Fin650:Project Appraisal
Lecture 3
Essential Formulae in Project Appraisal
1
Benefits and Cost Realized at Different Times



Benefits and costs realized in different times are
not comparable
Some benefits and costs are recurrent, while
some are realized only for a temporary period
Examples: Roads, built now at heavy costs, to
generate benefits later, Dams, entail
environmental costs long after their economic
benefits have lapsed, A life lost now entails cost
for at least as long into the future as the person
would have lived
2
Fundamentals in Financial Evaluation



Money has a time value: a $ or £ or € or Tk. today, is
worth more than a $ or £ or € or Tk. next year.
A risk free interest rate may represent the time value
of money.
Inflation too can create a difference in money value
over time. It is NOT the time value of money. It is a
decline in monetary purchasing power.
3
Moving Money Through Time


Investment projects are long lived, so we usually use
annual interest rates.
With compound interest rates, money moved forward
in time is ‘compounded’, whilst money moved
backward in time is ‘discounted’.
4
Financial Calculations



Time value calculations in capital budgeting usually
assume that interest is annually compounded.
‘Money’ in investment projects is known as ‘cash
flows’: the symbol is:
Ct is Cash flow at end of period t.
5
Financial Calculations
 The present value of a single sum is:
PV = FV (1 + r)-t
the present value of a dollar to be received at the end
of period t, using a discount rate of r.

The present value of series of cash flows is:
PV 
t

1
CFt
(1  r )
t
6
Financial Calculations:
Cash Flow Series
A payment series in which cash flows are Equally
sized and Equally timed is known as an annuity.
There are four types:
1. Ordinary annuities; the cash flows occur at the
end of each time period. (Workbook 5.10 and
5.11)
2. Annuities due; the cash flows occur at the start
of each time period.
3. Deferred annuities; the first cash flow occurs
later than one time period into the future.
(Workbook 5.10 and 5.11)
4. Perpetuities; the cash flows begin at the end of
the first period, and go on forever.
7
Evaluation of Project Cash Flows.




Cash flows occurring within investment projects
are assumed to occur regularly, at the end of
each year.
Since they are unlikely to be equal, they will not
be annuities.
Annuity calculations apply more to loans and
other types of financing.
All future flows are discounted to calculate a Net
Present Value, NPV; or an Internal Rate of
Return, IRR.
8
Decision Making With Cash Flow
Evaluations


If the Net Present Value is positive, then the
project should be accepted. The project will
increase the present wealth of the firm by the
NPV amount.
If the IRR is greater than the required rate of
return, then the project should be accepted. The
IRR is a relative measure, and does not measure
an increase in the firm’s wealth.
9
Calculating NPV and IRR With
Excel -- Basics.
1.
2.
3.
4.
Ensure that the cash flows are recorded with the
correct signs: -$, +$, -Tk, +Tk. etc.
Make sure that the cash flows are evenly timed:
usually at the end of each year.
Enter the discount rate as a percentage, not as a
decimal: e.g. 15.6%, not 0.156.
Check your calculations with a hand held calculator
to ensure that the formulae have been correctly set
up.
10
Calculating NPV and IRR With
Excel -- The Excel Worksheet.
11
Calculating MIRR and PB With
Excel.
 Modified Internal Rate of Return – the cash flow cell
range is the same as in the IRR, but both the required
rate of return, and the re-investment rate, are
entered into the formula: MIRR( B6:E6, B13, B14)
 Payback – there is no Excel formula . The payback
year can be found by inspection of accumulated
annual cash flows.
12
ARR and Other Evaluations With
Excel.


Accounting Rate of Return – there is no Excel formula.
Average the annual accounting income by using the
‘AVERAGE’ function, and divide by the chosen asset
base.
Other financial calculations – use Excel ‘Help’ to find
the appropriate function. Read the help information
carefully, and apply the function to a known problem
before relying on it in a live worksheet.
13
Calculating Financial Functions With
Excel -- Worksheet Errors.
Common worksheet errors are:
 Cash flow cell range wrongly specified.
 Incorrect entry of interest rates.
 Wrong NPV, IRR and MIRR formulae.
 Incorrect cell referencing.
 Mistyped data values.
 No worksheet protection.
14
Calculating Financial Functions
With Excel -- Error Control.
Methods to reduce errors:
 Use Excel audit and tracking tools.
 Test the worksheet with known data.
 Confirm computations by calculator.
 Visually inspect the coding.
 Use a team to audit the spreadsheet.
15
Essential Formulae -- Summary
1.The Time Value of Money is a cornerstone of finance.
2. The amount, direction and timing of cash flows, and relevant
interest rates, must be carefully specified.
3. Knowledge of financial formulae is essential for project
evaluation.
4. NPV and IRR are the primary investment evaluation criteria.
5. Most financial functions can be automated within Excel.
6. Spreadsheet errors are common. Error controls should be
employed.
7.To reduce spreadsheet errors: -document all spreadsheets, keep a
list of authors and a history of changes, use comments to guide
later users and operators.
8. Financial formulae and spreadsheet operation can be demanding.
16
Seek help when in doubt.
Discounting Future Benefits and Costs
Basic Concepts:
A.
Future Value Analysis
In general, the future value in one year of some amount X is given by:
FV= X(1+i)
where i is the annual rate of interest. This is simple compounding
B.
Present Value Analysis
In general, if the prevailing interest rate is i, then the present value
of an amount Y received in one year is given by:
PV 
Y
1 i
Discounting is the opposite of compounding.
17
18
Discounting Future Benefits and Costs
Net Present Value Analysis
The NPV of a project equals the difference between the present
value of benefits, PV(B), and the present value of the costs, PV(C):
NPV = PV(B)-PV(C)
Compounding and Discounting Over Multiple Years
Future value over multiple Years
In general, if an amount, denoted X, is invested for n years and
interest is compounded annually at i percent, then the future value
is:
FV = X(1+i)n
Present value over multiple years
In general, the present value of an amount received in n years,
denoted Y, with interest discounted annually at rate i percent, then
the present value is:
Y
PV 
(1  i ) n
The term 1/(1+i)n is called the discount factor
19
20
Discounting and Alternative Investment Criteria
Basic Concepts:
A.
Discounting

Recognizes time value of money
a. Funds when invested yield a return
b. Future consumption worth less than present
consumption
o
o
PVB = (B /(1+r) +(B 1/(1+r) 1+.…….+(Bn /(1+r) n
r
o
o
o
PVC = (C /(1+r) +(C 1/(1+r) 1+.…….+(Cn /(1+r)
r
o
n
o
NPV = (B o-C o)/(1+r) o+(B 1-C 1)/(1+r) 1+.…….+(B n-C n)/(1+r) n
r
21
Discounting and Alternative Investment Criteria
(Cont’d)
B. Cumulative Values


The calendar year to which all projects are
discounted to is important
All mutually exclusive projects need to be
compared as of same calendar year
1
If NPV r= (B o-C o)(1+r) 1+(B 1-C 1) +..+..+(B n-C n)/(1+r) n-1 and
NPV 3= (B o-Co)(1+r) 3+(B 1-C 1)(1+r) 2+(B 2-C 2)(1+r)+(B 3-C 3)+...(B n-C n)/(1+r) n-3
r
3
1
Then NPV r = (1+r) 2 NPV r
22
Examples of Discounting
Year
Net Cash Flow
0
-1000
1
200
2
300
3
350
4
1440
200 300
350 1440
NPV  1000



 676.25
2
3
4
1.1 (1.1)
(1.1) (1.1)
0
0.1
NPV01.1  1000(1.1)  200
300 350 1440


 743.88
2
3
1.1 (1.1)
(1.1)
350 1440
NPV  1000(1.1)  200(1.1)  300 

 818.26
1
2
(1.1) (1.1)
2
0.1
2
23
Alternative Investment Criteria
1.
2.
3.
4.
Net Present Value (NPV)
Benefit-Cost Ratio (BCR)
Pay-out or Pay-back Period
Internal Rate of Return (IRR)
24
Net Present Value (NPV)
1.
The NPV is the algebraic sum of the discounted values of the
incremental expected positive and negative net cash flows
over a project’s anticipated lifetime.
2.
What does net present value mean?

Measures the change in wealth created by the project.

If this sum is equal to zero, then investors can expect to
recover their incremental investment and to earn a rate of
return on their capital equal to the private cost of funds
used to compute the present values.

Investors would be no further ahead with a zero-NPV
project than they would have been if they had left the
funds in the capital market.

In this case there is no change in wealth.
25
Alternative Investment Criteria
First Criterion: Net Present Value (NPV)
 Use as a decision criterion to answer
following:
a. When to reject projects?
b. Select project (s) under a budget
constraint?
c. Compare mutually exclusive projects?
d. How to choose between highly profitable
mutually exclusive projects with different
lengths of life?
26
Net Present Value Criterion
a. When to Reject Projects?
Rule: “Do not accept any project unless it generates a positive
net present value when discounted by the opportunity cost of
funds”
Examples:
Project A: Present Value Costs $1 million, NPV + $70,000
Project B: Present Value Costs $5 million, NPV - $50,000
Project C: Present Value Costs $2 million, NPV + $100,000
Project D: Present Value Costs $3 million, NPV - $25,000
Result:
Only projects A and C are acceptable. The investor is made
worse off if projects B and D are undertaken.
27
Net Present Value Criterion (Cont’d)
b. When You Have a Budget Constraint?
Rule: “Within the limit of a fixed budget, choose that subset of the
available projects which maximizes the net present value”
Example:
If budget constraint is $4 million and 4 projects with positive
NPV:
Project E:
Costs $1 million, NPV + $60,000
Project F:
Costs $3 million, NPV + $400,000
Project G:
Costs $2 million, NPV + $150,000
Project H:
Costs $2 million, NPV + $225,000
Result:
Combinations FG and FH are impossible, as they cost too much. EG
and EH are within the budget, but are dominated by the
combination EF, which has a total NPV of $460,000. GH is also
possible, but its NPV of $375,000 is not as high as EF.
28
Net Present Value Criterion (Cont’d)
c. When You Need to Compare Mutually Exclusive
Projects?
Rule: “In a situation where there is no budget constraint but a
project must be chosen from mutually exclusive alternatives,
we should always choose the alternative that generates the
largest net present value”
Example:
Assume that we must make a choice between the following
three mutually exclusive projects:
Project I: PV costs $1.0 million, NPV $300,000
Project J: PV costs $4.0 million, NPV $700,000
Projects K: PV costs $1.5 million, NPV $600,000
Result:
Projects J should be chosen because it has the largest NPV.
29
Shortcut Methods for Calculating the Present
Value of Annuities and Perpetuities
1/2
Annuities and Perpetuities
An annuity is an equal, fixed amount received (or paid) each
year for a number of years.
A perpetuity is an annuity that continues indefinitely.
 Present value of an annuity

n
A
PV  
t
(
1

i
)
t 1
or PV = A x
Where
n
n i
i is the annuity factor,
a
a
a
n
1

(
1

i
)
ain 
i
n
The term
, which equals the present value of an annuity of
i
$/Tk. 1 per year for n years when the interest rate is i
percent, is called the annuity factor.
30
Shortcut Methods for Calculating the Present
Value of Annuities and Perpetuities
2/2
Present value of a perpetuity
PV = A/i, if i>0
 Present value of an annuity that grows or declines at
a constant rate
PV(B) = [B1/ (1+g)]x ai0n , i0 = 1-g/1+g
if i>g
If g is small, B1/1+g is approximately equal to B1,
and i0 = 1-g
 Present value of benefits (or costs) that grow or
decline at a constant rate in perpetuity
PV(B) = B1/ (1-g), if i>g

31
Long-Lived Projects and Terminal Values
It is generally assumed that projects have finite economic
Life.
For projects with infinite life, we may calculate NPV using

The formula:
NBt
NPV  
t
(
1

i
)
t 0
Assumes that the net benefits are constant or grow at a constant rate.
Not a very realistic assumption.
For most long lived projects, select a relatively short discounting
period (useful life of the project) and include a terminal value to
reflect all subsequent benefits and costs.
k
Where T(k) denotes the terminal value.
NBt
NPV  
 T (k )
t
t  0 (1  i )
32
Alternative Methods for Estimating Terminal
Values

Terminal Values Based on:




Simple Projections
Salvage or Liquidation Value
Depreciated Value, economic depreciation
Percentage of Initial Constructions Cost
Setting the Terminal Value equal to zero
Note: Accounting depreciation should never be included as
a cost (expense) in CBA

33
Comparing Projects with Different Time Frames

Two Methods for Comparing Projects with Different Time Frames
 Rolling Over the Shorter Project

Comparison between a cogeneration power plan and a hydroelectric
project
Equivalent Annual Net Benefit Method (EANB)
EANB of an alternative equals its NPV divided by the annuity
factor
That has the same life as the project

NPV
EANB  n
ai
Where
ain
is the annuity factor,
n
1

(
1

i
)
ain 
i
34
Real Versus Nominal Currency

Constant currency




Use CPI as the deflator
If benefits and costs are measured in nominal currency, use
nominal discount rate
If benefits and costs are measured in real currency, use real
discount rate
To convert a nominal interest rate i, to a real interest rate, r,
with an expected inflation rate, m, use the following
equation
im
r
1 m
If m is small, the real interest rate is approximately equals the
Nominal interest rate minus the expected rate of inflation:
r = i-m
35
Alternative Investment Criteria: Benefit
Cost Ratio

As its name indicates, the benefit-cost ratio (R),
or what is sometimes referred to as the
profitability index, is the ratio of the PV of the net
cash inflows (or economic benefits) to the PV of
the net cash outflows (or economic costs):
PV of Net Cash Inflows(or EconomicBenefits)
R
PV of Net Cash Outflows (or EconomicCosts)
36
Basic Rule
If benefit-cost ratio (R) >1, then the project
should be undertaken.
Problems?

Sometimes it is not possible to rank projects
with the benefit-cost Ratio

Mutually exclusive projects of different sizes

Not necessarily true that if RA>RB, that
project “A” is better than project “B”
37
Benefit-Cost Ratio (Cont’d)
Problem:The Benefit-Cost Ratio does not adjust for mutually exclusive
projects of different sizes. For example:
Project A:
PV0of Costs = $5.0 M, PV0 of Benefits = $7.0 M
NPVA = $2.0 M
RA = 7/5 = 1.4
Project B:
PV0 of Costs = $20.0 M,
PV0 of Benefits = $24.0
M
NPVB = $4.0 M
RB = 24/20 = 1.2
According to the Benefit-Cost Ratio criterion, project A should be chosen
over project B because RA>RB, but the NPV of project B is greater than
the NPV of project A. So, project B should be chosen
Conclusion: The Benefit-Cost Ratio should not be used to
rank projects
38
Alternative Investment Criteria
Pay-out or Pay-back period



The pay-out period measures the number of years it will
take for the undiscounted net benefits (positive net
cashflows) to repay the investment.
A more sophisticated version of this rule compares the
discounted benefits over a given number of years from
the beginning of the project with the discounted
investment costs.
An arbitrary limit is set on the maximum number of
years allowed and only those investments having
enough benefits to offset all investment costs within this
period will be acceptable.
39
Pay-Out or Pay-Back Period

Projects
with
shortest
payback
period
are
preferred by the criteria

Assumes all benefits that are produced by in
longer life project have an expected value of
zero after the pay-out period.

The criteria may be useful when the project is
subject to high level of political risk.
40
Alternative Investment Criteria
Internal Rate of Return (IRR)

IRR is the discount rate (K) at which the present
value of benefits are just equal to the present
value of costs for the particular project
Bt  Ct
0

t
i  0 (1  k )
t
Note: the IRR is a mathematical concept, not an
economic or financial criterion
41
Common uses of IRR:
(a)
If the IRR is larger than the cost of funds then the
project should be undertaken
(b)
Often the IRR is used to rank mutually exclusive
projects. The highest IRR project should be chosen
(c)
An advantage of the IRR is that it only uses
information from the project
42
Difficulties With the Internal Rate of Return Criterion
First Difficulty: Multiple rates internal rate of return for
Project
Bt - Ct
+300
Time
-100
-200
Solution 1: K = 100%;
NPV= -100 + 300/(1+1) + -200/(1+1)2 = 0
Solution 2: K = 0%;
NPV= -100+300/(1+0)+-200/(1+0)2 = 0
43
Difficulties With The Internal Rate of Return Criterion (Cont’d)
Second difficulty: Projects of different sizes and also strict alternatives
Year
0
Project A -2,000
Project B -20,000
1
2
3
...
...
+600
+4,000
+600
+4,000
+600
+4,000
+600
+4,000
+600
+4,000

+600
+4,000
NPV and IRR provide different Conclusions:
Opportunity
cost of funds = 10%
0
NPV A : 600/0.10 - 2,000 = 6,000 - 2,000 = 4,000
0
B
NPV : 4,000/0.10 - 20,000 = 40,000 - 20,000 = 20,000
0
B
Hence, NPV > NPV
0
A
IRR A : 600/K A - 2,000 = 0 or K A = 0.30
IRR B : 4,000/K B - 20,000 = 0 or K B = 0.20
Hence, K A>K B
44
Difficulties With The Internal Rate of Return Criterion (Cont’d)
Third difficulty:Projects of different lengths of life and strict alternatives
Opportunity cost of funds = 8%
Project A: Investment costs = 1,000 in year 0
Benefits = 3,200 in year 5
Project B: Investment costs = 1,000 in year 0
Benefits = 5,200 in year 10
NPV 0A : -1,000 + 3,200/(1.08)5 = 1,177.86
NPV 0B : -1,000 + 5,200/(1.08)10= 1,408.60
Hence, NPVB0 > NPVA0
IRRA : -1,000 + 3,200/(1+KA)5 = 0 which implies that KA = 0.262
IRRB : -1,000 + 5,200/(1+KB)10 = 0 which implies that KB = 0.179
Hence, KA>KB
45
Difficulties With The Internal Rate of Return Criterion (Cont’d)
Fourth difficulty: Same project but started at different times
Project A: Investment costs = 1,000 in year 0
Benefits = 1,500 in year 1
Project B: Investment costs = 1,000 in year 5
Benefits = 1,600 in year 6
NPV A : -1,000 + 1,500/(1.08) = 388.88
NPV B : -1,000/(1.08) 5 + 1,600/(1.08) 6 = 327.68
0
Hence, NPV A> NPV
0
B
IRR A : -1,000 + 1,500/(1+K A) = 0 which implies that K A = 0.5
IRR B : -1,000/(1+K B) 5+ 1,600/(1+K B) 6= 0 which implies that K B = 0.6
Hence, K B >KA
46
IRR FOR IRREGULAR CASHFLOWS
For Example: Look at a Private BOT Project from the perspective of the
Government
Year 
Project A
IRR A
0
1
2
3
4
1000
1200
800
3600
-8000
3600
-6400
10%
Compares Project A and Project B ?
Project B
IRR B
1000
1200
800
-2%
Project B is obviously better than A, yet IRR A > IRR B
Project C
IRR C
1000
1200
800
3600
-4800
-16%
Project C is obviously better than B, yet IRR B > IRR C
Project D
IRR D
-1000
1200
800
3600
-4800
4%
Project D is worse than C, yet IRR D > IRR C
Project E
IRR E
-1325
1200
800
3600
-4800
20%
Project E is worse than D, yet IRR E > IRR D
47
The Social Discount Rate: Main Issues



How much current consumption society is willing to give up
now in order to obtain a given increase in future
Consumption?
It is generally accepted that society’s choices, including the
choice of weights be based on individuals’ choices
Three unresolved issues




Whether market interest rates can be used to represent how
individuals weigh future consumption relative to present
consumption?
Whether to include unborn future generation in addition to
individuals alive today?
Whether society attaches the same value to a unit of
investment as to a unit of consumption
Different assumptions will lead to choice of different
discount rate
48
Does the Choice of Discount Rate Matter?


Generally a low discount rate favors projects with
highest total benefits, irrespective of when they
occur, e.g. project C
Increasing the discount rate applies smaller
weights to benefits or (costs) that occur further in
the future and, therefore, weakens the case for
projects with benefit that are back-end loaded
(such as project C), strengthens the case for
projects with benefit that are front-end loaded
(such as project B)
49
NPV for Three Alternative Projects
Year
Project A
Project B
Project C
0
-80,000
-80,000
-80,000
1
25,000
80,000
0
2
25,000
10,000
0
3
25,000
10,000
0
4
25,000
10,000
0
5
25,000
10,000
140,000
Total benefits
45,000
40,000
60,000
NPV (i=2%)
37,838
35,762
46,802
NPV (i=10%)
14,770
21,544
6,929
50
Appropriate Social Discount Rate in Perfect
Markets
•
•
•
•
As individuals, we prefer to consume immediate benefits
to ones occurring in the future (marginal rate of time
preference)
We also face an opportunity cost of forgone interest when
we spend money today rather than invest them for future
use (marginal rate of return on private investment)
In a perfectly competitive market:
rate of return on private investment = the market
interest rates = marginal rate of time preference (MRTP)
The rate at which an individual makes marginal trade-offs
is called an individuals MRTP
Therefore, we may use the market interest rate as the social
discount rate
51
Equality of MRTP and Market Interest Rate
52
Alternative Social Discount Rate in Imperfect
Markets
Six






potential discounting methods
Social discount rate equal to marginal rate of return on
private investment, rz
Social discount rate equal to marginal rate of time
preference, pz
Social discount rate equal to weighted average of pz, rz
and i , where i is the government’s real long-term
borrowing rate
Social discount rate is the shadow price of capital
A discount rate that declines over the time horizon of the
project
A discount rate SG, based on the growth in real per capita
consumption
53
Alternative Social Discount Rate in Imperfect
Markets
Using
the Marginal Rate of Return on Private
Investment


The government takes resources out of the private sector
Society must receive a higher rate of return compared to
the return in the private sector
Criticism

Too high





Return on private sector investment incorporates a risk
premium
Government project might be financed by taxes, displaces
consumption rather than investment
Project may be financed by low cost foreign loans
Private sector return may be high because of monopoly or
negative externalities
Government investment sometimes raises the private
return on capital
54
Alternative Social Discount Rate in Imperfect
Markets
Using

the Marginal Social Rate of Time Preference, pz
Numerical values of pz

Real after-tax return on savings, around 2 percent for the US
economy
Criticisms



Individuals have different MRTP
How to aggregate such individual MRTP
Market interest rate reflects MRTP of individuals currently alive
Using

the Weighted Social Opportunity Cost of Capital
WSOC= arz + bi + (1-a-b)pz
Numerical Value, 3 percent for the US economy
55
Harberger’s Social Discount Rate
Social
discount rate should be obtained by weighting rz and pz
by the relative size of the relative contributions that
investment and consumption would make toward funding the
project
s = arz + (1-a)pz,
where a = ΔI/(ΔI+ ΔC) and (1-a) = ΔC/(ΔI+ ΔC)
 Savings are not very responsive to changes in the interest
rate, ΔC is close to zero
 The value of the parameter a is close to one
marginal rate of return on private investment rz is a good
approximation of true social discount rate
The
56
Alternative Social Discount Rate in Imperfect
Markets
Criticisms


Use


of WSOC
Criticisms applicable to use of rz and pz applies
Different discount rates for different projects based on source
of financing
the Shadow Price of Capital
Strong theoretical appeal
Discounting be done in four steps




Costs and benefits in each period are divided into those that
directly affect consumption and those affect investment
Flows into and out of investment are multiplied by the shadow
price of capital θ, to convert them into consumption equivalents
Changes in consumption are added to changes in consumption
equivalents
Discounting the resultant flow by pz
57
Alternative Social Discount Rate in Imperfect
Markets

Shadow Price of Capital
(rz   )(1  f )

p z  rz f   (1  f )
Where rz is the net return on capital after depreciation, δ is
the depreciation rate of capital, f is the fraction of gross
return that is reinvested, and pz is the marginal social
rate of time preference



Numerical Values for the θ,SPC, 1.5-2.5 for the US
economy
Applying SPC in Practice
Criticism of calculation and use of the SPC
58
Alternative Social Discount Rate in Imperfect
Markets


Using Time-Declining Discount Rates
Conclusion, Social Discounting in Imperfect Markets



If all costs and benefits are measured as increments to
consumption, use MSRTP, pz, Boardman et. Al. suggests a
value of 2 percent, sensitivity 0-4 percent
If all costs and benefits are measured as increments to private
sector investment, use MRROI, rz, Boardman et. Al. suggests a
value of 8 percent, sensitivity 6-10 percent
If all costs and benefits are measured as increments to both
consumption and private sector investment, use SPOC, θ, to
increments in investment and then discount at MSRTP,
Boardman et. Al. suggests for SPOC, a value of 1.65 percent,
sensitivity 1.3-2.7 percent; and ΔI = 15 percent and, ΔC= 85
percent, in the absence of information
59
The Social Discount Rate in Practice




Many Government Agencies do not discount at all
Shadow Price of Capital is rarely used
Governments do not use time-varying discount
rates
Constant positive rate that varies from country to
country




US, 7-10 percent
Canada, 10 percent, sensitivity 5-15 percent
0-3 percent for Health and Environment Projects
ADB, EIRR of 10-12 percent
60