- Information
- AI Chat
INTERNAL RETIE OF RENTER
Strategic management (6112)
Addis Ababa University
Recommended for you
Preview text
INTERNAL RATE OF RETURN (INTERNAL RATE OF RETURN)
This is an important and legitimate alternative to the NPV method. It is the discount rate to make the discounted cash inflows (that is, the NPV is equal to zero). The IRR and the NPV techniques are closely related in that both involve discounting the cash flows from a project; thus, both account for the time value of money.
When we use the NPV method to evaluate a capital project, the discount rate is the rate of return required by investors for investments with similar risk, which is the opportunity cost of capital. The IRR is an analytical tool use to determine the rate of return associated with a project so as to be able to know whether this rate is higher or lower than the project opportunity cost of capital.
The IRR can also be defined as the discount rate that equates the present value of a project’s expected cash flow to the present value of the project’s outflows:
PV (Project’s future cash flows) = PV (Cost of the project)
In other words, we can describe the IRR as the discount rate that causes the NPV to equal zero.
The above formula can be re-write as:
NPV = NCF 0 + NCF 1 /(1+IRR) + NCF 2 /(11+IRR) 2 +.../(1+IRR)n
=
¿∑
t= 1
n
NCFt/( 1 +IRR)t= 0
CALCULATING THE IRR:
Traditionally, we solve problems related to IRR by a trial and error method taking note of the firm’s cost of capital.
Illustration:
Suppose that BFN Nig. Ltd has an investment opportunity with cash flows shown in the table below, and the cost of capital is 12%. Calculate the IRR for the project.
Year Cash flows # 0 (560) 1 240 2 240 3 240
The best place to start is the use of the firm’s cost of capital as the discount rate, it should be noted that when we discount the NCFs by the cost of capital, we are calculating the project’s NPV:
NPV 12% = NCF0 + {NCF 1 / (1+IRR)} + {NCF 2 / (1+ IRR) 2 + ...+ {NCFn /(1+ IRR) n} =
NPV 12% = -#560 + (#240/1) + {240/ (1) 2 } + {240+ (1) 3 } =#16.
It should be noted that the result we are looking for is zero. Now that we arrived at #16, we can conclude that the discount rate of 12% is too low, thus, we should try a higher rate. Let us try 13%:
NPV13% = -#560 + {#240/(1)} + {#240/(1)2} + {#240/(1)3}
= #6.
We are yet to arrive at the desired result thus let us try 14%
NPV 14% = -#560 + {240/(1)} + {240/(1)2} +{240/(1)3} = -#2.
With DCF of 14%, the NPV is negative. This means that the IRR lies between 13% and 14%. Let us try 13%
NPV13% = -#560 +{#240/(13)} + {#240/(13)2} +{240/(13)3} = 0
Interpretation:
The mean that the NPV of BFN Nig. Ltd capital project is zero at a discount rate of 13%. Her required rate of return is the cost of capital, which is 12%. Since the project’s IRR of 13% exceeds its cost of capital, the IRR criterion shows that the project should be accepted.
The project’s NPV is positive at #16 which also means that the firm should go ahead with the project. Thus, both the IRR and NPV have reached the conclusion.
Alternatively, we can calculate the IRR with the following formula:
IRR = A + {a/ (a - b) X (B - A)}
Where:
IRR = internal Rate of Return
A = discount rate that gives a positive NPV;
B = discount rate that gives the negative NPV;
a = positive NPV derived from discount rate A;
b = negative NPV derived from discount rate B.
IRR = 15 + (24 - 15) X {65,673/ (65,673 – (-76,189)}
15 + 9 X { 65,673 / (141,862)}
15 + 9 X 0.
15 + 4.
19.
∴ IRR = 19%
Agreement between IRR and NPV methods
In the illustrations above, both the IRR and NPV do agrees. The two methods will always agree so long we are dealing with ‘‘independent projects’’ and the cash flows of the projects are conventional in nature.
An independent project is one that can be selected with no effect on any other project, assuming the firm has unlimited financial resources. A project with conventional cash flow can be defined as a project with continuous cash inflows after the initial cash outflows for the project. In other words, after the initial capital is invested (cash outflow), every other forms of cash flows are inflows.
In examining the relationship between the NPV and the IRR, it will be ideal to adopt the use of graphical analysis. Here we graph NPV as a function of the discount rate. This type of graph is called the NPV profile and it shows the NPV of the project at various costs of capital.
The table below shows the NPV profile for the OlopadeEnt. Ltd projects. The NPVs are placed on the vertical axis (y-axis), while the discount rates are placed on the horizontal axis (x-axis). Calculations from the various examples were used with some additional NPVs calculated at various discount rates.
Discount Rate % NPV (# thousands) 0 450 5 310 10 168 15 65 20 - 25 - 30 -
From the above table, one can see that a discount rate of 0% corresponds with an NPV of #450,000; a discount rate of 5% corresponds with #310,352; and so forth.
As the discount rate increases, the value of NPV reduces.
N 200
N 150
N 100
N 50
$
- N 50
- N 100
- N 150
N160,
A
O
B = The NPV is negative at discount rates higher than 13%, so the project should be rejected if discount rate is higher than this value
O = The NPV profile intersects the x-axis at the point where the NPV = 0, which corresponds to an IRR of 13%.
Discount Rate
0% 5% 10% 15% 20% 25% 30%
The graph above shows that the relationship between the NPV and the discount rate is inverse, because as the discount rate increase, the NPV declines smoothly. The curve intersects the x-axis at the exalt point where NPV equals zero (0) and the IRR is 19%
The curve or the NPV profile explain why the NPV and IRR methods leads to identical accept- reject decisions for Olapade’s project. The IRR of 19% is the exalt point where the NPV changes from positive to a negative value.
WHEN THE NPV AND THE IRR METHODS DISAGREE
Now, we know that when we have an independent project and conventional cash flows, both IRR and NPV do agrees, however, there are some situations where we are faced with taking decisions when we have unconventional cash flows.
Unconventional cash flows are a situation whereby the cash flows pattern of a project fluctuates in a positive (inflows) and negative (outflows) order. In some situation, unconventional cash flows are the reversed of the conventional cash flows. For instance, when the initial cash flow is positive and every other cash flow are negative. The annuity payment on Lifetime Annuity received by Insurance Company from her clients for life insurance scheme is cash inflows, the subsequent payment made by the insurance company to the client over the lifetime of the client are cash outflows. Under this situation, we are to reverse the IRR decision rule and accept the project if the IRR is less than the cost of capital to make the IRR and the NPV agree.
The implication is that the Life Insurance Company is practically borrowing money from her clients and the IRR is a measure of the cost of that money. The cost of capital is the rate at which the insurance company can borrow elsewhere. When the IRR is less than the cost of capital, then it means that the lifetime annuity provides the insurance company with money at a lower cost than any alternative sources
IR R= 16%
0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65%
Discount Rate
Graphical explanations
From the table one can easily identify a problem, for instance the NPV is initially negative (-#5); then, at a discount of 20%, became positive i. #.56; and then, at 60% discount rate became negative again.
From the graph, we have two IRRs, one at 16% and the other one at 55%. It should be known that neither of these two IRRs is correct as the information provided by the IRR technique is suspicious.
The question we will have to ask ourselves here is that how many IRR solutions can we have in a given cash flow? The answer to this depends on the numbers of sign reversals in the cash flow stream. Projects with conventional cash flows are expected to have only one sign reversal, thus one IRR. In our oil extraction example, there two cash flow sign reversals, meaning, we have two cash flows.
In conclusion, it is impossible to calculate an IRR for some cash flow patterns, especially in a situation whereby the initial cash flow (t=0) is either a cash inflow or outflow and is followed by cash flows with two or more cash reversals. This type of situation is often experienced in the building industry whereby the pattern cash flows is in this other NCF0 = #30, NCF1 = -#45 and NCF2 =#40. This type of cash flow may occur when a contract is awarded, and the contractor is mobilized with #30 million, usually for materials and supplies after which he does the construction and pay labour cost with #45 million, and upon completion of the project receives his final payment of #40 million. It should be noted that when it became impossible to compute an IRR, the project either has a positive NPV or a negative NPV for all possible discount rates.
Mutually exclusive projects and the IRR
NVP (N
Million)
N2.
N1.
N0.
- N1.
- N2.
- N3.
- N4.
IR R= 16% IRR = 55%
Two projects are said to be mutually exclusive when accepting one naturally would mean rejecting the other. For instance, suppose you own a small store in the commercial nerves of Lagos, Nigeria. And you are looking at the two business opportunities, such as running a barber shop or opening an eatery centre.
It is evident that both projects cannot be pursued at the same time with the same facility; these two projects are classed as mutually exclusive as accepting either would amount to rejecting the other.
When faced with a choice between mutually exclusive projects, how does one make a choice? Under the NPV approach, the answer is cheap. All one need do is to accept the project that have the highest NPV because it will generate the highest level of value for the firm. However, in the case of IRR, one cannot easily say which mutually exclusive project is to be selected by mere observation of the project IRRs.
MODIFIED INTERNAL RATE OF RETURN (MIRR)
This is an analytical tool that helps to eliminate the reinvested rate assumption of the IRR method by converting each operating cash flow into a future value at the end of the project life in compounded order at the cost of capital. These values are then summed up to get the project’s terminal value. One of the major weaknesses of the IRR method in compares with the NPV method deals with the rate at which the cash flow generated by a project is reinvested. While the later assumed that the cash flows generated by a capital project are reinvested at the cost of capital, the later assumed that they are reinvested at the IRR. Decision as per which of these methods is the better assumption depends on which rate the firm can actually earn when they reinvest the project cash flow over time. Analysts generally believe that the cost of capital, which is always lower than the IRR, better reflects the rate at which firms are likely to earn. Thus, using the IRR rate may connotes being over optimistic.
The MIRR offer a better alternative. The MIRR is the interest rate that equates the project’s cost (PV cost), or cash outflows, with the future value of the project’s cash inflows at the end of the project (PV TV).
Here we eliminate the problems associated with the reinvestment rate by using the cost of capital as the interest rate to compute the future value of the investment.
Computing the MIRR formula:
First, let:
PV (cost of the project) = PV (Cash inflows)
PV cost =PVTV
PV cost =TV/ (1+MIRR)n
Two preliminary calculations are crucial to successful computation of MIRR. We begin to first of all calculate the value of PVcost, which is the present value of the cash outflows that make up the investment cost of the project. For most capital projects, the investment cash outflows are made at the beginning of the project, i. t=0, thus there may be no need to calculate the present value of such
The terminal value of N1250 million equals the sum of the #360 million in year one (1) compounded at 15% for two years plus the #360 million in year two (2) compounded at 15% for one (1) year plus the #360 million in year three (3). We can present our message mathematically as such:
TV = CF 1 (1 +k) n-1 + CF 2 (1 +k)n-2 +....+CFn (1 + k)n-n
= #360(1)2 + #360(1) +#
=#1250 million
Given that the cost of capital is #860 million, and the TV is #1250 million, we can now calculate the MIRR as follows:
PV cost = TV / (1 +MIRR)n
= #860= #1250/(1 + MIRR) 3
Cross multiply
(1 + MIRR)3 = #1250/#
(1 + MIRR)3 = 1.
Take the root cube of both sides
(1 + MIRR) =(1)1/
= 1.
MIRR = 1-1.
=0.
0 X 100
13%
At 13%, the MIRR is lower than the cost of capital at 15%,so the project should be rejected.
Decision Rule for MIRR:
When MIRR > Cost of capital = Accept the project; and,
When MIRR< Cost of capital = Reject the project.
Merits of MIRR
1 It is easy to understand; 2 It is based on the discounted cash flow technique; 3 It makes provision for a margin of errors, which is not well taken care of by even the NPV; 4 It enjoys almost all the advantages of an NPV method.
Demerits of MIRR
1 When faced with a non conventional cash flow, IRR approach will yield no meaningful or usable result; 2 When dealing with mutually exclusive projects, IRR can lead to incorrect investment decisions; 3 A lower IRR can be better if a cash flow is followed by cash outflows; 4 Making a choice between alternative investment options could at times be cumbersome.
PROBABILITY INDEX (PI)
This is an investment decision criteria used by analyst to prioritise investment opportunity in terms of their expected profitability. It is a tool used to ensure that the firm uses it limited resources for investment with the highest returns possible. PI is rarely use in appraising projects because it often yield the same decision with the NPV method.
However, the PI is more useful when dealing with Capital rationing related problems. The probability Index (PI), which is also called the benefit-cost ratio, is determined by dividing the net present value of future cash flow by the initial outlay. The mathematical equation of a PI could be derived as follows:
PI =
¿∑
t= 0
n
¿ ¿
NPV/ Io
Illustration:
BukolaBamidele& Co, a manufacturing firm is contemplating investing in a project proposal that requires an initial outlay of #700,000, which promises the following cash flows for the next five years:
When both the NPV and PI are positive, accept the project;
When both the NPV and PI are negative, reject the project;
For mutually exclusive projects, all the projects under consideration should be ranked, and the one with the highest PI should be selected.
Merits of PI
1 It has almost all the advantages of NPV method; 2 It is highly useful in ranking projects and for capital rationing; 3 It is easily appreciable by a layman.
Demerits of PI
1 it have almost the same disadvantages as the NPV; 2 It cannot be used without the rule of NPV.
INTERNAL RETIE OF RENTER
Course: Strategic management (6112)
University: Addis Ababa University
- Discover more from:
