FM09-CH 02

Ch. 2: Concepts of Value and Return

1

CHAPTER 2

CONCEPTS OF VALUE AND RETURN

Problem 1

Time preference (discount) rate 9%

(i) A. Investment 15,000

B. Period (years) 4

C. Compound value factor at 9% for 4 years 1.4116

D. Compound value at the end of 4 years: [A x C]

: 15,000 (1.09)4 = 15,000 x 1.4116

21,173.72

Now After 1 year

(ii) A. Investment 6,000 6,000

B. Period (end of year) 5 5

C. Compounding periods 5 4

D. Compound value factor (lump sum) 1.5386 1.4116

E. Compound value [A x D]

: 6,000 (1.09)5 = 6,000 x 1.5386

9,231.74

: 6,000 (1.09)4 = 6,000 x 1.4116 8,469.49

(iii) A. Annual investment (end of year) 18,000

B. Period (years) 8

C. Compound value factor (annuity) 11.0285

D. Compound value at the end of 8 years [A x C]

: 18,000 [(1.09)8 - 1]/0.09 = 18,000 x 11.0285

19,8512.53

(iv) A. Annual investment (beginning of year) 18,000

B. Periods (years) 8

C. Compound value factor (annuity due) 12.0210

D. Compound value at the end of 8 years [A x C]

: 18,000 [{(1.09)8 - 1}/0.09] (1.09) = 18,000 x 12.0210

216,378.66

Withdrawal Balance

(v) A. Annual investment for 4 years 18,000.00

B. Compound value at the end of 4 years:

: 18,000 x [{(1.09)4-1}/0.09]

82,316.32

C. Compound value at the end of 5 years:

: (82,316.32 x 1.09) - 12,000

89,724.79

12,000.00

77,724.79

C. Compound value at the end of 6 years:

: (77,724.79 x 1.09) - 12,000

84,720.02

12,000.00

72,720.02

D. Compound value at the end of 7 years:

: (72,720.02 x 1.09) -12,000

79,264.82

12,000.00

67,264.82

E. Compound value at the end of 8 years:

: (67,264.82 x 1.09) - 0

73,318.66

0.00

7,3318.66

I. M. Pandey, Financial Management, 9th Edition, New Delhi: Vikas.

2

Problem 2

Discount rate 13%

(i) A. Cash flow 2,000

B. Period 0

C. Present value factor 1

D. Present value (Rs): 2,000/(1.13)0 2,000

(ii) A. Cash flow 2,000

B. Period 1

C. Present value factor: 1/(1.13)1 0.8850

D. Present value (Rs): [A x C]: 2,000/(1.13)1 1,769.91

(iii) A. Cash flow 2,000

B. Period 2

C. Present value factor 0.7831

D. Present value (Rs) [A x C] 1,566.29

(iv) A. Cash flow 4,000

B. Period 3



(v) A. Cash flow 7,000

B. Period 3


D. Present value (Rs) [A x C]: 7,000/(1.13)3 4,851.35

(vi) A. Cash flow 3,000

B. Period 4



(vii) A. Cash flow 4,000

B. Period 5

C. Present value annuity factor:

[{(1.13)5-1}/{0.13(1.13)5}]

3.5172

D. Present value (Rs):[A x C]:

( )

PVt

t

t

=

=

∑4 000

1131

5,

.= 4,000 × 3.5172

14068.93

(viii) A. Cash flow 4,000

B. Period 5

C. Present value factor (annuity due)

[{(1.13)5-1}/{0.13(1.13)5}](1.13)

3.9745

D. Present value (Rs):

( )∑

=

=

4

0t

t

t

13.1

000,4PV = 4,000 x 3.9745

15,897.89


3

Problem 3

Discount rate 14%

Year 0 1 2 3 4 5 6

A. Cash flows 3,000.00 3,000.00 3,000.00 3,000.00 7,000.00 1,000.00

B. PVF at 14% 0.8772 0.7695 0.6750 0.5921 0.5194 0.4556

C. Present value [A x B] 12,832.30 2,631.58 2,308.40 2,024.91 1,776.24 3,635.58 455.59

( ) ( ) ( )PV

Rs

t

t

= + +

= × + × + × =

=

∑3 000

114

7 000

114

1 000

114

3 000 2 9138 7 000 0 5194 1 000 0 4556 12 832 30

1

4

5 6

,

.

,

.

,

.

, . , . , . , .

Problem 4

A. Rate of interest 15%

B. Sum received now (Rs) 100

C. Period (years) 10

D. Present value factor (annuity) at 15% 5.0188

E. Capital recovery factor (annuity) at 15%

: [1/D]: 1/5.0188

0.1993

F. Annual instalment (end of period) [B x E]

( )100

1

115

100 50188

100 5 0188 19 93

1

10

=

=

= =

=

∑A

A

A Rs

t

t .

.

/ . .

19.93

G. Present value factor (annuity due) at 15%:

: 5.0188 x 1.15

5.7716

H. Capital recovery factor (annuity due) at 15% [1/G] 0.1733

I. Annual instalment (beginning of period) [B x H] 17.33

Problem 5

A. Interest rate 10%

B. Debt payable now (Rs) 1,000

C. Period of instalments (years) 5


E. Capital recovery factor (annuity) at 10% 0.2638

F. Annual instalment (end of period): [B x E]

( )1 000

1

110

1 000 3 7908

1 000 1 3 7908 26380

1

5

,.

, / .

, ( / . ) .

=

=

= =

=

∑A

A

A Rs

t

t

263.80


4

Problem 6

A. Time value of money 12%

B. Payment now (Rs) 13,000



E. Capital recovery factor (annuity) at 12%: [1/D] 0.2774

F. Annual instalment (end of period): [B x E], 13,000 x 0.2774 3,606.33

G. Present value factor (annuity due) at 12%: [D x 1.12] 4.0373


I. Annual instalment (beginning of period) (Rs): [B x H], 13,000 x 0.2477 3,219.93

Problem 7

A. Discount rate 11%

B. Outlay now 10,000



E. Capital recovery factor (annuity) at 11% [1/D] 0.2706

F. Annual instalment (end of period) [B x E or B/D] 2,705.70

G. Present value factor (annuity due) at 11% 4.1024


I. Annual instalment (beginning of period) [B x H] 2,437.57

Problem 8

A. Discount rate 8%

B. Annual interest payment 150

C. Period (years) 30

D. Present value factor (annuity), 30 periods, 8% 11.2578

E. Present value of annual interest [B x D] 1,688.67

F. Maturity value at the end of 30 years 1,000

G. Present value factor, end of 30 years 0.0994

H. Present value of maturity value [F x G] 99.38

I. Present value of bond [E + H]:

: ( ) ( )PV

Rs

t

t

t

= +

= × + × =

=

∑150

108

1 000

108

150 112578 1 000 0 0994 1 788 04

1

30

30.

,

.

. , . , .

1,788.04

Problem 9

A. Discount rate 15%

B. Annual pension 10,000

C. Periods of pension 20

D. Present value factor, 20 years, 15% 6.2593

E. Present value of pension at the end of 20 years 62,593.31

F. Present value factor, end of 20 years 0.0611

G. Present value of pension now [E x F] 3,824.47


5

Problem 10

A. Interest rate 10%

B. Year 0 1-6 7 8 12

C. Cash flow -10,000 2,000 -1,500 1,600 2,500

D. Present value factor 1.0000 4.3553 0.5132 0.4665 1.4788

E. Present value (Rs) [C x D] -10,000 8,710.52 -769.74 746.41 3,696.91

F. Net present value (Rs) 2,384.11

Problem 11

IRR 0 1 2 3 4 5 6 7 8 9 10

( i ) Deposit and receive 14% -100 114

( ii ) Borrow and pay 12% 100 -112

( iii ) Borrow and pay 13% 1,000 0 0 0 0 0 0 0 0 0 -3,395

The following formula is used to compute IRR:

( )

NPVNCF

IRRC

t

t

t

n

=+

− =

=

∑1

0

1

0

Problem 12

Year 0 1 2 3 4 5 6 7 8 9

Bank deposit 13% -100 0 0 0 0 0 0 0 0 300

Problem 13

We can use the following formula in calculating the time period, n, in this problem:

( )

( )

( )

1 r 2

n ln 1 + r ln 2

n = ln 2 / ln 1 + r

n+ =

=

A. Investment 6,000 6,000 6,000 6,000

B. Interest rate (annual) 6% 10% 20% 30%

C. Expected amount after n years:

6000(1+r)n = 12,000 12,000 12,000 12,000 12,000

D. Compound value factor:(1.06)n

= 12,000/6,000

2

2

2

2

E. ln (1+r) 0.0583 0.0953 0.1823 0.2624

F. ln 2 0.6931 0.6931 0.6931 0.6931

G. n = ln 2/ln (1+r) 11.90 7.27 3.80 2.64

H. Interest rate (semi-annual) 0.03 0.05 0.1 0.15

I. ln (1+r/2) 0.0296 0.0488 0.0953 0.1398

J. n (half-years)= ln 2/ln (1+r/2) 23.45 14.21 7.27 4.96

Problem 14

A. Annual earnings in 19X1 45,000

B. Period (years) 7

C. Annual earnings in 19X8 67,550

D. (1+g)7 1.5011

E. 7 ln (1+g) 0.4062

F. ln (1+g) 0.0580

G. (1+g) 1.060


6

H. g (growth): 1.06 - 1 ( )

( )( )

g =

− = − = − = =

67 550

45 0001 15011 1 106 1 0 6 6%

1 71 7,

,. . .

//

0.060

Problem 15

A. Land price 40,000

B. Instalments 20

C. Annual instalment 8,213

D. Present value annuity factor required:40000/8213 4.8703

E. From present value of annuity table, r equals 20%. We can

write the formula for IRR ( r ) as follows:

( )

40 0008 213

11

20

,,

=+

=

∑ t

t

t r

20%

Problem 16

A. Needed future sum after 15 years 300,000

B. Period (years) 15

C. Interest rate 12%

D. Future value factor of an annuity, 15 years, 12% 37.28

E. Annuity value [A/D]:

( )A

112 1

01230 000

37 28

15.

.,

.

−

=

A = 30,000

A = 30,000 / 37.28 = Rs 8,047.27

8,047.27

Problem 17

A. Needed future sum at the age of 50 1,000,000



D. Future value factor of an annuity, 30 years, 10% 164.49

E. ( a ) Annuity value [A/D] 6,079.25

F. Future value factor of a lump sum, 30 years, 10% 17.4494

G. ( b ) Lump sum deposited now [A/F] 57,308.55

Problem 18

A. Savings today 80,000



D. Future value factor, 10 years, 10% 2.5937

E. Future value [A x D] 207,499.40

F. Future value of an annuity factor, 10 years, 10% 15.9374

G. Annual withdrawal [E/F] 13,019.63

H. Present value of annuity factor, 10 years, 10% 6.1446

I. Annual withdrawal [A / H] 13,019.63


7

Problem 19

A. Price of house 500,000

B. Cash payment 100,000

C. Balance 400,000

D. Instalment period 20

E. Interest rate 12%

F. Present value of an annuity factor, 20 years, 12% 7.4694

G. Annual instalment 53,551.51

The interest paid and principal repaid each year are as follows:

Principal

Years Balance Instalment Interest repaid

0 400,000.00

1 394,448.49 53,551.51 48000.00 5551.51

2 388,230.79 53,551.51 47333.82 6217.69

3 381,266.98 53,551.51 46587.70 6963.82

4 373,467.50 53,551.51 45752.04 7799.47

5 364,732.09 53,551.51 44816.10 8735.41

6 354,948.43 53,551.51 43767.85 9783.66

7 343,990.73 53,551.51 42,593.81 10,957.70

8 331,718.11 53,551.51 41,278.89 12,272.62

9 317,972.77 53,551.51 39,806.17 13,745.34

10 302,577.99 53,551.51 38,156.73 15,394.78

11 285,335.83 53,551.51 36,309.36 17,242.15

12 266,024.62 53,551.51 34,240.30 19,311.21

13 244,396.06 53,551.51 31,922.95 21,628.56

14 220,172.08 53,551.51 29,327.53 24,223.98

15 193,041.22 53,551.51 26,420.65 27,130.86

16 162,654.65 53,551.51 23,164.95 30,386.57

17 128,621.70 53,551.51 19,518.56 34,032.95

18 90,504.79 53,551.51 15,434.60 38,116.91

19 47,813.85 53,551.51 10,860.57 42,690.94

20 0.00 53,551.51 5,737.66 47,813.85

Problem 20

A. Price of flat 200,000

B. Down payment 40,000

C. Loan processing fee 5,000

D. Net amount of loan 155,000

E. Period of mortgage loan (years) 12

F. Loan instalment 28,593

G. Required factor [D/F] 5.421

H. Present value of an annuity factor at trial rate 14%, 12 years 5.660

I. Present value of an annuity factor at trial rate 16%, 12 years 5.197

J. Rate of return:

%52.14197.5660.5

421.5660.5%1%14 =

−

−×+

15%

(approx.)


8

Problem 21

Required rate 13%

End Beginning

Year Cash flow PVF PV Cash flow PV

0 0 1.0000 0.0 2000 2000

1 2000 0.8850 1769.9 2000 1769.9

2 2000 0.7831 1566.3 2000 1566.3

3 2000 0.6931 1386.1 1000 693.1

4 1000 0.6133 613.3 1000 613.3

5 1000 0.5428 542.8 1000 542.8

6 1000 0.4803 480.3 1000 480.3

7 1000 0.4251 425.1 0

6783.8 7665.7

Problem 22

A. Payment now 200,000

B. Annuity 25,000

C. Expected period for annuity (years) 20

D. Interest rate 0.12

E. Annuity factor 7.4694

F. Present value of annuity 186,736

Sundaram should prefer Rs 200,000 now.

Problem 23

A. Time value of money 10%

B. 30-year annuity 5,000

C. PVAF, 10%, 30 year 9.4269

D. Present value of 30-year annuity 47,134.6

E. 20-year annuity 6,600

F. PVAF, 10%, 20 year 8.5136

G. Present value of 20-year annuity 56,189.52

H. Cash right now 50,000

You should choose 20-year annuity of Rs 6,600 as it has highest

PV.

Problem 24

Interest rate 8%

( i ) Amount now or 80,000

10-year annuity 14,000

PVAF, 8%, 10 year 6.7101

Present value of 10-year annuity [14,000 × 6.7101] 93,941

Ms Punam should choose 10-year annuity offering higher PV.


9

( ii ) Amount now or 150,000


PVAF, 8%, 20 year 9.8181

Present value of 20-year annuity [14,000 x 9.8181] 137,454

Ms Punam should choose to have Rs 150,000

(iii) Amount now or 120,000


PVAF, 8%, 15 year 8.5595

Present value of 15-year annuity [14,000 × 8.5595] 119,833

Both alternatives are almost the same.

Problem 25

Required rate of return 14%

End Beginning

Cash flow PVF PV Cash flow PV

0 0 1.0000 0 2,000 2,000

1 2,000 0.8772 1,754 2,000 1,754

2 2,000 0.7695 1,539 2,000 1,539

3 2,000 0.6750 1,350 2,000 1,350

4 2,000 0.5921 1,184 2,000 1,184

5 2,000 0.5194 1,039 3,000 1,558

6 3,000 0.4556 1,367 3,000 1,367

7 3,000 0.3996 1,199 3,000 1,199

8 3000 0.3506 1,052 3,000 1,052

9 3,000 0.3075 923 6,000 1,845

10 6,000 0.2697 1,618 6,000 1,618

11 6,000 0.2366 1,420 6,000 1,420

12 6,000 0.2076 1,245 6,000 1,245

13 6,000 0.1821 1,092 6,000 1,092

14 6,000 0.1597 958 6000 958

15 6,000 0.1401 841 0 -

PV 18,581 21,182

Problem 26

Borrowing 50,000

Interest rate 10%

Annuity factor, 10%, 5 year 3.7908

Annual payment: 50,000/3.7908 13190

Year Outstanding Instalment Interest Repayment

0 50,000 0 0

1 41,810 13,190 5,000 8,190

2 32,801 13,190 4,181 9,009

3 22,892 13,190 3,280 9,910

4 11,991 13,190 2,289 10,901

5 0 13,190 1,199 11,991


10

Problem 27

Nominal rate of interest 12%

Period 1

Effective interest rate: annual compounding 12%

Half-yearly compounding:

Compounding period 2

Half-yearly rate [12%/2] 6%

Effective (annual) interest rate [(1.06)2-1] 12.36%

Quarterly compounding:


Quarterly rate [12%/4] 3%


Monthly compounding:


Monthly rate [12%/12] 1%


Problem 28

A. Face value of debenture 1,000

B. Current yield (annual) 18%

C. Half-yearly yield [18%/2] 9%

D. Period (years) 10

E. Compounding periods [10 x 2] 20

F. Half-yearly interest amount 75

G. PVAF, 9%, 20 periods 9.1285

H. Present value of 20-period annuity of Rs 75 [F x G] 684.64

I. PVF of a lump sum, 9%, 20 periods 0.17843

J. PV of maturity value of Rs 1000 [1,000 x 0.1784] 178.43

K. Present value of the debenture [H + J]

( ) ( )Value of bond =

75

1.09

t

t

t=

∑ +

1

20

20

1 000

109

,

.

863.07

Problem 29

A. Initial deposit 1,000

B. Interest rate (annual) 12%

C. Compounding period in a year 4

D. Quarterly rate [12%/4] 3%

E. Period 7.5

F. Total compounding periods [C x E] 30

G. FVF, 3%, 30 periods 2.42726

H. Future value [A x G]: [(1.03)30 x 1,000] 2,427.26


11

Problem 30

A. Half-yearly interest 50

B. Maturity (years) 7

C. Maturity value (at par) 1,000

D. Maturity value (at premium) 1,100

E. Required rate of return 12%

F. Present value annuity factor, 6%, 14 periods 9.2950

G. Present value factor, 6%, 14 periods 0.4423

H. Value of the bond (redeemed at par): 907.05

(a) Value of interest [A x F] 464.75

(b) Present value of maturity value [C x G] 442.30

I. Value of bond (redeemed at premium): 951.28

(a) Value of interest [A x F] 464.75

(b) Present value of maturity value [D x G] 486.53

Problem 31

10000

100

8%

2%

0.006667

The present value of your deposit is Rs 10,000 and you want to withdraw Rs 100 every month. Thus

You will be able to completely withdraw your deposit in about 14 years.

Current deposit (Rs)

Montly withdrawal

Annual interest rate

Quarterly rate

Monthly interest rate

( )

( )

( )( )

8.130.0066512

1.0986n

1.09860.0066512n3ln 1.00667 lnn12

0.300667.1

333.000667.010015000667.1

1

00667.100667.0

1150100

00667.100667.0

1

00667.0

1100000,10

n12

n12

n12

n12

=×

=

=×==

=

=×−=

×−=

×−×=

Problem 32

A. Preference share capital 800,000

B. Maturity period (years) 8

C. Required return 12%

D. Compound value annuity factor, 12%, 8 years 12.2997

E. Sinking fund factor, 12%, 8 years [1/D] 0.0813

F. Annual contribution in SF (end of the year) [A x E] 65,042.27

G. Compound value annuity factor (annuity due), 12%, 8

years

13.7757

H. Sinking fund factor (annuity due), 12%, 8 years [1/G] 0.0726

I. Annual contribution in SF(beg. of the year) [A x H] 58,073.46


12

Problem 33

A. Face (and maturity) value of bond 1,000

B. Interest rate (half yearly) 7%

C. Half yearly interest 70

D. Remaining life of bond (half years) 8

E. Required rate of return (half yearly) 6%

F. Present value annuity factor, 6%, 8 years 6.2098

G. Present value factor, 6%, 8 years 0.62741

H. Value of bond: 1,062.10

(a) Present value of interest [C x F] 434.69

(b) Present value of maturity value [A x G] 627.41

Problem 34

A. Annual payments 3,800

B. Period (years) 4

C. Principal 10,000

D. Internal rate of return:

( )

NPVr

t

t

=+

− =

=

∑3 800

110 000 0

1

4,

,

19.14%

By trial & error IRR is approx. 19%

Problem 35

A. Loan amount 10,000

B. Period (years) 8


D. Annual repayment 2,013

E. Internal rate of return:

( )

NPVrt

=+

− =

=

∑2 013

110 000 0

8

1

8,

, 12%

Interest rate charged by the bank and the internal rate of return are the same. 12% is the true rate of interest.

Loan amortisation schedule

Beg. Repayment End

Year balance Instalment Interest Principal balance

0 10,000

1 9,187 2,013 1,200 813 9,187

2 8,276 2,013 1,102 911 8,276

3 7,257 2,013 993 1,020 7,257

4 6,114 2,013 871 1,142 6,114

5 4,835 2,013 734 1,279 4,835

6 3,402 2,013 580 1,433 3,402

7 1,798 2,013 408 1,605 1,798

8 0 2,013 216 1,797 0


13

Problem 36

A. Amount deposited 1,000

B. Interest rate for years 1-5 (5 years) 10%

C. Interest rate for years 6-13 (8 years) 13%

D. Compound value for 13-year period:

[1,000(1.10)5 x (1.13)8] 4,281.45

E. Compound rate of interest:

[(4,281.45/1,000)1/13 - 1] 11.84%


14

CASE

Case 2.1: Divya Handtools Private Limited (DHPL)

This case is intended to discuss concepts of value. The students will get practice in computing present value under

different situations. The instructor should use this case to ensure that students understand the logic and concepts of

time value of money.

Capacity expansion

Year NCF PVF PV, 14%

0 -250 1.000 -250.0

1 45 0.877 39.5

2 45 0.769 34.6

3 45 0.675 30.4

4 68 0.592 40.3

5 68 0.519 35.3

6 68 0.456 31.0

7 68 0.400 27.2

8 68 0.351 23.8

9 30 0.308 9.2

10 85 0.270 22.9

NPV 44.2

Year 10 cash flows include salvage value.

Minimum savings each year from replacement

Cash outlay (Rs million) 50

Life (years) 10

PVFA 14%, 10 5.2161

Annuity (annual savings, Rs million) 9.6

This is a case of capital recovery

Annual instalment of SBI loan

Amount (Rs million) 200

Interest rate 14%

Period (years) 10

PVFA 14%, 10 5.2161

Annual instalment (Rs mn.) 38.3

Payment at maturity - SBI loan


Interest rate 14%

Period (years) 10

Future value factor 3.7072

Single payment (future value) 741.4

Quarterly instalment - FI loan


Annual interest rate 13.50%

Quarterly rate 3.375%

Quarterly periods 40

PVFA 3.375%, 40 21.7754

Quarterly instalment (Rs mn) 9.2

The company should borrow from FI since the annual interest rate is lower.


15

Lease


Period (years) 10

Lease rental (beginning of the year) 52

PVFA of annuity due (13.5%) 6.038

Value of lease rentals 314.0

The value of lease rentals is higher than the amount of borrowing (Rs 300 million). Hence, borrowing is heaper than

leasing.

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