Discounting
Discounting is a method for placing weights on future
values to convert them into present values so that they can
be combined to a single number in common units, the
“present value”.
0
(present)
t1
t2
time
The discount factor: determined
by the discount rate and t.
• discount rate, r:
o a parameter that determines the discounting
factor or weight, dt
o typically: zero to 10%
• discounting factor, dt:
o the weight placed on a value accrued in year t to
convert it to a present value in the present (year
0).
o typically <1
discount
rate, r
Discounting example: r=0.05 and an arbitrary stream of net benefits
0.05
0
0
5
10
15
discount
factor, d t
1
25
t
0
5
10
15
20
25
30
40
20
dt*NBt
0
0
5
10
15
20
25
30
40
PVNB t = dt * NBt
20
0
0
5
10
15
20
25
200
PVNB
30
d = 1/(1+r)t
0.5
0
Yearly pres. val. Yearly net
of NB, PVNBt
benefits, NBt
20
100
0
<--PVNB = the sum of the PVNBt terms across the horizon
0
time, t
30
Rationale 1:
We should discount to account for the
opportunity cost of investment.
• Public (environmental) sector investments
should generate returns at or above the level
available in the private sector.
– E.g. business loan interest rate, rate of return in the
stock market
• “Time value of money”, “opportunity cost of
capital”
Rationale 1:
We should discount because it is consistent with how
individuals make decisions in the face of tradeoffs over
time.
• Individuals are “impatient”.
• Individuals are typically indifferent between
payoff X at time t
and
payoff Y<X at time t’ < t.
– E.g., I will give you $100 in 1 year: What would you be willing to
accept today in lieu of this promise?
• Jargon: “pure rate of time preference”
A common approach to choosing a social discount
rate is to estimate the observed return to investment
(long-run, after-tax, risk-free)
• 7%: approx. the real return to investment in large companies (1926–
1990) (Newell and Pizer, 2004)
• But personal taxes (up to 50%) mean that return to investors, the
consumption rate of interest, is closer to 4%.
– And this 4% return includes a premium for risk (firms may fail).
– To separate the issue of risk from this analysis, consider relatively safe
government bonds: 4% nominal return2% after taxes.
• “The appropriate rate …. in the United States is generally taken to
be around a 3% real, riskless rate.” (Kopp et. al, 1997)
Federal guidelines prescribe assessing
results over a range of discount rates
• U.S. OMB 2003 :
– provide estimates of PVNB using discount rates of 3% and 7%
– 3%: consumption rate of interest (i.e. after tax return on
investments)
– 7%: OMB estimate of the opportunity cost of capital
• see: http://www.whitehouse.gov/omb/circulars_a004_a-4/
• “Many economists believe that this…range…is too high”
(Fraas and Lutter, 2011)
Specifying a social discount rate for long-run climate
policy analysis often employs the Ramsey framework
Ramsey (1928) optimal growth model:
Economy operates as if a “representative agent” selects consumption
and savings to max PV of the stream of utility from consumption over
time.
One implication of the Ramsey model is the following equation:
• r = ρ + ƞg
• r: return to capital (real, long-run)
• ρ: pure rate of time preference
“time discount rate”, due to “impatience”
• ƞ: elasticity of marginal utility w.r.t. consumption
• g: average growth in consumption per capita
Two different perspectives on parameterizing the Ramsey
discounting equation lead to very different results.
ρ: pure rate of time preference;
ƞ: elasticity of marginal utility w.r.t. consumption;
g: average growth in consumption per capita
1. Descriptive approach/Nordhaus & the DICE model
• Use economic data to estimate parameters:
• Nordhaus (2008):
• r = ρ + ƞg = 0.04 (average over the next century (Nordhaus, 2008, 10))
–
5.5% over first 50 years (61).
• Economic growth and population growth will slow, rate will fall over time.
2. Prescriptive approach/Stern & the Stern Review (2006)
• Argument: No ethical reason to discount future generations due to a
pure rate of time preference except for the possibility that we might not
be here at all (ρ reflects only ann. prob. of extinction). 1.3% growth
assumed.
• r = ρ + ƞg = 0.001 + 1*0.013 = 0.014
There is no consensus on a single
value for the social discount rate
Q: What discount rate
do you favor for
discounting long-term
environmental
projects?
Respondents-over
2,000 Ph.D. level
economists
Weitzman, M. L. (2001). “Gamma discounting.” American Economic Review 91, 260-271.
Image: http://www.dbj.jp/ricf/en/research/symposium199511.html
There is no consensus on a single
value for the social discount rate
Weitzman (2001):
“The most critical single problem with discounting future
benefits and costs is that no consensus now exists, or
for that matter has ever existed, about what actual rate
of interest to use.”
The considerations on which a discount rates are based
“are fundamentally matters of judgment or opinion, on
which fully informed and fully rational individuals might be
expected to differ.”
The lack of discount rate consensus
means that sensitivity analysis is critical
• “We should have less confidence in a project for which
– the sign of the PVNB is highly sensitive to
• the discount rate or to
• small changes in projected future benefits and costs,
compared with a project with a PVNB that is not very
sensitive to these elements.” (Goulder and Stavins 2002, p. 674)
There is a potential problem with
preference-driven parameterization when
the problem spans generations
• The justification for a pure rate of time preference is
based on the choices individuals make about payoffs in
their own lifetime.
• Some economists argue that at the societal level there
is no good ethical argument for using a pure rate of time
preference other than zero.
– Especially, across generations.
Note: Real versus nominal
• Discount rates are NOT meant to address
inflation (before discounting all values should be in real terms).
• Nominal value: expressed
in the money of the day
• Real value:
adjusted for inflation
Nominal
($ of the day)
Buying
power in
1982
$ 10
Buying
power in
2012
$ 10
Real
($ of 2012)
Buying
$ 20.38
power in
2012
$ 10
• In a BCA analysis, benefits and costs from each
year are stated in the same real units, e.g. in
2005 dollars.