Irr(1)

Topics Covered

Competitors to NPV The Payback Period The Book Rate of Return Internal Rate of Return

ALSO Consider Capital Rationing

Payback The payback period of a project is the number

of years it takes before the cumulative forecasted cash flow equals the initial outlay.

The payback rule says only accept projects that “payback” in the desired time frame.

This method is very flawed, primarily because it ignores later year cash flows and the the present value of future cash flows.

PaybackExample

Examine the three projects and note the mistake we would make if we insisted on only taking projects with a payback period of 2 years or less.

050018002000-C

018005002000-B

50005005002000-A

10% @NPVPeriod

PaybackCCCCProject 3210

PaybackExample

Examine the three projects and note the mistake we would make if we insisted on only taking projects with a payback period of 2 years or less.

502050018002000-C

58-2018005002000-B

2,624350005005002000-A

10% @NPVPeriod

PaybackCCCCProject 3210

Book Rate of ReturnBook Rate of Return - Average income divided by average book value over project life. Also called

accounting rate of return.

assetsbook

incomebook return of rateBook

The book rate of return It does recognise that capital is required to

earn income. However it has shortcomings as an

investment appraisal method It does not have a hurdle rate It does not recognise the time value of money Ambiguity regarding its definition.

The Internal Rate of Return Like the NPV this is a discounted cash flow

criterion. In the single period case where we are merely interested in accepting or rejecting a project it is exactly equivalent to the NPV rule. However in all other situations congruence of the IRR and NPV rules cannot be guaranteed. When a conflict does occur the NPV gives the correct answers.

Definition of IRR The IRR is the discount rate that when used to

discount a projects cash flow gives an NPV of zero.

IRR in Single Period Case

IRRC

C1

0

1

C

C-C = = IRR

C = )+(1C

+1C = C

0 = NPV IFi+1

C+C- = NPV

0

01

10

10

10

Proof

Example Cost of project 1000 at time 0 Payoff in time 1 is 1400 IRR = (1400/1000) – 1 = 40%

010001

1400

IRR

IRR Decision Rule Accept all projects with a rate of return that is

greater than the cost of capital.

IRR versus NPVFor accept/reject decisions in a single period

case the NPV and IRR decision rules give exactly the same answer.

For ranking decisions conflicts may occur

Structure of Analysis Single Period Case

Accept / Reject Decisions Ranking Decisions

Multi-period Case Accept / Reject Decisions Ranking Decisions

Project ACash Flows

Project BCash Flows

t0 (10) (100)

t1 20 130

IRR 100% 30%

NPV @ 10% (10) + 20/1.1 = 8.2

-100 + 130/1.1 =18.2

Single Period: Mutually Exclusive Projects – Differing Scales

Internal Rate of Return

IRR ignores the magnitude of the project. The following two projects illustrate that problem.

818,1175000,35000,20F

182,8100000,20000,10E

%10@NPVIRRCCProject t0

IRR in Multi-period Case The IRR generally gives the same answer in

accept/reject decisions. However, there may be some technical

difficulties with the IRR. First we will examine how to compute the

IRR in the multiperiod case.


Example

You can purchase a turbo powered machine tool gadget for $4,000. The investment will generate $2,000 and $4,000 in cash flows for two years, respectively. What is the IRR on this investment?


Example


0)1(

000,4

)1(

000,2000,4

21

IRRIRRNPV


Example


0)1(

000,4

)1(

000,2000,4

21

IRRIRRNPV

%08.28IRR


-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

10 20 30 40 50 60 70 80 90 100

Discount rate (%)

NP

V (

,000

s)

IRR=28%

IRR computation with Excel ® See IRRDR.xls

Multi-period Computation of IRR without a computer 1. Calculate the NPV at a high discount rate so that it is < 0. 2. Calculate the NPV at a low discount rate so that it is > 0.

(e.g. use 0%) 3. Divide the difference between the positive NPV and the

negative NPV by the change in the discount rate to get the approximate change in NPV for a one- percent change in the discount rate.

4. Using the information in 3 compute the required increase (decrease) in the rate to reduce (increase) the positive (negative) NPV to zero.

5. Use the rate computed in 4 to discount the cash flows. If the NPV is not equal to zero then alter your estimate of the IRR accordingly.

Calculation of IRR in multi-period caseTIME 0 1 2 3

CASH

FLOW

-350 110 121 200

The NPV @ 0% = 81The NPV @ 20% = (59)

Thus there is a spread of 140(7) in the NPV for a spread of 20%(1%) in the discount rate.

IRR Calculation Continued On the basis of this information we should

reduce the discount rate by (59/7)% = 8.43% from 20%.

If we try a rate of 12% the NPV = (13). Therefore we have to reduce by a further 2% or so to get NPV = 0.

Discounting at 10% gives an NPV of 0. Hence the IRR is 10%.


Accept / Reject Decisions: Technical Problem - Multiple Rates of Return

Certain cash flows can generate NPV=0 at two different discount rates.

The following cash flow generates NPV=0 at both (-50%) and 15.2%.

150150150150150800000,1

CCCCCCC 6543210


Pitfall 2 - Multiple Rates of Return Certain cash flows can generate NPV=0 at two different discount rates.

The following cash flow generates NPV=0 at both (-50%) and 15.2%. 1000

NPV

500

0

-500

-1000

Discount Rate

IRR=15.2%

IRR=-50%

Multiple IRRs

A project has as many IRRs as it has changes in the sign of its cash flows.

Number of IRRS

0 1 2 3

1 (100) 200 300 400

3 (100) 200 (100) 500


Mutually Exclusive Projects

Internal Rate of ReturnReinvestment rate assumption There is an implicit assumption that all intermediate cash inflows are reinvested at

the IRR It makes far more sense to assume that the intermediate cash flows are reinvested at

the opportunity cost of capital. We assume that discount rates are stable during the term of the project.

Profitability Index Profitability Index is PV/C0 The rule is to accept all projects which have a

PI > 1. One gets the same answers as the NPV for

accept/reject decisions. For ranking decisions conflicts arise in a

manner similar to the IRR case e.g. differences in the scale of the project.

The Profitability Index and Capital Rationing There is one case where the Profitability

Index is superior to the NPV. This is where the firm faces a limit on the amount it can invest in a single year and projects are divisible and there is no postponement. In this special case one should maximise the NPV per £1 invested.

Profitability Index When resources are limited, the profitability

index (PI) provides a tool for selecting among various project combinations and alternatives

A set of limited resources and projects can yield various combinations.

Project appraisal: capital rationing,

•Coping with investment appraisal in an environment of capital rationing, taxation and inflation

•More specifically:

– Explain why capital rationing exists and be able to use the profitability ratio in one-period rationing situations

Capital rationing

•Capital rationing occurs when funds are not available to finance all wealth-enhancing projects

•Soft rationing

•Hard rationing

•One-period capital rationing

– 1 Divisible projects

– 2 Indivisible projects

One-period capital rationing with divisible projects

•Capital at time zero has been rationed to £4.5m

Bigtasks plc (continued)

Ranking according to absolute NPV

Present value Profitability index = –––––––––––––––– Initial outlay

Net present value Benefit–cost ratio = –––––––––––––– Initial outlay

Bigtasks plc: Profitability indices and benefit–cost ratios

Bigtasks plc: Ranking according to the highest profitability index

Indivisible projects

NPV – the pros NPV is the theoretically correct criteria for

making investment decisions when maximisation of shareholder wealth is the objective

It recognises the time value of money It forces managers to consider their

projections carefully when estimating future cash flows

NPV – the pros continued It is generally easy to use It has a clear decision rule It can deal with multiple discount rates –

unlike IRR It is not affected by differences in scale –

unlike IRR and Profitability Index

NPV – the cons It does not help find value creating projects It does not lend itself to ex-post evaluation of

managers in a straightforward manner. But residual income can help here.

NPV vs. IRR

Single-Period

Case

Multi-Period

Case

Accept/Reject Always

Consistent

Can be technical

Problems with IRR

Non Independent

Projects (e.g. Ranking)

Conflicts Conflicts

IRR – the pros It is theoretically correct for accept/reject

decsions It recognises the time value of money

IRR – the cons It cannot be relied upon to signal the correct decision

for non-independent projects – e.g. mutually exclusive projects of differing scales.

Can have multiple IRRs Some projects have no IRR The re-investment assumptions of the rule do not

make economic sense

IRR – the cons continued It cannot cope with multiple discount rates In addition it has all the other the drawbacks

of the NPV criterion.

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Irr(1)