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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
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 0.6931
D. Present value (Rs) [A x C]: 7,000/(1.13)3 4,851.35
(vi) A. Cash flow 3,000
B. Period 4
C. Present value factor: 1/(1.13)4 0.6133
D. Present value (Rs): [A x C]: 3,000/(1.13)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}]
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
Ch. 2: Concepts of Value and Return
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
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]
( )1 000
1
110
1 000 3 7908
1 000 1 3 7908 26380
1
5
,.
, / .
, ( / . ) .
=
=
= =
=
∑A
A
A Rs
t
t
263.80
I. M. Pandey, Financial Management, 9th Edition, New Delhi: Vikas.
4
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]:
: ( ) ( )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
Ch. 2: Concepts of Value and Return
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
I. M. Pandey, Financial Management, 9th Edition, New Delhi: Vikas.
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
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
Ch. 2: Concepts of Value and Return
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.)
I. M. Pandey, Financial Management, 9th Edition, New Delhi: Vikas.
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.
Ch. 2: Concepts of Value and Return
9
( 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 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
I. M. Pandey, Financial Management, 9th Edition, New Delhi: Vikas.
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:
Compounding period 4
Quarterly rate [12%/4] 3%
Effective (annual) interest rate [(1.03)4-1] 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]
( ) ( )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
Ch. 2: Concepts of Value and Return
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
I. M. Pandey, Financial Management, 9th Edition, New Delhi: Vikas.
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
C. Interest rate 12%
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
Ch. 2: Concepts of Value and Return
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%
I. M. Pandey, Financial Management, 9th Edition, New Delhi: Vikas.
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
Amount (Rs million) 200
Interest rate 14%
Period (years) 10
Future value factor 3.7072
Single payment (future value) 741.4
Quarterly instalment - FI loan
Amount (Rs million) 200
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.
Ch. 2: Concepts of Value and Return
15
Lease
Amount (Rs million) 300
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.