A Simple Factoring Pricing Model
AA Simple Factoring Pricing Model
Ilaria Nava ∗ , Davide Cuccio, Lorenzo Giada and Claudio Nordio July 31, 2019
Working paper Abstract
In a simplified setting, we show how to price invoice non-recourse factoring taking into ac-count not only the credit worthiness of the debtor but also the assignor’s one, together withthe default correlation between the two. Indeed, the possible default of the assignor mightimpact the payoff by means of the bankruptcy revocatory, especially in case of undisclosedfactoring.
JEL
Classification codes: G12, G13, G33, G38
AMS
Classification codes: 91B25, 91B70
Keywords:
Factoring, Credit Risk, Bankruptcy, Default Correlation, Kendall’s Tau, Gum-bel Copula.
Conventional practices in non-recourse factoring (see Figure 1) rely on the debtor’s creditwor-thiness as a major determinant for pricing, or often consider it as the unique relevant parameterin the derivation of the expected payoff of an operation. In most cases, indeed, the default ofthe invoice debtor is assumed as the unique event with an actual impact on factoring proceed-ings. As a consequence, the standard pricing approach typically neglects the occurrence of othercircumstances which might condition the outcome of an operation, thus limiting the number offeatures considered in the computation of its fair value.According to this view, the problem of pricing in invoice factoring can be simply representedby defining a set of complementary probabilistic events, which are centered on the conditionsdeemed as crucial for the realization of the payoff. In the case of standard non-recourse factoring,therefore, such event corresponds to: τ 1) Issuance of the receivable to the debtor and creation of the credit claim with maturity T .2) Assignment of the receivable to the factor upon payment of the purchase price C (1 − α ) andtransfer of the enforcement power for the credit reimbursement to the assignee. where τ is the time to default of the debtor, and T the expected time of repayment of thereceivable (i.e. the operation’s maturity). When the default event occurs, i.e. τ Events : P ayof f s :(a) τ A <τ B 1, in thefirst term of this payoff, represents the discount charged by the assignee on the nominal amountof the invoice upon its purchase, as discussed above. This quantity αC should be put back to theaccount seller in case of application of a clawback action. As mentioned, this always goes alongwith the return of the credit claim towards the ceded debtor to the assignor, or, in case the claimhas already been cashed in, with the return of the proceeds collected by the assignee until thedefault of the seller. The second term of the payoff reflects instead the positive recovery that thefactor may obtain from the redistribution of the bankruptcy proceedings of A. As the assigneeis entitled to lodge a claim equal to its original financial exposure to the defaulted assignor, thisterm is expressed as r A C (1 − α ), where 0 < r A < C (1 − α ) the original exposure.It emerges, therefore, that the payoff foreseen in case of bankruptcy revocatory of the assignoris a function of the assignee’s purchase price itself:Price = C (1 − α ) . (8)We can sum up all the events associated to a certain payoff, in order to get a simpler viewof the actual events in scope. By following the same passages as in section 2, the assignee’sexpected payoff boils down to:Price = r B C · E ( τ B 1) Payment by the debtor of the due receivable amount C at the envisaged maturity T .2) Default of the assignor in τ A > T and return of the factoring proceedings C collected bythe assignee to the invoice seller, together with the fee gained on the credit, αC . During theredistribution of the bankruptcy proceedings of the assignor, the assignee recovers a fraction ofits financial exposure to the assignee, r A C (1 − α ) . which, in terms of the corresponding probabilities and of the relation in (8), returns :Price = r B C · P ( τ B < T, τ A > ∆) + C · P ( τ B > T, τ A > ∆) + ( r A C (1 − α ) − αC ) · P ( τ A < ∆)= r B C · P ( τ B < T, τ A > ∆) + C · P ( τ B > T, τ A > ∆) − C · P ( τ A < ∆)1 − (1 + r A ) · P ( τ A < ∆) . (10)Equation (10) shows therefore, how factoring operations could be priced when executed ina framework which envisions the application of revocatory actions in case of bankruptcy of theseller. The introduction of such regime makes the factor’s payoff sensitive to the probabilityof default of its assignor, as it also appears by comparing equations (10) and (7). This impliesthat, ceteris paribus, the lower the creditworthiness of the assignor, the higher its probabilityof defaulting within the bankruptcy suspect period, and thus the higher the probability of theassignee to incur a financial loss due to the revocatory procedure. As a consequence, the expectedpayoff of the factor tends do decrease as the the default probability of the assignor increases.This effect is only mitigated by the possibility for the assignee to recover a fraction r A of itsfinancial exposure in case of default of the seller, as also indicated by the denominator of (10).Vice versa, the higher the probability of the assignor not to default within such period, thehigher the probability of the assignee to receive a higher positive payoff (the debtor’s recoveryrate r B C or the full credit amount C ), jointly depending on the default probability of the debtor.This suggests that the correlation between the time to default of the assignor and the debtorplays now a considerable role in the definition of the price. As it will be better illustrated in thenext section, a larger correlation acts by shifting the expected payoff towards the full recoveryof the receivable amount. For more details, see the appendix at section A.1 Application In this section we will apply the pricing formula (10) to show how it works in a simple case, wherewe define a specific distribution for the random variables τ A , τ B . We assume, therefore, that thedefault times have a bivariate exponential distribution obtained combining exponential marginaldistributions with constant default intensity with a Gumbel copula, disallowing simultaneousdefaults. This particular choice of copula is common in the recent literature, and has theadvantage of being analytically tractable. The resulting joint survival probability reads: P ( τ A > t A , τ B > t B ) = e − [( λ A t A ) θ +( λ B t B ) θ ] θ (11)with θ ∈ [1 , ∞ ), while the constant default intensity λ x gives an exponential form to the marginalsurvival probability P ( τ x > t ) = e − λ x t . Observe that Kendalls Tau (a non-parametric measureof dependence between two random variables) τ K = 1 − θ , so that the independent casecorresponds to θ = 1, and the comonotonic (fully dependent) case to θ = ∞ . From eq.(11), thejoint cumulative probability function of the default times is: F A,B ( t A , t B ) = P ( τ A < t A , τ B < t B ) = e − [( λ A t A ) θ +( λ B t B ) θ ] θ − e − λ A t A + 1 − e − λ B t B (12)and the joint probability density function is: f A,B ( τ A , τ B ) = (cid:20) θ − (cid:16) λ θA τ θA + λ θB τ θB (cid:17) θ (cid:21) (cid:16) λ θA τ θA + λ θB τ θB (cid:17) ( θ − ) · λ θA λ θB τ θ − A τ θ − B e − [( λ A τ A ) θ +( λ B τ B ) θ ] θ . (13)Having set this out, we can simply convert the probabilities featuring the pricing equation(10) in terms of the specified joint distribution as: P ( τ B < T, τ A > ∆) = (cid:90) ∞ ∆ dτ A (cid:90) T f A,B ( τ A , τ B ) dτ B (14) P ( τ B > T, τ A > ∆) = (cid:90) ∞ ∆ dτ A (cid:90) ∞ T f A,B ( τ A , τ B ) dτ B (15) P ( τ A < ∆) = (cid:90) ∞ dτ B (cid:90) ∆0 f A,B ( τ A , τ B ) dτ A (16)Thus, the overall is results equal to :Price = r B C (cid:18) − e − [( λ A ∆) θ +( λ B T ) θ ] θ + e − λ A ∆ (cid:19) + C (cid:18) e − [( λ A ∆) θ +( λ B T ) θ ] θ (cid:19) − C (cid:0) − e − λ A ∆ (cid:1) e − λ A ∆ − r A (1 − e − λ A ∆ )= C (1 − r B ) e − [( λ A ∆) θ +( λ B T ) θ ] θ − e − λ A ∆ (1 + r B ) e − λ A ∆ (1 + r A ) − r A . (17)This formulation provides a better insight on how the correlation parameter θ enters intothe definition of the price. An increase of the default correlation will cause a reduction of theprobability of the factor to get the recovery payoff r B C , i.e. the probability that debtor defaults The constant default intensity λ x gives an exponential form to the survivalship probability P ( τ x > t ) = e − λ x t For more details, see the appendix at section A.2. θ . Furthermore, as stated before, the payoff will discount a higher reduction as theprobability of default of the assignor increases (eq. (16)), other things being unchanged .Below, we show these results assuming a suspect period of half the length of the factoringmaturity (∆ = 0 . T = 1), and an invoice nominal amount C = 100. For simplicity, weassume that the recovery rates from the default of A and B are both equal to 20%, so that r A = r B = 20%. The expected payoff of the assignee is first computed according to the conven-tional pricing formula in (7), i.e. considering only the debtor’s default probability. We assumethat the debtor’s default intensity is λ B = 0 . 1, which corresponds to a P D B = 9 . 5% - in linewith a single B rating - and a credit spread of 7.6% on an ordinary loan. The pricing equation(17) is then applied for comparison purposes, firstly taking the default intensity of the assignoras λ A = 0 . 1, and secondly as λ A = 0 . 2, corresponding to a credit spread twice as large as thefirst one. θ τ K Prices λ B =0.1 λ B =0.1, λ A =0.1 λ B =0.1, λ A =0.21 0.00 92.387 88.164 83.6292 0.50 92.387 91.011 88.0893 0.67 92.387 91.606 89.3094 0.75 92.387 91.796 89.8735 0.80 92.387 91.866 90.199Table 1: Application of pricing formulas (7) and (17) for different values of θ , and defaultintensities λ A , λ B ; ∆ = 0 . T = 1.In the next table we show the results when the time to repayment of the loan T = 0 . These results are obviously influenced by the specific choice of copula we used in this example, which impliesa symmetric correlation between τ A and τ B . Alternative copula models, as those implying a skewed correlationbetween the times to default, could be a valid substitute approach to the one discussed, and lead to differentresults, for example by increasing the correlation between default, without modifying that between survival. τ K Prices λ B =0.1 λ B =0.1, λ A =0.1 λ B =0.1, λ A =0.21 0.00 96.09835 87.42007 77.384722 0.50 96.09835 90.44666 80.953483 0.67 96.09835 91.07899 81.380434 0.75 96.09835 91.28085 81.450815 0.80 96.09835 91.35512 81.46387Table 2: Application of pricing formulas (7) and (17) for different values of θ , and defaultintensities λ A , λ B ; ∆ = 1 and T = 0 . Assessing the creditworthiness of a debtor is a key element for non-recourse invoice factoring, asit affects the evaluation by factoring facilities of the actual profitability of such operations. Asa matter of fact, the standard pricing techniques in this field often focus on the debtor’s defaultas the only relevant event for price determination, while neglecting the potential role played bythe default of the receivable assignor, or its correlation with the debtor’s one. These events, infact, turn out to be relevant whenever they exert a direct binding effect on the payoff of thefactor, as it happens, for instance, by means of the application of the bankruptcy revocatoryprocedure. Under this regime, the factor credit claim towards the debtor, or the proceedingsalready collected from the latter, are put back to the receivable seller in case of declaration of itsbankrupt, provided that the receivable sale contract was finalized in a period of time (usually sixmonths to one year) prior to the declaration of default. This determines, especially in the case ofnon notification factoring, the inability of the assignee to enforce its position towards the cededdebtor. Furthermore, it implies the obligation for the assignor to return the invoice discountoriginally retained upon purchase and thus a potential financial loss at the end of the revocatoryprocedure. The larger the assignor’ s probability of default, therefore, the higher the probabilityfor the assignee to fall into such revocatory framework, which translates, eventually, into a lowerexpected payoff. In addition to this, the assignee’s return does also depend on the joint riskinessof the assignor and the debtor, as the relative timing of their default events can substantiallyinfluence the payoff of the operation. In particular, the higher the default correlation betweenthe two, the higher, ceteris paribus, their probability to jointly survive until the time due for therepayment of the receivable, allowing the assignee to receive the value of the outstanding invoice.These considerations feature therefore a different pricing approach than the conventional one,as it admits, under bankruptcy revocatory, the inclusion of the creditworthiness of the assignorand its correlation with the debtor’s default as additional determinants of factoring prices. Acknowledgments We are grateful to Corrado Passera for encouraging our research.9 Appendix A.1 Derivation of equation (10) We briefly derive in this section the pricing equation (10), by substituting the relation in (9)into the assignee’ s expected payoff:Price = r B C · P ( τ B < T, τ A > ∆) + C · P ( τ B > T, τ A > ∆) + ( − αC + r A C (1 − α )) · P ( τ A < ∆) . As both the discount αC and the financial exposure of the assignee C (1 − α ) are clearly afunction of the purchase price of the invoice, it is necessary to make the latter explicit into theabove equation. Hence, replacing equation (9) into the previous one, we obtain:Price = r B C · P ( τ B < T, τ A > ∆) + C · P ( τ B > T, τ A > ∆) + (Price − C + r A Price) · P ( τ A < ∆)which, factoring out the price variable, returns:Price (1 − (1 + r A ) · P ( τ A < ∆)) = r B C · P ( τ B < T, τ A > ∆)+ C · P ( τ B > T, τ A > ∆) − C · P ( τ A < ∆)and eventually:Price = r B C · P ( τ B < T, τ A > ∆) + C · P ( τ B > T, τ A > ∆) − C · P ( τ A < ∆)1 − (1 + r A ) · P ( τ A < ∆) . (18) A.2 Derivation of equation (17) As stated in section 4, equation (17) is obtained as the sum of the three integrals in equations(14) - (16) multiplied by the respective payoffs.As for equation (14) one has: P ( τ B < T, τ A > ∆) = (cid:90) ∞ ∆ dτ A (cid:90) T f A,B ( τ A , τ B ) dτ B . (19)Making use of the well-known definition of the marginal and joint cumulative functions of τ A , τ B , i.e. F A ( τ A ), F B ( τ B ) and F A,B ( τ A , τ B ), we can more easily compute the above expressionas: P ( τ B < T, τ A > ∆) = F B ( T ) − F AB ( τ B = T, τ A = ∆)= (cid:16) − e − λ B T (cid:17) − (cid:18) e − [( λ A ∆) θ +( λ B T ) θ ] θ − e − λ A ∆ + 1 − e − λ B T (cid:19) = − e − [( λ A ∆) θ +( λ B T ) θ ] θ + e − λ A ∆ . (20)Accordingly, equation (15) gives: P ( τ B > T, τ A > ∆) = (cid:90) ∞ ∆ dτ A (cid:90) ∞ T f A,B ( τ A , τ B ) dτ B = 1 − F A (∆) − F B ( T ) + F AB ( τ B = T, τ A = ∆)= 1 − (cid:16) − e − λ A ∆ (cid:17) − (cid:16) − e − λ B T (cid:17) + e − [( λ A ∆) θ +( λ B T ) θ ] θ − e − λ A ∆ + 1 − e − λ B T = e − [( λ A ∆) θ +( λ B T ) θ ] θ , (21)10hich nicely agrees with the result reported in equation (11).Equation (16) is trivially: P ( τ A < ∆) = (cid:90) ∞ dτ B (cid:90) ∆0 f A,B ( τ A , τ B ) dτ A = F A (∆)= 1 − e − λ A ∆ (22)By summing up the contributions in equations (20) - (22) one immediately obtains the result inequation (17). References [1] L. Giada, C. Nordio , Breaking break clauses , RISK Magazine, March 2013[2] A. McNeil, R. Frey, P. Embrechts , Quantitative Risk Management , Princeton Uni-versity Press, 2005[3]