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

• Published in: Mathematics (Basel) 2024-03, Vol.12 (5), p.645
• Main Authors: Farmer, J. Doyne, Geanakoplos, John, Richiardi, Matteo G., Montero, Miquel, Perelló, Josep, Masoliver, Jaume
• Format: Article
• Language: English
• Subjects: Analysis; climate; Climate change; Complex systems; Costs; discount; Discount rates; Discounts; Economic growth; Environmental economics; Fourier transforms; Global temperature changes; Inflation (Finance); Interest rates; Inversions; Ornstein–Uhlenbeck; Parameter estimation; Prices; Pricing; Standard deviation; stochastic process; Time series; United Kingdom; Yield curve
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12050645

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.