Level dependent annuities : defaults of multiple degrees

Abstract

Motivated by the risk of stopped debt coupon payments from a leveraged company in financial distress, we value a level dependent annuity contract where the annuity rate depends on the value of an underlying asset-process. The range of possible values of the asset is divided into a finite number of regions. The annuity rate is constant within each region, but may differ between the regions. We consider both in finite and finite annuities, with or without bankruptcy risk, i.e., bankruptcy occurs if the asset value process hits an absorbing boundary. Such annuities are common in models of debt with credit risk in financial economics. Suspension of debt service under the US Chapter 11 provisions is one well-known real-world example. We present closed-form formulas for the market value of such multi-level annuities contracts when the market value of the underlying asset is assumed to follow a geometric Brownian motion.