1. 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

2.  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 V 0,V 1,...,V T 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 = V 0 + d 1 V 1 +... + d T V T, where d t = d t, d = (1+ r) −1, t = 0,1,...,T, is the discount factor and V denotes NPV.  Many aspects of discounting: spatial, temporal, credibility.

3. 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.

4.  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.

5. 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 Discounting in traditional sense 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) If the discount rates are time consistent, than the connection between the discounting factor and the risk free rate of capital returns is Investments/policies are usually evaluated through the present value of consumption utility,, …, where,

6. Role of explicit uncertainties in traditional discounting $ Two discounting scenarios: 1. the average discount rate of 4% (0.04) yields: present value = 2. the discount rate which with 0.5 probability implies 1% and with the same probability - 7% (which makes on average 0.04) yields: $ Example: the project yields 1000 USD in 200 years Discounting factor is The effect here comes from incorrect treatment of uncertainty of the interest rate for discounting:

7. 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

8. Induced discounting I Investments in mitigations to meet a catastrophe at generate a stream of positive or negative values,, …, … In standard growth models, equals a utility of consumption at time t, The aggregate value at time of a catastrophe is 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, :

9. Induced discounting II Time consistency with standard geometric discounting stems from the “memoryless” of the geometric distribution of random time horizon Assume a ( 1 / p ) years catastrophe, e.g., 100 - year flood, q = 1- p ( the probability not to occur at t = 0, 1, … ) Only geometric discounting has time consistency i.e., any two successive periods have the same discounting.

10. where,, r is a discount rate Infinite deterministic stream of values can represent a cash flow of a long-term investment activity. In economic growth and integrated assessment models, the value represents utility of an infinitely living agent or welfare of a society with n representative agents, utilities u and consumption x, welfare weights Main concerns: It is often assumed that a long-term investment activity has an infinitely long time horizon:

11.   for geometric discounting  The infinite time horizon of evaluations creates an illusion of truly long-term analysis. In reality, this evaluation accounts only for values V t from a finite random interval [0, ] defined by a random “stopping time” with probability P[ = t ] = pq t : For a modest market interest rate of 3.5%, r = 0.035, the expected duration of does not exceed 30 years. The expected duration of is for small

12. Advantages of using  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 properly addresses cross-generational perspectives We can think of as a random “stopping time” associated with the first occurrence of a “killing”. i.e., a catastrophic “stopping time” event. Recall:

13. 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.  Then the expected value Let us consider a criterion with discounted factor an assume that shock arrives at a random time moment θ from {0,1,...} with probability, where with,,  Therefore, the stopping time of the “shocked” evaluation is defined by. The discounted factor of this evaluation has the rate

14. Induced discounting: Dominating role of the minimal discounting factors 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%. 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:

15. 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 10 7 – 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.