Level-Dependent Annuities: Defaults of Multiple Degrees

Motivated by the effect on valuation of stopped or reduced debt coupon payments from a 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 this asset is divided into a finite number of regions, with constant annuity rates within each region. We present closed-form formulas for the market value of level-dependent annuities contracts when the market value of the underlying asset is assumed to follow a geometric Brownian motion. Such annuities occur naturally in models of debt with credit risk in financial economics. Our results are applied for valuing both corporate debt with suspended interest payments under the U.S. Chapter 11 provisions and loans with contractual level-dependent interest rates.