EAD Estimates for Facilities with Explicit Limits

Abstract

The estimation of exposure at default, EAD, for a facility with credit risk, has received a lot of attention, principally in the area of counterparty risk and has focused on situations where the variability of the exposure is due to: the existence of variability in the underlying variables of a derivative; the use of a fixed nominal amount not expressed in the presentation currency; or the existence of collateral whose value (variable over time), reduces the exposure. Less attention has been given to the case of loan commitments with explicit credit limits. In this case, the source of variability of the exposure is the possibility of additional withdrawals when the limit allows this. The implementation of Basel II is forcing credit institutions to address this problem in a rigorous, transparent and objective manner. Moreover, Basel II imposes a set of minimum conditions on the internal EAD estimates in order to allow the use of these as inputs in the calculation of the minimum capital requirement. Currently, credit institutions have problems meeting the requirements of both the data and the methodologies.

Appendices

Appendix A. Equivalence Between Two Minimisation Problems

Proposition: Consider a set of observations $O = {\left\{ {\left( {{x_i},{y_i}} \right)} \right\}_{i = 1, \ldots, n}}$ and the problem G.I given by:

$$ \begin{array}{llllll} Minimis{e_{g \in G}}\left[ {\sum olimits_{i = 1}^n {L\left( {{y_i} - g({x_i})} \right)} } \right] \hfill \\ {\hbox{\it Subject}}\,\,{{to}}\;f(x) \geq g(x) \geq \,h(x) \hfill \\\end{array} $$

(11.58)

where the error is measured in terms of the function L that satisfies:

$$ L(x + y) = L(x) + L(y)\quad if {\ }x \cdot y \geq 0 $$

(11.59)

then, g is a solution of Problem G.I if and only if it is a solution of Problem G.II given by:

$$ \begin{array}{llllll} Minimis{e_{g \in G}}\left[ {\sum olimits_{i = 1}^n {L\left( {Min\left[ {Max\left[ {{y_i},h({x_i})} \right],f({x_i})} \right] - g({x_i})} \right)} } \right] \hfill \cr Subjec t\,to\,\,f(x) \geq g(x) \geq h(x) \hfill \\\end{array} $$

(11.60)

Proof: The set O can be partitioned into three classes $O = {O^{+} }\coprod {O^{-} }\coprod {O^= }$, where:

$$ {O^{+} } = \{ ({x_i},{y_i})\left| {{y_i} > f({x_i})} \right.\}, \;{O^{-} } = \{ ({x_i},{y_i})\left| {{y_i} < h({x_i})} \right.\} $$

(11.61)

For observations in O ⁺:

$$ \left( {{y_i} - f\left( {{x_i}} \right)} \right) \cdot \left( \,{f\left( {{x_i}} \right) - g({x_i})} \right) \geq0 $$

(11.62)

Therefore, from (11.59) and (11.62), the error in Problem G.I associated with an observation in O ⁺ can be expressed in terms of the error in Problem G.II plus an amount independent of g:

$$ \begin{array}{ll} err\left[ {GI,\left( {{x_i},{y_i}} \right)} \right] = L\left( {{y_i} - g\left( {{x_i}} \right)} \right) = L\left( {{y_i} - f\left( {{x_i}} \right) + f\left( {{x_i}} \right) - g\left( {{x_i}} \right)} \right) \\ = L\left( {{y_i} - f\left( {{x_i}} \right)} \right) + L\left( {f\left( {{x_i}} \right) - g\left( {{x_i}} \right)} \right) \\ = L\left( {{y_i} - f\left( {{x_i}} \right)} \right) + L\left( {Min\left[ {Max\left[ {{y_i},\;h\left( {{x_i}} \right)} \right],\;f\left( {{x_i}} \right)} \right] - g\left( {{x_i}} \right)} \right) \\ = L\left( {{y_i} - f\left( {{x_i}} \right)} \right) + err\left[ {GII,\left( {{x_i},{y_i}} \right)} \right] \\\end{array} $$

(11.63)

But the O ⁺ set does not depend on the function$g$, therefore for these observations, and for all g, the error in Problem G.I can be decomposed in a fixed amount, independent of the g function, given by $\sum {L\left( {{y_i} - f\left( {{x_i}} \right)} \right)}$, where the index i applies at the observations in O ⁺ and the error in Problem G.II.

Similarly, for observations in O ⁻, the error in Problem G.I is equal to the error in Problem G.II plus the fixed amount$\sum {L\left( {h\left( {{x_i}} \right) - {y_i}} \right)}$.

Finally, for the observations in O ⁼ the errors in Problem G.I and in Problem G.II are the same.

Appendix B. Optimal Solutions of Certain Regression and Optimization Problems

Let X and Y be random variables with joint distribution given by F(x,y), then we get in the case of a quadratic loss function

$$ {d^*}(x) = E\left\langle {Y\left| X \right.} \right\rangle = \mathop {{Min}}\limits_{d(x)} \left\{ \mathop {E}\limits_{F*} \left\langle {{{\left( {Y - d(X)} \right)}^2}} \right\rangle \right\}. $$

(11.64)

In the case of the linear asymmetric loss function, with a > 0 and b > 0:

$$ L(x) = \left\{ {\begin{array} {lll} {a \cdot x\quad iff\,x \geq 0} \\{b \cdot x\quad iff\,x < 0} \\\end{array} } \right. $$

(11.65)

The following is found

$$ {d^*}(x) = Q\left\langle {Y\left| {X,\frac{b}{{a + b}}} \right.} \right\rangle = \mathop {{Min}}\limits_{d(x)} \left\{ \mathop {E}\limits_{F*} \left\langle {L\left( {Y - d(X)} \right)} \right\rangle \right\} $$

(11.66)

See, for example, Pratt et al. (1995, pp. 261–263).

Therefore, a solution for (11.28) can be obtained from (11.64), and taking into account:

$$ Y = \frac{{EAD - E}}{\omega };\;d\left( {X = RD} \right) = LEQ(RD) \cdot h(RD);\;{\hbox{where}}\;h(RD) = \frac{{L - E}}{\omega } $$

(11.67)

Then, d* is given by (11.64) and assuming that h(RD) = h(f) for observations in RDS(f):

$$ {d^*}\left( {X = RD(\,f)} \right) = E\left\langle {\frac{{EAD - E}}{\omega }\left| {{RD}} \right.} \right\rangle = \overline {LEQ} (RD(\,f)) \cdot \frac{{L(\,f) - E(\,f)}}{{\omega (\,f)}} $$

(11.68)

The result showed in (11.29) is obtained from the former equation.

Appendix C. Diagnostics of Regressions Models

11.1.1 Model II (Sect. 11.7.2.1)

• By using original data:

$$ \frac{{EA{D_i}}}{{L\left( {td - 12} \right)}} - \frac{{E\left( {td - 12} \right)}}{{L\left( {td - 12} \right)}} = 0.64 \cdot \left( {1 - \frac{{E\left( {td - 12} \right)}}{{L\left( {td - 12} \right)}}} \right) $$

(11.69)

• By using censored data:

$$ \frac{{EA{D_i}}}{{L\left( {td - 12} \right)}} - \frac{{E\left( {td - 12} \right)}}{{L\left( {td - 12} \right)}} = 0.7 \cdot \left( {1 - \frac{{E\left( {td - 12} \right)}}{{L\left( {td - 12} \right)}}} \right) $$

(11.70)

• By using a variable time approach:

$$ \frac{{EA{D_i}}}{{L\left( {tr} \right)}} - \frac{{E\left( {tr} \right)}}{{L\left( {tr} \right)}} = 0.49 \cdot \left( {1 - \frac{{E\left( {tr} \right)}}{{L\left( {tr} \right)}}} \right) $$

(11.71)

11.1.2 Model I (Sect. 11.7.2.2)

• By using Model I, variable time approach:

$$ LEQ(\,f) = - 0.82 + 1.49 \cdot \sqrt {{1 - E(\,f)/L(\,f)}} $$

(11.72)

The diagnostics for this regression model are:

11.1.3 Model III (Sect. 11.7.2.3)

• By using a variable time approach:

$$ \begin{array}{llllll} Median\left[ {EAD(\,f) - E(\,f)} \right] = 86.8 + 0.76 \cdot \left( {L(\,f) - E(\,f)} \right) \hfill \\Quantile\left[ {EAD(\,f) - E(f),0.666} \right] = 337.8 + 0.92 \cdot \left( {L(\,f) - E(\,f)} \right) \hfill \\\end{array} $$

(11.73)

With the diagnostics given by:

and for the quantile:

Appendix D. Abbreviations

AIRB	Advanced internal ratings-based approach
CCF	Credit conversion factor
CF	Conversion factor
EAD	Exposure at default
EAD i  = E(td)	Realised exposure at default associated with O i
EAD(f)	EAD estimate for f
ead i	Realised percent exposure at default, associated with O i
E(t)	Usage or exposure of a facility at the date t
e(t)	Percent usage of a facility at the date t
e i  = e(tr)	Percent usage associated with the observation O i={g, tr}
f	Non-defaulted facility
g	Defaulted facility
i = {g, tr}	Index associated with the observation of g at tr
IRB	Internal ratings-based approach
LEQ	Loan equivalent exposure
LEQ(f)	LEQ estimate for f
LEQ i	Realised LEQ factor associated with the observation O i
LGD	Loss given default
L(t)	Limit of the credit facility at the date t
O i	Observation associated with the pair i = {g, tr}
PD	Probability of default
Q a  = Q(x, a)	Quantile associated with the a% of the distribution F(x)
RDS	Reference data set
RDS(f)	RDS associated with f
RD	Risk drivers
S(tr)	Status of a facility at the reference date tr
t	Current date
td	Default date
tr	Reference date
td − tr	Horizon