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1 TrueIRR IRR and NPV redefined Chapters 1. Introduction 2. Cumulative Future value 3. Classification of series 4. Combination series and IRR 5. TrueIRR Technique . !ar"in #alue concept \$. %pplication of TrueIRR &. Reinvestment assumptions in IRR and !IRR '. True()#

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IRR concept

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• TrueIRRIRR and NPV redefinedChaptersIntroductionCumulative Future valueClassification of seriesCombination series and IRRTrueIRR TechniqueMargin Value conceptApplication of TrueIRRReinvestment assumptions in IRR and MIRRTrueNPV

• Chapter 1

Introduction

Limitations of IRR and NPVIllustration on Multiple IRRsReasons for multiple IRRs

• 1.Limitations of IRR and NPV Internal Rate of Return(IRR) is a commonly used capital budgeting technique. It is also used in many other areas. Several deficiencies in IRR technique. existence of multiple IRRsReinvestment Assumption etc.Financial theorists suggest NPV and MIRR techniques to overcome the limitations of IRR. However these techniques are also not free from limitations. Multiple IRRs: Some cash flow series have multiple IRRs. In such cases, the decision maker will be in dilemma as to which IRR has to be considered for decision-making. Further, in the case of such series, as we increase the discount rate, NPV oscillates from positive to negative and negative to positive. Therefore, the decision maker cannot rely upon the NPV either.

• 2. Illustration of Multiple IRRsIllustration 1:A Bank is offering a Recurring Deposit (R.D.) linked Loan scheme. Under the scheme, the Bank will give a loan of \$15000. The loan will have to be repaid along with the installments of R.D. scheme. The installments are given below. At the end of 6th year, the Bank will be repaying you \$35061 being the maturity amount of R.D. scheme. IRR of the Scheme is 4.06% and 27.2%.

Interestingly, the proposal has two IRRs. The customer doesnt know which IRR has to be considered for decision-making. Further, as we increase the Discount Rate, NPV is fluctuating from positive to negative and negative to positive. Therefore, customer cannot rely upon NPV either.

YearInstallments1613826138316138410000510000

• 3. Reasons for Multiple IRRs Why do the IRR and NPV behave in this fashion? When does such a phenomenon occur? Does it occur occasionally? Does the IRR formula has any inherent defects? Does the MIRR method provide the right answer? If it is because of the reinvestment assumption, then why does the NPV also behave in a strange manner?

Solution to Multiple IRRs A study has been made to find out an answer to the above questions. As a result, a new method has been developed which overcomes the deficiencies of IRR method without loosing the advantages of IRR. The new method is termed as TrueIRR. Further, it has been found that the MIRR method is not a solution to the multiple IRRs problem and the reinvestment assumption in IRR method is a misconception. To understand the TrueIRR method, we need to understand the following: Cumulative future value (CFV) Lending value and borrowing valueClassification of cash flow series

• Chapter 2

Cumulative future value

Cumulative future value (CFV)Illustration on CFVNet future value (NFV)Finding IRR using NFVLending Value and Borrowing Value

• 1. Cumulative future value

The Cumulative future value is the tool with the help of which we will be able to apply the TrueIRR technique and solve the problems of multiple IRR. Considering the time value of money, cash flow value expressed in terms of its value at the beginning of the proposal is present value.at the end of the proposal is terminal value or future value.

Similarly, we can express the value of a cash flow at any intermediate period. They are of two types, Future value (at kth period) Cumulative future value (at kth period).

Contd.

• 1. Cumulative future value (contd.)

Cash flow Value of a particular period, expressed in terms of its Value at any intermediate period, say kth period, may be termed as future value (at kth period).Future value (at kth period) of cash flow of jth period i.e., t j, k = C j * (1+ r)(k j)Where,t = Future value, j = Cash flow period, C = Cash flow, r = Interest rate, k = Period at which cash flow value is expressedThe sum of future values at any intermediate period, say kth period, of cash flow values of 0th to kth period may be termed as cumulative future value (at kth period). In other words, cumulative future value at a particular period is the sum of future values of cash flow Values up to that period expressed in terms of their Values at that period. Fori = 0,V0 = C0 Fori = 1 to n,Vi = Vi-1 *(1 + r) + CiWhere, i = Period, V = CFV, C = cash flow , r = Interest Rate, n = Terminal Period

• 2. Illustration on CFV

The future values of cash flow at various periods at 10% Interest Rate are shown in the following table.

Table 2.3 In the table No. 2.3, the vertical totals of future values are shown. These totals are Cumulative future values (CFVs) of the corresponding period. Interest Rate = 10%

Future values at 10% Interest Rate YearYearCash flow 01234010001000110012101331146412000-200022002420266223000--30003300363035000---5000550048000----8000Total1000310064101205121256

• 3. Net future value

We know that the Sum of all future values at terminal period is Net future value (NFV). Relationship between net future value and net present value: NFV= NPV*(1+r)nIf NPV is zero, then NFV will also become zero. Therefore IRR can also be defined as the Rate at which the NFV of a cash flow series is Zero. CFV of Terminal period is the sum of future values (at terminal period) of all cash flow Values. Therefore CFV of terminal period is NFV. With the help of CFV, we can find out NFV and with the help of NFV, we can find out IRR. The steps involved in finding out the IRR using the NFV are similar to finding out the IRR using the NPV. To understand the problem of multiple IRRs and to find a solution for the same, we need to understand the method of finding IRR using the CFV and NFV.

• 4. Finding IRR using NFV

The interpolation formula is as follows:IRR (NFV) = LR + |LNFV| / ( |LNFV| + |HNFV| ) * (HR LR)Where, LR = Lower Rate, HR = Higher Rate, |LNFV| = Absolute value of NFV at LR, |HNFV| = Absolute value of NFV at HR.

Illustration 2:Step 1: Let us calculate NFV at a trial interest rate, say, 10%. Interest Rate=10%

NFV=84 Table 2.4 Here, NFV is positive. Contd.

YearCash flow CFV0-3,000-3,00011,000-2,30021,000-1,53031,000-683483584

• 4. Finding IRR using NFV (Contd.)

Step 2: Now let us increase the interest rate to 12%. Interest Rate=12%

NFV=-106 Table 2.5Here, NFV is Negative.Step 3: Now we can find IRR by interpolation method. IRR (NFV) = LR + |LNFV| /(|LNFV| + |HNFV|) * (HR LR)LR = 10%, HR = 12%, LNFV = 84,HNFV = -106IRR (NFV) =10% + 84 /(84 + 106) * (12% - 10%) = 10.9%IRR (NFV) = 10.9% & IRR (NPV) = 10.9%

YearCash flow CFV0-3,000-3,00011,000-2,36021,000-1,64331,000-8404835-106

• 5. Lending Value and Borrowing Value

CFV may be classified into borrowing value and lending value depending upon its sign and significance. If CFV of a particular period is negative, we are yet to recover that much amount of lending at that point of time. Therefore, the negative CFV may be referred to as lending value. On the other hand, if the CFV of a particular period is positive, we are yet to return that much amount of borrowing at that point of time. Therefore, the positive CFV may be referred to as borrowing value. The sum of all lending values of a series may be referred to as total lending value (TLV). The sum of all borrowing values of a series may be referred to as total borrowing value (TBV). TLV and TBV are useful in classification of cash flow series.

Note: Hereafter, the word Lending has been used to convey the meaning of both Lending as well as Investment.

• Chapter 3

Classification of series Lending seriesBorrowing seriesCombination series Steps to identify the type of series

• 1. Lending series

The cash flow series may be basically classified into three categories, depending on their nature. Lending series Borrowing series Combination series

Lending series If all the CFVs of a series are negative, the series may be referred to as lending series. In such series, money is invested during the initial periods and returns occur during the later periods. Cash flow series of a lending proposal is an example of the lending series. In the case of a lending series, the IRR is the rate of return on lending. Therefore, the IRR of a lending series may be termed as internal rate of lending (IRL).

• 2. Borrowing series

If all the CFVs of a series are positive, the series may be referred to as borrowing series. In such series, cash inflows occur during the initial periods and cash outflows occur during the later periods. This type of series occur when a firm is borrowing money and returning the same with interest during later periods. If the nature of a series is not identified with its result, the decision maker may erroneously think that the proposal is an investment option, compare the result with the cost of capital and arrive at a wrong decision. Therefore, it is important to identify the type of the series along with the result. In the case of a borrowing series, the IRR is the cost of borrowing. Therefore, the IRR of a borrowing series may be termed as internal rate of borrowing (IRB).

• 3. Combination series

A series, which is a combination of lending series, and borrowing series may be referred to as combination series. In a combination series, some CFVs are positive and some are negative. For this purpose, CFV has to be calculated by taking the IRR as the interest rate. Combination series can be further classified into two categories. If a combination series is dominated by lending series, such a series may be referred to as combination series (lending). In a combination series (lending), the total lending value (TLV) will be greater than the total borrowing value (TBV). If combination series is dominated by borrowing series such a series may be referred to as combination series (borrowing). In a combination series (borrowing), the TBV will be greater than the TLV.

• 4. steps to identify the type of seriesCalculate the CFVs at interest rate equal to IRR.Calculate the total borrowing value (TBV) and total lending value (TLV). Now,If the TBV = 0, then lending series.If the TLV = 0, then borrowing series.If TLV < > 0 and TBV < > 0, then combination series. Further,

If the TLV > TBV, then combination series (lending) If the TBV > TLV, then combination series (borrowing).

• Chapter 4

Combination series and IRRInterpretation of IRR of Combination seriesImplicit Assumption in the case of IRR New approach to Combination series

• 1. Interpretation of IRR of Combination seriesIn case of lending series, IRR is the internal rate of lending (IRL). In other words, it is the rate of return on lending. The IRL is compared with borrowing rate or cost of capital and decision is taken. In case of borrowing series, IRR is the internal rate of borrowing (IRB). In other words, it is the rate at which the firm has to pay for borrowing money. The IRB is compared with expected rate of return on investment and decision is taken.

The combination series is the combination of lending series and borrowing series. How to interpret IRR of a combination series? Whether IRR of combination series has to be compared with borrowing rate or expected rate of return on investment?

• 2. Implicit Assumption in case of IRR We are trying to find out a single IRR for lending as well as borrowing part of a combination series. For the decision-making, the said IRR cannot be compared with both the borrowing rate and the rate of return on investment unless they are equal. Therefore, the application of IRR to a combination series, involves the assumption that the borrowing rate and the rate of return on investment are equal to the company, which can never be true. For every company, the borrowing rate will be different from the lending rate. As the assumption underlying the use of IRR is not true in real life situations, the application of IRR to a combination series is bound to provide irrational answer. Because of this false assumption, we may get multiple IRRs or we may not get an IRR. Even if we get a single IRR to a combination series, the same will not be the correct IRR.

• 3. New approach to Combination series A combination series contains at least one lending series and one borrowing series. In the case of a lending series, the objective is to maximize the return, whereas in the case of a borrowing series, the objective is to minimize the cost of borrowing. Even though we are using the same formula of IRR in the case of both types of series, objectives are different. Therefore, trying to find out a unified IRR is illogical in the case of a combination series.

The solution to the above problem would be to bifurcate the cash flow of the lending series and borrowing series, which are interwoven in a combination series, and then apply different interest rates for the two series. However, the bifurcation of a series is a difficult task considering the fact that the two series are interwoven with each other. Further, the size of the lending series and the borrowing series hidden in a combination series may vary depending on the interest rate. However, if we apply different interest rates to lending values and borrowing values, we will be able to overcome the problem of multiple IRRs.

• Chapter 5

TrueIRR TechniqueTrueIRR and combination seriesSteps in application of TrueIRR to Combination seriesCombination series (lending) & TrueIRR -illustrationBifurcation of Combination series

• 1 TrueIRR and combination seriesIn the case of a combination series, the borrowing rate is applied to all borrowing values and the lending rate is applied to all lending values. Out of the borrowing rate and lending rate, one is predetermined and another is found out. This new method is termed as TrueIRR. In the case of a combination series (lending), the objective is to find out the internal rate of lending (IRL). Therefore, we must predetermine the borrowing rate and apply the said rate to the borrowing part of the series. Thereafter, we can find out the lending rate which is the internal rate of lending (IRL) of the lending part of the series. Whereas, in the case of a combination series (borrowing), we must predetermine the lending rate and apply the said rate to the lending part of the series. Thereafter, we can find out the borrowing rate and this is the internal rate of borrowing (IRB) of the borrowing part of the series.

• 2. Steps in application of TrueIRR to Combination seriesStep 1: Find out the IRR as usual. Prepare the CFV table by taking the IRR as the interest rate and test whether the series is a combination series (lending) or combination series (borrowing). Then, determine the borrowing rate or the lending rate depending on the type of the series. Step 2.1: Prepare the CFV table. In the case of a combination series (lending), apply the borrowing rate for positive CFVs and a trial lending rate for negative CFVs. In the case of a combination series (borrowing), apply the lending rate for negative CFVs and apply a trial borrowing rate for positive CFVs. Step 2.2: Repeat the step 2.1 until we find out the two NFVs so that one is positive and another is negative. Step 3: Apply the interpolation formula and find out the IRL/IRB.Step 4: Prepare the CFV table at the IRL/IRB rate and confirm that the NFV is zero. Also, confirm that the series is a combination series (lending) or combination series (borrowing) by comparing TBV with TLV.

• 3. Combination series (lending) &TrueIRR -illustrationIllustration 6: Let us calculate IRL of a cash flow assuming 10% borrowing rate under TrueIRR technique:

Here, TLV > TBV. Therefore this is a combination series (lending). Its IRL is 7.34% p.a. at 10% borrowing rate.

IRL = 7.344% Borrowing Rate = 10% YearCash flow CFV015,00015,0001-6,13810,3622-6,1385,2603-16,138-10,3524-10,000-21,1125-10,000-32,662635,0610NFV = 0TBV = 30622 TLV = 64126

• 4. Bifurcation of Combination seriesWe can bifurcate the lending series and borrowing series hidden in a combination series. The steps to be followed in the bifurcation of a combination series are described hereunder.Step 1: Calculate the borrowing values and lending valuesStep 2: Calculate the cash flows of the borrowing series and lending series. The formula for finding out the cash flow of borrowing series is as follows: Ci = Vi Vi-1 * (1+B)Where, C = cash flow, i = Period, V = borrowing value, B = borrowing rate. The formula for finding out the cash flow of lending series is as follows: Ci = Vi Vi-1 * (1+L)Where, C = cash flow, i = Period,V = lending value, L = lending rate

• Chapter 6

Margin Value concept

Margin ValueRelationship between IRR and NPV

• 1. Margin Value

With the help of CFV, we can find out a new value, i.e., margin value. Margin value helps us to understand the relationship between the IRR and the NPV. NPV is the result of difference between the IRR and the discount rate. Therefore, NPV is the present value of the total margin from the proposal at a particular discount rate. Further, the margin at IRR is zero. Therefore, the difference between the IRR and the discount rate may be termed as margin rate. The margin of each period may be termed as margin value. The NPV computed under the margin method will be equal to the NPV computed under the present value method.NPV = Where, i = period, m = margin rate (or m = r-x), n = terminal period, U = CFV calculated by taking IRR as interest rate, x = discount rate, r = IRR

• 2. Relationship between IRR and NPVThe NPV is a function of the CFV, IRR and discount rate. CFV may be either lending value or borrowing value. Until now, we did not know the meaning and importance of CFV and this was the missing link. As we did not know the CFV, there was a lot of controversy as to which one is superior between the IRR and the NPV. Now, the controversy will be resolved. The relationship between the IRR and the NPV is clear. NPV is the sum of discounted margin values and the margin value is the difference between the IRR and the discount rate.

• Chapter 7

Applications of TrueIRRCombination series and TrueIRRComparison of proposals

• 1. Combination series and TrueIRRThe most important advantage of the TrueIRR method is that it overcomes the deficiencies of IRR and NPV methods in relation to combination series. In the case of a combination series, the IRR and NPV methods fail to provide correct result.Illustration: A hire purchase company offers a scheme, wherein, the company gives a vehicle costing of \$ 100,000 on hire purchase to the customer. The customer has to pay hire installments of \$9,000 every month for 12 months. The customer has to keep a Security Deposit of \$35,000 at the time of giving the hire purchase facility, which will be repaid at the end of 13 months with interest of \$3,500. The Manager (Finance) has computed the IRR of the proposal at 2.203% p.m. or 26.44% p.a. The minimum expected return on investment of the company is 25% p.a. As the IRR was more than the minimum expected return, the company has been running the scheme since 3 years. The total investment of the company in the scheme is \$4,000 million. The borrowing rate of the company was 15% p.a. during the last 3 years. Now, on coming to know that TrueIRR provides correct rate of return on investment, the company requests you to calculate TrueIRR. Also, estimate the loss incurred by the company due to application of IRR method.

• 1. Combination series and TrueIRR (contd.)Solution: This is a combination series (lending). At borrowing rate of 15% p.a., the IRL is 23.3% p.a. Difference between the IRR and the IRL is 3.14%. Total loss caused to the company is approximately \$126 million.

The above series looks like a conventional series. It has only one positive IRR. In spite of this, IRR and IRL of the series differ. Therefore, it may be concluded that we should test a series as to whether it is a combination series. If it is a combination series, we should apply the TrueIRR method only. If it is a lending series or borrowing series, then only, we can apply IRR method.

• 2. Comparison of Proposals

TLV and TBV help us to understand why NPV and IRR provide us contradictory results when comparing two or more exclusive proposals. We have already seen in Chapter 6 that TLV, TBV, IRR and NPV are closely related. NPV is a function of TLV, TBV, IRR and discount rate. We can find out IRR, NPV, TLV and TBV of proposals and take proper decision, as we know the relationship between them. TLV and TBV help us in understanding the reasons for contradictory results and arrive at a proper decision. In the case of appraisal of two or more mutually exclusive proposals, IRR as well as NPV of a cash flow series provide incorrect results, whereas, IRR and NPV along with TLV and TBV of the incremental cash flow provide correct results

• Chapter 8

Reinvestment Assumptions in IRR and MIRR Reinvestment Assumption in IRR and NPVReinvestment Assumption in IRR and NPV is a mythComparison of IRR and MIRR formulae

• 1. Reinvestment Assumption in IRR and NPV Financial theorists make the following arguments. IRR and NPV give conflicting results while ranking alternative proposals. This conflict occurs due to a reinvestment assumption implicit in all methods using discounted cash flow approach. Application of IRR involves the assumption that recovered funds are reinvested at a rate equal to internal rate of return. Further, it is argued that opportunities to invest recovered funds at internal rate of return, generally, do not exist. Therefore, the MIRR method has been developed. MIRR assumes cash flows are reinvested at cost of capital, while IRR assumes cash flows are reinvested at the IRR. Because reinvestment at the cost of capital is a better assumption, MIRR is an effective indicator of true profitability of a proposal.

• 2. Reinvestment Assumption in IRR and NPV is a myth

In the case of the IRR method, we are trying to find out the rate of interest that we earn on the investment. The cash inflow is set off against the balance of investment with interest. It is never assumed that we are investing the inflow. Interest is not calculated on the inflow. Further, the interest is not calculated on the investment to the extent of the inflow. Inflows may be utilized either for repayment of borrowing or for fresh investment in another proposal or for repayment of capital. The option to utilize the inflows for any of the above purposes is left to the management. In IRR method, we are simply finding out the compounded rate of interest and nothing more. Therefore, we can conclude that in the case of the IRR method, reinvestment assumption is not done at all, as there is no question of reinvestment of intermediate inflows. Similarly, in the case of the NPV method also, the question of reinvestment assumption does not arise.

• 3. Comparison of IRR and MIRR formulae

In the MIRR method, we are making the following exercises. We are finding out the future value of the cash inflow of the ith period at the terminal period by applying the reinvestment rate and again finding out the present value (at ith period) of the said future value at the MIRR rate. Then, we are multiplying with the discount factor at the MIRR rate as done in the case of the IRR method.This exercise is logical only when the inflow cannot be utilized for repayment of borrowing or any other purpose until the terminal period and it has to be reinvested until the terminal period. This assumption is seldom true. Therefore, MIRR method can never be applied as an alternative to IRR.

• Chapter 9

Combination series and NPV

Interpretation of NPVTrueNPVFinding TrueNPVCombination series and TrueNPV

• 1. Interpretation of NPVNPV and types of series: If the NPV of lending series is positive and its value is acceptable, the Decision maker will accept the lending proposal. If the NPV Borrowing series is positive and its value is acceptable, the Decision maker will accept the Borrowing proposal.

Combination series and NPV: In case of Combination series, NPV is fluctuating from Negative to positive and vice versa as we increase the Discount Rate. What is the criterion for taking the decision in case of Combination series?

• 2. TrueNPV A combination series contains at least one lending series and one borrowing series. In the case of a lending series, the objective is to earn at least up to the expected minimum lending rate. In the case of a borrowing series, the objective is to borrow at a minimum rate, at any cost, not more than the expected maximum borrowing rate. Even though, we are applying the same NPV formula to the both types of series, the objectives are different. In the case of a combination series, we are trying to attain two mutually contradictory objectives at the same time, which is impossible. The solution to the above problem is to find out the NPVs of lending series and borrowing series separately. Then, we will be able to overcome the problem of oscillating NPV. This new method is termed as TrueNPV.

• 3. Finding TrueNPVThe steps to find out the TrueNPV of a combination series are: Step 1: Apply the TrueIRR method and bifurcate the series.Step 2: Find out the NPV of the lending series and the same may be termed as NPV (lending). Step 3: Find out the NPV of the borrowing series and the same may be termed as NPV(borrowing). Alternatively, we can find the NPV of the lending part and borrowing part of the series by applying the margin method.

• 4. Combination series and TrueNPVIn the case of borrowing values of a combination series (lending), we will be applying the predetermined borrowing rate. Therefore, the question of finding out the NPV of borrowing part of the series does not arise. If the NPV (lending) is positive and its value is acceptable, the decision maker will accept the proposal. NPV (lending) does not oscillate like ordinary NPV. Further, its movement is similar to the movement of NPV of lending series.In the TrueNPV method, NPV of borrowing part of combination series (borrowing) is only relevant. In the case of lending values, we will be applying the predetermined lending rate. Therefore, there is no question of finding out the NPV of the lending part of the series. If the NPV (borrowing) is positive and its value is acceptable, the decision maker will accept the proposal. The NPV (borrowing) does not oscillate like an ordinary NPV. Further, its movement is similar to the movement of the NPV of the borrowing series.

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