Extreme events, discounting and
stochastic optimization
T. ERMOLIEVA
Y. Ermoliev, G. Fischer, M. Makowski, S. Nilsson, M. Obersteiner
IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU)
Robust Decisions, December 10-12 2007, IIASA, Laxenburg, Austria
 Many aspects of discounting: spatial, temporal, credibility.
 The aim of this talk is to analyze the implications of extreme events
(scenarios) on the choice of discounting for long-term decisions.
 How can we justify investments into catastrophic risk management, which
may possibly turn into benefits over long and uncertain time horizons in the
future?
 The traditional financial approaches often use the so-called net
present value (NPV) criteria to justify investments.
 An investment is defined as a cash flow stream V0 ,V1,...,VT over a time horizon
T , e.g., T = ∞. For example, a construction of a dike leads to maintenance costs,
losses (if dike breaks), associated repairs, insurance coverages, etc.
Assume that r is a constant prevailing market interest rate.
 Economic value of the dike or other alternative investments/projects are
estimated/compared by V = V0 + d1V1 + ... + dTVT , where dt = dt ,
d = (1+ r)−1, t = 0,1,...,T , is the discount factor and V denotes NPV.
Disadvantages of this criterion are well known. In particular, the NPV
critically depends on the prevailing interest rate which may not be easily
defined in practice.
NPV does not reveal the temporal variability of cash flow streams.
Two alternative streams may easily have the same NPV despite the fact
that in one of them all the cash is clustered within a few periods, but
in another it is spread out evenly over time. This type of temporal
heterogeneity is critically important for dealing with catastrophic losses
which occur suddenly as a “spike” after an extreme event. These two
issues are the main concern of the paper.
 Debates on proper discount rates for long term problems have
a longstanding history.
 Ramsey argued that to apply a positive discount rate r to discount
values across generations is unethical.
 Koopmans, contrary to Ramsey, argued that zero discount rate r would
imply an unacceptably low level of current consumption. The constant discount
rate has only limited justification. As a compromise between “prescriptive” and
“descriptive” approaches, Cline argues for a declining discount rate of 5% for the
first 30 years, and 1.5% beyond this.
 Weitzman proposed to model interest rates by a number of exogenous
time dependent scenarios. He argues for rates of 3 – 4% for the first 25
years, 2% for the next 50 years, 1% for the period 75–300 years and 0
beyond 300 years.
 Newell and Pizer analyzed the uncertainty of historical interest rates by
using data on the US market rate for long-term government bonds.
They proposed a different declining discount rate justified by a random
walk model.
Discounting in traditional sense
• Discounting is supposed to tell us for how much the profits/benefits/losses of a
program or a policy tomorrow (or in any time horizon) justify investments in it today
• Investments/policies are usually evaluated through the present value of consumption
utility
C0 , C1 , … Ct , where 0  q(t )  1,
U  U (C0 )  qU (C1 )  q 2U (C2 )  ...
• If the discount rates are time consistent, than the connection between
the discounting factor and the risk free rate of capital returns is
d (t )  q t 
1
(1  r )t
~ e rt
• Traditional approach, for example, for climate change policies, is to
set discounting rate equal to the risk free rate or to the average
capital market returns (Nordhaus, Manne)
Role of explicit uncertainties in traditional discounting
Example: the project yields 1000 USD in 200 years
Discounting factor is
d (t )  e rt
Two discounting scenarios:
1. the average discount rate of 4% (0.04) yields:
0.04200  0.34
present value = 1000 e
$
2. the discount rate which with 0.5 probability implies 1% and
with the same probability - 7% (which makes on average 0.04) yields:


0.5 1000 e0.01200  1000 e0.07 200  67.7 $
• The effect here comes from incorrect treatment of uncertainty of the
 rt
interest rate for discounting: E e
 e  Ert
Random time horizons
• A key question with discounting that, in fact, investments/savings
are linked to lifespans of assets/events and associated cash flows:
- cars, houses, pollutants (GHG)
• Risks (floods, criminals, terrorism) may reduce lifespans and, thus,
induce discounting related to the “stopping time” of a catastrophe.
In turn, applied discounting induces a time horizon of
evaluation
• Simple model
 ( 1/p ) year catastrophe, which may occur at t = 0, 1, … with
probability p (time invariant !)
 A 100 -, a 500 - , a 1000 - year event (flood, earthquake, etc.).
They may occur tomorrow, in two month, in 50 or 100 years.

- random time of its occurrence

Induced discounting I
• Investments in mitigations to meet a catastrophe at
stream of positive or negative values

generate a
v 0 , v1 , … vt , …
• In standard growth models,
time t ,
vt
equals a utility of consumption
Ct
at
vt  U (Ct )
• The aggregate value at time of a catastrophe

is
V  v0  v1  ...  v
• Proposition: The expected value of investments at  is the sum of
expected values conditional on its occurrence at t , t = 0, 1, 2, …
discounted by the tail probabilities that catastrophe occurs after
moment t , P   t :


EV  P  0Ev0  P  1Ev1  ...  P  t Evt  ...
Induced discounting II
• Assume a ( 1 / p ) years catastrophe, e.g.,
100 - year flood, q = 1- p ( the probability not to occur at t = 0, 1, … )
EV  Ev0  qEv1  q 2 Ev2  ...  q t Evt  ...
P  t   pq t  pq t 1  ...  pq t (1  q  q 2  ...) 
 pq t
1
 qt .
1 q
• Only geometric discounting has time consistency
EV  Ev0  qEv1  q 2 Ev2  ...  q t Evt  ... 


Ev0  q Ev1  qEv2  ...  q t 1 Evt  ...
i.e., any two successive periods have the same discounting.
• Time consistency with standard geometric discounting stems from
the “memoryless” of the geometric distribution of random time
horizon 
Main concerns: It is often assumed that a long-term investment activity has an
infinitely long time horizon:
V  t0 d t Vt
where, d t  d t
d  (1  r )1, r is a discount rate
Infinite deterministic stream of values
investment activity.
Vt
can represent a cash flow of a long-term
In economic growth and integrated assessment models, the value Vt represents
t
t
n
utility U ( x ) of an infinitely living agent or welfare Vt  i 1 i ui ( xi )
of a society with n representative agents, utilities u and consumption x, welfare weights
The infinite time horizon of evaluations
analysis.
Vt
creates an illusion of truly long-term
In reality, this evaluation accounts only for values Vt from a finite random interval
[0,  ] defined by a random “stopping time”


with probability P[  = t ] = pqt :
E t0 Vt  t0 P  t tk 0 Vk  t0 pq t tk 0 Vk


 t0 k t pq k Vt  t0 d tVt .

dt  P[  t ]

dt  q t
for geometric discounting
The expected duration of

is E  1 / p  1  1 / r  1 / r for small
For a modest market interest rate of 3.5%, r = 0.035, the expected duration
of  does not exceed 30 years.

Advantages of using
Recall:
E t  0 Vt
E t0 Vt  t0 P  t tk 0 Vk  t0 pq t tk 0 Vk


 t0 k t pq k Vt  t0 d tVt .
 Finite time horizon
 Stopping time can be associated with the arrival of potential
catastrophic event and not with the horizon of market interest rate
 The induced discounting dt  P[
cross-generational perspectives
 t]
properly addresses
We can think of  as a random “stopping time” associated with the
first occurrence of a “killing”. i.e., a catastrophic “stopping time” event.
Application of discounting for
the sensitivity of models w.r.t. “shocks”
 The sensitivity of models w.r.t. “shocks” is often assessed by introducing them
into discounted criteria. Previous Proposition demonstrates that this may lead to
serious miscalculations.
Let us consider a criterion with discounted factor an assume that shock arrives at a
random time moment θ from {0,1,...} with probability
P[  t ]   t ,  1    (1  p)
 Then the expected value
 , )V
E t 0 dtVt  E t 0 d t tVt  E tmin(
t
0
with q  d ,
, where
P[  t ]  pqt
p 1 q
 Therefore, the stopping time of the “shocked” evaluation E t 0 dtVt
defined by min(  , ) . The discounted factor of this evaluation
has the rate
(1  r )1(1   )1  (1  r   )1
d t t
is
1
Induced discounting:
Dominating role of the minimal discounting factors
• Proposition: Assume that random time horizon  corresponds to a
first catastrophe from a set of possible events (e.g., earthquakes,
floods, which may occur at different locations.
The induced discounting is dominated by the smallest discounting
rate.
Important implication of this proposition:
• Example: Coming back to our example with two scenarios of discounting rates
1% or 7% with probability 0.5. The dominating discount rate is 1%.
Example: Catastrophic Risk Management
The implications of Proposition for long-term policy analysis are
rather straightforward. Let us consider some important cases.
It is realistic to assume that the cash flow stream, typical for
investment in a new nuclear plant, has the following average time
horizons:
Without a disaster the first six years of the stream reflect the costs of
constructions and commissioning followed by 40-years of operating life
when the plant is producing positive cash flows and, finally, a 70-year
period of expenditure on decommissioning.
The flat discount rate of 5%, as Remark 1 shows, orients the analysis
on a 20-year time horizon. It is clear that lower discount rate places
more weight on distant costs and benefits. For example, the explicit
treatment of a potential 200-year disaster would require at least the
discount rate of 0.5% instead of 5%.
In fact, the mitigation of major nuclear plant disaster has to deal
with 107 – year event. A related example is investments in climate change
mitigations to cope with severe climate change related extreme events.
Definitely, a rate of 3.5%, as often used in integrated assessment
models can easily illustrate that climate change does not matter.