors
Rs.in million
Item 0 1 2 3 4 5
1. Equity funds (10)
2. Revenues 30 30 30 30 30 3. Costs (other than
depreciation and interest) 20 20 20 20 20 4. Loss of rental 1 1 1 1 1 5. Depreciation 3.75 2.813 2.109 1.582 1.187 6. Interest on working capital advance 0.70 0.70 0.70 0.70 0.70 7. Interest on term loans 1.20 1.125 0.825 0.525 0.225 8. Profit before tax 3.35 4.362 5.366 6.193 6.888 9. Tax 1.005 1.309 1.610 1.858 2.066 10. Profit after tax 2.345 3.053 3.756 4.335 4.822 11. Net salvage value
of fixed assets 5.000
12. Net salvage value
of current assets 10.000 13. Repayment of term term loans 2.000 2.000 2.000 2.000 14. Repayment of bank advance 5.000 15. Retirement of trade creditors 2.000 16. Initial investment (10) 17. Operating cash inflow 6.095 5.866 5.865 5.917 6.009 18. Liquidation and retirement cash flows (2.0) (2.0) (2.0) 6.00 19. Net cash flow (10) 6.095 3.866 3.865 3.917 12.009
Chapter 13
RISK ANALYSIS IN CAPITAL BUDGETING 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.
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 ‘xpected’ 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) Pessimistic Expected Optimistic
Initial investment 30000 30000 30000 Sale revenue 24000 42000 54000 Variable costs 16000 28000 36000 Fixed costs 3000 3000 3000 Depreciation 2000 2000 2000 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.
Pessimistic Expected Optimistic
Initial investment 30000 30000 30000 Sale revenue 28000 42000 70000 Variable costs 28000 28000 28000
Fixed costs 3000 3000 3000 Depreciation 2000 2000 2000 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.
Pessimistic Expected Optimistic
Initial investment 30000 30000 30000 Sale revenue 42000 42000 42000 Variable costs 56000 28000 21000 Fixed costs 3000 3000 3000 Depreciation 2000 2000 2000 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 3. 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
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 4. 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 5. Expected NPV 4 At
= - 10,000 …. (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
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
6. Given values of variables other than Q, P and V, the net present value model of Bidhan Corporation can be expressed as:
[Q(P – V) – 3,000 – 2,000] (0.5)+ 2,000 0 5 NPV + - 30,000 t =1 (1.1)t (1.1)5 0.5 Q (P – V) – 500 5 = --- - 30,000 t=1 (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
Valu e Pro b Cumulati ve Prob. Two digit random numbers Valu e Pro b Cumulati ve Prob. Two digit random
numbers Value Pro b Cumu -lative Prob. Two digit random numbers 800 0.1 0 0.10 00 to 09 20 0.4 0 0.40 00 to 39 15 0.3 0 0.30 00 to 29 1,00 0 0.1 0 0.20 10 to 19 30 0.4 0 0.80 40 to 79 20 0.5 0 0.80 30 to 79 1,20 0 0.2 0 0.40 20 to 39 40 0.1 0 0.90 80 to 89 40 0.2 0 1.00 80 to 99 1,40 0 0.3 0 0.70 40 to 69 50 0.1 0 1.00 90 to 99 1,60 0 0.2 0 0.90 70 to 89 1,80 0 0.1 0 1.00 90 to 99
Exhibit 2 Simulation Results
QUANTITY (Q) PRICE (P) VARIABLE COST (V) NPV
Ru n Rando m Numbe r Corres-ponding Value Random Number Corres-ponding value Rando m Number Corres-pondin g 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
QUANTITY (Q) PRICE (P) VARIABLE COST (V) NPV
Ru n Rando m Numbe r Corres-ponding Value Random Number Corres-ponding value Rando m Number Corres-pondin g 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
QUANTITY (Q) PRICE (P) VARIABLE COST (V) NPV
Ru n Random Number Corres -pondin g Value Random Number Corres-ponding value Rando m Number Corres-pondin g value 1.8955 Q(P-V)-31,895.5 37 89 1,600 86 40 59 20 28,761 38 94 1,800 00 20 25 15 -14,836 39 09 .800 15 20 29 15 -24,314 40 44 1,400 84 40 21 15 34,447 41 98 1,800 23 20 79 20 -31,896 42 10 1,000 53 30 77 20 -12,941 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
7. 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
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 = + - 30,0 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 of P1 = (PIj - PI) 2 P j
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 must should accept the investment.
9. 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.
10. The certainty equivalent co-efficients for the five years are as follows
Year Certainty equivalent coefficient
t = 1 – 0.06 t 1 1 = 0.94 2 2 = 0.88 3 3 = 0.82 4 = 0.76 5 = 0.70
The present value of the project calculated at the risk-free rate of return is : 5 (1 – 0.06 t) At t=1 (1.08)t 7,000 x 0.94 8,000 x 0.88 9,000 x 0.82 10,000 x 0.76 8,000 x 0.70 + + + + (1.08) (1.08)2 (1.08)3 (1.08)4 (1.08)5
6,580 7,040 7,380 7,600 5,600
+ + + +
(1.08) (1.08)2 (1.08)3 (1.08)4 (1.08)5 = 27,386
Net present value of the Project = (27,386 – 30,000 = Rs. –2,614
MINICASE Solution:
1. The expected NPV of the turboprop aircraft 0.65 (5500) + 0.35 (500) NPV = - 11000 + (1.12) 0.65 [0.8 (17500) + 0.2 (3000)] + 0.35 [0.4 (17500) + 0.6 (3000)] + (1.12)2 = 2369
2. If Southern Airways buys the piston engine aircraft and the demand in year 1 turns out to be high, a further decision has to be made with respect to capacity expansion. To evaluate the piston engine aircraft, proceed as follows:
First, calculate the NPV of the two options viz., ‘expand’ and ‘do not expand’ at decision point D2: 0.8 (15000) + 0.2 (1600) Expand : NPV = - 4400 + 1.12 = 6600 0.8 (6500) + 0.2 (2400) Do not expand : NPV = 1.12 = 5071
Second, truncate the ‘do not expand’ option as it is inferior to the ‘expand’ option. This means that the NPV at decision point D2 will be 6600
Third, calculate the NPV of the piston engine aircraft option. 0.65 (2500+6600) + 0.35 (800) NPV = – 5500 + 1.12 0.35 [0.2 (6500) + 0.8 (2400)] + (1.12)2 = – 5500 + 5531 + 898 = 929
3. The value of the option to expand in the case of piston engine aircraft
If Southern Airways does not have the option of expanding capacity at the end of year 1, the NPV of the piston engine aircraft would be:
0.65 (2500) + 0.35 (800) NPV = – 5500 + 1.12 0.65 [0.8 (6500) + 0.2 (2400)] + 0.35 [0.2 (6500) + 0.8 (2400)] + (1.12)2 = - 5500 + 1701 + 3842 = 43
Thus the option to expand has a value of 929 – 43 = 886
4. Value of the option to abandon if the turboprop aircraft can be sold for 8000 at the end of year 1
If the demand in year 1 turns out to be low, the payoffs for the ‘continuation’ and ‘abandonment’ options as of year 1 are as follows.
0.4 (17500) + 0.6 (3000)
Continuation: = 7857 1.12
Abandonment : 8000
Thus it makes sense to sell off the aircraft after year 1, if the demand in year 1 turns out to be low.
The NPV of the turboprop aircraft with abandonment possibility is
0.65 [5500 +{0.8 (17500) + 0.2 (3000)}/ (1.12)] + 0.35 (500 +8000) NPV = - 11,000 + (1.12) 12048 + 2975 = - 11,000 + = 2413 1.12
Since the turboprop aircraft without the abandonment option has a value of 2369, the value of the abandonment option is : 2413 – 2369 = 44
5. The value of the option to abandon if the piston engine aircraft can be sold for 4400 at the end of year 1:
If the demand in year 1 turns out to be low, the payoffs for the ‘continuation’ and ‘abandonment’ options as of year 1 are as follows:
0.2 (6500) + 0.8 (2400)
Continuation : = 2875 1.12
Abandonment : 4400
Thus, it makes sense to sell off the aircraft after year 1, if the demand in year 1 turns out to be low.
The NPV of the piston engine aircraft with abandonment possibility is: 0.65 [2500 + 6600] + 0.35 [800 + 4400] NPV = - 5500 + 1.12 5915 + 1820 = - 5500 + = 1406 1.12
Chapter 14
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.
