Long‐term risk with stochastic interest rates

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.