Chapter 2 - Theory - 3
Chapter 2 - Theory - 3
Chapter 2 - Theory - 3
Decisions
Session 6
Revision
• ARR (average rate of return)
• Payback
• NPV
• Profitability Index (PI Index)
INTERNAL RATE OF RETURN (IRR) METHOD
The IRR also called yield on investment, marginal efficiency of capital, marginal
productivity of capital, rate of return, time-adjusted rate of return
Similar to NPV, but in NPV there is a pre-determined rate (generally, cost of capital) and
the determinants of cost of capital are external to the project being considered
The IRR is defined as the discount rate (r) which equates the aggregate present value of
the operating CFTA received each year and terminal cash flows (working capital recovery
and salvage value) with aggregate present value of cash outflows of an investment
proposal.
The project will be accepted when IRR exceeds the required rate of return.
Here, CF = Cash flows, S = salvage value, W = Working capital release, CO = Cash flows
IRR FOR AN ANNUITY
The following steps are taken in determining IRR for an annuity.
Determine the pay back period of the proposed investment.
In Table A-4 (present value of an annuity) look for the pay back period that is equal to
or closest to the life of the project.
In the year row, find two PV values or discount factor (DF) closest to PB period but one
bigger and other smaller than it.
From the top row of the table, note interest rate (r) corresponding to these PV values
(DF).
Determine actual IRR by interpolation. This can be done either directly using Equation 1
or indirectly by finding present values of annuity (Equation 2).
PB – DFr
IRR = r - (Equation 1)
DFrL - DFrH
Where PB = Pay back period
DFr = Discount factor for interest rate r.
DFrL = Discount factor for lower interest rate
DFrH = Discount factor for higher interest rate.
r = Either of the two interest rates used in the formula
PVco – PVCFAT
Alternatively, IRR = r - × ∆r (Equation 2)
∆PV
Where PVCO = Present value of cash outlay
PVCFAT = Present value of cash inflows (DFr x annuity)
r = Either of the two interest rates used in the formula
∆ r = Difference in interest rates
∆ PV = Difference in calculated present values of inflows
Example 11
A project costs Rs 36,000 and is expected to generate cash inflows of Rs 11,200 annually for 5
years. Calculate the IRR of the project.
Example 11 contd.
A project costs Rs 36,000 and is expected to generate cash inflows of Rs 11,200 annually for 5
years. Calculate the IRR of the project.
Solution
(1) The pay back period is 3.214 (Rs 36,000/Rs 11,200)
(2) According to Table A-2, discount factors closest to 3.214 for 5 years are 3.274 (16 per cent
rate of interest) and 3.199 (17 per cent rate of interest). The actual value of IRR which lies
between 16 per cent and 17 per cent can, now, be determined using Equations 1 and 2.
Substituting the values in Equation 1 we get: IRR = 16+ [(3.274-3.214)/(3.274-3.199)] = 16.8
per cent.
Alternatively (starting with the higher rate), IRR = 17 – [(3.214-3.199)/(3.274-3.199)] = 16.8 per
cent.
Instead of using the direct method, we may find the actual IRR by applying the
interpolation formula to the present values of cash inflows and outflows (Equation 2).
Here, again, it is immaterial whether we start with the lower or the higher rate.
1 2 3 4 5 6 7
1 Rs 14,000 0.847 Rs 11,858 Rs 22,000 0.826 Rs 18,172
2 16,000 0.718 11,488 20,000 0.683 13,660
3 18,000 0.609 10,962 18,000 0.564 10,512
4 20,000 0.516 10,320 16,000 0.467 7,472
5 25,000 0.437 10,925 17,000 0.386 6,562
Total PV 55,553 56,018
Less: Initial inv. -56,125 -56,125
NPV (572) (107)
Contd.
• Since NPV is negative for both the machines, lower IRR
• A: Difference of 572/- and B: difference of 107/-
• Let new IRR = 17% and 20% (decrease by 1%)
• New PV and NPV:
• A: Rs 56,978 , Rs 853 and B: Rs 57,174 , Rs. 1,049
• Hence,
• A: 17% and 18% are consecutive discount rates with positive and negative
NPVs and hence the interpolation gives:
IRR = 17 + [56,978 - 56,125 / 56,978 – 55,553] =17.6%
• Similarly for B, IRR = 20.9%
Merits
• Considers Time value of money (TVM)
• Considers all benefits and costs
• Easier to understand for non-technical people (as compared to NPV)
• Cost of capital not used (atleast not directly)
• Consistent with the overall objective of maximizing shareholder
wealth.
Demerits
• Tedious calculations!!
• It produces multiple rates sometimes [Later]
• Sometimes may give different answers than NPV (Few slides ahead)
• IRR assumes all intermediate CashFlows are reinvested at IRR (Next to
next class)
• Can there be anything more ridiculous than assuming different reinvestment
rates for same year like dollars have different colors!
Example 9 (Book) Contd.
• A company is considering an investment proposal to install new milling controls
at a cost of Rs. 50,000. The facility has a life expectancy of 5 years and has no
salvage value. The tax rate is 35%. Assume the firm uses SLM of depreciation and
the same is allowed for tax purposes. The estimated cash flows before
depreciation and taxes (CFBT) from the investment proposal is as follows:
(+)
x
0 8
4 12 16 20
(b)
Discount rate (K)
(-)
Eg: A purchase or lease decision the more profitable out of the two will be selected, If there are
resource constraints, a firm will be forced to select that project which is the most profitable
rather than accept all projects which exceed a minimum acceptable level
The different ranking given by the NPV and IRR methods can be illustrated under the following
heads:
1. Size-disparity problem;
2. Time-disparity problem; and
3. Unequal expected lives.
1. Size-disparity problem
Size disparity arises when the initial investment in mutually exclusive projects is different.
Example
Particulars Project A Project B Project B-A
Cash outlays (Rs 5,000) (Rs 7,500) (Rs 2,500)
Cash inflows at the end of year, 1 6,250 9,150 2,900
IRR (%) 25 22 16
k 10
NPV 681.25 817.35
Example
Year Cashflows
Project A Project B
0 Rs 1,05,000 Rs 1,05,000
1 60,000 15,000
2 45,000 30,000
3 30,000 45,000
4 15,000 75,000
IRR (%) 20 16
NPV (0.08) 23,970 25,455
• We find on the basis of a comparison of the internal rate of returns that project A
is better, but the NPV method suggests that project B is better.
• Given the cost of capital is 8 per cent, given the objective of the firm to maximise wealth,
project B is definitely better.
• Under the time-disparity problem it is the cost of capital which will determine the
ranking of projects.
• If we take k = 0.10, we shall find project A is better as its net present value would be Rs
19,185 compared to Rs 18,435 of B.
• Its IRR is also more than that of B.
• Both the methods give identical prescription.
• But it does not imply that the IRR is superior to the NPV method, as the NPV is
giving the same ranking as the IRR. In the event of conflicting rankings, the firm
should rely on the rankings given by the NPV method.
3. Projects With Unequal Lives
Another situation in which the IRR and NPV methods would give a conflicting ranking to
mutually exclusive projects is when the projects have different expected lives.
Example
There are two projects A and B. A has a service life of one year, while B’s useful life is five years.
The initial cash outlay for both the projects may be assumed to be Rs 20,000 each. The cash
proceeds from project A (at the end of the first year) amount to Rs 24,000. The cash generated
by project B at the end of the fifth year is likely to be Rs 40,200. Assume that the required rate
of return is 10 per cent. Compute the NPV and the IRR of the two projects.
Example contd.
IRR and NPV of Projects A and B
Project IRR (per cent) NPV
A 20 Rs 1,816
B 15 4,900
Obviously, the ranking given by the IRR and NPV methods is different.
According to the IRR method, the recommendation would favour project A while the
NPV method would support project B.
The conflict in the ranking by the two methods in such cases may be resolved by
adopting a modified procedure.
There are two approaches to do this:
(a) Common time horizon approach
(b) Equivalent annual value/cost approach
(1) Common time horizon approach
Common time horizon approach makes a comparison between projects that
extends over multiples of the lives of each.
Example
Particulars Project A Project B
Initial outlay Rs 10,000 Rs 20,000
Cash inflows after taxes
Year-end 1 8,000 8,000
2 7,000 9,000
3 Nil 7,000
4 Nil 6,000
Service life (years) 2 4
Required rate of return 0.10
Solution:
Project A
Year Cash flows PV factor Total present value
0 Rs 10,000 1.000 (Rs 10,000)
1 8,000 0.909 7,272
2 7,000 0.826 5,782
3 (10,000)a 0.826 (8,260)
3 8,000 0.751 6,008
4 7,000 0.683 4,781
NPV 5,583
a Machine replaced at the end of year 2.
Project B
Year Cash flows PV factor Total present value
0 Rs 20,000 1.000 Rs 20,000
1 8,000 0.909 7,272
2 9,000 0.826 7,434
3 7,000 0.751 5,257
4 6,000 0.683 4,098
Net present value 4,061
Decision Project A should be preferred to project B because of its larger NPV.
(If we had compared the two projects without incorporating the
consequences of replacing the machine at the end of year 2, the decision
would have been the reverse, because the net present value of project A
then would be Rs 3,054 [Rs 7,272 + Rs 5,782 – Rs 10,000])
Limitations:
- In real life we may have project lives of 15/20 years (distant future
estimates)
- Assumptions of same technologies, price, operating costs and revenues
(2) Equivalent annual value (EANPV)/cost approach (EAC).
The EANPV/EAC is a better approach.
The EANPV is determined dividing the NPV of cash flows of the project by the annuity factor corresponding
to the life of the project at the given cost of capital.
While the maximisation of EANPV is the decision-criterion in the case of revenue-expanding proposals, the
minimisation of EAC is the guiding criterion for cost reduction proposals.
Example (Revenue-expanding Investment Proposal)
A firm is considering to buy one of the following two mutually exclusive investment projects:
Project A: Buy a machine that requires an initial investment outlay of Rs 1,00,000 and will generate the CFAT of Rs
30,000 per year for 5 years.
Project B: Buy a machine that requires an initial investment outlay of Rs 1,25,000 and will generate the CFAT of Rs
27,000 per year for 8 years.
Which project should be undertaken by the firm? Assume 10 per cent as cost of capital.
Solution
(i) Determination of NPV of Projects A and B
Project Years CFAT PV factor (0.10) Total PV NPV
A 1-5 Rs 30,000 3.791 Rs 1,13,730 Rs 13,730
B 1-8 27,000 5.335 1,44,045 19,045
However, on the basis of EANPV, project A becomes more desirable, with higher EANPV.
In fact, acceptance of project A would be a right decision.
Session 8:
NPV, IRR – Why?
NPV is better than IRR
• Projects have different cash outlays (Size-disparity)
• Pattern of cash flows is different (Time-disparity)
• The service lives of the projects are unequal (Unequal lives)
The IRR method implicitly assumes that the cash flows generated from the projects
are subject to reinvestment at IRR.
In contrast, the reinvestment rate assumption under the NPV method is the cost of
capital. The assumption of the NPV method is conceptually superior to that of the
IRR as the former has the virtue of having a uniform rate which can consistently be
applied to all investment proposals.
Example
The superficiality of the reinvestment rate under the IRR method can be demonstrated by
comparing the following two investment projects.
Project Initial investment Cash inflows
Year 1 Year 2
A Rs 100 Rs 200 0
B 100 0 Rs 400
• Under the IRR method, both projects have a rate of return of 100 per
cent.
• Under the IRR method, both projects have a rate of return of 100 per cent.
• If Rs 100 were invested for one year at 100 per cent, it would grow to Rs 200, and if invested
for two years, to Rs 400. Since both the projects have the same IRR, the firm should be
indifferent regarding their acceptability, if only one of two projects is to be picked up as both
the projects are equally profitable.
• However, for this to be true, it is necessary that Rs 200 received at the end of year 1 in case of
project A should be equal to Rs 400 at the end of year 2.
• In order to achieve this, it necessarily follows that the firm must be able to reinvest the first year’s earnings at
100 per cent.
• If not, it would be unable to transform Rs 200 at the end of the first year into Rs 400 at the end of the second.
• And if it cannot transform Rs 200 into Rs 400 in a year’s time, the two projects A and B cannot be ranked equal.
There is no reason to believe that a firm can find other investment opportunities at precisely the
required rate.
In contrast, the present value method does not pose any problem.
Let us calculate the present value assuming cost of capital (k) as 10 per cent.
The PV method indicates that project B is preferable to project A as its net present value is greater.
The reinvestment rate in the PV method seems more realistic and reasonable. It assumes that earnings are reinvested
at the same rate as the market cost of capital.
Modified IRR (MIRR)
• However, the Internal Rate of Return (IRR) can be modified assuming the cost of
capital to be the reinvestment rate.
• The intermediate cash inflows will be compounded by using the cost of capital.
• The compounded sum so arrived at and the initial cost outflows can be used as
the basis of determining the IRR.
• Thus, the assumption regarding the reinvestment rate of the cash inflows
generated at the intermediate stage is theoretically more correct in the case of
NPV as compared to the IRR.
Modified IRR Method
Compounded sumn
CO0 = --------------------------
(1 + MIRR)n
Where,
.The compounded sum is done at cost of capital (k) – Eg. on next slide
.CO = cash outflows
. MIRR = Modified IRR
Example (same as IRR):
_________________________________________________________________
Year CFAT Compounded factor at 10% for n – 1years Compounded sum
_________________________________________________________________
1 Rs 14,000 1.464 (for 4 years) Rs 20,496
2 16,000 1.331 (for 3 years) 21,296
3 18,000 1.210 (for 2 years) 21,780
4 20,000 1.110 (for 1 year) 22,200
5 25,000 No compounding 25,000
Total compounded sum at year-end 5 1,10,772
_________________________________________________________________
Note: Cost of capital is 10 per cent (Compounded factors are as per Table A-1.)
Rs 1,10,772
Rs 56,125 = --------------------
(1 + MIRR)5
1. Dividing the compound sum/terminal value (Rs 1,10,772) by the initial outlay (Rs
56,125), we have growth factor (1.9737).
2. In Table A-1, the factors closet to 1.9737 for 5 years are 1.925 (at 14%) and 2.011
(at 15%).
Based on interpolation:
Example
CO0 = Rs 1
CFAT1 2
CO2 2
Where subscripts 0, 1, 2 refer to respective time periods, CFAT = cash
inflows, CO = cash outflows
The required equation to solve the IRR is:
2 2
1 , which leads to r 2 1
1 r 2 1 r
Clearly, the value of IRR is intermediate.
Comparing with NPV
On the other hand, the NPV of this project, given k as 10 per cent, can be easily
ascertained. This would be negative (Rs –0.834), as shown below:
Year Cash flows PV factor Total present value
0 Rs (1) 1.000 Rs 1.000
1 +2 0.909 1.818
2 (2) 0.826 (1.652)
(0.834)
b) Multiple Rates of IRR
Another serious computational deficiency of IRR method is that it can yield
multiple internal rates of return.
Example
Initial cost Year 0 (Rs 20,000)
Net cash flow 1 90,000
Net cash flow 2 (80,000)
Rs 90,000 Rs 80,000
The required equation is : Rs 20,000
1 r 1 r 2
Let (1 + r) be = X and divide both sides of equation by Rs 10,000,
2 = [(9/X) – (8/X)2] = 0
Multiplying by X 2, we can transform the equation into the quadratic form, 2X 2 – 9X + 8 = 0
Such an equation with a variable to the second power has 2 roots which can be identified as:
b b 2 4ac
X (2)
2a
where a = coefficient of the variable raised to the second power
b = coefficient of the variable raised to the first power
c = constant or coefficient of the variable raised to the zero power
Substituting the values for a, b, and c into the quadratic formula produces value for X of 1.21.
Since X = (1 + r), the internal rates for this project are 21.9 and 228 per cent.
Which rate should be used for decision-making purposes?
Conclusion
• To conclude the discussion relating to the comparison of NPV and IRR
methods, the two methods would give similar accept-reject decisions
in the case of independent conventional investments.
• They would, however, rank mutually exclusive projects differently.
• The ranking by the NPV decision criterion would be theoretically
correct as it is consistent with the goal of maximisation of
shareholders’ wealth.
NPV vs PI
Net Present Value Vs. Profitability Index
In most situations, the NPV and PI, as investment criteria, provide the same accept and
reject decision, because both the methods are closely related to each other.
Under the PI method, the investment proposal will be acceptable if the PI is greater than
one; it will be greater than one only when the proposal has a positive net present value.
Likewise, PI will be less than one when the investment proposal has negative net present
value under the NPV method.
However, while evaluating mutually exclusive investment proposals, these methods may
give different rankings.
Example
Year Project A Project B
0 (Rs 50,000) (Rs 35,000)
1 40,000 30,000
2 40,000 30,000
Present value of cash inflow (0.10) 69,440 52,080
NPV 19,440 17,080
PI 69,440/50,000 = 1.39 52,080/35,000 = 1.49
Thus, project A is acceptable under the NPV method, while project B under the PI method.
Which project should the firm accept?
NPV is superior
• The NPV technique is superior and so project A should be accepted.