Implied fractional hazard rates and default risk distributions

Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. By their definition, they imply a unique probability density function. The applications of default probability distributions are varied, including the risk premium model used to price default bonds, reliability measurement models, insurance, etc. Fractional probability density functions (FPD), however, are not in general conventional probability density functions (Tapiero and Vallois, Physica A,. Stat. Mech. Appl. 462:1161–1177, 2016). As a result, a fractional FPD does not define a fractional hazard rate. However, a fractional hazard rate implies a unique and conventional FPD. For example, an exponential distribution fractional hazard rate implies a Weibull probability density function while, a fractional exponential probability distribution is not a conventional distribution and therefore does not define a fractional hazard rate. The purpose of this paper consists of defining fractional hazard rates implied fractional distributions and to highlight their usefulness to granular default risk distributions. Applications of such an approach are varied. For example, pricing default bonds, pricing complex insurance contracts, as well as complex network risks of various granularity, that have well defined and quantitative definitions of their hazard rates. C.S. Tapiero ( ) Topfer Chair Distinguished Professor of Financial Engineering and Technology Management, Department of Finance and Risk Engineering, New York University Tandon School of Engineering, 6 Metro Tech, 22201 Brooklyn, New York, USA e-mail: [email protected] P. Vallois Universite de Lorraine, Institut de Mathematiques Elie Cartan de Lorraine, INRIA.BIGS, CNRS UMR 7502 BP 239, F-54506, Vandoeuvre-Les-Nancy, France Page 2 of 14 C.S. Tapiero and P. Vallois Introduction Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. Their applications are varied including the risk premium model used to price default bonds, reliability measurement models, insurance, etc. A hazard rate implies a unique conventional probability density and its cumulative distribution. By the same token, a fractional hazard rate, defined by the application of a fractional operator, implies as well a conventional probability density and its cumulative distribution. As a result, implied fractional hazard distributions can be used to enrich the family of default probability distributions we use to analyze data of various granularities in finance and insurance. In other words, fractional statistical models providing a granularity index may be used to define implied and complete probability distributions. Unlike the definition of a fractional probability distribution based on Liouville’s and Caputo’s fractional operators that generally define “incomplete” distributions (see Tapiero and Vallois (2016) on fractional randomness), a fractional distribution implied by its fractional hazard rate provides a complete distribution. The purposes of this paper are to define fractional hazard rates and their properties as well as their implied distributions which we apply to several examples associated with insurance and risk models. Fractional hazard rates unlike conventional hazard rates are functions of their mathematical granularity. They include for example, the choice of a fractional operator applied such as that of Riemann-Liouville, 1832, Grunwald (1867), Letnikov (1868), Caputo (1967), consisting in a theoretical (and computational) approach to calculating the integral of a distribution (or a derivative). For example, if time is sampled in time intervals (t) , 0 < H < 1 rather than 0 < t < 1, then, necessarily (t) > t . These differences have statistical and informational implications that lead necessarily to computational differences. By the same token, at their infinitesimal limits when continuous time integrals are calculated, their outcomes may also differ. Some of their computational and theoretical differences have been pointed out to the implications of a “speed of convergence by the Berry-Esseen (1941) lemma (see also recent developments pointing out to a greater computational precision, Korolev and Shevtsova 2010a,b, Shevtsova 2007, 2008). These studies consider the approximation of a continuous function based on the definition of its approximating sums. Applications of fractional operators also lead to “memory models” that depend on the definition of the fractional derivative and therefore depend on the computational properties of such derivatives. Are these forward, or backward derivatives etc. When a derivative is defined at a given instant of known time relative to another past instant then of course, past memory is implied in current estimates. When “memory models” are based on future times, current estimates are auto-correlated with future states. These relationships have led to the definition of a “long run memory” which is an expression of autocorrelation since a granular time interval (t) > t is necessarily greater than intervals of time in the Riemannian calculus when the index 0 < H < 1 is applied (see also Baillie 1996, Beran 1992). For example, the future price of a stock at a future time t has a price correlated with that of its current price, (say at time τ , with t > τ ). These properties are Probability, Uncertainty and Quantitative Risk (2017) 2:2 Page 3 of 14 well known and have been pointed out repeatedly by Mandelbrot (1963). However, it seems that all fractional models have such a property thereby reflecting a property commonly encountered in data analyses of various granularities. Autocorrelation or long run memory, however may be obtained in several manners. Mandelbrot used a fractional volatility to define a fractional Brownian Motion whose variance is auto-correlated, Laskin (2003) applied a Liouville fractional operator to the set of differential-equations that defined Poisson processes (see also Baleanu et al. 2010, Podlubny, 1999)., Meltzer and Klafter 2004 applied fractional operators to Fokker Planck partial differential equations, and so on. In Tapiero and Vallois (2016), we have used a fractional operator to define a fractional random variable and its associated distributions (which was proved to be unconventional). In this paper we suggest a fractional hazard rate to define conventional fractional distributions and suggest that the fractional hazard rate is a reasonable and consistent approach to define fractional default distributions and therefore useful in the definition of risk models that define the increased (or decreased) risks that occur due to model granularity. Applications of this approach are considered as well, including default bonds reliability as well as insurance. Fractional hazard rates Let f : [0,∞[→ [0,∞[ be a default probability density function (PDF) and its cumulative distribution function (CDF), F(t) := ∫ t 0 f (τ)dτ , t ≥ 0. Let h(t) be the hazard rate at time t: h(t) = f (t) 1− F(t) , t ≥ 0, with F(t) = ∫ t 0 f (τ)dτ < 1, ∀t ≥ 0. (1) A continuous-time hazard rate is therefore the derivative of − ln (1− F(t)): h(t) = − d dt [ ln (1− F(t)) ] , t ≥ 0. (2) Note that with h > 0, and ∫ t 0 h(τ)dτ < ∞, for any t > 0 and ∫ t 0 h(τ)dτ = +∞. Given h,the default probability density function and its cumulative distribution function are uniquely defined by: f (t) = h(t) exp ( − ∫ t