Ch. 2: Concepts of Value and Return
CHAPTER 2 CONCEPTS OF VALUE AND RETURN Problem 1
Time preference (discount) rate (i)
9%
A. Investment Investment
15,000
B. Period (years)
4
C. Compound value factor at 9% for 4 years D.
1.4116
Compound value at the end of 4 years: years: [A x C] : 15,000 (1.09) 4 = 15,000 x 1.4116
21,173.72 Now
(ii)
A. Investment
After 1 year
6,000
6,000
B. Period (end of year)
5
5
C. Compounding periods
5
4
1.5386
1.4116
D. Compound value factor (lump sum) E. Compound value [A x D] : 6,000 (1.09) 5 = 6,000 x 1.5386 : 6,000 (1.09) 4 = 6,000 x 1.4116
9,231.74
(iii) A. Annual investment (end of year)
18,000
8,469.49
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] C. Compound value at the end of 5 years: : (82,316.32 x 1.09) - 12,000 C. Compound value at the end of 6 years: : (77,724.79 x 1.09) - 12,000 D. Compound value at the end of 7 years: : (72,720.02 x 1.09) -12,000 E. Compound value at the end of 8 years: : (67,264.82 x 1.09) - 0
82,316.32
1
89,724.79
12,000.00 77,724.79
84,720.02
12,000.00 72,720.02
79,264.82
12,000.00 67,264.82
73,318.66
0.00 7,3318.66
I. M. Pandey, Financial Management, 9 th Edition, New Delhi: Vikas.
Problem 2 Discount rate (i)
13%
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
C. Present value factor: 1/(1.13)
3
0.6931
D. Present value (Rs): [A x C]: 4,000/(1.13)
3
2,772.20
(v) A. Cash flow
7,000
B. Period
3
C. Present value factor: 1/(1.13)
3
D. Present value (Rs) [A x C]: 7,000/(1.13)
0.6931 3
4,851.35
(vi) A. Cash flow
3,000
B. Period
4
C. Present value factor: 1/(1.13)
4
D. Present value (Rs): [A x C]: 3,000/(1.13)
0.6133 4
1,839.96
(vii) A. Cash flow
4,000
B. Period
5
C. Present value annuity factor: [{(1.13)5-1}/{0.13(1.13)5}] D. Present value (Rs):[A x C]: 5
PV =
∑ (113 . )
3.5172 14068.93
4,000 t
= 4,000 × 3.5172
t
t =1
(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) D. Present value (Rs): 4
PV
=
∑ (1.13)
4,000 t t
3.9745
15,897.89 = 4,000 x 3.9745
t =0
2
Ch. 2: Concepts of Value and Return
Problem 3
Discount rate
14%
Year
0
A. Cash flows
1 0.8772
C. Present value [A x B] 4
,000 ∑ (3114 . )
+
t
t =1
=
3
7,000 5
. ) (114
0.4556 455.59
0.6750
1,000 . ) (114 ×
6
0.5194
+
1,000
×
0 .4556
A. Rate of interest
=
Rs12 ,832 .30
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 F. Annual instalment (end of period) [B x E]
0.1993 19.93
1 ∑ A (115 . ) t
t =1
100 = 5.0188A A = 100 / 5.0188 = Rs19 .93 G. Present value factor (annuity due) at 15%: : 5.0188 x 1.15 H. Capital recovery factor (annuity due) at 15% [1/G] I. Annual instalment (beginning of period) [B x H]
5.7716 0.1733 17.33
Problem 5
A. Interest rate
10%
B. Debt payable now (Rs)
1,000
C. Period of instalments (years)
5
D. Present value factor (annuity) at 10%
3.7908
E. Capital recovery factor (annuity) at 10%
0.2638
F. Annual instalment (end of period): [B x E]
263.80
5
1000 , =
6
0.5194
+
0.7695
Problem 4
10
5
12,832.30 2,631.58 2,308.40 2,024.91 1,776.24 3,635.58
3,000 × 2.9138 + 7 ,000
100 =
4
3,000.00 3,000.00 3,000.00 3,000.00 7,000.00 1,000.00
B. PVF at 14%
PV =
2
1 ∑ A (110 . ) t
t =1
1,000 = A / 3.7908 A = 1,000(1 / 3.7908 ) = Rs26380 .
3
0.5921
I. M. Pandey, Financial Management, 9 th Edition, New Delhi: Vikas.
Problem 6
A. Time value of money
12%
B. Payment now (Rs)
13,000
C. Period of instalments (years)
5
D. Present value factor (annuity) at 12%
3.6048
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
H. Capital recovery factor (annuity due) at 12% [1/G]
0.2477
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
C. Period of instalments (years)
5
D. Present value factor (annuity) at 11%
3.6959
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
H. Capital recovery factor (annuity due) at 11% [1/G]
0.24357
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]: 30
:
PV =
∑ (108 . ) 150 t
t
t =1
=
+
1,788.04
1,000
(108 . )
30
150 × 112578 . + 1,000
×
0.0994
=
Rs1 ,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
4
Ch. 2: Concepts of Value and Return
Problem 10
A. Interest rate
10%
B. Year C. Cash flow
0
1-6
7
8
12
-10,000
2,000
-1,500
1,600
2,500
1.0000
4.3553
0.5132 0.4665
1.4788
-10,000 8,710.52 -769.74 746.41
3,696.91
D. Present value factor E. Present value (Rs) [C x D] F. Net present value (Rs)
2,384.11
Problem 11
IRR
0
1
-100
114
( i ) Deposit and receive
14%
( ii ) Borrow and pay
12%
( iii ) Borrow and pay
13% 1,000
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0 -3,395
100 -112 0
The following formula is used to compute IRR: n
NPV =
∑ (1 t =1
NCFt +
IRR)
t
−
C0
=
0
Problem 12
Year Bank deposit
0
1
2
3
4
5
6
7
8
13% -100
0
0
0
0
0
0
0
0 300
9
Problem 13 We can use the following formula in calculating the time period, n, in this problem: n
(1 + r ) = 2 n ln (1 + r ) = ln 2 n =ln 2 / ln (1+ r ) A. Investment B. Interest rate (annual)
6,000
6,000
6,000
6,000
6%
10%
20%
30%
C. Expected amount after n years: 6000(1+r)n = 12,000 D. Compound value factor:(1.06) = 12,000/6,000 E. ln (1+r) F. ln 2
12,000 12,000 12,000 12,000 n
2 2 2 2 0.0583 0.0953 0.1823 0.2624 0.6931 0.6931 0.6931 0.6931
G. n = ln 2/ln (1+r) H. Interest rate (semi-annual) I. ln (1+r/2) J. n (half-years)= ln 2/ln (1+r/2)
11.90
7.27
3.80
2.64
0.03
0.05
0.1
0.15
0.0296 0.0488 0.0953 0.1398 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 D. (1+g)
10
67,550
7
1.5011
E. 7 ln (1+g)
0.4062
F. ln (1+g)
0.0580
G. (1+g)
1.060
5
I. M. Pandey, Financial Management, 9 th Edition, New Delhi: Vikas.
H. g (growth): 1.06 - 1
67,550 45000 ,
0.060
(1 / 7 )
g=
−
1 = (15011 . )
(1 / 7 )
−
1 = 106 . − 1 = 0.6
=
6%
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: 20
40,000 =
∑ (1 t =1
20%
8,213 t +
r)
t
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]: 15 (112 . ) − 1 = 30,000 0.12
A
8,047.27
3728 . A = 30,000 A = 30,000 / 37.28 = Rs 8,047.27 Problem 17
A. Needed future sum at the age of 50
1,000,000
B. Period (years)
30
C. Interest rate
10%
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
B. Period (years)
10
C. Interest rate
10%
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
6
Ch. 2: Concepts of Value and Return
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% G. Annual instalment
7.4694 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: 5.660 − 5.421 14% + 1% × = 14.52% 5.660 − 5.197
15% (approx.)
7
I. M. Pandey, Financial Management, 9 th Edition, New Delhi: Vikas.
Problem 21
Required rate
13% End
Year Cash flow 0
Beginning PVF
PV
Cash flow
PV
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 B. Annuity C. Expected period for annuity (years) D. Interest rate E. Annuity factor F. Present value of annuity
200,000 25,000 20 0.12 7.4694 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 G. Present value of 20-year annuity
8.5136 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.
8
Ch. 2: Concepts of Value and Return
( ii ) Amount now or
150,000
20-year annuity
14,000
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
15-year annuity
14,000
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 End
14% 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
9
I. M. Pandey, Financial Management, 9 th Edition, New Delhi: Vikas.
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% 2
Effective (annual) interest rate [(1.06) -1]
12.36%
Quarterly compounding: Compounding period
4
Quarterly rate [12%/4]
3% 4
Effective (annual) interest rate [(1.03) -1]
12.55%
Monthly compounding: Compounding period
12
Monthly rate [12%/12]
1%
Effective (annual) interest rate [(1.01) 12-1]
12.68%
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] 20
Value of bond =
∑ (1.09) 75t
t =1
t
+
1000 , . ) (109
863.07
20
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
10
Ch. 2: Concepts of Value and Return
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 Current deposit (Rs)
10000
Montly withdrawal
100
Annual interest rate
8%
Quarterly rate
2%
Monthly interest rate 0.006667 The present value of your deposit is Rs 10,000 and you want to withdraw Rs 100 every month. Thus
1
10,000 = 100 ×
0.00667
−
100 = 150 −
1
12n 0.00667 × (1.00667 ) 1
1 12n
0.00667 × (1.00667 )
=
(150 − 100) × 0.00667 = 0.333
1.00667
=
3.0
12n ln 1.00667
=
ln 3 = 12n × 0.00665
(1.00667 )12n 12n
n
=
1.0986 12 × 0.00665
=
=
1.0986
13.8
You will be able to completely withdraw your deposit in about 14 years.
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 H. Sinking fund factor (annuity due), 12%, 8 years [1/G]
13.7757 0.0726
I. Annual contribution in SF(beg. of the year) [A x H]
58,073.46
11
I. M. Pandey, Financial Management, 9 th Edition, New Delhi: Vikas.
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: 4
NPV =
, ∑ (13800 r) +
t 1 =
t
−
10,000 = 0
19.14%
By trial & error IRR is approx. 19% Problem 35
A. Loan amount
10,000
B. Period (years)
8
C. Interest rate
12%
D. Annual repayment
2,013
E. Internal rate of return: 8
NPV =
∑ (12,013r) +
t =1
8
−
12%
10,000 = 0
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. Year balance Instalment
Repayment Interest
End
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
12
Ch. 2: Concepts of Value and Return
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%
13
I. M. Pandey, Financial Management, 9 th Edition, New Delhi: Vikas.
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 0 1 2 3 4 5 6 7 8 9 10
PVF
-250 45 45 45 68 68 68 68 68 30 85
PV, 14%
1.000 0.877 0.769 0.675 0.592 0.519 0.456 0.400 0.351 0.308 0.270
-250.0 39.5 34.6 30.4 40.3 35.3 31.0 27.2 23.8 9.2 22.9
NPV Year 10 cash flows include salvage value.
44.2
Minimum savings each year from replacement Cash outlay (Rs million) Life (years) PVFA 14%, 10 Annuity (annual savings, Rs million)
50 10
5.2161 9.6
This is a case of capital recovery Annual instalment of SBI loan Amount (Rs million) Interest rate Period (years)
200 14% 10
PVFA 14%, 10 Annual instalment (Rs mn.)
5.2161 38.3
Payment at maturity - SBI loan Amount (Rs million) Interest rate Period (years) Future value factor Single payment (future value)
200 14% 10 3.7072 741.4
Quarterly instalment - FI loan Amount (Rs million) Annual interest rate Quarterly rate Quarterly periods
200 13.50% 3.375% 40
PVFA 3.375%, 40 Quarterly instalment (Rs mn)
21.7754 9.2
The company should borrow from FI since the annual interest rate is lower.
14
Ch. 2: Concepts of Value and Return
Lease
Amount (Rs million) Period (years) Lease rental (beginning of the year) PVFA of annuity due (13.5%) Value of lease rentals
300 10 52 6.038 314.0
The value of lease rentals is higher than the amount of borrowing (Rs 300 million). Hence, borrowing is heaper than leasing.
15
