Long-term risk with stochastic interest rates

Abstract

In constant-rate markets, the average stochastic discount factor growth rate coincides with the instantaneous rate. When interest rates are stochastic, this average growth rate is given by the long-term yield of zero-coupon bonds, which cannot serve as instantaneous discount rate. We show how to reconcile the stochastic discount factor growth with the instantaneous relations between returns and rates in stochastic-rate markets. We factorize no-arbitrage prices and isolate a rate adjustment that captures the short-term variability of rates. The rate-adjusted stochastic discount factor features the same long-term growth as the stochastic discount factor in the market but has no transient component in its Hansen–Scheinkman decomposition, capturing the long-term interest rate risk. Moreover, we show how the rate adjustment can be used for managing the interest rate risk related to fixed-income derivatives and life insurances.