Prasanna Chandra Chapter 6 Solution

Chapter 6 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 2 3 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.

6.

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)

= =

5,000 5,000 / 1000 = 5

From the tables we find that FVIF (16%, 10 years) = FVIF (18%, 10 years) =

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.

9.

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

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 / 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) PVIFA (18%, 10 years)

= =

5.019 4.494

Using linear interpolation we get: 5.019 – 5.00 r = 15% + ---------------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

Frequency of compounding 6 times

4 times

24 12 times

(1 + 0.12/6)6- 1 (1+0.24/4)4 –1 (1 + 0.24/12)12-1

Effective rate (%)

Difference between the effective rate and stated rate (%) 20.

24

= 12.6

= 26.2

= 26.8

0.6

2.2

2.8

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 8 th 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 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 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.

26.

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 30 per cent of the pension amount is

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 = P x 1.817 = P 27.

=

Rs.149,980 Rs.149,980 Rs.149,980 ------------ = Rs.82,540 1.817

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) = PVIFA (2%, 24) =

21.244 18.914

Using a linear interpolation 21.244 – 20.000 r = 1% + ---------------------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 (10 %, n) = Rs.100,000 / Rs.20,000 = 5.000 From the tables we find that PVIFA (10%, 7 years) = PVIFA (10%, 8 years) =

4.868 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 ------------500000 398415 282608 150588

Annual installment --------------171585 171585 171585 171585

Interest ----------70000 55778 39565 21082

Principal Remaining repaid balance ------------------------101585 398415 115807282608 132020 150588 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) PVIFA (13%, n)

= =

1,500,000 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 tone of iron = Rs.300 million x

= Rs.300 million x

33

34.

35.

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

1 – (1.06)15 / (1.16)15 0.16 – 0.06

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

PV = c/(r – g) = 12/[0.12 – (-0.03)] = Rs.80 million

(b)

1+g n 1 - ------1+r PV = A(1+g) ----------------r- g

= 12 x 0.9725 / 0.15 = Rs.77.8 million

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 PV = A(1+g) -----------------

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 1 +

52

- 1 = 1.0832 - 1 = 8.32 percent 52 MINICASE--1

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 =

FVIFA (10%, 15 years)

= Rs.48,338 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 46 1

400,000(1.12) 2 1.12

1– 1.08 = 400,000

400,000(1.12)14 15

15

0.08 – 0.12 = Rs.7,254,962 MINICASE--2 Solution: 1) Re.1 deposit each at the end of month 0 1

becomes

2

3

4

Rs.3.0402

5

6

9

12

40

Rs.3.0402 Rs.3.0402 Rs.3.0402

Rs.3.0402

44

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)

Year end 0 19 20

What deposit? 12L

1

2

3

4

5

6

7

8

Annuity Payments

9

10

11

12

12L 12L 12L

Annuity Period 13 14 15

16

17

18

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