Long-Term Risk with Stochastic Interest Rates

Investors with heterogeneous trading horizons require compensation for the exposure to different risks. The no-arbitrage valuation over increasing horizons is described by the evolution of stochastic discount factors (SDFs). Each of them exhibits a multiplicative decomposition into deterministic growth term, permanent and transient component, provided by Hansen and Scheinkman (2009). In particular, the growth rate captures the deterministic discounting for risks that are relevant in the long term. When interest rates in the market are constant, the SDF growth rate coincides with the instantaneous rate. On the contrary, when rates of interest are stochastic, the SDF growth rate is given by the long-term yield of zero-coupon bonds, which is unsuitable for instantaneous no-arbitrage valuation.

We show how to reconcile the long-run properties of the SDF with the instantaneous relations between returns and rates in stochastic-rate markets. In particular, we introduce a rate adjustment in pricing that isolates the short-term variability of rates. No-arbitrage prices are then factorized into rate-adjusted prices and a rate adjustment that is absent when interest rates are constant. Rate-adjusted prices employ constant yields to maturity for discounting future payoffs and constitute indifference prices for investors that neglect the term structure of rates. The rate-adjusted SDF features the same long-term growth rate of the SDF in the market but has no transient component in its Hansen-Scheinkman decomposition. Therefore, rate-adjusted prices provide the proper valuation for long-term interest rate risk and shed light on the impact of temporary and long-lasting interest rate shocks on security prices.