2.4 Elementos que componen una Micro-Central Hidroeléctrica
2.4.1 Toma de Agua
3.3.1.1 Expanded Single-Risk Factor Model
In this model, the recovery capability Ci,td is set to be a function of the latent variables Xtd
(the state of economy at the default year),Xtd+1, . . . , and Xtd+T (the state of economy at the
resolution year), where T denotes the workout duration. However, the impact of each latent variable towards the recovery capabilityCi,td is unknown.
The issue is that the LGD is observable only at the end of the workout process. In the course of the workout process, costs and recoveries can be realised, but most of the components remain uncertain until the resolution time, such as recoveries from unsold collaterals or future legal fees. Thus, the final LGD reflects the accumulated impact of the systematic factor during the whole workout duration. To isolate the systematic influence towards the LGD for each workout year is a difficult task, especially if only the final LGD is observable. The simplest possible model which incorporates the time-series of latent variables, would be
Ai,td =p·Xtd+ p 1−p2·ZA i, Ci,td =qtd·Xtd+. . .+qtd+T·Xtd+T+ q 1− ||q||2 2·Z C i , where 0<p<1 andZiA,ZCi ∼
N
(0,1). (M2)The coefficientq= (qtd, . . . ,qtd+T)is an element inside the(T+1)-dimensional unit circle ex-
cluding the origin. The vanilla model M1 is a special case of the expanded model M2. The spe- cific model M1 would be an LGD model by vintage of default, in particular forq= (1,0, . . . ,0). Unlike other latent variable based models in the literature (see section 3.2.3), we do not impose any assumption on the dependence structure of the latent variable time series(Xt)t∈N, i. e. how Xt andXsare dependent to each other for any given yeartands. Note that this model still relies on a single-factor, since(Xt)t∈Ndescribes a time-series of one systematic risk factor.
The coefficientpdescribes the sensitivity of the asset valueAi,td towardsXtd. The restriction
a higher default rate. Similarly, the coefficientsqtd, . . . ,qtd+T describe the sensitivity ofCi,td
towards the latent variables during the workout years. The restriction ofqt to be positive (for eacht) is not necessary from a technical point of view. A negative qt only implies negative correlation betweenXt andCi,td. This case may be rational from an economical perspective for
a large gap betweent andtd. The signs and magnitudes ofqtd, . . . ,qtd+T give hints on which
vintage models have the most explanatory power.
There are economic arguments supporting a highqtd as well as those supporting highqtd+T.
Loosely speaking, the coefficientqgives information, which year within the workout duration is the most "responsible" for the systematic effects in the realised LGD. It is not clear beforehand, how the coefficientsq will behave when the model is fed with workout LGD data. Different arguments supporting different propositions exist:
1. High systematic sensitivity at the default year(in line with the vintage of default). The empirical evidence on a high PD-LGD correlation (such as Frye (2003); Altman et al. (2004)) ties a loan’s LGD to its default time rather than to the rest of the workout periods. The fact that the default occurred in a downturn year contributes to the low market value of the collateral and the low cure chance of the defaulted obligor. This translates directly into a highqtd. This proposition is related closely to the plain vanilla model M1, which
performs well for the market-based LGD.
2. High systematic sensitivity at the resolution year(in line with the vintage of resolu- tion/recovery). The largest portion of a typical bank loan portfolio consists of secured loans. The collateral usually accounts for the predominant share of the recoveries and the workout process often stops soon after the collateral is sold. This implies that the LGD is tied mostly to the resolution time, which means a highqtd+T.
3. High systematic sensitivity near the default year(a weaker version of the vintage of default). Alternatively, one may argue that cash inflows (but also outflows) are relevant factors for calculating LGD. These transactions occur most often in the first years after the default event, which implies a highqtd orqtd+1. Eventually, this parameter continues
3.3. METHODS AND DATA 61
to decrease as the default gets older.
The model developed in this essay can principally cover each conceivable time pattern of re- coveries. The previous three recovery structures are likely to be the most common ones. We emphasise that the model does not exclude the market-based LGD. The model is intended to cover a typical bank, which may have a mixed portfolio containing exposures with workout LGDs and exposures with market-based LGDs.
3.3.1.2 Estimation Techniques
It is necessary to find out the latent variables impact on the LGD values to design an adequate downturn LGD estimation based on a latent variable approach. The coefficient q decodes in which workout year the LGDs are particularly sensitive towards the latent variables. There are two central issues regarding any estimation technique of q: 1) the uncertainty about the dependence structure of(Xt)t∈N, and 2) the uncertainty about the relationship betweenX and
LGD (or equivalently betweenCi,td and LGD). In this essay, both uncertainties will remain open
to avoid any unintentional influence on the result.
The maximum likelihood method, similar to Frye (2000b), is applied to estimatep. Accord- ing to Gordy and Heitfield (2002), the maximum likelihood method produces less bias (coming from a lack of data) than the method of moments. The likelihood function can be derived through the theoretical distribution of the conditional PD, i. e. the distribution ofgA(Xtd):=E[Di|Xtd](see
E5). We have established that the functiongA is invertible and differentiable inX. The change- of-variable technique produces the density function ofPDX =E[Di|Xtd], which is
fPDX(y) =fX(g −1 A (y))· dg−A1(y) dy =ϕ Φ−1(PDi)− p 1−p2Φ−1(y) p · p 1−p2 p · dΦ−1(y) dy , (E6)
where ϕ is the density function of the standard normal distribution. If the parameter PDi is known, the estimation of p is reduced to a one-dimensional problem. The maximum of the
likelihood function can be approximated numerically, which yields the estimatedp. By applying the estimatedpin the equation E5, the impliedXt can be calculated for each yeart.
A similar approach to estimateqwould require the information on the joint distribution of
(Xt)t∈N, referring to the first uncertainty mentioned above. It seems unrealistic and overly sim-
plified to assume that the latent variables are intertemporally independent or follow a particular Markovian process, i. e. today’s state is only influenced by that of yesterdays. A non-Markovian behaviour of the(Xt)t∈Nseems to be more realistic. However, specifying one can be challenging
and testing whether it is true even more difficult.
By using the realisations of E[LGDi|Xtd, . . . ,Xtd+T] and (Xt)t∈N, the coefficient q can be
estimated via a regression methodology. At this point, we are confronted with the second uncertainty mentioned above. In contrast to the function gA, the exact form of the function
gC(Xtd, . . . ,Xtd+T):=E[LGDi|Xtd, . . . ,Xtd+T]is not known
2. Choosing a particular form ofg
C
seems arbitrary. With different asset and exposure types, various jurisdictions, or even the insti- tution’s internal strategy, the correct form is most likely complex.
Even thoughgCremains unknown, we can safely assume that this function is locally smooth, i. e. (at least one time) partially differentiable, at a chosen valuex:= (xtd, . . . ,xtd+T). The idea
is to construct its Taylor series representation at the chosen value x. This value x can serve both as an evaluation point as well as a conservative value representing a downturn event. Two possibilities emerge on the function behaviour at the evaluation point: either 1) the functiongC is linear (or similar to one) or 2) the function is substantially different from a linear function. Avoiding this step by taking an assumption may substantially simplify many things, but it puts our analysis in the same bucket as the currently existing models in the literature. Since this essay aims to offer an alternative to the current downturn LGD methodology, which potentially results in a fatal LGD underestimation, this amount of thoroughness is necessary.
In the first case, the Taylor series representation only contains the first partial derivative and 2A simple linear relationship or restrictions on possible LGD values are the typical assumptions in the literature
3.3. METHODS AND DATA 63
the conditional LGD can be written as
E[LGDi|Xtd, . . . ,Xtd+T] =gC(xtd, . . . ,xtd+T) + td+T
∑
s=td ∂gC(xtd, . . . ,xtd+T) ∂Xs (Xs−xs) and E[LGDi|Xtd, . . . ,Xtd+T] =µ−σ qtdXtd+. . .+qtd+TXtd+T+ q 1− ||q||2 2Z C i | {z } Ci,td . (E7.1)Both parametersµ and σare intended to be the sum of the constant terms and a scaling
factor to fulfil the unit circle requirement ofq. Note thatE[ZCi |Xtd, . . . ,Xtd+T] =E[Z
C
i ] =0. The mean and standard deviation of the portfolio LGDs may serve as estimators for bothµandσ.
However, this should not be confused with the LGDs at the loan level.
By looking at the structure of both equations in E7.1, the OLS regression method would require data samples, in which the idiosyncratic factorZC is zero or minimal. The OLS method consequently produces an indirect estimation of q, which is σcqt for allt. This requirement can be achieved by constructing samples ofE[LGDi|Xtd, . . . ,Xtd+T] from a large portfolio. In
a large (fine-grained) portfolio, the idiosyncratic risk converges to zero and is intertemporally uncorrelated. Thus, the property that the OLS estimator is unbiased (fromσqt) is guaranteed by the Gauss-Markov theorem.
In the other case, where the functiongC is believed to be non-linear in at least one of its parameter at the evaluation point, then the Taylor series representation would produce a non- zero rest termR(Xtd, . . . ,Xtd+T).
E[LGDi|Xtd, . . . ,Xtd+T] =gC(xtd, . . . ,xtd+T) + td+T
∑
s=td ∂gC(xtd, . . . ,xtd+T) ∂Xs (Xs−xs)+ R(Xtd, . . . ,Xtd+T) (E7.2)The information on the rest termR(Xtd, . . . ,Xtd+T)(in short: R) should reside in the OLS residu-
als and the intercept. A problem occurs if the downturn impact onRis far stronger than the linear effect, then the coefficientsqdo not hold much information weight for the conditional LGD. In this particular case, the coefficientqmay be biased and the result (in form of a downturn LGD
estimation) will likely to fail the performance tests (see section 3.5).