Standard risk measures evaluate a portfolio at a single future time horizon. But real risk involves multiple horizons — the same position may be held for a day, a week, or a month, and the risk assessment should account for the possibility that the relevant horizon is itself uncertain.
The paper builds risk measures that incorporate horizon risk alongside interest rate uncertainty. The tool is backward stochastic differential equations (BSDEs), which define the risk measure dynamically: the risk at time t depends on the future evolution, which is itself uncertain.
Cash subadditivity replaces cash additivity. A cash-additive risk measure says: adding $1 of cash reduces risk by exactly $1. This assumes a known risk-free rate. With uncertain interest rates, adding $1 of cash reduces risk by an amount that depends on which horizon materializes and what rates prevail — generally less than $1 at longer horizons and more at shorter ones.
The hq-entropic risk measure demonstrates the framework's content. It belongs to the family of generalized shortfall risk measures (quantifying the expected cost of exceeding a threshold) but is not a certainty equivalent (not derivable from a utility function). Classical entropic risk measures are all three — BSDE, shortfall, and certainty equivalent — but the horizon-risk generalization breaks the equivalence. The three characterizations that agree in the single-horizon case diverge when the horizon is uncertain.
Risk at an uncertain horizon is not the same as risk at a fixed horizon with uncertain returns. The horizon is a separate source of risk, and its interaction with interest rate uncertainty creates new mathematical structure.