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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.

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 1

From the tables we find that FVIFA (20%, 6 years) = FVIFA (24%, 6 years) =

9.930 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) 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.

The present value of Rs.10,000 receivable after 8 years for various discount rates (r ) are: r = 10% PV = 10,000 x PVIF(r = 10%, 8 years) = 10,000 x 0.467 = Rs.4,670 r = 12%

PV

= 10,000 x PVIF (r = 12%, 8 years) = 10,000 x 0.404 = Rs.4,040

r = 15%

PV

= 10,000 x PVIF (r = 15%, 8 years) = 10,000 x 0.327 = Rs.3,270

9.

Assuming that it is an ordinary annuity, the present value is: 2,000 x PVIFA (10%, 5years) = 2,000 x 3.791 = Rs.7,582

10.

The present value of an annual pension of Rs.10,000 for 15 years when r = 15% is: 10,000 x PVIFA (15%, 15 years) = 10,000 x 5.847 = Rs.58,470

2

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The amount that can be withdrawn annually is: 100,000 100,000 A = ------------------ ------------ = ----------- = Rs.10,608 PVIFA (10%, 30 years) 9.427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) Using linear interpolation we get: 5.019 – 5.00 r = 15% + ---------------5.019 – 4.494

= =

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 3

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 Stated rate (%)

B 12

24

Frequency of compounding 6 times Effective rate (%)

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

C

4 times

24 12 times

(1 + 0.12/6)6- 1 (1+0.24/4)4 –1 (1 + 0.24/12)12-1 = 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%, ∞ ) 4

= 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 Rs.100,000 Rs.100,000 = = Rs.40,388 PVIF(12%, 8 years) 2.476 21.

now is:

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

5

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 = 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) =

21.244 6

= 20

PVIFA (2%, 24)

=

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 (15%, 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+

5.000 – 4.868 ----------------5.335 – 4.868

x 1 = 7.3 years

7

30.

Equated annual installment

= 500000 / PVIFA(14%,4) = 500000 / 2.914 = Rs.171,585 Loan Amortisation Schedule

Beginning Year amount ------ ------------1 500000 2 398415 3 282608 4 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 price escalation of 6% per annum in the price per tonne of iron = Rs.300 million x

= Rs.300 million x

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 8

assuming a

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 =

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)14

400,000(1.12)

2

15

9

1.12

15

1– 1.08 = 400,000 0.08 – 0.12 = Rs.7,254,962

10

Chapter 8 VALUATION OF BONDS AND STOCKS 1. P =

5  t=1

11

100 +

(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 7 t=1 (1.12) (1.12)

Note that when the discount rate and the coupon rate are the same the value is par value. 3.

equal to

The yield to maturity is the value of r that satisfies the following equality. 7 120 1,000 Rs.750 =  + = Rs.100 t 7 t=1 (1+r) (1+r) 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 Thus the value of r at which the RHS becomes equal to Rs.750 lies between 18% and 20%. 11

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. 80 =

10 14 100  + 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:

Yield to maturity

82 - 80 = 18% + ----------- x 2% 82 – 74.9 = 18.56%

5. P =

12  t=1

6

100 +

(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: Bond A 12

Bond B

*

Post-tax interest (C )

12(1 – 0.3) =Rs.8.4

*

Post-tax maturity value (M) 100 [ (100-70)x 0.1] =Rs.97

10 (1 – 0.3) =Rs.7 100 [ (100 – 60)x 0.1] =Rs.96

The post-tax YTM, using the approximate YTM formula is calculated below

Bond A :

Post-tax YTM = =

Bond B :

Post-tax YTM =

=

8.4 + (97-70)/10 -------------------0.6 x 70 + 0.4 x 97 13.73% 7 + (96 – 60)/6 ---------------------0.6x 60 + 0.4 x 96 17. 47%

7. P =

14  t=1

6

100 +

(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 price at the end of the 2nd year will be: P2

= =

Po x (1 + g)2 = Rs.35.33 (1.06)2 Rs.39.70

13

market

9. 10.

Po

Po

= =

D1 / (r – g) = Do (1 + g) / (r – g) Rs.12.00 (1.10) / (0.15 – 0.10) =

=

D1 / (r – g)

Rs.32 = g = 11.

Po Do So 8

= =

Rs.264

Rs.2 / 0.12 – g 0.0575 or 5.75% D1/ (r – g) = Do(1+g) / (r – g) Rs.1.50, g = -0.04, Po = Rs.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: B: C:

Present value of the dividend stream for the first 4 years Present value of the dividend stream for the next 4 years 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 = =

B=

C

1.68/(1.14) + 1.88 / (1.14)2 + 2.11 / (1.14)3 + 2.36 / (1.14)4 Rs.5.74

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

=

P8 / (1.14)8 P8 = D9 / (r – g) =

3.21 (1.05)/ (0.14 – 0.05) = Rs.37.45

So C

=

Thus, Po = =

Rs.37.45 / (1.14)8 = Rs.13.14 A + B + C = 5.74 + 4.89 + 13.14 Rs.23.77 14

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.

A=

2.00 (1.15) 2.00 (1.15)2 2.00 (1.15)3 2.00(1.15)4 2.00 (1.15)5 ------------- + ------------- +-------------- + ------------- + ------------(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 B=

4.02(1.10) 4.02 (1.10)2 4.02(1.10)3 4.02(1.10)4 4.02 (1.10)5 ------------ + ---------------- + ------------- + --------------- + --------------(1.12)6 (1.12)7 (1.12)8 (1..12)9 (1.12)10

=

4.42 --------(1.12)6

=

Rs.10.81

C= =

4.86 5.35 5.89 6.48 + -------------- + --------------- + ------------- + ------------(1.12)7 (1.12)8 (1.1.2)9 (1.12)10

D11 1 6.48 (1.05) -------- 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 Redemption value = 1,000

15

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 900 (1 + r)4 r 15.

= 1709.24 = 0.1739 or 17.39%

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

16.

=

2.36 x

=

Rs.74.80

- 0.10801 ----------- + 62.05 - 0.02

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 Rs.80

16

equation

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 r...