Uncertain growth and the value of the future

For environmental problems such as global warming future costs must be balanced against present costs. This is traditionally done using an exponential function with a constant discount rate, which reduces the present value of future costs. The result is highly sensitive to the choice of discount rate and has generated a major controversy as to the urgency for immediate action. We study analytically several standard interest rate models from finance and compare their properties to empirical data. From historical time series for nominal interest rates and inflation covering 14 countries over hundreds of years, we find that extended periods of negative real interest rates are common, occurring in many epochs in all countries. This leads us to choose the Ornstein-Uhlenbeck model, in which real short run interest rates fluctuate stochastically and can become negative, even if they revert to a positive mean value. We solve the model in closed form and prove that the long-run discount rate is always less than the mean; indeed it can be zero or even negative, despite the fact that the mean short term interest rate is positive. We fit the parameters of the model to the data, and find that nine of the countries have positive long run discount rates while five have negative long-run discount rates. Even if one rejects the countries where hyperinflation has occurred, our results support the low discounting rate used in the Stern report over higher rates advocated by others.Comment: 8 pages, 4 figure

Statistical analysis and stochastic interest rate modeling for valuing the future with implications in climate change mitigation

High future discounting rates favor inaction on present expend- ing while lower rates advise for a more immediate political action. A possible approach to this key issue in global economy is to take historical time series for nominal interest rates and inflation, and to construct then real interest rates and finally obtaining the resulting discount rate according to a specific stochastic model. Extended periods of negative real interest rates, in which inflation dominates over nominal rates, are commonly observed, occurring in many epochs and in all countries. This feature leads us to choose a well-known model in statistical physics, the Ornstein-Uhlenbeck model, as a basic dynamical tool in which real interest rates randomly fluctuate and can become negative, even if they tend to revert to a positive mean value. By covering 14 countries over hundreds of years we suggest different scenarios and include an error analysis in order to consider the impact of statistical uncertainty in our results. We find that only 4 of the countries have positive long-run discount rates while the other ten coun- tries have negative rates. Even if one rejects the countries where hyperinflation has occurred, our results support the need to consider low discounting rates. The results provided by these fourteen countries significantly increase the prior- ity of confronting global actions such as climate change mitigation. We finally extend the analysis by first allowing for fluctuations of the mean level in the Ornstein-Uhlenbeck model and secondly by considering modified versions of the Feller and lognormal models. In both cases, results remain basically unchanged thus demonstrating the robustness of the results presented

Discounting the Distant Future

If the historical average annual real interest rate is m \u3e 0, and if the world is stationary, should consumption in the distant future be discounted at the rate of m per year? Suppose the annual real interest rate r ( t ) reverts to m according to the Ornstein Uhlenbeck (OU) continuous time process dr ( t ) = α[ m – r ( t )] dt + kdw ( t ), where w is a standard Wiener process. Then we prove that the long run rate of interest is r ∞ = m – k 2 /2α 2 . This confirms the Weitzman-Gollier principle that the volatility and the persistence of interest rates lower long run discounting. We fit the OU model to historical data across 14 countries covering 87 to 318 years and estimate the average short rate m and the long run rate r ∞ for each country. The data corroborate that, when doing cost benefit analysis, the long run rate of discount should be taken to be substantially less than the average short run rate observed over a very long history

Chaos and chaotic phase mixing in cuspy triaxial potentials

This paper investigates chaos and chaotic phase mixing in triaxial Dehnen potentials which have been proposed to describe realistic ellipticals. Earlier work is extended by exploring the effects of (1) variable axis ratios, (2) `graininess' associated with stars and bound substructures, idealised as friction and white noise, and (3) large-scale organised motions presumed to induce near-random forces idealised as coloured noise with finite autocorrelation time. Three important conclusions are: (1) not all the chaos can be attributed to the cusp; (2) significant chaos can persist even for axisymmetric systems; and (3) introducing a supermassive black hole can increase both the relative number of chaotic orbits and the size of the largest Lyapunov exponent. Sans perturbations, distribution functions associated with initially localised chaotic ensembles evolve exponentially towards a nearly time-independent form at a rate L that correlates with the finite time Lyapunov exponents associated with the evolving orbits. Perturbations accelerate phase space transport by increasing the rate of phase mixing in a given phase space region and by facilitating diffusion along the Arnold web that connects different phase space regions, thus facilitating an approach towards a true equilibrium. The details of the perturbation appear unimportant. All that matters are the amplitude and the autocorrelation time, upon which there is a weak logarithmic dependence. Even comparatively weak perturbations can increase L by a factor of three or more, a fact that has potentially significant implications for violent relaxation.Comment: 17 pages, 17 figures -- revised and extended manuscript to appear in Monthly Notices of the Royal Astronomical Societ

The value of the distant future: Discounting in random environments

We analyze how future costs must be balanced against present costs. This is traditionally done using an exponential function with a constant discount rate. The choice of discount rate can dramatically effect the question on what is the value of the future. This is specially critical for environmental problems such as global warming, and it has generated a controversy as to the urgency for immediate action (Stern, 2006; Nordhaus, 2007a,b). We briefly review the issue for the nonspecialist and take into account the randomness of the economic evolution by studying the discount function of three widely used processes for the dynamics of interest rates: OrnsteinUhlenbeck, Feller and log-normal. We also outline our previous empirical survey on 14 countries over time spans ranging up to more than 300 years We estimate the parameters of one of the models studied (the Ornstein-Uhlenbeck process) and obtain the long-run discount rate for all these countries. The long-run discount obtained for stable countries (countries that have not suffered periods of destabilizing inflation) supports the low discounting rate proposed by Stern (2006) over higher rates that have been advocated by others (Nordhaus, 2007a,b)

Discounting the Distant Future: What Do Historical Bond Prices Imply about the Long-Term Discount Rate?

We present a thorough empirical study on real interest rates by also including risk aversion through the introduction of the market price of risk. From the viewpoint of complex systems science and its multidisciplinary approach, we use the theory of bond pricing to study the long-term discount rate to estimate the rate when taking historical US and UK data, and to further contribute to the discussion about the urgency of climate action in the context of environmental economics and stochastic methods. Century-long historical records of 3-month bonds, 10-year bonds, and inflation allow us to estimate real interest rates for the UK and the US. Real interest rates are negative about a third of the time and the real yield curves are inverted more than a third of the time, sometimes by substantial amounts. This rules out most of the standard bond-pricing models, which are designed for nominal rates that are assumed to be positive. We, therefore, use the Ornstein–Uhlenbeck model, which allows negative rates and gives a good match to inversions of the yield curve. We derive the discount function using the method of Fourier transforms and fit it to the historical data. The estimated long-term discount rate is 1.7% for the UK and 2.2% for the US. The value of 1.4% used by Stern is less than a standard deviation from our estimated long-run return rate for the UK, and less than two standard deviations of the estimated value for the US. All of this once more reinforces the need for immediate and substantial spending to combat climate change

Discounting the distant future: What do historical bond prices imply about the long term discount rate?

We present a thorough empirical study on real interest rates by also including risk aversion through the introduction of the market price of risk. With the view of complex systems science and its multidisciplinary approach, we use the theory of bond pricing to study the long term discount rate. Century-long historical records of 3 month bonds, 10 year bonds, and inflation allow us to estimate real interest rates for the UK and the US. Real interest rates are negative about a third of the time and the real yield curves are inverted more than a third of the time, sometimes by substantial amounts. This rules out most of the standard bond pricing models, which are designed for nominal rates that are assumed to be positive. We therefore use the Ornstein-Uhlenbeck model which allows negative rates and gives a good match to inversions of the yield curve. We derive the discount function using the method of Fourier transforms and fit it to the historical data. The estimated long term discount rate is $1.7$ \% for the UK and $2.2$ \% for the US. The value of $1.4$ \% used by Stern is less than a standard deviation from our estimated long run return rate for the UK, and less than two standard deviations of the estimated value for the US. All of this once more reinforces the support for immediate and substantial spending to combat climate change.Comment: 27 pages, 6 figure

Collective Decision-Making in Ideal Networks: The Speed-Accuracy Tradeoff

We study collective decision-making in a model of human groups, with network interactions, performing two alternative choice tasks. We focus on the speed-accuracy tradeoff, i.e., the tradeoff between a quick decision and a reliable decision, for individuals in the network. We model the evidence aggregation process across the network using a coupled drift diffusion model (DDM) and consider the free response paradigm in which individuals take their time to make the decision. We develop reduced DDMs as decoupled approximations to the coupled DDM and characterize their efficiency. We determine high probability bounds on the error rate and the expected decision time for the reduced DDM. We show the effect of the decision-maker's location in the network on their decision-making performance under several threshold selection criteria. Finally, we extend the coupled DDM to the coupled Ornstein-Uhlenbeck model for decision-making in two alternative choice tasks with recency effects, and to the coupled race model for decision-making in multiple alternative choice tasks.Comment: to appear in IEEE TCN

Value of the future: discounting in random environments

We analyze how to value future costs and benefits when they must be discounted relative to the present. We introduce the subject for the nonspecialist and take into account the randomness of the economic evolution by studying the discount function of three widely used processes for the dynamics of interest rates: Ornstein- Uhlenbeck, Feller, and log-normal. Besides obtaining exact expressions for the discount function and simple asymptotic approximations, we show that historical average interest rates overestimate long-run discount rates and that this effect can be large. In other words, long-run discount rates should be substantially less than the average rate observed in the past, otherwise any cost-benefit calculation would be biased in favor of the present and against interventions that may protect the future