Expected net present value, expected net future value, and the Ramsey rule

Expected net present value, expected net
future value, and the Ramsey rule
Christian Gollier∗
Toulouse School of Economics (LERNA and IDEI)
June 13, 2009
Abstract
Weitzman (1998) showed that when future interest rates are un-
certain, using the expected net present value implies a term structure
of discount rates that is decreasing to the smallest possible interest
rate. On the contrary, using the expected net future value criteria im-
plies an increasing term structure of discount rates up to the largest
possible interest rate. We reconcile the two approaches by introducing
risk aversion and utility maximization. We show that if the aggregate
consumption path is optimized and made flexible to news about future
interest rates, the two criteria are equivalent. Moreover, they are also
equivalent to the Ramsey rule extended to uncertainty.
Keywords: Discount rate, Ramsey rule, climate change, cost-benefit
analysis.
∗This paper benefited from the financial support of the Chair "Sustainable finance and
responsible investment" at TSE. The research leading to these results has also received
funding from the European Research Council under the European Community’s Seventh
Framework Programme (FP7/2007-2013) Grant Agreement no. 230589.
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1
Introduction
The emergence of the public awareness about the long term sustainability of
our growth has been accompanied over the last decade by an important effort
from many prominent economists to clarify the way future generations are
treated in the standard cost-benefit methodology. Indeed, this methodology
is often criticized for sacrificing distant generations. This raises the ques-
tions of the level and of the term structure of the discount rate. There is no
consensus in our profession about the rate that should be used to discount
long term costs and benefits. This implies that economists disagree funda-
mentally about the intensity of our effort to improve the distant future. This
is best illustrated in the field of climate change, but other applications exist
in relation to nuclear wastes, biodiversity, genetically modified organisms, or
to the preservation of natural resources. Stern (2006), who uses implicitly a
discount rate of 1.4% per year, estimated the current social cost of carbon
around 85 dollars per ton of CO2. Nordhaus (2008) criticizes the low dis-
count rate used by Stern, and recommend alternatively a discount rate of
5%, which leads to an estimation of 8 dollars per ton of CO2. This huge
difference in the estimation of the social cost of carbon yields radical discrep-
ancies in the policy recommendations related to global changes. This is due
to the exponential nature of the impact of a change of the discount rate on
net present values (NPV). For example, using a rate of 5% rather than 1.4%
to discount a cash flow occurring in 200 years reduces its NPV by a factor
1340.
A standard arbitrage argument justifies using the rate of return of capital,
hereafter denoted ρ, in the economy as the socially efficient discount rate.
Indeed, diverting productive capital to invest in environmental projects with
an internal rate of social return below ρ would reduce the welfare of future
generations. Another argument is based on the NPV. At equilibrium, the
interest rate in the economy equals ρ. Suppose that one borrow today at
rate ρ an amount equaling to the present value of future cash flows, which
implies that these cash flows will be just enough to reimburse the initial loan.
This has the effect to transfer all future costs and benefits to today. It implies
that the project should be implemented only if its NPV discounted at rate ρ
is positive. However, this argument requires to know the returns of capital
for the different maturities of the cash flows of the environmental projects
under scrutiny. Obviously, they are highly uncertain for the time horizons
that we have in mind when we think about climate change or nuclear wastes
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for example. Weitzman (1998, 2001) has developed a simple argument based
on this fact to recommend a smaller rate to discount distant impacts. The
NPV is a decreasing convex function of the interest rate, and the degree
of convexity of this function is increasing in the time horizon. Therefore,
by Jensen’s inequality, introducing an uncertain permanent shock on future
interest rates raises the expected NPV, and this effect is increasing in the
maturity. It implies that the uncertainty on ρ has an effect on the expected
NPV that is equivalent to a sure reduction in ρ. It also implies that the term
structure of this reduction is increasing. Newel and Pizer (2003) and Groom,
Koundouri, Panopoulou and Pantelidis (2007) estimated the impact of the
uncertainty about future interest rates on the socially efficient discount rate
for different time horizons. More recently, Gollier, Koundouri and Pantelidis
(2008) estimated the term structure based on the uncertainty of the GDP-
weighted world interest rates. They obtained discount rates of 4.2%, 2.3%
and 1.8% respectively for the short term, 100 years and 200 years.
We confront this approach to two other ones. The first alternative ap-
proach is exactly symmetric to the one proposed by Weitzman (1998). It
measures the impact of the uncertainty on the expected net future value
(NFV), as described for example by Gollier (2004), Hepburn and Groom
(2007) and Buchholz and Schumacher (2008). Rather than assuming that all
costs and benefits of the environmental project are transferred to the present,
it assumes that they are all transferred to the terminal date of the project.
This alternative approach yields the opposite results: the uncertainty on fu-
ture interest rates raises the discount rate in an increasing way with the time
horizon. A third approach consists in making no intertemporal transfer of
costs and benefits, which implies that consumption is modified to compensate
them in real time. Under this approach, one needs to compare the impact
of uncertain changes in consumption at different dates on the intertemporal
welfare. Ramsey (1928) derived a simple formula that equalizes the efficient
discount rate net of the rate of pure preference for the present to the prod-
uct of the growth rate of GDP by the index of aversion to (intertemporal)
consumption inequality. Gollier (2002, 2008) and Weitzman (2007) extended
this formula to take account of the uncertainty on the growth rate of GDP,
which is itself linked to the uncertainty of future interest rates. These au-
thors show that the term structure of the efficient discount rate is flat when
shocks on the growth rate are temporary, and is decreasing when shocks have
a permanent component.
These results are very heterogeneous, and look rather unrelated to each
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others. The aim of this paper is to unify them in an single framework under
uncertainty. We show that these approaches are equivalent once risk aversion
is properly integrated into the model. We make explicit the investor’s ob-
jective function, which is assumed to be the standard Discounted Expected
Utility criterion. We show how to generalize the discount rates based on the
NPV and NFV rules to the case of risk aversion. In fact, we show that the
expectations considered above must be based on risk-neutral probabilities,
which are proportional to the marginal utility of consumption at the evalua-
tion date. This is in line with the standard consumption-based methodology
in finance to estimate risk premia.
Introducing risk aversion alone does not solve the puzzle. Another step
must be made, in which the consumption path is optimized. Under the
condition that the investor optimizes her consumption plan contingent to
the observed interest rate, we show that the NPV and NFV approaches lead
to exactly the same term structure of discount rates, which is decreasing and
tends to the lowest possible interest rate. Moreover, we show that the two
equivalent approaches are then also perfectly compatible with the well-known
Ramsey rule.
This paper is mostly non-technical, contrary to Gollier (2009). This com-
panion paper also shows that the decreasing nature of the term structure
obtained in this framework depends heavily upon the assumption that shocks
are permanent. If they are purely transitory, the term structure of discount
rates should be flat. In section 2, we present the discount rates examined
respectively by Weitzman (1998) and Gollier (2004) under risk neutrality.
Section 3 is devoted to add risk aversion into the picture, and to introduce
the Ramsey rule. Finally, we show in section 4 that the three approaches are
equivalent once we recognize that agents adapt optimally their consumption
plan to news about the future rate of return of capital in the economy. We
also illustrate this with a numerical example.
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2
Net present value and net future value un-
der uncertainty
Consider a simple investment that generates a sure payoff Z at date t per
euro invested at date 0.1 If ρ is the continuously compounded interest rate
during the period, it is optimal to undertake the project if its Net Present
Value
N P V = −1 + Ze−ρt
is positive. This NPV rule is sustained by a simple arbitrage argument:
implementing the project and borrowing Ze−ρt at date 0 until date t would
generate the sure payoff N P V today, with no other net payoff along the
lifetime of the project. Suppose now that the interest rate eρthatwillprevail
between dates 0 and t is constant but uncertain. It is unknown at the time
the investment decision must be made, but the uncertainty eρisfullyresolved
at date t = 0. Because the investment opportunity cost is uncertain, it is
likely to affect the optimal decision. Weitzman (1998, 2001) assumes that the
optimal decision criterion in that context is to invest if the expected NPV is
positive, i.e., if −1+ZEe−ρt is positive. Obviously, this is equivalent to using
a discount rate rp such that e−rpt = Ee−ρt, where rp is defined as follows:
1
rp(t) = − lnEe−ρt.
(1)
t
It is easy to check that rp(t) is less than Eeρ,andthatittendstoitslowest
possible rate for large t. The discount rate rp is sustained by a strategy in
which the project is implemented and in which the investor borrows at date
0 a random amount Ze−ρt at interest rate eρuntilt. Alltheriskisthusborne
at date 0.
One could alternatively consider another arbitrage strategy, in which the
investor borrows one euro at date 0 to finance the project. In that case,
only one net payoff is generated. It takes place at date t and is equal to
the expected Net Future Value (NFV) −Eeρt + Z. Investing in the project
is optimal if the expected NFV is positive. This is equivalent to using a
discount rate equaling
1
rf (t) =
ln Eeρt.
(2)
t
1 Any sure investment project can be decomposed into a portfolio of investment projects
with a single future cash flow occuring at different dates. So this assumption is made
without loss of generality.
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It is again easy to check that rf (t) is larger than Eeρ, andthat it tends to
its largest possible rate when t tends to infinity. The discount rate rf is
sustained by an arbitrage strategy in which the initial investment cost is
financed by a loan at rate eρbetween0andt. Alltheinvestmentriskisthus
borne at the future date t in this case.
In the absence of uncertainty, it is obvious that rf (t) = rp(t), which means
that the NPV and NFV criteria are equivalent in that case. This is not the
case under uncertainty, as initially observed by Pazner and Razin (1975),
and then by Gollier (2004), Hepburn and Groom (2007) and Buchholz and
Schumacher (2008). This shows that, in general, the choice of the discount
rate cannot be disentangled from how the investment is financed, and from
how the risky payoff of the project is allocated through time. Hepburn and
Groom (2007) proposed an explanation for the paradox, namely that the
certainty-equivalent social discount rate does fall with the passage of time,
but it increases as the evaluation date for the investment moves further into
the future. In a model with a risk-neutral planner, this provides interesting
insights to this paradox, but it leaves us with another unsatisfactory con-
clusion, namely that the evaluation date is arbitrary and thus one cannot
objectively decide which discount rate should be used. They show that the
decision criterion is highly sensitive to the arbitrary evaluation date, noticing
that "a fuller analysis is required", and concluding that "in the murky waters
of intergenerational policy, any theoretical advance providing a ray or two of
light is to be welcomed."
This paper is also related to a recent paper by Buchholz and Schu-
macher (2008), who also recognize the necessity to introduce risk aversion
into the analysis. They propose an interesting criterion in which invest-
ing at the discount rate rbs(t) is defined in such a way that it yields the
same expected utility as investing at the uncertain rate of return of capital:
u(exp rbst) = Eu(exp eρt). They conclude that the discount rate rbs is de-
creasing or increasing with the time horizon t depending upon the intensity
of risk aversion. In particular, a relative risk aversion less than unity yields
an increasing term structure. Our approach differs much from Buchholz and
Schumacher’s one mostly because we use the more standard marginalist ap-
proach to asset pricing.
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3
Three discount rates with risk aversion
In this section, we introduce risk aversion into the picture. Consider a repre-
sentative investor with an increasing and concave von Neumann-Morgenstern
utility function u. In order to determine whether an investment project is
socially desirable or not, we define the intertemporal welfare function as fol-
lows:
T
W = Xe−δtEu(
t=0
e
ct).
Parameter δ is the investor’s rate of pure preference for the present, whereas
T is the (potentially arbitrarily long) time horizon of the planner. We as-
sume that consumption at date t, which is denoted ect,isarandomvariable.
Because the discount factor on utils is exponential, this discounted expected
utility model is compatible with a time-consistent behavior. Finally, let ρ
denote the rate of return of capital per period. It is a random variable eρat
the time of the investment decision prior to t = 0, but the uncertainty is
fully revealed at date 0. Using a standard arbitrage argument, this implies
that the socially efficient discount rate ex post is ρ. We hereafter determine
three different ways to characterize the socially efficient discount rate prior
to the realization of eρ.
As in the previous section, consider an investment that generates Z = ert
euros with certainty at date t per euro invested at date 0, where r is the
sure internal rate of return of the project. If r is large enough, the project is
socially desirable, i.e., it increases W . We define the discount rate associated
to maturity t as the critical r such that investing a small amount ε in the
project would have no effect on welfare W . However, measuring the impact of
the implementation of the investment project on W requires determining first
how the benefit of the project is transformed into changes in consumption at
different dates.
Let b denote the share of the initial investment cost that is financed
through a reallocation of capital in the economy. The remaining share 1 − b
corresponds to a reduction of consumption at date 0. Thus, b characterizes
the way the net benefit is allocated in the lifetime of the project. We assume
that at the terminal date t of the project, the borrowed capital and the
interest are paid back and reinvested in the economy. It implies that the the
net increase in consumption at date t equals Z − beρt. Given the financing
strategy b which may be made contingent to the realization ρ at date 0, the
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impact of a marginal investment in the project on the intertemporal social
welfare W equals
∆W = −Eu0(ec0)(1−b)+e−δtEu0(ct)(ert−beρt).
(3)
Because b may be contingent to eρ,whichmayitselfbecorrelatedtoec0andect,
this formula cannot be simplified. For each b, there is a unique r that solves
the optimality condition ∆W = 0. We examine three particular specifications
for b.
A natural candidate for b is b = 0 : the cash flows of the project are
consumed when they occur. In that case, condition ∆W = 0 yields the
Ramsey discount rate r = rr(t) with
1
Eu0(
rr(t) = δ − ln
.
(4)
t
e
ct)
Eu0(
For example, assume that relative risk aversio ec0)
n is constant, i.e., u0(c) = c−γ,
and that the growth rate of consumption is certain, i.e. ct = c0egt. In that
particular case, equation (4) implies that rr(t) = δ + γg, which is the well-
known Ramsey rule: the discount rate net of the rate of impatience equals
the product of the growth rate of consumption by the index of relative risk
aversion.
An alternative candidate is b = Ze−ρt = e(r−ρ)t. This is the NPV case
in which all costs and benefits are transferred to the initial date. Condition
∆W = 0 yields r = rp(t), with
1
Eu0(
rp(t) = − ln
.
(5)
t
e
c0)e−ρt
Eu0(
This is the "NPV" discount rate under risk aec0)
version. The ratio in the right-
hand side of this equality is a weighted expectation of e−ρt. Because the
overall risk of the strategy is allocated to consumption at date 0, the weights
are proportional to the marginal utility at that date. In the finance litera-
ture, this distortion of objective probabilities is referred to as "risk-neutral
probabilities". Notice that the NPV discount rate characterized by equation
(5) boils down to the Weitzman discount rate (1) when the representative
agent is risk neutral, or when consumption ec0 at date 0 is independent of
the interest rate eρ. Introducing risk aversion does not change the shape of
the term structure, which is decreasing and tends to the smallest possible
interest rate for large maturities.
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A third approach consists in assuming that the initial cost of the project
is entirely financed by a transfer of capital from other projects (b = 1). This
leads to the "NFV" discount rate. Equalizing ∆W to zero for such a b defines
the NFV discount rate r = rf (t) :
1
Eu0(
rf (t) =
ln
.
(6)
t
e
ct)eρt
Eu0(
This NFV discount rate generalizes the d ect)
iscount rate (2) to the case of
risk aversion. The risk-neutral probability distribution weights the objec-
tive probabilities to the marginal utility of consumption at date t. Without
any information about the statistical relationship between ectandeρ,onecan-
not characterize the term structure of the NFV discount rate, except in the
case of risk neutrality.
As observed by Hepburn and Groom (2007), one could define an infinite
number of such discount rates by choosing arbitrarily the evaluation date.
The translation of this observation into the framework of this paper is that the
socially efficient discount rate depends upon the way the project is financed.
This demonstrates that, as suggested by Hepburn and Groom (2007), both
Weitzman (1998) and Gollier (2004) are right. Whether one should use a NPV
or a NFV approach depends upon who will benefit from the investment.
4
Unifying field: Optimization of the con-
sumption strategy
The conclusion of the previous section is problematic. Standard cost-benefit
analysis states that the way one evaluates an investment project should not
depend upon the way the project is financed. There should be only one
term structure of discount rates. This property is one of the cornerstones
of public finance. In this section, we show that the three discount rates rr,
rp and rf are in fact all equal if we recognize that economic agents optimize
their consumption path.
From date 0 on, the investor knows the rate of return on capital ρ. Reduc-
ing consumption by ε at date t1 allows the agent to increase consumption by
εeρ(t2−t1) at date t2. Therefore, the optimal consumption path conditional to
ρ must be such that u0(ct ) equals e(ρ−δ)(t2−t1)u0(c ), otherwise transferring
1
t2
consumption from one date to the other would increase the intertemporal
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welfare W . This implies in particular that conditional to ρ, we must have
that
u0(c0) = e(ρ−δ)tu0(ct).
(7)
Observe that it implies that optimal consumption plans in general depend
upon ρ.
This optimality condition implies that rr(t) = rp(t) = rf (t), and more
generally that the socially efficient discount rate is independent upon the
evaluation date, or upon the way (marginal) projects are financed. Indeed,
using condition (7), we have that
1
Eu0(
1
Eu0(
rr(t) = δ − ln
=
ln
= r
t
e
ct)
Eu0(
and
1
Eu
e
c
−
p(t),
0)
t
e
c0)e−ρt
Eu0(
1
Eu
e
c0)
0(
0(
rf (t) =
ln
=
ln
= r
t
e
ct)eρt
Eu0(
Proposition 1 (Equivalence ec
−
p(t).
t)
t
e
c0)e−ρt
Eu0(
Property) Under the ec0)
assumption that consumers
can make their consumption plan flexible to changes in the return of capital,
the socially efficient discount rate is independent of how the uncertain net
benefit of the environmental project is allocated through time. It means in
particular that rr(t) = rp(t) = rf (t).
This is in fact an application of the envelop theorem. Thus, we have
reconciled the various approaches that have recently been used to discuss
whether the discount rate should be decreasing with the time horizon. Under
this framework in which the shock on interest rates is permanent, the socially
efficient discount rate is decreasing and tends to the smallest possible interest
rate.
In order to illustrate these properties, let us consider an economy in which
the infinitely lived representative agent with a unit initial wealth has a rate
of impatience equaling δ = 2% and an index of relative risk aversion equaling
γ = 2. It implies that, conditional to ρ, consumption ct equals (ρ−g)exp(gt),
where g = (ρ−δ)/γ is the rate of growth of consumption.2 Suppose first that
the rate of return of capital is risk free and equal to ρ = 5%. In that case,
2 The condition g < ρ is necessary and sufficient for the existence of a solution to the
consumption problem with an infinite horizon. Condition γ > 1 guarantees that this
condition holds.
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