controlling Risk in Accounts Receivable Management

1、controlling Risk in Accounts Receivable ManagementControlling Risk in AccountsReceivable ManagementJAMES C. T. MAO, andProfessor of Finance, University of British ColumbiaCARL ERIK SARNDALProfessor of Management Science, University of British ColumbiaA critical aspect in the granting of trade credit

2、 is the control of risk ofbad debt losses. Financial writers have generally measured this risk bythe expected value of the probability distribution of these losses. KalmanJ. Cohen and Frederick S. Hammer i, for example, define the returnon credit sales as the difference between gross profit and expe

3、cted baddebt losses. Similarly, Dun and Bradstieet Inc. 2 suggests that a firmshould make a credit sales to a customer only if the operating profiton the sales exceeds the expected bad debt losses. This focus on expectedlosses raises an important question: how well does this risk measurepredict actu

4、al bad debt losses? The purpose of this paper is to answerthis question by studying the variance of the ratio of these expectedlosses to sales under various assumptions regarding the number and sizedistribution of customers, as well as the degree of correlation betweentheir default risks.Basic Assum

5、ptionsSuppose a firm is selling on credit to n customers each of whom has anon-zero probability of default. Let Si stand for the sales to customerI and let Pi stand for customer Is probability of defeault. Then, jsbad debt, Bi, has a probability distribution given by:8D _f<Si with probability Pi,

6、.*o with probability i-PiDenote -B* and Si by fi and T respectively. Since EBi) = SiPi, theJ. Business Finance & Accounting i, 3. Printed in Great Britain396 J. C. T. Mao and C. E. Sarndalratio of total expected bad debt losses to sales is accordingly given by theexpression:- .(2)To determine ho

7、w well EBIT) predicts bad debt losses, we need toknow the variance of (BIT).To calculate the variance of the bad debt ratio BIT, we must alsospecify the correlation between the amounts of bad debt, Bt. Betweenthe extreme cases of complete independence and perfect correlation,various degrees of corre

8、lation are possible. We shall discuss both theextreme and the intermediate situations. To simplify the analysis, weshall assume that pi, the coefficient of correlation, is positive andconstant between the amounts of bad debt Bi and Bj for any pair ofcustomers i and ?.Derivation of FormulasBecauseB=

9、tuthe variance of B is given by the expression:Var B)= t Var (5<) + Yl Cov (B<5) . (3)t = li?0?jUsing the probability distribution of Bt given by (i), and introducingPij to denote the probability that customers i and / default jointly, wecan derive the variances and covariances in equation (3)

10、 as follows:YzrBi) = at = Pii-Pi)Si .(4)Cov Bu Bj) = cTi = EBtBj) - E(Bt)EBj)= Pi-PiPi)SiSi .(5)It follows that the correlation pij between the amounts of bad debtBi and Bj isThis formula clarifies the relation between p< and Pij. For example, ifBi and Bj aie uncorrelated, then pij = o, and Pij,

11、the probability of jointControlling Risk in Accounts Receivable Management 397default, must equal PiPj. On the other hand, if Bi and Bj are perfectlycorrelated, then pi)=i, and if P< and Pj are both small, then Pij isapproximately equal toNext substitute (4) and (5) into (3). Invoking the assumpt

12、ion of aconstant correlation, pi = p, for all pairs of customers i,j 3,Var B=andVarfIj = (I - p)|: P<(i - Pi).(7)Equation (7) has rather simple interpretations. If p = i, equation (7)becomes:.(8)Note that -v/Pi(i -Pi) is the standard deviation of bad debt per dollarof sales to customer i. Hence,

13、in the case of perfect correlation, Var BIT)is the square of the weighted average of these per-doUar standarddeviations of the individual jBis, using the relative sizes of the customers,SilT, as weights. Alternatively, Var (BIT) can be viewed as the squareof the sum of the standard deviations of the

14、 Pis, divided by the squareof total sales.If the bad debts, Bi and B, for any pair of customers i and j areindependent, then p = o, and (7) becomes:Var BIT)= tPii -Pi) (SilT) . (9)In this case, to calculate Var (BIT), we again multiply the per-dollarstandard deviation of each Bi by the relative size

15、 of customer t,398 y. C. T. Mao and C. E. SamdalBut, instead of adding these products, as we would in taking a weightedaverage, we first square each product and then add to get the value ofVar BIT). Alternatively, Var BIT) can be viewed as simply the sumof the variances of the JBis, divided by the s

16、quare of total sales.If there is some degree of correlation (but less than perfect) betweenthe Bis, then o<p<i, and (7) cannot be simplified any further. In thatcase, Var BIT) is equal to the weighted average of the values calculatedaccording to equations (8) and (9), using p and i /> as we

17、ights.Further simplifications of the above formulas are possible if allcustomers are of the same size. If 151 = 52= . =<Sn, then SilT= i/n.In equation (8), the case of perfect correlation, a weighted average isnow a simple average. In equation (9), the case of independence, theformula for Var BjT

18、) becomes:whereandVar (P) = if: (Pi-7r)2ni=iare respectively the mean and variance of the Pis. Observe that if n isfixed in equation (10), Var BIT) varies inversely with the variance ofthe Pis. That is, for a given average risk of default, the Var BjT) willbe greater if all customers have identical

19、Pis than if they have differingPis.An ApplicationIn interpreting the results, we shall consider first a firm whose clienteleconsists of a large number of relatively small customers under threeassumptions concerning the degree of association between their defaultControlling Risk in Accounts Receivabl

20、e Management 399risks: independent, some degree of correlation, and perfect correlation.Then we shall explain how risk will be altered if the firm also sells to asmall number of large customers.Case IConsider a firm which has extended $200 of trade credit to each of its500 customers. For each of its

21、 customers, the firm has estimated aprobability of default. Several customers, /?0?0, may have the same esti-mated probability of default, P?0?0. Thus, according to columns i and 2 inTable i, there are/i=5o customers each with a 0-1% probability ofdefault, /a=2oo customers each with 0-5% probability

22、 of default, etc.Foi this set of data, EBjT), the ratio of expected bad debt to total salesis equal to 0-0271.TABLE ICOMPUTATION OF VARIANCE OFTHE RATIO OF BAD DEBT TO SALESNumber Salesof Probability perForp = o,Forp=i,customers of default customercomputecomputefiPiSi%500-0012001,998-00316-072000-00

23、520039,800-002821-35o-oio20059,400-002984-96150500-01520029.550-001215-520-05020028,500-0015653-8310O-IOO20036,000-00600-00100-20020064,000-00800500-00476-9750-45020049,500-00497-9750-50020050,000-00500-00Total500404,248-0010,866-67p=o: Var BIT)-P<)S?0?02 = 0-0000404400 J. C. T. Ma

24、o and C. E. SarndalTo calculate Var BjT) under the independence assumption 4, wefirst calculate total values of T and/<Pi(i -Pi)Si (see columns 3 and 4of Table i). Dividing X!/tPi(r -Pi)Si by T Ogives us a value of 0-0000404for Var BjT). The standard deviation of BIT equals 0-0064, which isless t

25、han one-fourth of the expected value of 0-0271. We are assuminghere that the clientele consists of a large number of small customers withindependent default risks. In this situation, EBjT) provides a reason-ably accurate prediction of what the actual value of BIT will be. Notethat if these 500 custo

26、mers had identical probabilities of defaultP< = 77=0-0271, then Var (P) would be zero and Vat BjT) would beslightly higher, namely, equal to 77(1 7r)/n = 0-0000527 (see equationfio). This illustrates the inverse relationship between Var (P) andVar (BIT), which we observed earlier.Next, suppose th

27、e customers are relatively small, but perfectlycorrelated. To calculate Var (BIT), we first compute /i/P<(i - Pi) 5<(see column 5 of Table i). Dividing the square of ZfiVPiti-Pi) Siby r gives us a value of o-oi 18074 for Var (BIT). To appreciate thisthe entire business of a firm is coming from

28、 one customer. With riskmagnitude, one need only observe that the standard deviation of BITequals 0-1087, which is over four times as large as 0-0271, the expectedvalue of BIT There is, therefore, a high probability that BjT willdeviate widely from E(BIT). This situation is equivalent to one whereth

29、us correlated, E(BIT) does not provide an accurate prediction ofwhat the actual value of BIT will be.Knowing the values of V(BjT) under the extreme cases of independ-ence and perfect correlation enables us to compute the value of V(BIT)under the realistic assumption of less than perfect correlation.

30、 Suppose,for example, p has a value of 0-3. We find that Var (BjT) has a value of0-0035705, which, using equation (7), is obtained as a weighted averageof the extreme values associated with independence and perfect corre-lation:Var (P/7) = (o-oii8o74)o-3 + (0-0000404)0-7 = 0-0035705The power of E(BI

31、T) as a predictor of actual losses is thus seen to varyinversely with the value of p.Case IIThe above analysis assumes that the firms customers are all relativelysmall. But what if the firm also sells to a few relatively large customers.?Controlling Risk in Accounts Receivable Management 401For exam

32、ple, suppose instead of the original distribution, sales arenow distributed among customers in the manner outlined in Table 2.Total sales still equal $100,000, but there are now three large customersmaking credit purchases of $5000, 3io,ooo and $20,000. The remaining$65,000 are distributed equally a

33、mong 325 small customers eachaccounting for $200 of sales. This new distribution is derived from thatof Table i by viewing a certain percentage of small customers as a singlefirm in those customer groups with the three lowest probabilities ofdefault.TABLE 2NEWDISTRIBUTION OF SALES AMONGCUSTOMERSNumb

34、er ofcustomersProbabilities of defaultSales per customer/*PiSi$25o-ooi200Io-ooi5,0001000-005100I0-00520,000100O-OIO200IO-OIO10,000500-015200150-050200IOO-IOO200100-20020050-350-20050-45020050-500200Without going through the computation, we present the followingvalues for the expectation and variance

35、 of BIT:pE(BjT)Var (BjT)00-02710-00033690-30-02710-0037780I0-0271o-oi 18074For p = o and /> = o-3, the new values of Var (BjT) are larger than thecorresponding values when the firms clientele consisted of only small402 J. C. T. Mao and C. E. Sarndalcustomers. The merging of smaller customers into

36、 larger economicunits has the effect of raising the value of /> to i among the merged units,and this explains the higher values of Var (BjT). It should be noted thatgiven the value of p, the increase in Var (BjT) depends on the relativesize of the large customers and the probabilities of these cu

37、stomersdefaulting. If p=i to begin with, however, the value of Yar (BjT) isalready as large as it could be. This explains why merging smallercustomers into larger units does not, in this case, alter the value ofVar (BjT).SummaryThe purpose of this paper is to examine the conditions under which theex

38、pected bad debt ratio may be used to provide an accurate predictionof what the actual bad debt will be. The approach we have used hasbeen to derive formulas for the variance of the bad debt ratio underdifferent assumptions regarding the number and size distribution ofcustomers, as well as the degree

39、 of correlation between their defaultrisks. The results indicate that the prediction is best if a firms customersare small and there is a wide distribution of sales among them. Theprediction will not be accurate if the default risks are closely correlated,for example, when sales are concentrated in a few large customers.Referencesi Kalman J. Cohen and Frederick S. Hammer, A