TrueIRR IRR and NPV redefined

1
TrueIRRIRR and NPV redefined

• Chapters
• Introduction
• Cumulative Future value
• Classification of series
• Combination series and IRR
• TrueIRR Technique
• Margin Value concept
• Application of TrueIRR
• Reinvestment assumptions in IRR and MIRR
• TrueNPV

2
Chapter 1Introduction

• Limitations of IRR and NPV
• Illustration on Multiple IRRs
• Reasons for multiple IRRs

3
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 IRRs
• Reinvestment 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.

4
2. Illustration of Multiple IRRs

• Illustration 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.

5
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 value
• Classification of cash flow series

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Chapter 2 Cumulative future value

• Cumulative future value (CFV)
• Illustration on CFV
• Net future value (NFV)
• Finding IRR using NFV
• Lending Value and Borrowing Value

7
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.

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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
  expressed
• The 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.
• For i 0, V0 C0
• For i 1 to n, Vi Vi-1 (1 r) Ci
• Where, i Period, V CFV, C cash flow ,
  r Interest Rate, n Terminal Period

9
2. Illustration on CFV
Interest Rate 10

• 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.

10
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(1r)n
• If 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.

11
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 Rate10
• NFV84

12
4. Finding IRR using NFV (Contd.)

• Step 2 Now let us increase the interest rate
  to 12.
• Interest Rate12
• NFV-106
• Table 2.5
• Here, 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 -106

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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.

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Chapter 3Classification of series

• Lending series
• Borrowing series
• Combination series
• Steps to identify the type of series

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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).

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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).

17
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.

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4. steps to identify the type of series

• Calculate 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 lt gt 0 and TBV lt gt 0, then combination
  series. Further,
• If the TLV gt TBV, then combination series
  (lending)
• If the TBV gt TLV, then combination series
  (borrowing).

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Chapter 4Combination series and IRR

• Interpretation of IRR of Combination series
• Implicit Assumption in the case of IRR
• New approach to Combination series

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1. Interpretation of IRR of Combination series

• In 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?

21
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.

22
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.

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Chapter 5TrueIRR Technique

• TrueIRR and combination series
• Steps in application of TrueIRR to Combination
  series
• Combination series (lending) TrueIRR
  -illustration
• Bifurcation of Combination series

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1 TrueIRR and combination series

• 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.

25
2. Steps in application of TrueIRR 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
  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.

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3. Combination series (lending) TrueIRR
-illustration

• Illustration 6 Let us calculate IRL of a cash
  flow assuming 10 borrowing rate under TrueIRR
  technique
• Here, TLV gt TBV. Therefore this is a combination
  series (lending). Its IRL is 7.34 p.a. at 10
  borrowing rate.

27
4. 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
• Ci Vi Vi-1 (1B)
• 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 (1L)
• Where, C cash flow, i Period,V lending
  value,
• L lending rate

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Chapter 6Margin Value concept

• Margin Value
• Relationship between IRR and NPV

29
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

30
2. 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.

31
Chapter 7Applications of TrueIRR

• Combination series and TrueIRR
• Comparison of proposals

32
1. 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.

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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.

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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

35
Chapter 8Reinvestment Assumptions in IRR and
MIRR

• Reinvestment Assumption in IRR and NPV
• Reinvestment Assumption in IRR and NPV is a myth
• Comparison of IRR and MIRR formulae

36
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.

37
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.

38
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.

39
Chapter 9Combination series and NPV

• Interpretation of NPV
• TrueNPV
• Finding TrueNPV
• Combination series and TrueNPV

40
1. Interpretation of NPV

• NPV 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?

41
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.

42
3. Finding 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.

43
4. Combination series 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.
• 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.