Characterizing and Forecasting Market Bid Functions in ... - CiteSeerX

Characterizing and Forecasting Market Bid Functions in Treasury Bill Auctions*

Raphaële PREGET and Patrick WAELBROECK** CREST LEI and EUREQua

February 2001

Abstract We use the results of 125 discriminatory French Treasury bill auctions to determine the factors that affect the shape of the aggregate demand functions. We proceed in two steps. In the first step, we estimate the parameters of a logistic specification. In the second step, we explain fluctuations across auctions by economic variables. Estimated parameters are used to forecast the results of incoming auctions.

*

We thank Michael Landsberger, Michael Visser and Bernard Lebrun for helpful comments and suggestions on

earlier versions of this article. Dataset was kindly provided by the French Treasury for academic purposes only. **

CREST LEI, 28 rue des Saints Pères, 75007 Paris, France.

1. Introduction

The aim of this article is to study the empirical properties of market bid functions in French Treasury bill auctions during the year 1995. We define the market bid function as the aggregate demand curve of auction bidders as a function of the interest rates. This empirical study does not test a theoretical model of auction. On the contrary, we try to underline stylized facts about the market bid function for Treasury bills. The main reason that justifies this empirical approach is the difficulty to model Treasury bills auctions from a theoretical point of view. Constructing a sophisticated theoretical model to account for the complex environment of Treasury bills auctions is a real challenge. Indeed, Treasury bill auctions concern a divisible good and bidders can ask for variable quantities. Moreover, these auctions take place while other markets offer perfectly substitutable products. Finally, other special features of French Treasury bill auctions, such as non-competitive bids, increase the difficulty to obtain a relevant theoretical model. However, theoretical and empirical research on auctions has been recently particularly active and there are more and more studies on auctions of divisible goods. Back and Zender (1993), Viswanathan and Wang (1997), Ausubel and Cramton (1998) and Wang and Zender (1999) provide recent theoretical models on Treasury auctions. Nevertheless, theoretical models only give a limited answer to the multiple questions arising from Treasury bill auctions. This article is also motivated by the availability of rich data sets resulting from auction tables. Data from Treasury bill auctions are rich and interesting. They provide complete and reliable information. We believe that it is important to exploit the richness of these data sets for a better understanding of the mechanisms underlying such auctions even with limited

2

theoretical background. In general, the empirical literature on Treasury bill auctions concerns either the relative performance of different auction mechanisms, or the comparison of prices that result from the auction with prices that prevail on other government security markets. Most empirical studies compare the discriminatory auction with the uniform price auction.1 One such studies is due to Bolten (1973), who asked the following question: « Should the discriminatory auction be maintained? » Considering that participation to Treasury bill auctions is larger with the uniform price mechanism, the author concludes that the answer is negative if the noncompetitive demand remains constant. Conclusions from the empirical study of Umlauf (1993) on Mexican Treasury bill auctions are similar: the uniform price auction is better than the discriminatory auction. However, the revenue difference for the Treasury comes from the empirical fact that collusion can not be maintained in the uniform price auction. On the contrary, Simon (1994), using data from the U.S. Treasury during the 1970s, finds that the uniform price mechanism increases the borrowing cost for the U.S. government. Moreover, the study by Heller and Lengwiler (1997) on Swiss bonds shows that the cost difference between the two mechanisms is not significantly different from zero. However, these authors base their conclusions on a specific theoretical model that does not take account of strategic interactions between bidders. Boukai and Landsberger (1998) and AtleBerg, Boukai and Landsberger (1998) (BL and ABL in the remaining of the article) are the only empirical studies on market bid functions we are aware of. Using Israeli and Norwegian data sets, these authors find that market bid

1 These two types of auctions only differ in the payment rule. Indeed, the bid transmission mechanism and the allocation rule are identical. However, in the uniform price auction, winners pay the stop-out price, which is the lowest accepted price. In the discriminatory auction, each bidder pays the proposed prices.

3

functions differ across auctions. However, their analysis of more than 300 auctions reveals that the chaotic aspect of these bid functions can partly be eliminated when they normalize auction rates by those prevailing on the secondary market. Moreover, they identify for almost all auctions an S shaped form that can well be estimated using a logistic growth curve. They believe that fluctuations from one auction to the other can be explained by random perturbations on the parameters of the logistic curve. Hence, they construct standard market bid functions in order to compare the discriminatory auction with the uniform price auction. ABL study market bid functions for Norwegian Treasury bills in a discriminatory auction and the market bid functions for Norwegian Treasury bonds in a uniform price auction. They show that, under some hypotheses, the Norwegian Treasury could have saved a lot of money by using discriminatory auctions also for bonds. Our study follows the approach initiated by BL and ABL. We proceed in two steps. In the first step, we estimate parameters of the market bid functions using a logistic function. We can justify this specification on three grounds. First, we observe market demand functions that are S-shaped. Second, the logistic functional form is flexible, since it can capture both convex and concave features. Third, this specification can be interpreted as the integral of a bell-shaped density function of interest rates, in which demanded quantities are higher for averages rates than for extreme rates. It is appealing to think that bids are concentrated around a mean rate and that rates are therefore distributed with a single mode. We can directly compare our results on French Treasury bills to those found by BL and ABL with their respective data set. In the second step, we determine economic variables that explain the variations of the logistic curves across auctions. This step extends the methodology of BL. As a matter of fact, we do not try to construct a standard bid function for French Treasury bills (as BL and ABL have done with their data set). On the contrary, we explain fluctuations from one auction to the other (that are considered by BL as purely random) by economic variables. 4

We obtain the following results. First, the estimation of the parameters of the logistic curves gives fitted functions that are remarkably close to the observations. This confirms the logistic nature of the market bid functions. Second, we show that fluctuations across auctions are not random, but can instead be explained by a number of economic variables. These results are interesting because explanatory variables that determine the shape of the market bid function can be used for forecasting. The remainder of the article is organized as follows. In section 2, we describe the institutional framework of French Treasury bill auctions as well as the data set. The econometric methodology is detailed in section 3. We comment on the results in section 4. In section 5, estimated parameters are used to forecast the results of incoming auctions. Finally, section 6 concludes the article and gives a new perspective on the debate about the choice of the best auction mechanism.

2. Data and Institutional Framework

We consider the 125 auctions of BTFs that took place during 1995.2 BTFs are zerocoupon discount Treasury bills with maturity smaller or equal to one year. Bids for BTFs are submitted in terms of rates and not prices. The French Treasury uses a sealed bid discriminatory auction mechanism in which securities are supplied at the effective rate tendered by the bidder, as opposed to the marginal rate. These auctions regularly take place every Monday or the next working day. Each Thursday, the French Treasury announces the exact volume that will be offered for each maturity. Each bidder may propose any quantity at

2

This paragraph is partly based on the Annual Report on Treasury securities (1995) published by the French

Ministry of the Economy.

5

different rates. The Banque de France then aggregates the bids and anonymously transmit them to the French Treasury which determines the equilibrium rate (the stop-out rate) : only yields below this reference level are served until the volume announced is satisfied. Results from the different auctions are communicated through various information channels less than 20 minutes after the end of the auction. Securities are delivered and financial transactions are executed a couple of days after the auction. The 125 auctions can be divided into four categories (see Table 1). The amount of BTF auctioned in 1995 exceeds 1088 billion French Francs. It is clear that quarterly BTF auctions are more important in number (42% of the total number of auctions) but also in volume (66% of the total volume). Table 1 Distribution of BTF maturity in 1995 Monthly

Quarterly Semi-annual

Yearly

Total

Number of auctions

22

52

28

23

125

Aggregate volume*

148 072

724 233

119 840

96 273

1 088 418

* in millions of Francs

We have obtained from the French Treasury the tables of auction bids for each auction. These tables are very reliable since the Treasury uses a computerized remote bidding system (TELSAT). We construct our data set as follows. First, we determine the aggregate bid function by combining the cumulative quantities and the corresponding rates. Second, we construct the variables that are specific to the auction, such as the number of bidders, the number of bids, the number of observations for the aggregate bid function (number of points), the volume of transaction, the stop-out rate, the weighted average rate and the percentage of bidders that have been served. Next, we complete our data set with the amount of non-competitive bids

6

for each auction, market rates for different maturities and stock market information. We also compute the auction coverage that we define as the ratio of the total bids to the amount served (including non-competitive bids), and the scatter of each auction, defined as the gap between the stop-out rate and the average rate in basis points. Table 2 gives the mean, the standard deviation, the minimum and the maximum for the most important variables. Table 2 Mean

Std. Dev.

Min.

Max.

Volume announced*

8 136

5 272

3 000

22 000

Total amount of competitive bids*

23 359

14 022

6 840

58 900

Auction coverage

3.18

1.23

1.15

7.87

Scatter of auctions

0.02

0.02

0.00

0.13

41

77

0

517

567

708

0

2 982

Number of primary dealers (SVT )

19.12

1.20

16

21

Number of bids

53.47

11.01

36

95

Average number of bids by SVT

2.79

0.51

1.85

4.79

Number of points

12

4

6

24

Pct. of SVT with completely served bids

12

15

0

75

Pct. of SVT with partially served bids

55

19

10

95

Pct. of SVT who obtain nothing

33

20

5

90

ONC1* ONC2* 3

*in millions of Francs

ONC1s and ONC2s refer to non competitive bids. ONC1s are submitted at the same time as competitive bids, and ONC2s after the auction, once the price is determined. We do not consider individual bid functions for two reasons. On the one hand, there are few observations to estimate aggregate or market bid functions (12 on average) and even less to estimate individual bid functions (2.79 on average) from anonymous bidders. On the other

3

SVT stands for Spécialiste en Valeurs du Trésor who are the French primary dealers. Roughly speaking they

are the bidders.

7

hand, we consider the point of view of the Treasury, which is interested in market bid functions. It is clear that auction rates and secondary market rates are related to each other. However, market rates are not always available. To deal with missing observations, we have constructed a secondary rate market that extrapolates missing observations.4 Many North American empirical studies, such as Cammack (1991), Spindt and Stolz (1992), Jegadeesh (1993) and Goldreich (1997) find that the auction price is significantly lower than prices prevailing on secondary markets. We can not confirm this result in our data set. Our data suggest that auction rates are on average 2 basis points lower than secondary market rates, but the standard deviation of 5.3 basis points is too large to conclude that the difference is significant. Nevertheless, there exists a strong relationship between rates resulting from Treasury auctions and secondary market rates. A description of the explanatory variables can be found in Appendix 1.

3. Methodology

We proceed in two steps to estimate the effects of economic variables on market bid functions. 5 In the first step, we assume that market bid functions can be specified as logistic curves defined by three parameters. We estimate parameter values for each auction by the method of nonlinear least squares. In the second step, we formulate a system of equations that we use to explain variations of the parameters from one auction to the other. Parameters of

4

The details of the construction of the market rates are available upon request.

5

All MatLab programs are available on request.

8

this system are estimated by the Seemingly Unrelated Regressions (SUR) method of Zellner (1971).

3.1. First step: estimation of the parameters of the logistic curve We specify a market bid function as a logistic curve defined by three parameters (a, λ, τ): y=

a æ (x − τ )ö 1 + expç − ÷ è λ ø

(1)

where y denotes the aggregate bid at rate x > 0. The three parameters of the logistic curve can be interpreted as follows. • a is the value of y when x goes to infinity. This parameter corresponds to the market saturation level. This parameter measures the height of the curve: the larger the value of a, the larger the total demand for the security. • λ is a scale parameter. It is related to the elasticity of the bid function. It is also a measure of the steepness of the curve or equivalently the degree of dispersion of the bids. • τ determines the position of the inflection point of the logistic curve on the x-axis. Increasing τ shifts the curve to the right. Auctions are indexed by i (i = 1, ..., I) while observations on the aggregate demand curve of an auction are indexed by j (j = 1, ..., ni). We have I = 125 auctions and the number of observations for each auction ranges from ni = 6 to ni = 24. The three parameters that define the logisitic curve of auction i are denoted by θi = (ai, λi, τi)′. Scalars xij and yij denote the coordinates of the jth observation in auction i. Since we consider that the demand on each auction has a logistic shape, we use the following specification: 9

yij = f(xij, θi) + uij

(2)

where f(.,.) corresponds to the logistic function defined in (1). We estimate the parameters of the logistic curve by the method of nonlinear least squares. The principle is to minimize the sum of squared residuals. We use the estimates of a linear transformation of the logistic function as starting values for the iterative algorithm.6 The derivatives needed to compute the standard errors can be found in Appendix 2.7 To illustrate the goodness of-fit of the estimation procedure we compare fitted to observed stop-out rates as well as fitted to observed weighted average rates for every auction. The stop-out rate ( xsi ) is such that f ( xsi , θ i ) = Qi , where Qi is the volume issued by the Treasury. It is the solution to the following equation:

6

To estimate the parameters of the curve, we use an iterative procedure that requires initial values. It is

important to give initial values that are close to the true global minimum to avoid convergence of the minimization algorithm to a local minimum. To obtain these preliminary parameter estimates, we use a transformation of the logistic curve (1) that yields a linear equation that can be written as:

æ y ö τ 1 logç ÷ =− + x λ λ è a − yø To use this specification, we must assign a value to a. For each auction, we use the highest observation, i.e. yin . i

However to avoid division by zero, we use exactly 101% of this value. We can then estimate parameters

b1i = −τ i / λi and b2i = 1 / λi for each auction by the ordinary least squares method. We would like to stress that we can only obtain two parameters of the logistic curve with this approach:

λi = 1 / b2 i and τ i = −b1i / b2i . There is another limitation. We restrict parameter ai to a value that is not reliable. Indeed, some bidders do not need securities during a particular auction, but they nevertheless bid in order to respect their commitment to participate in a significant and regular way to Treasury bill auctions. These bids occur at high rates to minimize the probability to obtain the securities. By fixing ai to the last observations, we weight these outliers too much in the estimation procedure. To overcome these shortcomings, we apply the method of nonlinear least squares to simultaneously estimate the three parameters of the logistic curve. 7

Estimated coefficients of all 125 auctions are available on request.

10

æ a ö xsi = τi − λi logç i − 1÷ è Qi ø

(3)

The weighted average rate ( xei ) is determined as follows : 1 xei = Qi

( (

) dx )]

x ai exp − ( xi − τi ) / λi ∂yi 1 x x dx = x ( ) ò i ∂xi i i Q xò i  x λi 1 + exp − ( xi − τi ) / λi x si

0i

s

0i

[

2

i

(4)

where x0i is the rate for which the demand reaches 1 million (the smallest bid allowed).

3.2. Second step: explaining the shape of the logistic curve by economic variables

In this second step, we explain fluctuations of the estimated parameters from one auction to the other. We believe that these fluctuations, which do not invalidate the logistic specification, are not completely random and that they can be explained by economic variables. To test this assumption, we specify a set of equations and we estimate the associated parameters using Zellner's SUR method. Let a = (a1, ..., aI)′, λ = (λ1, ..., λI)′ and τ = (τ1, ..., τI)′ denote the vectors of parameters of the logistic curve of all auctions. Let θ = (a′, λ′, τ′)′ be the vector of stacked parameters. We denote by w1i, w2i and w3i the vectors of respective dimension k1, k2 et k3 of variables explaining each parameter ai, λi and τi in auction i. We stack these vectors in matrix W of dimension 3I × (k1 + k2 + k3): éW1 0 ê W = ê 0 W2 êë 0 0

0ù ú 0ú W3 úû

where W1 = (w11, ..., w1i, ..., w1I)′, W2 = (w21, ..., w2i, ..., w2I)′, et W3 = (w31, ..., w3i, ..., w3I)′ are matrices of dimension I × k1, I × k2 and I × k3 respectively. We can write the model in a standard SURE form: 11

θ = Wβ + ε,

(5)

where β = (β1′, β2′, β3′)′ is the vector of coefficients to be estimated, ε = (ε1′, ε2′, ε3′)′ is the vector of error that includes the prediction errors of the first step procedure and independent errors from the SUR specification. We assume that for each observation, the error terms have zero mean and covariance matrix Σ and are independently distributed from W as well as across observations. Parameters from this model can be estimated using the generalized least square method by observing that the matrix of covariance of the residuals is simply Ω = E[εε′] = Σ ⊗ II, where II is the identity matrix of dimension I×I.8 From an econometric point of view, these two steps can be combined. However, the assumptions underlying this specification are slightly different, and it is not clear that a single estimation procedure deliver better results. Nevertheless, the single step estimation based on the method of nonlinear least squares is presented in appendix 2.

4. Results

We obtain two important results. First, using estimates from the first step of the two step procedure, we find that the logistic specification accurately describes demand in Treasury auctions. Second, we can explain a good part of fluctuations of the parameters of the logistic curve across auctions by economic variables. The estimation procedure of the first step yields an excellent quality of fit. To illustrate this

8

To apply the feasible generalized least square method, we first estimate each equation separately by the method

of ordinary least squares and then use the resulting covariance matrix of the residuals in the generalized least squares formula.

12

result, we have reported in Appendix 3, the first 8 estimated logistic curves together with the observations for the first eight auctions. The comparisons of the estimated to the observed stop-out rates as well as the estimated to observed weighted average rates for every auctions, also illustrate the validity of our approach. In Figures 1 and 2, observed rates lie on the x-axis

Figure 1

Figure 2

C o m p a r is o n o f o b s e r v e d a n d e s tim a t e d s to p -o u t r a te s

C o m p a r is o n o f o b s e r v e d a n d e s tim a t e d w e ig h te d a v e r a g e r a t e s

8 .8 0

8 .8 0

8 .3 0

8 .3 0

7 .8 0

7 .8 0

7 .3 0

6 .8 0

6 .3 0

Estimated weighted average rates

Estimated stop-out rates

while estimated rates lie on the y-axis.

7 .3 0

6 .8 0

6 .3 0

5 .8 0

5 .8 0

5 .3 0

5 .3 0

4 .8 0 4 .8 0 5 .3 0 5 .8 0 6 .3 0 6 .8 0 7 .3 0 7 .8 0 8 .3 0 8 .8 0 O b s e rv e d s to p -o u t ra te s

4 .8 0 4 .8 0 5 .3 0 5 .8 0 6 .3 0 6 .8 0 7 .3 0 7 .8 0 8 .3 0 8 .8 0 O b s e rv e d w e ig h te d a v e ra g e r a te s

We use all 125 auctions to obtain estimates of the effects of economic variables on the estimated parameters of the logistic curve. We find that a substantial number of explanatory variables significantly influence the shape of the market bid functions. Hence fluctuations across auctions can not be solely attributed to random shocks. In our model, we assume that the parameters of the logistic curves depend on variables that are known at the time of the auction. We can use this model to forecast shapes of market bid functions regardless of the result of the auction. Although the estimation method that we use allow different explanatory variables to

13

enter each equation, we use the same variables for all equations.9 The model in which we only keep variables with associated estimated coefficients that are significantly different from zero gives almost identical results. Table 3 gives the estimates for each parameter. Coefficients that are different from zero at the five percent level are presented in bold. We note that to obtain coefficients of the same order of magnitude10 we have divided parameter a by 10000 and multiplied λ by 100, while we have adjusted the explanatory variables in each table. We stress that our methodology uses aggregate demand functions, without reference to a model of individual bids. Therefore, coefficients should be interpreted as estimated values that we use for the forecasting exercise in the next section. Indeed, although we can give some interpretations of parameters a and τ, parameter λ is much more difficult to interpret without a model of aggregation. Table 3 s. d. lambda*100 coeff s. d. s. d. a/10000 coeff tau coeff Intercept *100 0,1540 0,0419 *10 0,1470 0,0308 0,5754 0,1639 Maturity (in weeks) /100 -0,3968 0,7064 /100 0,2997 1,1139 /100 0,9351 0,5903 Settlement date /10 -0,8699 0,3623 -0,4161 0,5752 /100 -0,4935 0,3048 Number of auctions /10 -0,5590 0,1814 -0,7444 0,2836 /10 0,5886 15,0260 Total liquidity demand /100000 0,2360 0,1387 /10000 0,4525 0,2224 /100000 0,1794 0,1179 Volume announced /1000 0,1830 0,0166 /10000 -0,8439 0,2619 /1000000 -0,1402 0,1388 Secondary market rate -0,6274 0,1606 /10 0,7180 2,2383 0,9813 0,0119 Quoted on the secondary market 0,2119 0,1117 0,3448 1,7606 /10 0,2075 0,0933 Stock market volume/index volume/10 -0,8931 0,3113 index/100 -0,6487 0,1361 index/1000 -0,2051 0,0725 Short term spread /10 -0,2764 0,1422 /10 -0,4296 0,1772 /10 0,1762 0,0940 Long term spread -0,1983 0,3305 -0,1044 0,0530 /10 -0,8629 0,2807

The most significant variable that influences parameter a is the volume announced by the French Treasury. Indeed, the larger the volume, the larger is the aggregate demand. However, it is difficult to determine what is the cause and what is the consequence. On the one hand, if

9

The only exception is the stock market variable. For the equation related to parameter a, we use the stock

market volume, while for the other two parameters, we use a stock market index. 10

This is required in the minimization procedure with such a large number of parameters to estimate.

14

the Treasury announces a large volume, bidders can buy more securities which pushes the bid function upward. On the other hand, the Treasury can also anticipate a large demand and consequently increase the volume it announces. Indeed, the Treasury has strong incentives to offer securities for which the market demand is the strongest. BL and ABL also observe that the height of the logistic curve is positively correlated with the volume announced and they normalize quantities by this variable: they assume that bidders bid for a fraction of the total amount announced. Four other variables have a significant impact on parameter a, they all affect a negatively. There are the settlement date, the number of auctions, the secondary market rate and the stock market volume. The negative coefficient for the market rate variable might be due to the fact that the Treasury offers lower volume when the rates are high. The negative impact of the stock market volume is naturally due to an arbitrage relationship. Although the result is not significant, a high value of the long term spread reduces the value of a. This can also be explained by an arbitrage relation: when the long term spread increases, the relative attractiveness of short term bills is reduced as well as the total demand for BTFs. Concerning parameter λ, our results show that the number of auctions, the volume announced, the stock market index and the short term spread significantly negatively affect parameter λ, whereas, the total liquidity demand variable has a significant positive impact. The impact of the secondary rate on parameter τ is so strong that few other variables add any explanatory power. As expected the secondary market rate is strongly related to this parameter value that can be interpreted as the mean of the distribution of the rates during the auction. This is a simple arbitrage relation that states that primary rates can not be too different from secondary market rates for perfectly substitutable securities. This result is in line with the study of BL. There are three other variables that significantly affect τ. First,

15

when a security is not quoted on the secondary market, there is no alternative to the auction and bidders have to use an aggressive strategy and push rates down. On the contrary, when the BTF is quoted on the secondary market, there is more competition on the supply side which increases τ. Second, stock market evolution reduces the value of τ, as there exists a substitution effect between stocks and bills: when the stock market goes up, bill prices go up and rates go down. Finally, the shape of the yield curve significantly influences this parameter. The intuition is straightforward. When the spread is small, short term bills are more attractive than long term bonds, which pushes BTF (short term securities) rates upward. So when the long term spread increases, BTFs are less attractive, and τ decreases.

5. Forecasts

In this section we forecast the expected demand for French Treasury Bills using the preceding specification. We use the estimated coefficients to construct the parameters of the logistic curve in each auction. Then we use these parameters to construct the stop out rate and the weighted average rate using equations (3) and (4). However to comply with existing rules, we calculate these rates using a discrete grid of rates just as the Treasury does. So, for the stop-out rate we round the resulting number to the upper basis point and use the rationing rule applied by the Treasury (the resulting percentage is such that supply is equal to demand). Computing the weighted rate is slightly more challenging. We exactly reproduce the procedure used by the Treasury. To understand this procedure, we use Figure 3.

16

Figure 3 Quantities y5 y4

Q

y3 y2 y1 1 x0

x1

x2

x3

xs x4 x5

Rates

The logistic function represents the expected demand. Quantity Q determines the stopout rate xs. Now, we assume that the grid of the x-axis corresponds to one basis point increment and that bidders can only bid using this grid (just as in the reality). We do not observe the whole demand curve but only points on this curve. Hence, in this figure the stopout rate is not xs, but x5. The computation of the weighted average rate only uses the first five points, just as the procedure of the Treasury: xe = [x1 y1 + x2 (y2 - y1) + x3 (y3 - y2) + x4 (y4 - y3) + x5 (Q - y4)] / Q To determine the first point on the curve, we compute the first rate x0 for which demand reaches one million French francs and we label x1 the value of x0 rounded to the upper 1% basis point. We start the forecasting exercise with in sample forecasts to check the robustness of the methodology. Results show that the forecasts are fairly accurate. So next, we estimate the coefficients from the first 100 auctions and we use these values to forecast results of the last 25 auctions. Results from theses out of sample forecasts are reported in Table 4. The forecast values should be compared to the observed ones.

17

Table 4 Out of sample forecast for the last 25 auctions a

mean stand. dev. validity at 20% lambda mean stand. dev. validity at 20% tau mean stand. dev. validity at 1% stop out rate mean stand. dev. validity at 1% weighted average rate mean stand. dev. validity at 1%

observed 19 665 10 512 observed 0,017 0,009 observed 5,85 0,64 observed 5,82 0,64 observed 5,82 0,64

forecast 23757 9825 40% forecast 0,019 0,007 24% forecast 5,86 0,66 91% forecast 5,82 0,65 76% forecast 5,82 0,65 80%

On average results are accurate. We were only disappointed by the forecasts of parameter λ. This is certainly due to the poor explanatory power of our model with respect to this parameter. In contrast, forecasts of the stop-out rate and the weighted average rate are quite good. In Table 4, validity at 20% for a and λ, and at 1% for the rates indicates the proportion of expected values that fall into an interval of respectively 20% and 1% around the observed values. The expected weighted average rate (the relevant variable for the Treasury) is close to the observed value: 20 out of 25 forecasts fall into an interval of 1% around the true value.

18

6. Conclusions and perspectives

We have empirically analyzed 125 auctions of BTFs during 1995. We confirm the findings that market bid functions can be specified as logistic curves. The quality of the fit is remarkable. Moreover, we have shown that the fluctuations of the parameters of this curve from one auction to the other are not completely random and can be accounted for by economic variables. In our model we only considers variables that are known prior to the auction, that allows us to forecast the results of a particular auction according to relevant variables such as the volume announced, the settlement date, the secondary market rate, the short and long term interest rates spread... This model explain the intuitive fact that some economic variables have a significant impact on the demand for BTF. This study extends the work of BL and ABL in an interesting way since its allows us to propose a new perspective on the debate about the choice between the discriminatory auction and the uniform price auction. This debate has been going on at least since Friedman (1960) who proposed to change the discriminatory auction by the uniform price mechanism in the U.S. At this time, theoretical models have contributed little to a definitive answer to which type of auctions should be used to minimize financial costs. However, this recurrent question is at the heart of the empirical and theoretical literature on Treasury auctions. Indeed, the amount of money at stake is considerable. Our empirical approach opens new ways to answer questions that are difficult to handle from a theoretical point of view, since the complexity of the environment of these auctions is such that no satisfactory theoretical model could be reasonably used to give a definite answer. We think that the relative performance of a specific type of auction depends on economic variables that influence market bid functions. Thus, we believe that it is possible that in some situations the discriminatory auction mechanism dominates the uniform price 19

mechanism, but that the converse can be true in other economic context. It seems therefore relevant to determine empirically which economic variables affect market bid functions for both mechanisms. To compare both auction mechanisms we need to construct a model using data from uniform price auctions since the explanatory variables that significantly influence the shape of the market bid functions can be different from one mechanism to the other. Interestingly, ABL found that demand in uniform price auctions could also accurately be described by a logistic specification. We could then use the methodology that we have developed here for this type of auctions. Then, using estimated coefficients, we can forecast the expected weighted average rate under both procedures in a specific economic environment. Unfortunately this is not possible in France since the Treasury only uses discriminatory auctions. It should be therefore interesting and rewarding to pursue this approach to other countries. Our empirical analysis is especially relevant in the U.S. where the Treasury changed from the discriminatory to the uniform price mechanism for auctions of the same securities, and therefore where data are available for both types of auctions.

20

References ATLE BERG, Sigbjorn, BOUKAI, Benzion and LANDSBERGER, Michael, 1998, "Bidding for Treasury Securities Under Different Auction Rules : The Norvegian Experience", Working Paper. AUSUBEL, Lawrence M. and CRAMTON, Peter C., 1998, "Auctioning Securities", University of Maryland, 7:3, miméo. BACK, Kerry M. and ZENDER, Jaime F., 1993, "Auctions of Divisible Goods: On the Rationale for the Treasury Experiment", Review of Financial Studies, 6:1, p. 733-764. BOLTEN, Steven, 1973, "Treasury Bill Auction Procedures: An Empirical Investigation", Journal of Finance, p. 577-585. BOUKAI, Benzion and LANDSBERGER, Michael, 1998, "Market Bid Functions For Treasury Securities As Logistic Growth Curves", Working Paper. CAMMACK, Elizabeth B., 1991, "Evidence of Bidding Strategies and the Information in Treasury Bill Auction", Journal of Political Economy, 99:11, p. 100-130. FRIEDMAN, Milton, 1960, "A Program for Monetary Stability", Fordham University Press, New York. GOLDREICH, David, 1997, "Underpricing in Treasury Auctions", Working Paper, Institute of Finance and Accounting, London Business School. HELLER, Daniel and LENGWILER, Yvan, 1997, "The Auctions of Swiss Government Bonds: Should the Treasury Price Discriminate or Not?", Working Paper.

21

JEGADEESH, Narasimhan, 1993, "Treasury Auction Bids and the Salomon Squeeze", Journal of Finance, 48, p. 1403-1419. MINISTRY of the ECONOMY, 1995, Annual Report on Treasury securities. SIMON, David P., 1994, "The Treasury’s Experiment with Single-Price Auctions in the Mid1970s: Winner’s or Taxpayer’s Curse?", Review of Economics and Statistics, 76, p. 754-760. SPINDT, Paul A. and STOLZ, Richard W., 1992, "Are U.S. Treasury Bills Underpriced in the Primary Market?", Journal of Banking and Finance, 16, p. 891-908. UMLAUF, Steven R., 1993, "An Empirical Study of the Mexican Treasury Bill Auction", Journal of Financial Economics, 33, p. 313-340. VISWANATHAN, S. and WANG, James J.D., 1997, "Auctions with When-Issued Trading: A Model of the U.S. Treasury Markets", Miméo, Fuqua School of Business, Duke University. WANG, J. and ZENDER, J., 1999, "Auctioning Divisible Goods", Miméo, Fuqua School of Business, Duke University. ZELLNER, A., 1971, "The « seemingly unrelated » regression model", in An Introduction to Bayesian Inference in Econometrics, ed. Wiley Classics Library, p. 240-246.

22

Appendix 1: Description of the explanatory variables

We consider ten predetermined variables. 1. BTF maturity (in weeks); 2. Settlement date (in days starting from the auction day); 3. A binary variable that determines if there are at least two other auctions during the same day (number of auctions); 4. Total liquidity demand (×10-3), i.e. the total amount demanded by the Treasury for all auctions of the same day; 5. Announced volume (×10-3) is the total amount demanded during a single auction; 6. Market rate is in fact the secondary market rate when it is available or value computed by linear interpolation, or from the benchmark rate of the Banque de France; 7. A binary variable that accounts for the existence of a rate that is quoted on the secondary market; 8. Stock market index (×10-2) and volume are respectively the closing price and the volume of the CAC 40 the trading day preceding the auction; 9. Short term spread is the difference between the one year interest rate and the one month interest rate; 10. Long term spread is the difference between the 10 year rate and the one year interest rate. These last two variables take the shape of the yield curve into account.

23

Appendix 2: Single step estimation

We make slightly different assumptions on the structure of the error term to estimate the coefficient in a single step. We implicitly assume that the variables that we use perfectly explain the parameters of the logistic curve, which is less flexible than the two step procedure. However, we increase the efficiency of the estimation procedure. We are aware that both methods are still imperfect. On the one hand, there are necessarily explanatory variables that we have not taken into account. On the other hand, even though we believe that these explanatory variables explain most of the variations across auctions, we still think that some parts of these fluctuations are purely random. Nevertheless, we assume in this single step procedure that all fluctuations on the parameters of the logistic curve across auctions can be perfectly accounted for by the explanatory variables. We start from model (2). But now, we assume that ai = w1i′β1, λi = w2i′β2 and

τi = w3i′β3. To estimate parameters β1, β2, β3 in a single step, we use the method of nonlinear least squares by minimizing the sum of square residuals: S(β1, β2, β3) =

å å I

ni

i =1

j =1

(yij − f(xij, w1i, w2i, w3i, β1, β2, β3))2

We can write the model in matrix notation. Let y be the vector of stacked quantities y = ( y1,1 ,..., y1,n ,..., yi ,1 ,..., yi ,n ,..., y I ,1 ,... y I ,n ) . 1

i

I

Similarly, let x denote the vector of stacked rates. Let

N = åi =1 ni denote the total number of observations. Let X1 denote the N×k1 matrix of I

[

]

stacked elements of the form w1i ⊗ 1ni for i = 1 to I, where w1i is the vector corresponding to the ith line of W1 and 1ni is a vector of ni ones. We construct in the same way X 2 = [wi,2] and X

3

= [wi,3], two matrices of respective dimensions N×k2 and N×k3. Finally,

β = (β1′, β2′, β3′)′ is a vector of dimension (k1+k2+k3). Then, we can rewrite S(β) as:

24

æ ö W1β1 ÷ S ( β) = çç y − 1 + exp(− ( x − W3β 3 ) W2β 2 ) ÷ø è

′

æ ö W1β1 çy − ÷ ç 1 + exp( − ( x − W3β 3 ) W2β 2 ) ÷ø è

where we use element by element divisions. Now, we solve βNLS = argminβ S(Z, β) using an iterative method and initial values of the coefficients obtained by the previous two step method.

25

Appendix 3: Estimation of the logistic curve for the first 8 auctions

In these figures, we can observe the logistic structure of the market bid functions. Rates lie on the x-axis, while quantities lie on the y-axis. The horizontal line corresponds to the volume issued by the Treasury. 2.5

x 10

4

BTF001

BTF002 8000 7000

2 6000 5000

1.5

4000 1

3000 2000

0.5 1000 0 5.91

3.5

x 10

5.92

5.93

4

5.94

5.95

5.96

5.97

0 6.83

6.86

6.87

6.88

6.89

6.9

6.91

6.92

6.255

6.26

BTF004 14000

12000

2.5

10000

2

8000

1.5

6000

1

4000

0.5

2000

0 5.65

x 10

6.85

BTF003

3

2.5

6.84

5.66

5.67

5.68

4

5.69

5.7

5.71

5.72

5.73

5.74

0 6.21

6.215

6.22

6.225

6.23

BTF005

6.235

6.24

6.245

6.25

BTF006 14000

12000 2 10000 1.5

8000

6000

1

4000 0.5 2000

0 5.57

3.5

x 10

5.58

5.59

5.6

4

5.61

5.62

5.63

5.64

5.65

5.66

0 6.59

6.61

6.62

BTF007

6.63

6.64

6.65

6.66

6.67

BTF008 14000

3

12000

2.5

10000

2

8000

1.5

6000

1

4000

0.5

2000

0 5.51

6.6

5.52

5.53

5.54

5.55

5.56

5.57

5.58

5.59

5.6

0 6.01

6.02

6.03

6.04

6.05

6.06

6.07

6.08

6.09

6.1

26