Net Present Value Analysis
Avinash Kishore([email protected])
Based on Notes from Andrew Foss
February 04, 2011Review Section
Agenda
Fundamental Theories of Welfare Economics
Static Efficiency
Dynamic Efficiency (NPV)
Internal Rate of Return
Equivalent Annual Net Benefits
Readings on Benefit-Cost Analysis
Practice Problem(s)
Excel Workbook Embedded Here:
2
Microsoft Office Excel 97-2003 Worksheet
Fundamental Theories of Welfare Economics:Pareto Criterion and Pareto Optimality
Pareto Criterion: A policy change is an improvement if at least some people are made better off and no one is made worse off
Pareto Optimality: No other feasible policy could make at least one person better off without making anyone else worse off
3Adam’s Payment
Beth’s Payment
StatusQuo
Policy A
Policy B
Policy C
Policy D
Feasibility Frontier
Possible Payments to Adam and Beth Which satisfy Pareto Criterion?
‒ Policy A does ‒ Policy B does not‒ Policy C does not‒ Policy D does‒ All policies in light gray triangle
Which satisfy Pareto Optimality?
‒ Policy A does not ‒ Policy B does not‒ Policy C does‒ Policy D does‒ All policies on feasibility frontier (because nothing “better” from there)
$25 $100
$25
$100
Fundamental Theories of Welfare Economics:Kaldor-Hicks Criterion
Kaldor-Hicks Criterion: A policy change is an
improvement if the “winners” could fully compensate the
“losers” and still be better off themselves
– Also known as Potential Pareto Improvement Criterion
Kaldor-Hicks Criterion rules out policies with total
benefits smaller than total costs (that is, policies with
negative net benefits, where NB = TB - TC)
When the Kaldor-Hicks Criterion is used to compare all
feasible policy options, the best is that which maximizes
net benefits
– If all policies have negative net benefits, keep the status quo4
0
5
10
15
20
25
0 1 2 3 4 5 6 7 8 9 10
Ma
rgin
al B
en
efi
ts o
r M
arg
ina
l Co
sts
Quantity of Pollution Control
0
20
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60
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0 1 2 3 4 5 6 7 8 9 10
To
tal B
en
efi
ts o
r T
ota
l C
os
ts
Quantity of Pollution Control
Static Efficiency (Single time period)
Undertake policy to the point at which TB – TC is the
highest
marginal benefit = marginal cost
5
Total Benefits
Total Costs
Marginal Benefits
Marginal CostsNet Benefits
Q*
Total Benefits and Total Costs Marginal Benefits and Marginal Costs
Q*
Dynamic Efficiency: when B and C in different time periods
To achieve dynamic efficiency (multiple time periods),
undertake policy with highest net present value
If all policies have negative NPV, keep the status quo
Discount rate should reflect social opportunity cost
U.S. Office of Management and Budget (OMB)
published guidance on discount rate and benefit-cost
analysis in Circular A-4 (September 2003):http://www.whitehouse.gov/omb/assets/regulatory_matters_pdf/a-4.pdf
6
T
tttt
r
CB
0 )1(NPV
OMB Guidelines on C-B Analysis
“For transparency’s sake, you should state in your
report what assumptions were used, such as the time
horizon for the analysis and the discount rates
applied to future benefits and costs.
It is usually necessary to provide a sensitivity analysis
to reveal whether, and to what extent, the results of
the analysis are sensitive to plausible changes in the
main assumptions and numeric inputs”.
7
Why do we need discounting? (Circular A4, OMB)
Benefits or costs that occur sooner are generally more
valuable– Resources invested earn a positive return, so current consumption is more
expensive than future consumption, since you are giving up that expected
return on investment when you consume today. (Opportunity Cost).
– Postponed benefits also have a cost because people generally prefer
present to future consumption. (Positive time preference).
– Also, if consumption continues to increase over time, as it has for most of
U.S. history, an increment of consumption will be less valuable in the future
than it would be today (Principle of diminishing marginal utility).
8
What is the appropriate discount rate?
“a real discount rate of 7 percent should be used as a
base-case for regulatory analysis”.
Why?
– “The 7 percent rate is an estimate of the average before-tax
rate of return to private capital in the U.S. economy.
– the returns to real estate and small business capital as well as
corporate capital.
– It approximates the opportunity cost of capital
– it is the appropriate discount rate whenever the main effect of
a regulation is to displace or alter the use of capital in the
private sector”.
9
The appropriate discount rate ?
The effects of regulation do not always fall exclusively or primarily
on the allocation of capital.
When regulation primarily and directly affects private
consumption (e.g., through higher consumer prices for goods and
services), a lower discount rate is appropriate.
The alternative most often used is sometimes called the social
rate of time preference
…the rate at which society discounts future consumption flows to
their present value. – the rate that the average saver uses to discount future consumption as a
measure
Real rate of return on long-term government debt ( = 3%)
10
How does discounting work?
Reciprocal of compounding
Benefits and costs far in the future are more sensitive to discount
rate than near-term benefits and costs
– Run discounting program in Excel workbook
11
r = 3% → NPV = $29M r = 10% → NPV = -$10M
-$250
-$200
-$150
-$100
-$50
$0
$50
$100
0 1 2 3 4 5 NPV
Net
Ben
efit
s (m
illi
on
s)
Undiscounted Net Benefits Discounted Net Benefits
-$250
-$200
-$150
-$100
-$50
$0
$50
$100
0 1 2 3 4 5 NPV
Net
Ben
efit
s (m
illi
on
s)
Undiscounted Net Benefits Discounted Net Benefits
How does discounting work?
When costs are incurred up front and benefits occur in
the future, low discount rates result in higher NPVs than
high discount rates
12
Relationship between Discount Rate and NPVwith Upfront Costs and Future Benefits
-$50
-$40
-$30
-$20
-$10
$0
$10
$20
$30
$40
$50
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Net
Pre
sen
t V
alu
e (m
illio
ns)
r
Dynamic Efficiency:Power Plant Example
You are a special assistant to Gov. Schwarzenegger of
California. He wants to shut down a coal-fired power plant
and replace it with either a hydropower plant or a natural gas-
fired plant. He asks you to analyze the options.
Assumptions (unrealistic…)
– Both plants can be built in 1 year and operate for 5 years
– Both plants yield annual benefits of $50M relative to coal
– Hydropower plant has upfront fixed costs of $100M and
annual operating costs of $5M
– Natural gas plant has upfront fixed costs of $40M and
annual operating costs of $20M
– Discount rate is 7 percent, but also try 3 and 10 percent
13
Dynamic Efficiency:Power Plant Example
Hydropower has a slightly higher NPV than natural gas
at 7 percent discount rate, but lower at 10 percent
14
$0
$20
$40
$60
$80
$100
$120
r = 3% r = 7% r = 10%
Ne
t Pre
se
nt
Va
lue
(m
illio
ns
)
Hydropower Natural Gas
Discounting & Climate Change: The Stern Report
A time scale of centuries, so the brute power of compound interest– Manhattan purchase: 60 guilders ($ 1000) in 1626
– @ 7%: ~$80 trillion; @ 3%: ~ 15 million (5 million times)
The Stern Report conclusions are driven mainly by the low assumed discount
rate : (r = 1.4%)
– If r = 6%, the PDV of global-warming loss 100 years hence is <1/100 th
Question: Is it worthwhile to sacrifice costs C =1% of GDP now to
remove damages D = 5% of GDP a 100 years from now?
Stern’s B/C ratio = 4.5 (upper bound 5 if zero discount rate is chosen)
B/C ratio (@ r = 6%) = 0.1
Prof Martin Weitzman: “In fact, it is not an exaggeration to say that the
biggest uncertainty of all in the economics of climate change is the
uncertainty about which interest rate to use for discounting”
15
Health related costs and benefits
Question: is discounting even appropriate?
– lives saved today cannot be invested in a bank to save more
lives in the future.
Answer: Yes!
– People have been observed to prefer health gains that occur
immediately to identical health gains that occur in the future.
– If future health gains are not discounted while future costs
are, then what happens?
– an attractive investment today in future health improvement
can always be made more attractive by delaying the
investment
16
Internal Rate of Return:Overview
Internal rate of return answers the question, “What
discount rate would make NPV zero?”
When costs are incurred up front and benefits occur in
the future, undertake project if IRR > r
In the first discounting example (p. 7), IRR ≈ 8 percent
Internal rate of return is less robust than NPV, and it
should not be used to rank projects when constraints
make it impossible to undertake them all
17
0)1(
NPV0
T
tt
tt
IRR
CB
Internal Rate of Return:Power Plant Example
A nuclear power plant can be built in 1 year for $100M, can operate
for 5 years, yields annual benefits of $55M relative to coal, has
annual operating costs of $5M, and has decommissioning costs of
$155M in Year 6
18-$6
-$5
-$4
-$3
-$2
-$1
$0
$1
$2
$3
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Net
Pre
sen
t V
alu
e (m
illio
ns)
r
IRR??
IRR is not useful in this case because there are costs in the future
Equivalent Annual Net Benefits
Suppose the hydropower plant replacing the coal plant
in California can operate for 10 years and the natural
gas plant can still only operate for 5 years
– At r = 7 percent, NPVhydro = $216M and NPVgas = $83M
– At r = 32* percent, NPVhydro = $32M and NPVgas = $30M
* This is an unusually high discount rate, but it illustrates the point for the example numbers
Calculate equivalent annual net benefits to compare
these projects of different duration
– At r = 7 percent, EANBhydro = $27M and EANBgas = $16M
– At r = 32 percent, EANBhydro = $8M and EANBgas = $9M 19
Trr
rNPVEANB
)1(1
Readings on Benefit-Cost Analysis:Arrow et al. (1996)
Benefit-cost analysis is a important framework for
making regulatory decisions
– Careful consideration of benefits and costs
– Common unit of measurement for disparate impacts (dollars)
– Useful tool for improving effectiveness of regulation
– Techniques for incorporating uncertainty
But benefit-cost analysis should not be the sole basis
for making regulatory decisions
– Consideration of distributional impacts as well
– Perhaps not necessary to perform benefit-cost analysis for
minor regulations
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Readings on Benefit-Cost Analysis:Goulder and Stavins (2002)
Discounting does not shortchange the future, so long as
an appropriate discount rate is used
– It simply puts current values and future values of benefits and
costs in equivalent monetary terms; apples-to-apples
comparison
– It accounts for time value of money (interest) and not inflation:
rnominal ≈ inflation + rreal
When the “winners” of a policy do not actually
compensate the “losers,” the Kaldor-Hicks criterion
carries less weight
Lowering the discount rate to increase NPV is
problematic because it mixes efficiency and equity 21
Private Goods and Public Goods:Beekeeper and Farmer Example
A beekeeper and a farmer are neighbors. The bee-
keeper’s bees help pollinate the farmer’s orchard.
The beekeeper’s marginal benefit from Q beehives is
MBbeekeeper(Q) = 10 – Q
The beekeeper’s marginal cost is constant at
MCbeekeeper(Q) = 7
The farmer’s marginal benefit from Q beehives is
MBfarmer(Q) = 5 – Q
If the beekeeper ignores impacts on the orchard, how
many beehives will the beekeeper have? What if the beekeeper takes impacts on the orchard into account?
23
Private Goods and Public Goods:Beekeeper and Farmer Example
If the beekeeper treats the beehives as a private good,
the beekeeper will have 3 beehives
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0
5
10
15
0 5 10 15
MB
or
MC
Quantity of Beehives
MB_beekeeper MC_beekeeper
Q*
MCbeekeeper
MBbeekeper = MCbeekeeper
10 – Q = 7
Q* = 3 beehives
MBbeekeeper
0
5
10
15
0 5 10 15
MB
or
MC
Quantity of Beehives
MB_beekeeper MC_beekeeper MB_farmer MB_society
Private Goods and Public Goods:Beekeeper and Farmer Example
If the beekeeper treats the beehives as a public good,
the beekeeper will have 4 beehives
– Public goods are underprovided by private decision-making
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Q*
MBbeekeeper
MCbeekeeper
MBsociety = MBbeekeeper + MBfarmer
(vertical sum of MBs for public goods)
For 0 ≤ Q ≤ 5 (where both MBs ≥ 0),
MBsociety = (10 – Q) + (5 – Q) = 15 – 2Q
MCsociety = MCbeekeeper
MBsociety = MCbeekeeper
15 – 2Q = 7
Q* = 4 beehives
MBsociety
MBfarmer
Private Goods and Public Goods:Beekeeper and Farmer Example
If the beekeeper and farmer can negotiate without
transaction costs, what outcome would we expect?
26
Increasing the number of beehives from 3 to 4 gives the
beekeeper extra benefits of $6.50 (area under MBbeekeeper)
but extra costs of $7 (area under MCbeekeeper), so the
beekeeper’s profit decreases by $0.50
Increasing the number of beehives from 3 to 4 gives the
farmer extra benefits of $1.50 (area under MBfarmer) and
does not impose extra costs on the farmer
By the Coase Theorem, the farmer could give the bee-
keeper between $0.50 and $1.50 to have 4 beehives