Chapter 6
TIME VALUE OF MONEY
1. Value five years hence of a deposit of Rs.1,000 at various interest rates is as follows: r = 8% FV5 = 1000 x FVIF (8%, 5 years) = 1000 x 1.469 = Rs.1469 r = 10% FV5 = 1000 x FVIF (10%, 5 years) = 1000 x 1.611 = Rs.1611 r = 12% FV5 = 1000 x FVIF (12%, 5 years) = 1000 x 1.762 = Rs.1762 r = 15% FV5 = 1000 x FVIF (15%, 5 years) = 1000 x 2.011 = Rs.2011 2. Rs.160,000 / Rs. 5,000 = 32 = 25
According to the Rule of 72 at 12 percent interest rate doubling takes place approximately in 72 / 12 = 6 years
So Rs.5000 will grow to Rs.160,000 in approximately 5 x 6 years = 30 years 3. In 12 years Rs.1000 grows to Rs.8000 or 8 times. This is 23 times the initial deposit. Hence
doubling takes place in 12 / 3 = 4 years.
According to the Rule of 69, the doubling period is: 0.35 + 69 / Interest rate
Equating this to 4 and solving for interest rate, we get Interest rate = 18.9%.
4. Saving Rs.2000 a year for 5 years and Rs.3000 a year for 10 years thereafter is equivalent to saving Rs.2000 a year for 15 years and Rs.1000 a year for the years 6 through 15.
Hence the savings will cumulate to:
2000 x FVIFA (10%, 15 years) + 1000 x FVIFA (10%, 10 years) = 2000 x 31.772 + 1000 x 15.937 = Rs.79481. 5. Let A be the annual savings.
A x FVIFA (12%, 10 years) = 1,000,000
A x 17.549 = 1,000,000
So, A = 1,000,000 / 17.549 = Rs.56,983. 6. 1,000 x FVIFA (r, 6 years) = 10,000
FVIFA (r, 6 years) = 10,000 / 1000 = 10 From the tables we find that
FVIFA (20%, 6 years) = 9.930
FVIFA (24%, 6 years) = 10.980
Using linear interpolation in the interval, we get: 20% + (10.000 – 9.930)
r = x 4% = 20.3%
(10.980 – 9.930)
7. 1,000 x FVIF (r, 10 years) = 5,000
FVIF (r,10 years) = 5,000 / 1000 = 5
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
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 / FVIF (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
Using linear interpolation we get: 5.019 – 5.00
r = 15% + --- x 3% 5.019 – 4.494
= 15.1%
16. PV (Stream A) = Rs.100 x PVIF (12%, 1 year) + Rs.200 x PVIF (12%, 2 years) + Rs.300 x PVIF(12%, 3 years) + Rs.400 x PVIF (12%, 4 years) + Rs.500 x PVIF (12%, 5 years) +
Rs.600 x PVIF (12%, 6 years) + Rs.700 x PVIF (12%, 7 years) + Rs.800 x PVIF (12%, 8 years) + Rs.900 x PVIF (12%, 9 years) + Rs.1,000 x PVIF (12%, 10 years) = Rs.100 x 0.893 + Rs.200 x 0.797 + Rs.300 x 0.712 + Rs.400 x 0.636 + Rs.500 x 0.567 + Rs.600 x 0.507 + Rs.700 x 0.452 + Rs.800 x 0.404 + Rs.900 x 0.361 + Rs.1,000 x 0.322 = Rs.2590.9 Similarly, PV (Stream B) = Rs.3,625.2 PV (Stream C) = Rs.2,851.1 17. FV5 = Rs.10,000 [1 + (0.16 / 4)]5x4 = Rs.10,000 (1.04)20 = Rs.10,000 x 2.191 = Rs.21,910 18. FV5 = Rs.5,000 [1+( 0.12/4)] 5x4 = Rs.5,000 (1.03)20 = Rs.5,000 x 1.806 = Rs.9,030 19 A B C
Stated rate (%) 12 24 24
Frequency of compounding 6 times 4 times 12 times
Effective rate (%) (1 + 0.12/6)6- 1 (1+0.24/4)4 –1 (1 + 0.24/12)12-1
= 12.6 = 26.2 = 26.8
Difference between the effective rate and stated
rate (%) 0.6 2.2 2.8
20. Investment required at the end of 8th year to yield an income of Rs.12,000 per year from the end of 9th year (beginning of 10th year) for ever:
Rs.12,000 x PVIFA(12%, ∞ ) = Rs.12,000 / 0.12 = Rs.100,000
To have a sum of Rs.100,000 at the end of 8th year , the amount to be deposited now is:
Rs.100,000 Rs.100,000
= = Rs.40,388 PVIF(12%, 8 years) 2.476
21. The interest rate implicit in the offer of Rs.20,000 after 10 years in lieu of Rs.5,000 now is:
Rs.5,000 x FVIF (r,10 years) = Rs.20,000 Rs.20,000
FVIF (r,10 years) = = 4.000
Rs.5,000 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 after 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 2025 to 2029, evaluated as at the beginning of 2024 (or end of 2023) 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 2020 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 2015 to 2020 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
0.30 x Rs.6000 = Rs.1800
Assuming that the monthly interest rate corresponding to an annual interest rate of 12% is 1%, the discounted value of an annuity of Rs.1800 receivable at the end of each month for 180 months (15 years) is:
Rs.1800 x PVIFA (1%, 180)
(1.01)180 - 1
Rs.1800 x --- = Rs.149,980 .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.149,980 P x 1.817 = Rs.149,980 Rs.149,980 P = --- = Rs.82,540 1.817 27. Rs.3000 x PVIFA(r, 24 months) = Rs.60,000 PVIFA (r,24) = Rs.60000 / Rs.3000 = 20 From the tables we find that:
PVIFA(1%,24) = 21.244
PVIFA (2%, 24) = 18.914
Using a linear interpolation
21.244 – 20.000
r = 1% + --- x 1% 21.244 – 18,914
= 1.53%
Thus, the bank charges an interest rate of 1.53% per month. The corresponding effective rate of interest per annum is
[ (1.0153)12 – 1 ] x 100 = 20%
28. The discounted value of the debentures to be redeemed between 8 to 10 years evaluated at the end of the 5th year is:
Rs.10 million x PVIF (8%, 3 years) + Rs.10 million x PVIF (8%, 4 years) + Rs.10 million x PVIF (8%, 5 years) = Rs.10 million (0.794 + 0.735 + 0.681) = Rs.2.21 million
If A is the annual deposit to be made in the sinking fund for the years 1 to 5, then
A x FVIFA (8%, 5 years) = Rs.2.21 million A x 5.867 = Rs.2.21 million
A = 5.867 = Rs.2.21 million
A = Rs.2.21 million / 5.867 = Rs.0.377 million
29. Let `n’ be the number of years for which a sum of Rs.20,000 can be withdrawn annually.
Rs.20,000 x PVIFA (10%, n) = Rs.100,000
PVIFA (10 %, 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 5.335 – 4.868
30. Equated annual installment = 500000 / PVIFA(14%,4) = 500000 / 2.914
= Rs.171,585
Loan Amortisation Schedule
Beginning Annual Principal Remaining
Year amount installment Interest repaid balance
--- --- --- --- --- ---
1 500000 171585 70000 101585 398415
2 398415 171585 55778 115807282608
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
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 tone 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 33 (a) PV = Rs.500,000 (b) PV = 1,000,000PVIF10%,6yrs = 1,000,000 x 0.564 = Rs.564,000 (c ) PV = 60,000/r = 60,000/0.10 = Rs.600,000 (d) PV = 100,000 PVIFA10%,10yrs = 100,000 x 6.145 = Rs.614,500 (e) PV = C/(r-g) = 35,000/(0.10-0.05) = Rs.700,000
Option e has the highest present value viz. Rs.700,000 34. (a) PV = c/(r – g) = 12/[0.12 – (-0.03)] = Rs.80 million 1+g n 1 - ---(b) 1+r PV = A(1+g) --- = 12 x 0.9725 / 0.15 = Rs.77.8 million r - g
35. It may be noted that if g1 is the growth rate in the no. of units and g2 the growth rate in price per unit, then the growth rate of their product, g = (1+g1)(1+g2) - 1 In this problem the growth rate in the value of oil produced, g = (1- 0.05)(1 +0.03) - 1 = - 0.0215
Present value of the well’s production = 1+g n 1 - 1+r
r - g
= (50,000 x 50) x ( 1-0.0215)x 1 – (0.9785 / 1.10)15
0.10 + 0.0215
= $ 16,654,633 36.
The growth rate in the value of the oil production g = (1- 0.06)(1 +0.04) - 1 = - 0.0224
Present value of the well’s production =
1+g n 1 - 1+r PV = A(1+g) --- r - g = (80,000 x 60) x ( 1-0.0224)x 1 – (0.9776 / 1.12)20 0.12 + 0.0224 = $ 30,781,328.93
37. Future Value Interest Factor for Growing Annuity, ( 1+ i )n – ( 1 + g)n
FVIFGA =
i - g
(1. 09)20 – ( 1.08)20 So the value of the savings at the end of 20 years = 100,000 x
0.09 – 0.08 = Rs. 9,434,536
38
Assuming 52 weeks in an year, the effective interest rate is 0.08 52
1 + - 1 = 1.0832 - 1 = 8.32 percent 52
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 2 15 1.12 15 1 – 1.08 = 400,000
0.08 – 0.12 = Rs.7,254,962
MINICASE--2 Solution: 1)
Re.1 deposit each at the
end of month 0 1 2 3 4 5 6 9 12 40 44
becomes Rs.3.0402 Rs.3.0402 Rs.3.0402 Rs.3.0402 Rs.3.0402 Rs.3.0402
MBA expenses for year I at present = 20 lakhs. After 10 years it would be = 20(1+0.05)10 = 32.58 lakhs
MBA expenses for year II at present = 25 lakhs. After 11 years it would be = 25(1+0.05)11 = 42.76 lakhs
At the end of 3 months, each 1 Rupee deposited in the RD account becomes = FVIFA(0.08/12,3) = [{(1+0.08/12)3 -1} / (0.08/12)] x (1+0.08/12) = {(1.00667)3-1}/0.00667 x 1.00667 = Rs.3.0402 which when compounded quarterly becomes at the end of 10 years = 3.0402 x [(1+0.08/4)4x10 -1]/ (0.08/4)
= 3.0402 x [(1.02)40 – 1] / 0.02 = Rs. 183.634
For a RD maturity value of Rs.183.634 if the deposit to be made is Rs.1, for a maturity value of Rs.32.58 lakhs, the monthly deposit to be made will be = 32,58,000/183.634 = Rs.17,742 Similarly for a maturity value of Rs.42.76 lakhs the monthly deposit needed .will be = 42,76,000 / [3.0402 x {(1.02)44 – 1} / 0.02] = Rs. 20,236
2)
Amount required for Jasleen’s marriage at the end of 20 years = Rs.300 lakhs
Cumulative fixed deposit to be made now to get the above amount = 300,00,000 / (1+0.08/4)4x20 = Rs.61,53,292
3)
Annuity Period
Year end 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
What deposit? Annuity Payments 12L 12L 12L 12L 12L 12L 12L 12L 12L 12L
Annuity needed per annum at the beginning of each year in real terms after 10 years = Rs.12 lakhs
With inflation at 5 percent, in nominal terms, this may be considered as a growing annuity for
10 years at a growth rate of 5 percent and discount rate of 10 percent. Present value of the annuity , as at the beginning of the 10th year from now
= 12,00,000 x (1+0.05)[ 1 –(1+0.05)/(1+0.10)10 /(0.10-0.05)] = Rs.93,74,163 Amount to be deposited in cumulative fixed deposit now, to have a maturity value of Rs.93,74,163 at the end of 9 years = 93,74,163/(1+0.08/4)4x9 = Rs.45,95,432
