Non-Linear Discounting and Default Compensation: Valuation of Non-Replicable Value and Damage: When the Social Discount Rate may become Negative

Abstract

In this short note we develop a model for discounting.A focus of the model is the discounting, when discount factors cannot be derived from market products. That is, a risk-neutralizing trading strategy cannot be performed. This is the case, when one is in need of a risk-free (default-free) discounting, but default protection on funding providers is not traded. For this case, we introduce a default compensation factor (exp(+ λT )) that describes the present value of a strategy to compensate for default (like buying default protection would do).In a second part, we introduce a model, where the survival probability depends on the required notional. This model is different from the classical modelling of a time-dependent survival probability (exp(−λT )). The model especially allows that large liquidity requirements are instantly more likely do default than small ones.Combined the two approaches build a framework in which discounting (valuation) is nonlinear.The non-linear discounting presented here has several effects, which are relevant in various applications:• If we consider the question of default-free valuation, i.e., factoring in the cost of default protection, the framework can will lead to over-proportional higher values (or cost) for large projects (or damages). The framework can lead to the effect that discount-factors for very large liquidity requirements or projects are an increasing function of time. It may even lead to discount factors larger than one. This may have relevance in the assessment of event like climate change.• For the valuation of defaultable products, e.g., like a defaultable swap, the framework leads to the generation of a continuum of (defaultable) par rate curves (interest rate curve) and the valuation of a payer and a receiver swap differs by more than just a sign.