( )
( ) ( )
*
0 0
* *
1 0
*
, -1600
, 1600 1 10, 000 ( 0) 8400 1600
F r k CF
F r k r F
r
= =
= − + + <
= − −
Since r* = 5.25 ( imin) ⇒1600r* < 8400
i.e., F r k1
( )
*, ≥ 0and it is compounded at k in the next step.( ) (
* *) ( )
2 , 8400 1600 1 10000
F r k = − r +k −
(iii)
( )
( ) ( )
* 2
*
*
, 0
8400 1600 1 10000 0 5.25 6.25
1 F r k
r k
r k
=
⇒ − + − =
⇒ = −
+
r* can be obtained for different values of k. Following points can be observed:
(i) r*, the return on the invested capital increases with k, the imputed cost of surplus funds;
(ii) r* →iminasymptotically; and (iii) If k = r*, then k = r* = i*.
5.25 6.25 1
(1 ) 5.25(1 ) 6.25 0.25, 4.00
k k
k k k
k
= −
+
⇒ + = + −
⇔ =
a predetermined required rate of return (the cost of capital). The projects with positive NPV have IRR greater than the cost of capital (k).
Consider the following figure, which depicts the relationship between NPV and the discount rate:
If k = 0, NPV is the highest. However, this situation is very unlikely. NPV is inversely proportional to
the figure, NPV is zero when k = 20%. This is the IRR (r) of the project. Now, let k = 10%. In this
Again, for k = 25%, NPV is negative (equal to OB) so when NPV is negative, IRR (20%) is less than
Thus the acceptance and rejection criteria of both the methods remain the same.
The situation will remain the same if the projects to be examined are independent. However if the k and decreases as k increases. The point where NPV is zero is the IRR. After this point, NPV will be negative and the project will be unacceptable.
B
IRR
Discount rate (%)
. .
0 0
.
5 25
.
1 1
.
2 5
NPV (Rs.)
A
O
Fig. 10.6
In
case, NPV is positive (equal to OA). For positive NPV, IRR (20%) is more than the cost of the capital (10%).
the cost of the capital (25%).
projects are mutually exclusive, this situation may change. The exclusiveness of the projects may be on two accounts.
(i) Technical exclusiveness Technical exclusiveness refers to the situation when the alternatives to be examined have different profitability and the most profitable alternative is to be chosen. For example buy or manufacture decisions, purchase or lease decisions etc.
(ii) Financial exclusiveness Financial exclusiveness refers to the situation when the alternatives are subject to the financial constraints. The most profitable alternatives are to be selected from a give (profitable) set of alternatives. Due to limited funds all alternatives cannot be selected. This situation is also referred to as capital rationing.
Consider the following projects:
Table 10.30 Project Initial outlay
(Rs.)
Annual cash flows (Rs)
Life of the project (years)
NPV (@ 10%) (Rs.)
IRR (%)
A 14,000 2745 20 8158 19
B 19,000 3550 20 10203 18
The two methods are giving conflicting results. If the projects were independent or the firm had sufficient funds, both could have been selected. But if only one project is to be selected, then the decision maker is in a dilemma.
The conflicting ranking by the NPV and the IRR method is attributed to some of the following situations.
(i) Size disparity – When the projects involve different cash outlays;
(ii) Time disparity – When the timings and the pattern of cash flows are different.; and (iii) Life disparity - When the projects involve different lives
(i) Size disparity If the initial outlay of mutually exclusive projects under consideration are different, then the NPV and IRR criteria are likely to provide conflicting rankings.
Consider the following projects.
Example 7: Two assembling systems manufactured by two different companies are being considered by a manufacturing firm. The system A would cost Rs. 50,000, has expected life of 5 years and would generate cash inflows of order Rs. 17,500 per year. The system B would cost Rs. 30,000, has expected life of 5 years, and would generate cash inflows of order Rs. 11,500 per year. Rank the two projects on the basis of NPV and IRR.
Table 10.31
Particulars Project A Project B
Initial outlay (Rs.) 50,000 30,000
Life of the project (years) 5 5
Annual cash flows (Rs.) 17500 11500
NPV (Rs.) 14853 12358
IRR (%) 22.11 26.50
Thus NPV criterion is leading to selection of project A whereas the IRR criterion is leading to selection of project B. Hence the two methods are giving conflicting results.
Reason for conflict The reason for the conflict lies in the fact that NPV method assumed that the intermediate cash (in) flows are reinvested at a rate equal to the cost of the capital which is a fairly reasonable assumption ensuring the minimum opportunity rate on the intermediate flows. However, IRR method assumed that the intermediate flows are reinvested at a rate equal to the IRR of the project.
This expectation is on the higher side of what actually is the rate of return. Liquid cash cannot generate such a high rate of return. Thus the results of IRR method are upward biased.
Resolving the conflict In such situations, NPV method is invariably better than the IRR method.
The reason being that the objective of the NPV method is maximization of the shareholders' wealth, which is ensured by a project earning the highest NPV. The conflict can be resolved by modifying IRR.
The approach to modify IRR is called the incremental approach.
According to this approach, in case of mutually exclusive projects with different outlays, if the IRR of both the projects exceed some predetermined required rate, calculate IRR on the difference of the outlays of the two projects. This difference of outlays of the two projects is called the incremental outlay. If the IRR on this incremental outlay is more than the required rate of return, accept the project with the bigger outlay.
In our case,
Table 10.32
Particulars Values Incremental outlay (Rs.) 20,000
Incremental cash flows (Rs.) 6,000
NPV (Rs.) 2,495.20
IRR (%) 15.24
Since incremental IRR (15.24%) is more than the cost of the capital (10%), project A should be selected, which the decision reached at by NPV method also.
The justification of this approach lies in the fact that if IRR of the incremental outlay exceeds the cost of the capital, then the firm is earning profits of the smaller project and an additional profit on the incremental outlay.
The incremental outlay approach would always yield results identical to those obtained by the NPV method.
(ii) Time disparity This situation arises when the mutually exclusive projects are exhibiting different cash flow patterns although their initial outlays may be the same.
Consider the following projects.
Table 10.33
Particulars Project A Project B
Initial outlay (Rs.) 16800 16800
Life of the project (years) 3 3
Cash flows (Rs.) Years
1 2 3
14000 7000 1400
1400 8400 15500
NPV (Rs.) 2,512.94 2,782.05
IRR (%) 22.79 17.59
Under such situations, the cost of capital is the key determinant in the ranking of projects. The following table gives the NPV of the projects for different values of k.
Table 10.34
k (%) Project A (Rs.) Project B (Rs.)
0.05 3,897.06 5,277.97
0.08 3,033.06 3,705.87
0.1 2,512.94 2,782.05
0.175851 0.01
0.2 448.30 (691.74)
0.2279 0.41
0.25 (322.56) (1,894.40)
0.3 (962.71) (2,844.30)
0.5 (2,627.16) (5,027.16)
0.6 (3,108.64) (5,537.23)
Following is the graph of the NPVs of the two projects. Any discount rate till NPV of project A is equal to the NPV of project B would support project B and any other discount rate would support project A.
Fig. 10.7
.
0.05
.
0.1
.
0.2
.
0.3
.
0.4
.
0.5
.
0.6 4000
.
3000
.
2000
.
1000
.
-1000
.
-2000
.
-3000
.
Discount rate 5000
.
IRR (A) IRR (B)
NPV (Rs.)
Project B
Project A
(iii) Life disparity – projects with unequal lives Consider the following proposals
A person has been allotted a piece of land for which he would get possession at the end of the year. He has to deposit an amount of Rs. 10 lacs today in order to own land at the end of the year. Now, real estate is a flourishing business and he has been offered a premium of Rs. 8 lacs if he sells the land as soon as he gets its possession. However a well -wisher of him suggested that if he retains the plot for 5 years, he would be able to earn a premium of Rs. 28 lacs. If the cost of capital is 10%, what should be his decision?
In this case we calculate the NPV and IRR of both the projects
Table 10.35
Particulars Project A Project B
Initial outlay (Rs. in lacs) 10 10
Life of the project (years) 1 5
Cash flows (Rs.) Years
1 2 3 4 5
18 - - - -
- - - - 38
NPV (Rs.) 5.79 12.36
IRR (%) 80 31
Again the results by the two methods are different.
Resolving the conflict There are two ways of resolving this conflict.
(i) Common time horizon approach
In this approach, the projects are made to have equal lives by repeating them a number of times so that their multiple life periods become equal. In our case, this can be done as follow:
Table 10.36
Project A Project B
Years
Cash flows (Rs. in lacs)
PV factor
Total present value (Rs. in lacs)
Cash flows (Rs. in lacs)
PV factor
Total present value (Rs. in lacs)
0 -10 1.000 -10 -10 1.000 -10
1 18 0.909 16.362 -
2 -10 0.909* -9.09 -
3 18 0.826 14.868 -
4 -10 0.826* -8.26 -
5 18 0.751 13.518 38 22.36
17.4 12.36
*The cash inflow will be reinvested at the same time period.
Then we find the NPV of the new (multiple) project A is higher than the NPV of project B. Then project A should be selected.
The implicit assumption of this approach is that the investment which is being replaced will have the similar pattern of cash flows in future as it had in past.
This approach works when the projects have short lives. For large projects having lives say 15 and 20 years, the common life period would have been 60 years, which is very large, and estimates over this period may have very large errors.
(ii) Annual capital charge (ACC) This drawback of common life horizon approach can be overcome by annual capital charge approach. Annual capital charge of an investment is the cost on an annual basis of the initial outlay and operating costs associated with that investment, the time value of money taken into account. Annual capital charge is also referred to as the equivalent annual cost.
The annual capital charge can be determined as follows:
(i) Find the present value of the initial outlay and operating costs.
(ii) Apply suitable capital recovery factor to convert this present value into the annual capital charge.
Consider the following project:
Table 10.37
Particulars Values Initial outlay (Rs.)
Operating costs (Rs.) Years
1 2 3 4 5
10,00,000
2,00,000 2,50,000 3,00,000 3,50,000 4,00,000 If the cost of capital is 10%, we want to determine the ACC.
Sol: (i) The present value of costs =
1 2 3 4
2, 00, 000 2, 50, 000 3, 00, 000 3, 50, 000 4, 00, 000 10, 00, 000
(1 0.10) (1 0.10) (1 0.10) (1 0.10) (1 0.10) Rs. 21, 01, 220
+ + + + +
+ + + + +
=
5
(ii) The capital recovery factor (inverse of PVIFA (n=5, k= 10%)) = 1
0.2638 3.7908= ACC 21, 01, 220 0.2638
Rs. 5, 54, 302
⇒ = ×
=
This method is quite useful in the area of public price regulation of utilities. For example, the construction cost and the operating cost of a power station may be converted into an annual capital charge, which serves as basis for the determination of the tariff structure. The tariff structure may be so determined as to recover the annual capital charge.
Example 8: A firm is examining following two proposals for installing an electronic security system:
Table 10.38
Particulars System A System B
Initial cost (Rs.) 50,000 75,000
Expected life (years) 6 10
Running costs (Rs.) 15,000 12,000
Salvage value (Rs.) 3000 10,000
The depreciation is charged on straight-line basis. If the tax rate is 35%, which system should be installed?
Sol: To calculate the annual capital charge, first of all we calculate the present value of the cash flows associated with the two projects.
Table 10.39
Costs (Rs.) Adjusted PV (Rs.)
Particulars
Machine A Machine B
PV factor
(@10%) Machine A Machine B
Initial cost (Rs.) 50,000 75,000 1.000 50,000 75,000
Operating costs:
1-6 years (A) 1-10 years (B)
7008.33
5525
4.355 6.145
1,83,127.66
3,39,511.25 Less: salvage value
6th year (A) 10th year (B)
3,000
10,000
0.564 0.386
1692
3860
Present value of the total costs 231435
410651.25 Capital recovery factor
A B
4.355
6.145
ACC 53,142.36 66,826.89
The operating costs can be determined as follows:
Table 10.40
Particulars Machine A Machine B
Running cost (Rs.) Less: Tax (@35%) Less:
Tax shield on depreciation : Initial cost - Salvage value
0.35 Life of the machine ×
Effective operating cost (Rs.)
15,000 5250
2741.67
7008.33
12000 4200
2275
5525
Since machine A has lower ACC than machine B, so machine A should be opted.
Net present value (NPV) versus profitability index (PI) The NPV method and the PI method in general will provide the similar results. If NPV is positive, PI will be greater than one. However, in evaluating mutually exclusive projects, the two may give differing results.
Consider the following projects
Table 10.41
Project A Project B
Year Present value
factor (k=0.10) Cash flow (Rs.) Present value (Rs.) Cash flow (Rs.) Present value (Rs.) 0
1 2 3
1.00 0.909 0.826 0.751
(60,000) 42,000 50,000 50,000
(60,000) 38178 41300 37550
(40,000) 35,000 32,000 30,000
(40,000) 31815 26432 22530
NPV 57028 40777
PI 1.95 2.02
In this case, the project selected by the NPV method must be chosen for the reasons already discussed.
Capital rationing
Capital rationing is the process of choosing investment proposals when there are financial constraints in the form of limited capital expenditure budget.
There may be a large number of proposals before the management of a firm, all of which could have been selected if the firm had unlimited funds or sufficient funds to have undertaken all the proposals.
But this is not the case in reality; and in reality, firms have limited funds available for capital expenditure. Then from the given set of acceptable proposals, the firm has to choose the most profitable combination of proposals. Capital rationing aims at selection of such combinations.
.
Thus the process of capital rationing involves two steps.
(i) Identification of acceptable proposals; and
(ii) Selection of the most profitable combination of proposals.
(i) Identification of acceptable proposals We have discussed several criteria that can be used to identify acceptable projects. Generally IRR or PI is used for this purpose.
(ii) Selection of the most profitable combination of proposals At this stage, we ought to choose those combinations of proposals, which have the highest NPV.
At this stage, the proposals may be accepted/ rejected partially or in their entirety. In fact, on the basis of whether or not the projects are divisible, the projects may be classified into two categories:
(a) Divisible projects Those projects, which can be accepted or rejected in parts (e.g., investment in mutual funds), are called the divisible projects.
(b) Indivisible projects Those projects, which are to be accepted or rejected in their entirety (e.g., installation of new machinery), are called the indivisible projects.
Example 9: A firm has following proposals before it, all of which are divisible. The firm has a limited fund of Rs. 10.5 lacs. What should be the decision of the management?
Table 10.42
Particulars Project A Project B Project C Project D Initial investment (Rs. in lacs) 4.5 3.0 2.6 9.0
NPV (Rs. in lacs) 0.90 0.75 2.25 2.70
PI 1.20 1.25 1.60 1.30
Sol: Stage (i): Rank the projects in descending order of PI
Table 10.43
Rank Project PI Initial investment (Rs. in lacs) NPV (Rs. in lacs) 1
2 3 4
C D B A
1.60 1.30 1.25 1.20
2.6 9.0 3.0 4.5
2.25 2.70 0.75 0.90
Stage (ii) NPV (Full C + part of D) = 3.565 NPV (Full D + part of C) = 4.47
Project D should be selected in its entirety along with a part of C.
Project selection is a bit simpler in case of indivisible projects. Consider the following example.
Example 10: A firm is considering the following proposals. It has a limited fund of Rs. 75 crores.
If the projects under consideration are indivisible, what should be the decision of the firm?
Table 10.44
Projects Initial outlay (Rs. in crores) NPV (Rs. in crores)
A 10 7 B 24 17 C 35 25 D 16 20 E 15 18 F 8 15
Sol: We arrange the projects in descending order of NPV
Table 10.45
Projects Initial outlay (Rs. crores) Cumulative outlay (Rs. crores) NPV (Rs. crores)
C 35 35 25
D 16 51 20
E 15 66 18
B 24 90 17
F 8 98 15
A 10 108 7
With the given financial restraint, NPV will be maximized by undertaking projects C, D, E and F. Rs.
If the firm adopts IRR method, then the optimal investment policy suggests the acceptance of all projects till the IRR is equal to the marginal cost of capital. Graphically this situation may be represented as follows
IRR/ cost of capital
IRR k
Acceptance limit
Budget Fig. 10.8
The firms cannot raise unlimited capital at a constant rate. After a certain level of borrowing, the cost of capital (k) will rise. In such situations, some marginal projects with IRR close to the increased cost of capital may no longer be acceptable. We arrange the projects in descending order of IRR and till IRR is equal to the cost of capital, the projects are accepted.
Consider the following example.
Example 11: For the following investment proposals, what should be the decision of a firm if the minimum required rate of return is 10%? The projects are mutually independent.
Table 10.46
Project Investment (Rs.) Life of the project (years) Net cash flow (per year) (Rs.)
A -12000 5 4281
B -10000 5 4184
C -17000 10 5802
Sol: We form all possible combinations in which projects can be selected:
Table 10.47
Combination Project Total investment (Rs.) Net cash flow (per year) (Rs.)
1 A 12000 4281 (1-5 years)
2 B 10000 4184 (1-5 years)
3 C 17000 5802 (1-10 years)
4 AB 22000 8465 (1-5 years)
5 AC 29000 10083 (1-5 years)
5802 (6-10 years)
6 BC 27000 9986 (1-5 years)
5802 (6-10 years)
7 ABC 39000 14267 (1-5 years)
5802 (6-10 years)
In order of the size of the investments, the combinations are rearranged as follows:
Table 10.48
Combination Project Total investment (Rs.) Net cash flow (per year) (Rs.)
2 B 10000 4184 (1-5 years)
1 A 12000 4281 (1-5 years)
3 C 17000 5802 (1-10 years)
4 AB 22000 8465 (1-5 years)
6 BC 27000 9986 (1-5 years)
5802 (6-10 years)
5 AC 29000 10083 (1-5 years)
5802 (6-10 years)
7 ABC 39000 14267 (1-5 years)
5802 (6-10 years)
Now, we calculate the incremental rate of return.
Table 10.49 Project Incremental investment
(Rs.)
Incremental flow (Rs.) IRR (%) NPV (Rs.) (k =15%) None
B 10,000 4184 31 3500
A-B 2,000 97 < 0 < 0
C-A 5,000 1521 16 85
2663 (1-5 years) AB-C 5,000
-5802 (6-10 years)
< 0 < 0
1521(1-5 years) BC-AB 5,000
5802 (6-10 years)
42 8494
97(1-5 years) AC-BC 2,000
0 (6-10 years)
< 0 < 0
4281(1-5 years) ABC-AC 10,000
0 (6-10 years)
31 3608
If the budget allows then the combination of project B and C is the best.
Example 12: The capital expenditure budget of a company is Rs. 25 lacs. The management has found the following (independent) projects to be feasible.
Table 10.50 Project Life of the
project (years)
Initial outlay (Rs.) Present value of the cash flows occurring from the project (Rs.)
A B C D
5 5 5 5
10,00,000 7,50,000 8,75,000 7,50,000
12,50,000 12,50,000 14,25,000 15,00,000
The cost of capital to the company is 19%. Any unutilized amount can be invested to earn an interest of 6%, which is risk-free. If the projects are indivisible, what should be the decision of the firm?
Sol: We calculate the NPV of the feasible projects.
Table 10.51
Project Initial outlay (Rs.) Present value of the cash flows occurring from the project (Rs.)
NPV (Rs.)
A B C D
10,00,000 7,50,000 8,75,000 7,50,000
12,50,000 12,50,000 14,25,000 15,00,000
2,50,000 5,00,000 5,50,000 7,50,000
At the most three projects can be chosen under given budgetary constraint. Combinations ABC and ACD are not feasible. We analyze remaining two projects.
Table 10.52
Combination Total outlay (Rs.) Unutilized sum (Rs.) NPV (Rs.)
ABD 25,00,000 0 15,00,000
BCD 23,75,000 1,25,000 19,00,912.5*
* NPV of this combination is equal to the NPV of the projects + NPV of the unutilized sum which can be calculated as follows.
Future value of . 1, 25, 000 after 5 years @ 6% interest 1, 25, 000 6 5 = . 1, 25, 000
100 = . 1,62,500
Present value of . 1,62,500 @ Rs
Rs Rs Rs
+ × ×
15% = . 1,62,500 0.621 = . 1,00,912.5
of the combination = ( ) ( ) ( ) of the unutilized sum . 18, 0
k Rs
Rs
NPV NPV B NPV C NPV D PV
Rs
= ×
⇒ + + +
= 0, 000 . 1, 00, 912.5 . 19, 00, 912.5
Rs Rs
+
=
The firm should select combination BCD.
Linear programming model for capital rationing Linear programming model (particularly integer programming models) can be efficiently used for allocating funds to different feasible projects in case of budgetary constraints. In this case the objective function is to maximize the net present value of different combinations of projects subject to constraints of limited budgets, mutually exclusive alternatives and project divisibility. A general formulation of the problem can be shown as
( )
( )
1 0
1
1
(1 )
subject to
; (Budgetary constraint)
m; 0 1 (Project divisibility constraint)
m n
t
tj j
j t
m
tj j t
j
m
j j
j
Max Z CF k X
IO X B
X X
−
= =
=
=
= +
≤
≤ ≤ ≤
∑∑
∑
∑
where
Cash flows of the project at the period;
Number of projects under consideration;
life of the project;
Cost of capital;
th th
tj
th
CF j t
m
n j
k
=
=
=
=
Decision variable corresponding to the divisibilty of the project;
Initial outlay the project at time ; and Total budget resources at time .
th j
th tj
t
X j
IO j t
B t
=
=
=
Problems
1. What is capital budgeting? Discuss its importance to the management of a firm.
2. On the basis of accounting rate of return, which project does you find more suitable:
Table 10.53
Project X Project Y
Year
Book value (Rs.)
Depreciat ion (Rs.)
Profit after tax
(Rs.)
Cash flow (Rs.)
Book value (Rs.)
Deprecia tion (Rs.)
Profit after tax
(Rs.)
Cash flow (Rs.)
0 1,00,000 0 0 0 (1,00,000) 0 0 (1,00,000)
1 75,000 25,000 40,000 65,000 70,000 25,000 10,000 35,000 2 50,000 25,000 30,000 55,000 50,000 25,000 20,000 45,000 3 25,000 25,000 20,000 50,000 20,000 25,000 30,000 55,000
4 0 25,000 10,000 35,000 0 25,000 40,000 65,000
3. A company is considering replacement of a manual operation by a mechanized one. The cost of this change is Rs. 50,000. The new system has an expected useful life of 5 years and no salvage value. The estimated cash flows before depreciation and tax (CFBT) from the proposal are as follows:
Table 10.54
Year CFBT (Rs.)
1 10,000 2 11,260 3 12,500 4 19,000 5 23,255
Assume the tax rate to be 35% and straight-line depreciation method (depreciation equally spread over the useful life of the system) and no salvage value, calculate
(i) Payback period;
(ii) ARR;
(iii) NPV (k=10%) (iv) IRR; and (v) PI (k=10%).
4. In the above exercise, if the cash flows after tax and depreciation are reinvested at rates 6% for the first two years, 8% for the third year and 10% for the remaining years, calculate the present value of the system.
5. If an equipment costs Rs. 5,00,000 and lasts for eight years, what should be the minimum annual cash inflow before it is worth to purchase the equipment. Assume the cost of capital to be 10%.
6. How much can be paid for a machine which brings an annual cash inflow of Rs. 25,000 for 12 years at a discount rate of 15%?
7. A company is considering the purchasing and installation of a new welding machine and is evaluating the following proposals:
(a) The company can buy a second hand machine for Rs. 1,00,000 which has an expected life of 5 years and salvage value of Rs. 25,000. After its useful life the machine can be replaced by another second hand machine having the expected cost Rs. 1,25,000, working life of 5 years and salvage value Rs. 40,000.
(b) A new machine can be bought for Rs. 3,00,000 which would have an expected working life of 10 years and salvage value Rs. 50,000.
Both the proposals would render the same service. If the cost of capital is 10%, which proposal should be accepted?
8. The cash flow stream of a project is given below: