Module 6 IRR and Payback Period

Module 6: Internal Rate of Return and Payback Period

SI-4251 Ekonomi TeknikMuhamad Abduh, Ph.D.

Outline Module 6 Rate of Return Internal Rate of Return (IRR) Cash Flow with Single Rate of Return Cash Flow with Multiple Rate of Return Payback Period Discounted Payback Period

Muhamad Abduh, Ph.D.6-2 SI-4251 Ekonomi Teknik

Rate of Return Rate of Return (ROR) is the rate of interest

paid on all unpaid balance of borrowed money, or the rate of interest earned on the unrecovered balance of an investment so that the final payment or receipt brings the balance to zero with interest considered

Internal Rate of Return (IRR) is the interest rate that causes the equivalent receipts of a cash flow equal to the equivalent payments of a cash flow


i* PR = PP

EUAWR = EUAWP

FR = FP

PR – PP = 0FR – FP = 0EUAWR – EUAWP = 0

Finding Internal Rate of Return1. Draw cash flow diagram2. Convert all receipts into present, future or EUAW3. Convert all payments into present, future or EUAW4. Subtract (3) from (2) and set it equal to zero find i*5. Find i* that satisfies (4) by trial and error interpolation


PR= I1(P/F, i*, 22) + SV(P/F, i*, 36)

0 1 2 3

A1

SVI1P O1

PP= P + A1(P/A, i*, 36) + O1(P/F, i*, 8)

PR - PP = I1(P/F, i*, 22) + SV(P/F, i*, 36) – [P + A1(P/A, i*, 36) + O1(P/F, i*, 8)] = 0

Rate of Return The relationship between Rate of Return and the Present Worth

Amount can be graphically describe as follows:

Cash flow shown above assures a single rate of return that:


interest rate, i

Present worth,

PW

0i*

PW (i) > 0 for i < i*PW (i) = 0 for i = i*PW (i) < 0 for i > i*

Cash Flow with Single Rate of ReturnFor a cash flow will have a single rate of return, if satisfies the following conditions:Test # 1:1.F0 < 0

(the first non-zero cash is a disbursement)

2.One change in sign in the sequence F0, F1, F2, … Fn (the cash flow has an initial disbursement or a series of disbursement followed by a series of receipts)

3.PW(0) > 0 (the sum of all the receipts is greater than the sum of all the disbursements)

Test # 2:1.F0 < 0

(the first non-zero cash is a disbursement)

2.Find rate of return, i*, for the cash flow; for unknown i*, total unrecovered balance, Ut < 0 for t = 0, 1, 2, 3, … n-1


Cash Flow with Multiple Rate of ReturnFor a cash flow will have a multiple rate of return, if not satisfies Test #1 Test #2


interest rate, i

Present worth,

PW

0i* i*

Payback Period In business and industry it is common to evaluate alternatives (assets) in terms of

their payback or payout period. Payback without interest the length of time required to recover the first cost of

an investment from the net cash flow produce by that investment for interest rate of zero

This method has disadvantages as it fails to consider: the time value of money the consequences of the investment following the payback period

Payback with interest or discounted payback period this method determines the length of time required until the investment’s equivalent receipts exceed the equivalent capital outlays


'n

t

tt iF

0

01

n

ttF

0

0

n’ = the smallest value that satisfies the equation

Payback Period without InterestExample:Three alternatives of investment are being considered without interest:


end of year

A B C

0 - 1.000.000,- - 1.000.000,- - 700.000,-1 500.000,- 200.000,- - 300.000,-2 300.000,- 300.000,- 500.000,-3 200.000,- 500.000,- 500.000,-4 200.000,- 1.000.000,- 0,-5 200.000,- 2.000.000,- 0,-6 200.000,- 4.000.000,- 0,-

Present worth, i = 0

PW(i)A = 600.000,-

PW(i)B = 7.000.000,-

PW(i)C = 0,-

Payback period

3 years 3 years 3 years

Discounted Payback PeriodExample:Alternative A is being considered at the rate of return 15%


end of year

A

0 - 1.000.000,-1 500.000,-2 300.000,-3 200.000,-4 200.000,-5 200.000,-6 200.000,-

Ft = - 1M + 500K(P/F, 15%, 1) + 300K(P/F, 15%, 2) + 200K(P/F, 15%, 3) + 200K(P/F, 15%, 4) + 200K(P/F, 15%, 5) > 0

Project Balance Evaluation of a proposed project is often done by analyzing the project balance. A project balance describes the equivalent of loss or profit of the project cash flow

as a function of time. At any given time, a project balance can be calculated as:

Visual description of a project balance project balance diagram Four important characteristics of project balance diagram are:

The net future worth of investment The time when the equivalent cash flow switch from negative to non-negative, or vice

versa The net equivalent cash flow exposed to risk of loss (the area where PB(i) is negative) The net equivalent cash flow earned (the area where the PB(i) is positive)


1

0

)1()(

TT

ttT iFiPB

Project Balance


0 1 2 3 4 5

10M

1M5M

8M

6M

3M

+-

PB(20)1= -11M

PB(20)0 = -10M

PB(20)2= -8.2M

PB(20)3= -1.84M

PB(20)4= 3.79M

PB(20)5= 7.55M

Project cash flow @ 20% discount rate

Homework #61. Work for alternative B and C of the example

(slide #8). Determine the discounted payback period for each alternative and suggest your selection.

2. For the same problem investment (#1), calculate the project balance for each alternative, draw the project balance diagram, and identify the future worth and exposure to risk of loss


Module 6 IRR and Payback Period

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