Loss Given Default Estimations in Emerging Capital Markets

Abstract

This paper proposes an approach to decompose the RR/LGD model development process with two stages, specifically, for the RR/LGD rating model, and to calibrate the model using a linear form that minimizes residual risk. The residual risk in the recovery of defaulted debts is determined by the high uncertainty of the recovery level according to its average expected level. Such residual risk should be considered in the capital requirements for unexpected losses in the loan portfolio. This paper considers a simple residual risk model defined by one parameter. By developing an optimal RR/LGD model, it is proposed to use a residual risk metric. This metric gives the final formula for calibrating the LGD model, which is proposed for the linear model. Residual risk parameters are calculated for RR/LGD models for several open data sources for developed and developing markets. An implied method for updating the RR/LGD model is constructed with a correction for incomplete recovery through the recovery curve, which is built on the training sets. Based on the recovery curve, a recovery indicator is proposed which is useful for monitoring and collecting payments. The given recommendations are important for validating the parameters of RR/LGD model.

The Estimation Procedure of the Calculated Standard Error for the Average Marginal Share of Repayment

The solution of problem (5) gives the optimal values of the recovery period T and the limiting recovery R_∞. The error of the values depends on the quality statistics of the approximation of the cumulative recovery of the recovery curve (4). The linear problem of the parameter estimation question θ = {R, T} for the non-linear regression problem (τ) = ρ_τ(θ) + δ_τ ∙ ε_τ , near the optimal solution θ of problem (5) is given a linear regression relation for the error Δθ = θ − θ in the standardized form:

$$ \frac{RR\left(\tau \right)-{\rho}_{\tau}\left(\theta \right)}{\delta_{\tau }}=\frac{\partial_{\theta }{\rho}_{\tau }}{\delta_{\tau }}\ \Delta \theta +{\varepsilon}_{\tau }, $$

(42)

where ∂_θρ_τ is composed by the n × 2 partial derivatives matrix $\left[\frac{\partial\ }{\partial R}{\rho}_{\tau}\left(R,T\right),\frac{\partial\ }{\partial T}{\rho}_{\tau}\left(R,T\right)\right],{\varepsilon}_{\tau }$ assumed to be normal uncorrelated random variable with unknown variance for each recovery period τ , of which there are n. Apparently, for an optimal solution in the sense of equation (5) for θ, the solution of problem (A1) for Δθ will be obvious Δθ = 0. However, the error Δθ will be expressed through the covariance matrix according to the well-known formula (see, for example, Strutz 2016):

$$ \mathit{\operatorname{cov}}\left(\Delta \theta \right)={\left({\left[\frac{\partial_{\theta }{\rho}_{\tau }}{\delta_{\tau }}\right]}^T\times \left[\frac{\partial_{\theta }{\rho}_{\tau }}{\delta_{\tau }}\right]\right)}^{-1}\bullet \frac{RSS}{n-2}, $$

(43)

where for (A1):

$$ RSS={\sum}_{\tau}\frac{1}{{\delta_{\tau}}^2}{\left( RR\left(\tau \right)-{\rho}_{\tau}\left(\theta \right)\right)}^2. $$

Denoting the partial derivatives as:

$$ {\rho}_{\tau }=R\bullet \left(1-{e}^{-\frac{\tau }{T}}\right); $$

$$ {\partial}_R{\rho}_{\tau }=1-{e}^{-\frac{\tau }{T}}; $$

(44)

$$ {\partial}_T{\rho}_{\tau }=-R{e}^{-\frac{\tau }{T}}\frac{\tau }{T^2}, $$

and according for the estimation error R, the only the upper diagonal element of the matrix cov(Δθ), it is needed to obtain

$$ \delta {R}^2=\frac{1}{n-2}\bullet \frac{-{\sum}_{\tau}\frac{\partial_R{\rho}_{\tau}\bullet {\partial}_T{\rho}_{\tau }}{{\delta_{\tau}}^2}\bullet {\sum}_{\tau}\frac{{\left( RR\left(\tau \right)-{\rho}_{\tau}\right)}^2}{{\delta_{\tau}}^2}}{\sum_{\tau}\frac{{\partial_R{\rho}_{\tau}}^2}{{\delta_{\tau}}^2}\bullet {\sum}_{\tau}\frac{{\partial_T{\rho}_{\tau}}^2}{{\delta_{\tau}}^2}-{\left({\sum}_{\tau}\frac{\partial_R{\rho}_{\tau}\bullet {\partial}_T{\rho}_{\tau }}{{\delta_{\tau}}^2}\right)}^2}. $$

(45)

To estimate the error R_∞ as the measure for the standard deviation δR_∞, it is necessary in formula (45) to substitute the solution of problem (5) as R—the limiting recovery R_∞, the time for recovery T, and δ_τ² = δRR_Avg²(τ) or δRR_w²(τ), these replacements depend on the calculation of the recovery curve.