Rate of Return Analysis

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CHAPTER 7

• Rate of Return Analysis

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Internal Rate of Return (IRR)

• By definition
• IRR is the interest rate that makes the NPW of
  the investment equal to zero.
• IRR is the interest rate earned on the unpaid
  balance of an amortized loan.
• IRR is the interest rate charged on the uncovered
  project balance of the investment such that, when
  the project terminates, the uncovered project
  balance will be zero.

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Rate of Return Analysis

• Suppose you have the following cash flow stream.
  You invest 700, and then receive 100, 175,
  250, and 325 at the end of years 1, 2, 3 and 4
  respectively. You ask yourself if this is a good
  investment.
• One way to answer the question is to determine
  the implicit rate of return you receive for your
  700. You find i to satisfy 
• 700 100/(1i) 175/(1i)2 250/(1i)3
  325/(1i)4. 
• That is, you find the interest rate for which
  the present worth of the future CFS
  (100,175,250,325) is equal to 700. It turns out
  that i 6.09107 .

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Internal Rate of Return

• Motivating Example. Banks 1 and 2 offer you the
  following Deals 1 and 2 respectively (draw CF
  charts )
• Deal 1. Invest 2,000 today. At the end of
  years 1, 2, and 3 get 100, 100, and 500 in
  interest at the end of year 4, get 2,200 in
  principal and interest.
• Deal 2 Invest 2,000 today. At the end of years
  1, 2, and 3 get 100, 100, and 100 in interest
  at the end of year 4, get 2,000 in principal
  only.
• Question. Which deal is best? ( interest is
  unknown, use different interest to compare?
  sensitivity study?)

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Internal Rate of Return

• One approach
• Deal 1 Find out the implicit interest rate you
  would be receiving that is, solve for
• 2000 100/(1i)1 100/(1i)2 500/(1i)3
  2200/(1i)4
• Solution i 10.7844 . This is the interest
  rate for the PV of your payments to be 2,000.
• Deal 2 We find i for which
• 2000 100/(1i)1 100/(1i)2 100/(1i)3
  2000/(1i)4
• Solution i 3.8194.
• Which deal would you prefer?
• Would either deal be attractive?

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Internal Rate of Return

• Internal Rate of Return. Let (x0,x1, , xn) be a
  cash flow stream. The internal rate of return
  (IRR) of this stream is a number i satisfying 
• 0 x0 x1 /(1i)1 x2 /(1i)2 xn/(1i)n
• Remark. We use the word internal because the
  rate of return depends only on the stream. It is
  not defined with reference to the financial
  world. It is the rate for which the PV of the
  CFS is zero.
• Remark. Excel has a function to compute IRR.

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Internal Rate of Return

• Main Result for IRR. Suppose the cash flow
  stream (x0,x1, , xn) has x0 lt 0, and xk ? 0 for
  k 1, , n with at least one of x1, , xn
  positive. Then there is a unique positive root
  to the equation
• f(c) x0 x1 c x2 c2 xn cn
• Further, if x0 xn gt 0 (the total return
  then exceeds the initial investment, x1 xn
  gt -x0), then the corresponding IRR, i (1/c) 1
  is positive. ( draw a graph and explain )
• The example illustrates all the ideas in the
  first conclusion.  
• For the second conclusion, note
• f(1) x0 x1 1 x2 12 xn 1n x0 x1
  x2 xn gt 0.
• This means the root c0 for which f(c0) 0
  satisfies c0 lt 1, since f is strictly increasing
  and continuous. Since the root i0 satisfies c0
  1/(1i0), 1 gt c0 1/(1i0) gives i0 gt 0

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Internal Rate of Return

• Remark. One can make up cash flow streams where
  no positive real root exists, or even where roots
  are complex numbers. To do so you must violate
  the hypotheses of the IRR Main Result. These
  hypotheses are usually reasonable.
• Note. Some entries in a CFS can be 0.
• Example. Buy a Merrill Lynch Ready Assets Trust
  for 5,000. Sell it two years later for 5926.
  Then
• CFS (-5000,0,5926), and IRR 8.8669.
• Critical Observation. The internal rate of
  return is the interest rate at which the benefits
  are equal to the costs.

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Calculating Rate of Return

• To calculate a rate of return on an investment,
  we convert the consequences of the investment
  into a CFS. Then we use the CFS to solve for the
  unknown value of i, say i, which is the internal
  rate of return. Five equivalent forms of the
  cash flow equation are
•  PW of benefits PW of costs 0
• (PW of benefits)/(PW of costs) 1 
• Net Present Worth 0 
• EUAB EUAC 0 
• PW of costs PW of benefits.

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Calculating Rate of Return

• These five equations represent the same concept
  in different forms. They all relate costs and
  benefits with i as the only unknown.
• Ways to find the IRR
•   Compound Interest Tables
•  Interest Tables and Interpolation
•  Graphically (see the graph of f(c))
•  Numerically (Excels IRR, or other root finding
  methods).

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Calculating Rate of Return

• Remark. The graph of NPW versus i is typical of
  a CFS representing an investment (-P) followed by
  benefits (positive) from the investment. Such a
  graph will have the same general form as below.
  It will decrease at a decreasing rate and have a
  value 0 at some unique value i. Where the graph
  has a value 0 defines the IRR.

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Calculating Rate of Return

• Remark. If we have a CFS with borrowed money
  involved, e.g., (P,-A,-A,-A), the NPW plot would
  be flipped around and would look something like
  the following one
• Remark Linear interpolation gives an
  overestimation of IRR.

NPW
i
i
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Internal Rate of Return

• Interest Convention. If a lender says she is
  receiving 11, it might seem reasonable to you to
  say that the borrower is faced with -11
  interest. This is not the way interest is
  discussed. 
• Interest is referred to in absolute terms without
  associating a positive or negative sign with it.
  A banker might say she pays 5 interest on
  savings accounts, and charges 11 on personal
  loans no signs are associated with the rates.
• We implicitly recognize interest as a charge for
  the use of someone elses money and as a receipt
  for letting others use our money. In determining
  the interest rate in a particular situation, we
  solve for a single unsigned value of it. We then
  view this value in the customary way, either as a
  
•  charge for borrowing money, or
•   as a receipt for lending money.

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Rate of Return Analysis

• Example statements about a project 
• a) The net present worth of the project is
  32,000. 
• b) The equivalent uniform annual benefit is
  2,800.
•   c) The project will produce a 23 rate of
  return.
• The third statement is perhaps most widely
  understood.
• Rate of return analysis is probably the most
  frequently used exact analysis technique in
  industry. Its major advantage is that it
  provides a figure of merit that is readily
  understood.

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Rate of Return Analysis

• Rate of return analysis has another advantage.
  With NPW or EUAB one must choose an interest rate
  for use in the calculations. This choice may
  possibly be difficult or controversial. With RR
  analysis no (exterior) interest rate is
  introduced into the calculations. Instead, we
  compute a RR from the CFS.
• Warning. Relying only on RR is not always a good
  idea.

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?CFS Analysis
1. We have two CFSs. 2. Number them CFS1 and
CFS2, with CFS1 having the largest year 0 cost
(in absolute value). 3. Compute ?CFS CFS1
CFS2. (Its year 0 entry must be negative.) 4.
Find the IRR for ?CFS, say ?ROR . 5. If ?ROR ?
MARR, choose CFS1. If not, choose CFS2. For
example, the CFSs (-10,15) and (-20,28) would be
numbered CFS2 and CFS1 respectively. We have ?CFS
(-10,13) and ?ROR 30. MARR 6, so ?ROR gt
MARR and we choose CFS1 (-20,28).  
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?CFS Analysis

• More generally, suppose you must choose between
  projects A or B. We can rewrite the CFS for B as
• B A (B A).
• Thus B has two CFS components (1) the same CFS
  as A and (2) the incremental component (B A).
  Thus the only situation in which B is preferred
  to A is when the RR on (BA) exceeds the MARR.
  Thus, to choose between B and A, RR analysis is
  done by computing the IRR on the incremental
  investment (B-A) between the projects.

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?CFS Analysis

• Since we want to consider increments of
  investment, we compute the cash flow for the
  difference between the projects by subtracting
  the cash flow for the lower investment-cost
  project (A) from that of the higher
  investment-cost project (B).
• In summary, we compute the CFS for the difference
  between the projects by subtracting the cash flow
  for the lower investment-cost project (A) from
  that of the higher investment-cost project (B).
  Then, the decision rule is as follows
• IF IRRB-A gt MARR, select B
• IF IRRB-A MARR, select either A or B
• IF IRRB-A lt MARR, select A.
• Here, B-A is an investment increment.