SOCIAL DISCOUNTING AND THE LONG RATE OF INTEREST
Dorje C. Brody and
Lane P. Hughston
Mathematical Finance, 2018, vol. 28, issue 1, 306-334
The wellâ€ known theorem of Dybvig, Ingersoll, and Ross shows that the long zeroâ€ coupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the longâ€ 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 â€œsimpleâ€ interest rate (or Libor rate) is finite, then this rate (unlike the zeroâ€ 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 behavior of the resulting discount bond systems. The conditions necessary for the existence of such â€œhyperbolicâ€ and â€œgeneralized hyperbolicâ€ long rates are those of soâ€ called social discounting, which allow for longâ€ term cash flows to be treated as broadly â€œjust as importantâ€ as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrageâ€ free valuation framework for the costâ€ benefit analysis and risk management of longâ€ term social projects, such as those associated with sustainable energy, resource conservation, and climate change.
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