Social Discounting and the Long Rate of Interest

Abstract

The well-known theorem of Dybvig, Ingersoll and Ross shows that the long zerocoupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the long-term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long "simple" interest rate (or Libor rate) is finite, then this rate (unlike the zero-coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. New interest rate models are constructed, under this hypothesis and certain generalizations thereof, that illustrate explicitly the good asymptotic behaviour of the resulting discount bond systems. The conditions necessary for the existence of such "hyperbolic" and "generalized hyperbolic" long rates are those of so-called social discounting, which allow for long-term cash flows to be treated as broadly "just as important" as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrage-free valuation framework for the costbenefit analysis and risk management of long-term social projects, such as those associated with sustainable energy, resource conservation, and climate change.

Long-Term Risk: A Martingale Approach

“…We note that for pricing kernels with λ = 0, the limiting result in Theorem 3.3 degenerates as the limiting exponential yield vanishes, since in this case discounting at the exponential rate is too fast. In particular, consider the case where P t 0 = O(t −γ ) (see Brody and Hughston (2016) for their model of social discounting). In this case, we have a similar limiting result for the power yield.…”

The Long-Term Swap Rate and a General Analysis of Long-Term Interest Rates

“…In particular we focus on the long-term yield and the long-term simple rate, which have been already defined in the literature (cf. [27] and [7]). The long-term yield can be defined in different ways.…”

“…In [7] it is suggested to consider a particular model for the long-term simple rate for the discounting of cashflows occuring in a distant future. By using exponential discount factors the discounted value of a long-term project, that will be realised over a long time horizon, in most cases will turn out to be overdiscounted, hence too small to justify the overall project costs.…”

“…It was first shown in [14] and then discussed in [22], [25], [26], [28], and [34]. According to [7] the DIR theorem ultimately implies that discounted cashflows with higher time-to-maturity are over-penalised, so that the use of this long-term interest rate becomes unsuitable for the valuation of projects having maturity in a distant future. To overcome this issue, in [7] the authors propose to use for discounting the long-term simple rate, which is defined as the simple spot rate with an infinite maturity.…”

Theory of Cryptocurrency Interest Rates

“…Beginning with the work of Constantinides [6], finance theorists have learned to think about interest-rate modelling in a more general way, in terms of so-called pricing kernels. The pricing kernel method avoids some of the technical issues that arise with the introduction of the risk neutral measure and the selection of a preferred numeraire asset in the form of a money market account, and at the same time it leads to interesting new classes of interest rate models (see, e.g., [7][8][9][10][11] and references cited therein). By a pricing kernel, we mean an {F t }-adapted càdlàg semimartingale {π t } t≥0 satisfying (a) π t > 0 for t ≥ 0, (b) E [ π t ] < ∞ for t ≥ 0, and (c) lim inf t→∞ E [π t ] = 0, with the property that if an asset with value process {S t } t≥0 delivers a bounded F T -measurable cash flow H T at time T and derives its value entirely from that cash flow, then…”