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8/13/2019 LectureIII

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Lecture III: Finish Discounted Value Formulation

I. Internal Rate of ReturnA. Formally defined: Internal Rate of Return is that interest rate which

reduces the net present value of an investment to zero.

1. Finding the internal rate of return : The solution of the internal rate ofreturn does not analytically exist. Methods for finding the internalrate of return are, thus, numerical search techniques or iterative

processes.a. Simple approach: Numeric gradient

(i.) Compute the NPV of an investment at two points, or fortwo interest rates. {The solution technique works best ifone point has a positive NPV and the other a negative

NPV}.(ii.) Compute the line between the two points and find where

the line equals zero. Assume i2>i1 and NPV 1>NPV 2.

1 23 1 1

1 2

1 23 2 2

1 2

or

PV NPV i i ii i

PV NPV i i i

i i

=

=

NPV

i

i2

i1

i3

(iii.) Decide whether the NPV is close enough to zero-Stop if

yes, go back to step (ii) if no. {If the NPV at i3 is positive-replace i1, if it is negative, replace i2}.


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AEB 6145 Lecture IIIProfessor Charles B. Moss

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(iv.) Problems: The procedure does not have a goodconvergence {it may take a long time}. It may bedifficult to find a good starting place-you need twointerest rates that bound the zero (one positive NPV andanother with a negative NPV).

(v.) Advantage: It is robust- it will always work. b. Newtons Method( )( )1

k

k k k

i i

NPV ii i

NPV ii

+

=

=

(i.) For a reference to Newtons method see Burington, R. S. Handbook of Mathematical Tables and Formulas FifthEdition (New York: McGraw-Hill Book Company,1973): 189. See also my lecture notes on NewtonsMethod from AEB 6533.

http://128.227.113.61/chuck/aeb6533.mathprogramming/lecture9.pdf .

( )( )

( )( )

( )

0

11

11

1

1

1

N t t

t

N t t

t

N t

t t

NCF NPV i

i

NPV i CF t

i i

tNCF

i

=

=

=

=+

=

+

= +

(ii.) Choose an initial i.(iii.) Apply the formula to compute a new i.(iv.) Check to see if the NPV is close to zero. If no reiterate, if

yes stop.(v.) This technique is similar to the first, except an analytical

slope is substituted for the numerical slope.


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NPV

ii2i1 i3

(vi.) Comments: This method is relatively quick and is robust

if the net cash flows are smooth. Unfortunately, it is possible to stick the algorithm and it may break down if NPV has multiple optimum.

B. Why is it wrong to IRR.1. Remember that the IRR is simply the i that reduces the NPV to zero.

I want to show that:a. Under certain conditions IRR and NPV yield the same results, i.e.

in the case of conventional investments. b. I can show that under certain conditions IRR yields an inferior

investment decision that NPV correctly identifies.c. Thus, NPV dominates IRR as an investment criteria.

2. Assume that Big Green manufacturing comes out with a lease systemthat charges lease payments for 3 years and then requires the investorto purchase the machine. This machine generates revenue with cashflows. The total cash flows for the investment are presented in table1.

Table 1. Cash Flows for Big GreenMachinery Investment

Year Cash Flow0 7501 7502 1,0003 1,2504 -15,000

The IRR for this investment is .7014 and the NPV at a discount rate of5% is 898.14. Next, I want to generate the unrecovered balance for


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the investment in each year. The unrecovered balance is defined likea savings account. Departing from the investment example in table 1,consider an investment with a normal cash flow pattern as depicted intable 2.

Table 2. Unrecovered Balance for

Investment with Normal CashFlow Patterm

YearCashFlow

Intereston

BalanceUnrecovered

Balance0-15000.00 -15000.001 5000.00 -750.00 -10750.002 5000.00 -537.50 -6287.503 5000.00 -314.38 -1601.884 5000.00 -80.09 3318.03

As presented in table 2, the year 0 cash flow for this investment is 15,000 (corresponding with an initial investment). At the end of year0 (beginning of year 1) this negative cash flow would have accrued aninterest chare of 750. Adding the positive cash flow in period 1(5,000) to the interest charge and the unrecovered balance of 15,000yields an unrecovered balance at the beginning of period 1 of 10,750.Applying this technique to cash flows in table 1, but using two interestrates (5% as in NPV and .7014 as in the IRR).

Table 3. Unrecovered Balance Using Two Interest Rates.

Year Cash FlowUnrecovered

Balance .7014UnrecoveredBalance .05

0 750 1,276 7881 750 3,447 1,6142 1,000 7,566 2,7453 1,250 15,000 4,1954 -15,000 0 -10,805

The higher the ending balloon payment, the higher the IRR and thelower the NPV also the more undesirable the investment.

3. There exist certain legitimate investment questions for which the IRRdoes not exist.a. For the IRR to exist both positive and negative cash flows must

occur. However, several important and relevant investments mayexist that have all negative or all positive cash flows.

b. Examples:(i.) All negative cash flows: Suppose a farmer is evaluating

the purchase of two tractors. Also assume that he is notcurrently constrained in tractor time so that the additionalrevenue of either tractor is zero. The farmer wants atechnique that chooses the tractor with the least costthrough time.


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Table 4. Comparison of Tractor Costs

Year Tractor I Tractor II0 -15,000 -10,0001 -250 -100

2 -250 -2003 -700 -1,0004 -1,250 -1,5005 -1,500 -2,000

NPV at 5% -18,950 -14,800(ii.) No negative cash flows and the extremely profitable

investment.Table 5. No Negative Cash Flows and

an Extremely ProfitableInvestment

Year No Negative

Cash Flows

Extremely

Profitable0 0 -1001 100 2002 200 4003 300 6004 400 800

IRR 8 268.63 NPV 864.88 1,629.75

C. Conclusions1. IRR is defined as the discount rate that reduces the NPV of an

investment to zero.2. Problems:

a. It does not handle mixed investment flows appropriately. b. It may not exit or may have several IRRs.

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