TrueIRR

This is the summary of the Book 'TrueIRR'.

1.  Introduction [1]  

Internal rate of return (IRR) is a commonly used capital budgeting method.  The IRR method suffers from many deficiencies such as multiple IRRs, reinvestment assumption etc. Financial theorists suggest the application of net present value (NPV) and Modified IRR (MIRR) methods to overcome the limitations of the IRR.  However, these methods 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. 

Illustration: A Bank is offering a Recurring Deposit linked loan scheme.  Under the scheme, the Bank will give a loan of $15000.  The loan will have to be repaid in 3 yearly installments of $6138.  At the end of third year, customer has to open a deposit account of 3-year term with the Bank and deposit $10000 per year for 3 years.  At the end of the term, the Bank will be repaying an interest of $5061 along with the principal.  Now, as the deposit and loan are interlinked, a Customer requests you to calculate IRR and NPV of the whole proposal. 

Solution: Interestingly, the proposal has two IRRs i.e., 4.1% and 27.2%. 

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 value

¨        Classification of cash flow series

2.  Cumulative Future Value

 ‘Cumulative future value (CFV)’ is the tool with the help of which we will be able to apply the TrueIRR method and solve the problem of multiple IRRs.  Considering the time value of money, cash flow value expressed in terms of its value at the beginning of the proposal is ‘present value’ and 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 k^th period) 

¨        cumulative future value (at k^th period). 

Cash flow value of a particular period, expressed in terms of its value at any intermediate period, say k^th period, may be termed as “future value (at k^th period)”.

Future value (at k^th period) of cash flow of j^th period   i.e.,  t _{j, k} = C _j ´ (1+ r)^{(k – j)} 

Where,     t = Future value,     j = cash flow period,     C = cash flow

k = Period at which cash flow value is to be expressed     r = interest rate,

The sum of future values at any intermediate period, say k^th period, of cash flow values of 0^th to k^th period may be termed as “cumulative future value (at k^th period)”.  In other words, cumulative future value at a particular period is the sum of future values of the cash flow values up to that period expressed in terms of their values at that period. 

For          i = 0,                        V₀ = C₀

For          i = 1 to n,                  V_i  = [V_i-1  ´ (1 + r)]   + C_i 

                                Where, i = Period, V = CFV,   C = cash flow,   r = interest rate,  n = Terminal period

 

Example: Calculations of the CFVs can be easily done in the following table format with the help of calculators.

        We know that the sum of all future values at the terminal period is “net future value” (NFV).  Relationship between the net future value and the net present value is as follows:

                         NFV = NPV ´ (1+r)ⁿ       

If NPV is zero, then, NFV will also become zero.  Therefore, the IRR can also be defined as the rate at which the NFV of a cash flow series is zero.  CFV of the terminal period is the sum of future values of all cash flow values at the terminal period.  Therefore, CFV of the terminal period is the NFV [2] . 

      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 [3] .  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.

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 [4] 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. 

3.  Classification of Cash Flow Series

Cash flow series may be 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’). 

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’). 

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. 

The steps to identify the type of Series

Calculate the CFVs at an interest rate, which is equal to the IRR.  Also, calculate the TBV and TLV.

¨        If TBV = 0, then it is a lending series. 

¨        If TLV = 0, then it is a borrowing series.

¨        If both TLV as well as TBV are not zero, then it is a combination series.  Further, 

Ø      If TLV > TBV, then it is a combination series (lending)

Ø      If TBV > TLV, then it is a combination series (borrowing). 

4.  Combination Series and IRR

In the case of a lending series, the IRR is compared with the cost of capital and an appropriate decision is taken.  In the case of a borrowing series, the IRR is compared with the expected rate of return on investment.  A combination series is a combination of lending series and borrowing series.  Whether we have to compare the IRR of a combination series to the borrowing rate or to the expected rate of return on investment? 

Implicit Assumption in the case of Application 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.

        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. 

5.  TrueIRR

In 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. 

The steps in the application of the TrueIRR method to combination series:

Step 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 [5] or the lending rate [6] 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. [7]  

Example :                    Lending rate (IRL) = 7.344%  

                                    Borrowing rate = 10%

TBV = 30622    

TLV = 64126

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

Bifurcation of Combination Series

We 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 values                          

Step 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:

                  C_i = V_i –V_i-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:

                  C_i = V_i –V_i-1 ´ (1+L)

   Where,          C = cash flow, i = Period,       V = lending value, L = lending rate

6.  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

Relationship between IRR and NPV

The 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. 

7.  Applications of TrueIRR

Combination Series and TrueIRR

The 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. 

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. 

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. [8]

8.  Reinvestment Assumption and MIRR

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.” 

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. [9]   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. 

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 i^th period at the terminal period by applying the reinvestment rate and again finding out the present value (at i^th 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. 

9.  Combination series and NPV

We have already resolved the problems in the application of the IRR method to combination series by applying the TrueIRR method.  Now, let us understand the application of the NPV method to different types of series and resolve the problem of oscillating NPV.

NPV and types of Series

If NPV of a lending or borrowing series is positive and its value is acceptable, the decision maker will accept the proposal.  In the case of a combination series, the NPV is fluctuating from negative to positive and vice versa as we increase the discount rate.  What is the criterion for taking a decision in the case of combination series?  Why does the NPV behave in this fashion? 

        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”. 

TrueNPV: The 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.         

Combination Series (Lending) and TrueNPV    

In 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.

Combination series (Borrowing) and TrueNPV         

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. [10]

The references given here relate to the Book 'TrueIRR'.

Algebraic proofs of all formulae and assertions are given in Chapter 11 of the Book.

[1] For detailed explanation on any topic, you can refer to the particular chapter of the Book 'TrueIRR, based on the serial number of each topic.  For example, detailed explanation on the topic “1. Introduction” is given in Chapter 1 of the Book.

[2] The algebraic proof is given in Chapter 11.

[3] See illustration 2.3 in Chapter 2.

[4] In this Book, the word ‘lending’ has been used to convey the meaning of both ‘lending’ as well as ‘investment’.

[5] Borrowing rate may be expected cost of borrowing or cost of capital.

[6] Lending rate may be expected return on investment or expected lending rate.

[7] To understand these steps, see the illustration No. 5.1 and 5.3 given in Chapter 5.

[8] See illustration 7.3 given in Chapter 7.

[9] See illustration 8.2 given in Chapter 8.

[10] Certain typical series and application of trueIRR technique to these series are given in Chapter 10.