AGEC 608 Lecture 06, p. 1

AGEC 608: Lecture 6• Objective: Outline approach for discounting benefits

and costs that accrue at different points in time

• Readings: – Boardman, Chapter 6

• Homework #2: Chapter 3, problem 1Chapter 3, problem 2Chapter 4, problem 3due: March 13

• Homework #3: Chapter 4, problem 2Chapter 5, problem 1Chapter 6, problem 4due: March 27

AGEC 608 Lecture 06, p. 2

BCA in a timeless world

Dam construction

Costs:

Materials = $500,000

Labor = $600,000

Total Cost = $1,100,000

AGEC 608 Lecture 06, p. 3

BCA in a timeless world

Dam construction

Benefits:

Recreation = $400,000

Flood control = $300,000

Electricity = $500,000

Total Benefit =$1,200,000

AGEC 608 Lecture 06, p. 4

BCA in a timeless world

Dam construction

Total Benefit =$1,200,000

Total Cost = 1,100,000

Net Benefit = 100,000

Benefit exceeds cost, so dam appears to be a good investment

AGEC 608 Lecture 06, p. 5

Undiscounted Measures of Project Worth

• Ranking by inspection – – Sometimes it is possible to ascertain whether a

project is worthwhile simply by looking at the flows, or to compare candidate projects.

• Payback period– Defined as the length of time from the beginning of

the project until the net returns equals the value of the initial investment.

AGEC 608 Lecture 06, p. 6

Undiscounted Measures ofProject Worth

• Proceeds per unit of outlay – Total net value of the incremental output

divided by the total investment.

• Average annual proceeds per unit of outlay – Compute total net value of incremental output, divide this by

the number of years of production, and then divide the resulting figure by the capital investment.

• Average income on book value of investment– Ratio of the average income to the book value of assets (the

value after subtracting depreciation) stated in percentage terms.

AGEC 608 Lecture 06, p. 7

Time and Discounting

Often the benefits and costs of a project accrue at different times. The technique used to deal with this issue is discounting.

AGEC 608 Lecture 06, p. 8

Discounting

Discounting is a technique used to convert all benefits and costs to a common point in time, usually the present.

The value of a project, expressed in terms of the present, is called the Present Value.

AGEC 608 Lecture 06, p. 9

Discounting

Discounting is based on the premise that a dollar of benefit received today is worth more than a dollar of benefit received in the future.

The bias arises because current resources can be invested.

Discounting is the opposite of compounding.

AGEC 608 Lecture 06, p. 10

Discounting

The rate at which a current value is compounded is called the interest rate.

The rate at which a future value is discounted is called the discount rate.

AGEC 608 Lecture 06, p. 11

Interest Formulas

1) Present value of a variable time stream

T

t tr

tB

PV1 1

T

t tr

tB

PV1 5.01

r

trertetB

t

t

rteT

ttBPV

)1(T

1t11

a) Discrete end of year:

b) Discrete mid-year:

c) Continuous:

AGEC 608 Lecture 06, p. 12

Interest Formulas

2) Compound interest

rtPVeFV

a) Discrete annual compounding: 1 trPVFV

b) Discrete monthly compounding:tr

PVFV12

121

c) Continuous compounding:

AGEC 608 Lecture 06, p. 13

Interest Formulas

3) Annuity whose present value is PV (capital recovery factor):

tr

rPVA

)1(1

1

As t approaches infinity, this formula becomes rPVA

AGEC 608 Lecture 06, p. 14

Computing a present value

PV = Pt / (1 + r)t

PV = present value

Pt = value at time t

r = interest (discount) rate

t = year in which Pt is realized

AGEC 608 Lecture 06, p. 15

BCA with discounting

Dam revisited

Total Benefits accrue when dam is finished (t = 1)

Total Costs accrue at start of construction (t = 0)

Discount rate = 10% Should the dam be built?

AGEC 608 Lecture 06, p. 16

Total Benefits accrue when dam is finished (t = 1), soPt = $1,200,000 and PV of benefit is:

$1,200,000 / (1+0.10)1 = $1,090,909

Total Costs accrue at start of construction (t = 0), so Pt = $1,100,000 and PV of benefit is:

$1,100,000 / (1+0.10)0 = $1,100,000

PV(B) < PV(C) The dam should not be built.

Dam construction revisited

AGEC 608 Lecture 06, p. 17

Total Benefits accrue in the future (i.e. when dam is finished). The process of discounting reduces the value of those benefits because they occur in the future.

Why the reversal?

Because the merit of a project can hinge on the choice of discount rate, it can be a source of debate.

There is no simple rule for choosing a discount rate. Often a “well known” interest rate is used.

AGEC 608 Lecture 06, p. 18

Discounted Measures of Project Worth

T

tt

tT

tt

tT

tt

t

r

C

r

B

r

NBNPV

111 )1()1()1(1) Net present value (NPV)

2) Internal rate of return - the rateof interest that equates NPV to 0.

IRRii

NBT

ttt

1 )1(0

Acceptance criterion: NPV > 0

Acceptance criterion: IRR > social discount rate

AGEC 608 Lecture 06, p. 19

Discounted Measures of Project Worth

3) Benefit-cost ratioThe ratio of discounted

benefits to discounted costs

T

tt

tT

tt

t

r

C

r

BCB

11 )1()1(

Acceptance criterion: 1CB