Lem

Keywords: A1, D4 diamonds, discontinuity in price, hedonic equation, unconditional quantile regressions

This paper investigates the relationship between the selling price of diamonds and their weight in carats. For this purpose, we use a unique sample of 112,080 certified diamonds collected from www.info-diamond.com during the first week of July 2011. We find substantial differences in pricing depending on cut shape. The price of diamonds increases markedly with the carat weight, with a price elasticity equal to 1.94. However, estimates from unconditional quantile regressions show that the price-weight elasticity is not constant since it rises along the price distribution of diamonds. Finally, we observe the existence of significant increases in prices for diamonds featured with round weights compared to gems just below these threshold weights.

Introduction

When demand for a good durably exceeds its supply, economists expect a price increase of that good. The increasing trend should persist or even become more marked as long as no valid substitute appears on the corresponding market, as demand increases and as supply remains either constant or decreases. The diamond market is typical of such a situation. In 2003, world demand for rough diamonds (US$9.5 billion) was significantly above the diamond supply (US$8.2 billion), so that the excess demand had to be satisfied from producers' existing stockpiles. To absorb the difference between quantities supplied and demanded, a significant increase in exploration should occur to meet the increasing demand for diamonds. However, such a pattern does not seem possible because of technical difficulties.1

To explain the growing demand for diamonds over the last decades, [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF] stress that consumption of these durable goods is not only related to satisfaction provided by their intrinsic characteristics [START_REF] Lancaster | Consumer demand: A new approach[END_REF], but also to the social aspect that consumption of diamonds implies, namely conspicuous consumption [START_REF] Veblen | The theory of the leisure class. An economic study in the evolution of institutions[END_REF]. In other words, it is likely that diamonds are bought for the impression their consumption makes on others, and they are purchased because they are expensive (see [START_REF] Braun | A simple microeconomic model illustrating rising diamond prices and the durable goods problem[END_REF]. As a consequence, diamonds are likely to be bought at prices much above producers' marginal costs.

In the field of international economics, diamonds are classified as homogeneous goods (Rauch, 1999, Javorcik and[START_REF] Javorcik | Differentiated products and evasion of import tariffs[END_REF].2 This is a somewhat striking feature as there are plenty of objective characteristics like cut shape, weight, color, clarity, certification, polish, symmetry and fluorescence that are expected to have a strong influence on the diamond price. Hence, diamonds sound much more like a differentiated product. Curiously, despite the high value of the diamond market, we are aware of only one study so far that has attempted to assess the determinants of diamonds pricing. Drawing on a hedonic analysis of price à la [START_REF] Rosen | Hedonic prices and implicit markets: Product differentiation in pure competition[END_REF], [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF] investigate the impact of diamonds' characteristics on their price.

In their insightful contribution, [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF] reach a surprising conclusion. They show that buyers would be willing to pay premiums upward of 18% for a diamond that is one-half carat rather than buy a slightly less than one-half carat diamond and between 5% and 10% more for a one-carat diamond rather than buy a slightly less than one-carat diamond. These variations, analyzed from the classical economic viewpoint as pricing anomalies by the authors, may be explained by a whole-number effect. In other words, consumers would perceive a categorical difference between diamonds smaller than a carat and one carat or larger. Of course, there are other plausible explanations. Consumers could rely on rule-of-thumb decision rules, thereby introducing an artificial jump in demand at 1.0 carat; or diamonds considered as a durable good could be purchased for speculation.

In this paper, we further document the price setting of diamonds using a unique database comprising more than 100,000 certified diamonds. Detailed information on diamonds was collected by the authors in July 2011 from the website www.info-diamond.com. We focus especially on the relationship between the price of a diamond and its weight in carat net of the role played by the other characteristics of the gems. Specifically, we estimate hedonic price equations and attempt to address the three following issues. Firstly, what is the contribution of the carat weight and other features when explaining price variation in diamonds? Secondly, is price-weight elasticity constant for low and high price levels, or does it vary along the price distribution? Thirdly, are there any jumps in price when nearing round weights?

We rely on a hedonic price framework to bring answers to these questions. We find substantial differences in pricing depending on cut shape. The price of diamonds increases markedly with the carat weight, with a price elasticity equal to 1.94. However, estimates from unconditional quantile regressions following [START_REF] Firpo | Unconditional quantile regressions[END_REF] show that price-weight elasticity is not constant since it rises along the price distribution of diamonds. Moreover, like [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF], we observe the existence of significant increases in prices for diamonds featured with round weights compared to gems just below these threshold weights.

The remainder of the paper is organized as follows. In Section 2, we describe our onlinecollected data on diamonds. We estimate hedonic equations of diamond price in Section 3. Using unconditional quantile regressions, we investigate whether price elasticity remains constant along the price distribution in Section 4. The presence of discontinuity in prices at round weights is examined in Section 5. Finally, concluding comments are placed in Section 6.

Data and descriptive statistics

We study the determinant of diamond prices using a unique data set collected by the authors during the first week of July 2011. Data are available online and come from the website www.infodiamond.com, defined as "The universe of diamond, from the mine to the jeweller's"; the corresponding information is thus publicly available. This website includes a diamond exchange service that enables every visitor to buy a certified diamond directly from the world's largest diamond merchants. For the sake of transparency, Info Diamond claims that they do not own all these diamonds, but they have permission from their diamond merchant partners to display their stock on their website. It follows that Info Diamond offers access to one of the largest diamond exchanges on the internet.3

We construct a database comprising exactly 112,080 certified diamonds after deleting seven observations.4 For each stone, the Info Diamond website provides very detailed characteristics of the product on a specific webpage. In Table 1, we present some descriptive statistics of the most important features of the diamonds.

Insert Table 1 The first step in choosing a diamond involves selecting a shape, which is one of the only factors in diamond grading that is controlled by human hands. Cut shapes include Asscher, Round Brilliant, Cushion, Emerald, Flanders Heart, Marquise, Oval, Pear, Princess, Radiant and Triangular (see Figure 1 for a description). The Round Brilliant shape is the most popular one and is the most readily available in various qualities and sizes. The Princess shape is also becoming popular because it is both brilliant and unique. In our sample, the Brilliant and Princess are the most frequently observed shapes, with respectively 56.9% and 13.2% of stones. As emphasized by experts, the various shapes have an inherent difference in terms of the physics of light. Longer shapes like Pear, Oval, Marquise or Heart have a small zone in the center where light leaks out through the bottom, creating a darker area in the shape of a bow tie.5 3 [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF] also consider online data. They consider the online inventory of Blue Nile, Union and Amazon respectively and collected their data from July 6 to July 8, 2005.

4 From our original sample of 112,087 diamonds, we chose to delete three diamonds with missing information on cut shape, two Flanders cut diamonds as there were only 2 diamonds with this particular cut shape, and the two most expensive diamonds (whose price was above 2 million euros) as they sound more like outliers.

5

See, for instance, advice from Robert Hensley, President of Diamond Helpers (http://www.diamondhelpers.com/ask/0026-cutshape.shtml).

Insert Figure 1 The unit of weight for a diamond is the metric carat. One carat corresponds to 0.2 gram. In almost all diamond-trading countries, it should be noted that the diamond's weight can only be rounded up to the next hundredth from nine thousandths of a carat. When turning to the Info Diamond data, we find that the average weight of the diamonds for sale is equal to 1.16 carats, with a standard deviation of 0.86. This means there is substantial heterogeneity in the size of the diamonds. The median weight is equal to 1 carat. The 90th and 99th percentiles are respectively equal to 2.02 and 5.01 carats and only 68 diamonds (0.06% of the sample) are above 10 carats.

The color of a diamond is determined by using a certified set of master stones and a colorimeter, a device specifically designed to grade the color of a polished diamond. Colors range from letters D (the best grade) to Z (the worst grade). Categories D (exceptional white +) and E (exceptional white) are diamonds that disappear almost in the ultra white pre-folded cards. A light tint is perceptible on the side with F (rare white +) and G (rare white). H (white) occurs when a tint is perceptible, but difficult to see in front view. A tint is visible in front view for I and J (slightly tinted white), easily visible for K and L (tinted white), and readily visible in front view for colors ranging from M to Z. In our data set, most of the selected diamonds belong to the upper color category: 28.2% are exceptional white (D-E), 34.8% rare white (F-G), 13.7% white (H), 16.4% slightly tinted white (I-J), and 6.9% tinted white or tinted color (K-Z).

Another feature of the diamond is its clarity. The clarity scale includes 12 grades, ranging from FL (best grade) to I3 (worst grade). The FL grade (Flawless) is for diamonds having no imperfections either inside or outside under the magnification of a 10-power loupe. Stones classified IF (Internally Flawless) have no inclusions under a loupe. VVS1 and VVS2 (Very Very Slightly Included) concern stones having very small inclusions which are difficult to see under a loupe. Stones having small inclusions, which are slightly difficult to see under a loupe, are VS1 or VS2 (Very Slightly Included). SI1, SI2 and SI3 (Slightly Included) are stones with inclusions fairly easy to see under a loupe or even visible to the naked eye. Finally, I1, I2 and I3 (Included) are diamonds having flaws visible to the naked that may decrease the brilliance. In our sample, 4.2% of diamonds have no visible inclusion (FL-IFLS) and 17.9% have very small inclusions which are difficult to see (VVS1-VVS2).

A certificate delivered by a gemological laboratory describes each diamond. This is a grading report that describes objective factors defining each stone, such as weight, clarity, color or measurement, among others. Any defect of the stone's structure is noted on the certificate. In other words, such a certificate provides qualitative information. The most important gemological laboratories in the world are respectively the Gemological Institute of America (GIA), the European Gemological Laboratory (EGL), the American Gem Society (AGS), the International Gemological Institute (IGI) and the Hoge Raad voor Diamant (HRD). Among the diamonds for sale in our data, 73.2% have a GIA certificate and 20.8% an EGL certificate.

Polish and symmetry shed light on the finish of a diamond. The polish result depends on the presence of more or less polish lines and the presence of strips, while the symmetry outcome corresponds to good alignment of the facets, their symmetry and the centering of both the culet and the table. Both outcomes are rated on a scale of five: excellent, very good, good, fair, and poor. We aggregate the two lowest categories, as there are very few diamonds with the poor grades. About 80% of diamonds have either an excellent or very good polish result, and 70% excellent or very good symmetry.

Fluorescence characterizes diamonds that produce a visible reaction when they are exposed to ultraviolet radiation. Fluorescence is described using one of the following values: none, faint, medium, strong, very strong. Diamonds with a strong or very strong fluorescence are expected to be cheaper on average.6 In our database, 75.8% of diamonds have no fluorescence and 12.5% faint fluorescence. The last three variables (not reported in Table 1) are related to the ideal cut of a diamond. The depth percentage of a round diamond is calculated by taking the total diameter of the diamond and dividing it by its total depth or height.7 The table on a diamond corresponds to the flat facet on the top of the stone. Expressed as a percentage, it is defined by the total diameter of the diamond divided by its table width. For round diamonds, the range that is considered as ideal should fall between 55% and 58%.8 Finally, there are several gradations of the culet size: no culet (preferable), pointed, very small, small, medium, large, very large and extremely large.

All these objective characteristics are expected to influence or even make the price of diamonds. In our database, the mean price of a diamond (whatever its weight) is 10,230 euros, with a standard deviation of 25,880 euros. The median price is 4,440 euros. The cheapest diamond is sold for 465 euros and the most expensive one for 1,841,303 euros. As shown in Figure 2, there is a large dispersion in diamond prices: 20.9% of diamonds cost less than 2,000 euros, 34% cost between 2,000 and 5,000 euros. In the upper part of the price distribution, 4.6% cost between 25,000 and 50,000 euros, 2.0% between 50,000 and 100,000 euros, and 1% of diamonds cost more than 100,000 euros.

Insert Figure 2

We now focus on the relationship between the diamond's price and its weight measured in carats. We plot the quantity of carats (in log) supplied at different price levels (in log) in Figure 3. An upward-sloping curve is expected as larger stones are much more rare than smaller ones. Our results provide strong support for a positive slope of the diamond supply curve. Interestingly, we observe from the scatter diagram that the various price-weight points are grouped into a clear linear shape. This means that the carat weight is a very strong predictor of diamond price. Estimation of a simple OLS regression expressing the log price as a function of the log carat weight leads to a price elasticity of 1.697, with a t-test of 736.24. The global fit of the linear model is very good as 82.9% of variability in diamond prices is attributed to variability in carats.

Insert Figure 3 Of course, these preliminary findings have to be cautiously interpreted as they do not take into account the role of the other observable features of the diamonds, including cut shape, color and clarity. In what follows, we turn to a more rigorous econometric framework and estimate hedonic diamond price functions.

A hedonic diamond price function

To study the impact of the diamond's characteristics on its price, we first estimate a hedonic price equation using Ordinary Least Squares. This means that we focus on the average price of a diamond available in our database. Denoting by the price of a diamond , the model we seek to estimate may be expressed as:

ln = + * ln + * + (1)
where ln is the log of carat weight, is a vector of diamond's features, , and are parameters to estimate, and is a random perturbation. The different covariates introduced in the regression in addition to carat weight are cut shape, color, clarity, certificate, polish, symmetry and fluorescence.

The corresponding estimates are reported in Table 2.

While the interpretation of the estimated coefficients is straightforward in the case of continuous covariates (like carat weight), the situation is slightly more complex in the case of the estimated coefficients of dummy variables. Assuming that is normally distributed, a consistent and almost unbiased estimator of the proportional impact of a dummy variable on the price is (see [START_REF] Kennedy | Estimation with correctly interpreted dummy variables in semilogarithmic equations[END_REF]):

= ( ) [ . * ( )] -1 (2)
where is the OLS estimator of the estimated coefficient associated to and #( ) is its estimated variance.9 We correct accordingly the marginal impact of the various dummy variables introduced in our hedonic price function in order to comment on unbiased estimates.

In a first specification (column 1, Table 2), we consider the whole sample comprising 112,080

diamonds. The fit of the model appears impressive since 97.6% of the price variation is explained by our selected covariates. The various explanatory variables are all highly significant, which is not really surprising given the large size of our sample. Net of the influence of the various diamond features, we obtain a convex profile for the carat-price relationship. Specifically, a one percent increase in weight increases the price of the diamond by 1.94%. Larger stones are thus much more expensive on average. Evaluated at the means of the sample, the average price of a 3 carats diamond is 44,721 euros. For a 4.5 carats diamond (a 50% increase), the average price rises to 98,204 euros (so an increase of 120%).

There are substantial differences in diamond prices depending on cut shape. As shown in Table 2, brilliant diamonds are much more expensive on average. Heart and marquise diamonds are about 34% cheaper, and asscher, cushion, oval, pear and princess diamonds are 36-37% cheaper. The decrease ranges from 38 to 40% for radiant, triangular and emerald diamonds. Our data show that the whiter the color, the higher the value of the diamond. With respect to an exceptional white + diamond (D grade), the price is reduced by 7.7% for an E grade, 12% for an F grade and 18.7% for a G grade. Tinted diamonds are much cheaper, respectively -55.2% for a tinted white and even -66.7%

for a tinted color diamond.

The clarity grade of a diamond strongly affects the price. Diamonds that are almost pure are rare and therefore more expensive. With respect to clarity flawless stones (FL-IFL), the reduction in price amounts to 10.2% for grade VVS1, 18.2% for grade VVS2, 24.6% for grade VS1 and 31.5% for grade VS2. The fall ranges from 49.1% to 56.9% for S1-S3 grades and even 62.3% for (I1-I3) with visible inclusions. Diamonds with excellent polish and excellent symmetry also fetch premium prices, respectively +2.4% and +4.9%. According to our estimates, fluorescence is perceived as a defect.

Compared to diamonds having a strong or very strong fluorescence, diamonds with no fluorescence are sold 7.7% higher.10

Finally, our estimates indicate that there are significant differences in the average price of diamonds depending on their certificates. Stones certified by GIA are 10.7% higher, while stones certified by EGL sell at a 14.7% discount. GIA is considered as the most reputed laboratory for gem evaluation, as it has developed the first internationally accepted Diamond Grading System. It should be noted that these differences in certification are net of the weight effect. Indeed, the GIA laboratory issues reports on a majority of high quality diamonds over one carat in size.

As there are significant differences in the average price between brilliant stones and the other cut shapes, we estimate a hedonic price function for brilliant round diamonds in column 2.

Again, price is very predictable since the R² is equal to 0.977. In fact, there is very little difference when comparing the estimates reported respectively in columns 1 and 2. For instance, the price elasticity with respect to weight is 1.96 for brilliant stones instead of 1.94 when considering all types of cut shape. As previously found, the price of a diamond decreases when color is poor, when there are inclusions and when it is fluorescent. Conversely, diamonds with excellent polish, excellent symmetry and a GIA certification have a higher value on average.

We add three additional variables pertaining to the ideal cut of a diamond in column 3, the sample still being restricted to brilliant stones. According to the OLS estimates, the price increases by 3.3% when the depth is optimal (comprised between 59 and 62.5%), but only 0.04% when the table is optimal (comprised between 55 and 58%). The characterization of the culet also significantly affects the price of a stone. The price increases by 3.3% when the stone has no culet and by 2.6% when the information is not available. However, despite their significance, these additional covariates do not improve the global fit of the model that remains nonetheless remarkably high (0.977).

As they stand, our results show that there is little unobserved heterogeneity within the diamond price levels. This undoubtedly explains why there are so many diamond price calculators available on the Internet.11 Interestingly, in many cases very few characteristics are proposed to determine a price. On the website of Info Diamond for instance, users have to select only a cut shape, a weight in carat, a clarity and a color to get an estimate of the diamond price. This suggests that these covariates are the most influential predictors of the stone's price. We decided to test the relevance of this assertion by drawing on a decomposition methodology described in [START_REF] Fields | Accounting for income inequality and its change: A new method, with application to the distribution of earnings in the United States[END_REF].

The following technique allows us to assess the respective contribution of the various diamonds' characteristics to their price. Consider the linear model ln = ∑ + , with % = 1, … , (. Then, the variance of the log price may be decomposed as:

#(ln ) = ∑ cov( , ln ) + cov( , ln ) (3)
The relative contribution of each covariate and of the residual is given by the following equality ∑ s( ) + -( ) = 100%, with s( ) = cov( , ln )/#(ln ) and -( ) = cov( , ln )/ #(ln ). Each component s( ) provides a measure of the weight of the regressor on the outcome ln , so that ∑ s( ) is simply equal to the conventional R². In Table 3, we present results from decomposition respectively for the whole sample and for brilliant diamonds.

Our main finding is that carat weight has an enormous influence on the price of a diamond:

the contribution of carat weight to price amounts to 94.7% and even 95.2% for brilliant diamonds.

Interestingly, many websites suggest that carat weight is not enough to price a diamond accurately.

Strictly speaking, this statement is not false as, from a statistical viewpoint, many other factors significantly affect the value of a diamond as highlighted in Table 2. However, we also need to put this statement into perspective as the impact of the other diamond features appears much more marginal (and most often negligible). Among the remaining covariates, it is essentially clarity (2.5%) and to a lesser extent cut shape (0.9%) that play a role.

Insert Table 3 The next step is to assess the relative contribution of the diamond features net of its weight.

For that purpose, we estimate a set of size-specific OLS regressions for stones weighing respectively 0.5, 1 and 1.5 carats. As shown in Table 3, the fit of the various linear models is always very good since the R² is never lower than 0.9. Our results are twofold. First, when considering all cut shapes, we find that cut shape, color and clarity are the three main predictors of price since their contribution explains more than 80% of the total price variation. By comparison, symmetry, polish, fluorescence and certificate have much less influence. Secondly, we note that the respective contribution of the selected stone's features depends on carat weight. For instance, clarity explains 36.4% of price for a 0.5-carat diamond instead of about 28% for 1-1.5 carats diamonds. Similarly, the contribution of color seems to increase with weight whatever the diamond's cut shape.

Thus our results suggest that the effect of the diamond's characteristics is likely to be different at different points of the price distribution. We further investigate this issue in the next section, using a quantile regression framework.

Does price elasticity remain constant along the price distribution?

We estimate quantile regressions to determine whether the effect of carat weight and other diamond features remains constant along the price distribution. Quantile regressions focus on specific parts of the conditional distribution of the dependent variable [START_REF] Koenker | Regression quantiles[END_REF].

These regressions provide estimates that are robust to misspecification errors related to nonnormality and heteroskedasticity. Let us first focus on the relationship between price and carat weight. For that purpose, we estimate a set of conditional quantile regressions:

1 2 (ln ) = (3) + (3) * ln (4)
with 1 2 the 3th conditional quantile of the log price. The coefficients obtained for the carat weight respectively for the whole sample and for brilliant stones are represented in Figure 4.

Insert Figure 4 Interestingly, we observe a non-linear, increasing profile of the carat coefficient, meaning that the price elasticity with respect to weight is rising along the price distribution. Specifically, the profile is slightly increasing till the 50th percentile, then the price elasticity remains fairly constant until the 80th percentile, rising sharply after that point. The carat weight coefficient, for instance, is equal to 1.70 at the 80th percentile, 1.82 at the 90th percentile, 1.94 at the 95th percentile and even 2.16 at the 99th percentile (the 90th percentile corresponds to a weight of 2.02 carats).12 There is thus a rising premium for the largest stones. Since these diamonds are the rarest, they are also the most coveted by wealthy investors. The increased demand, essentially attributed to a pure wealth effect around the world, tends to drive prices up even higher.

More generally, we expect all the diamonds' characteristics to have a different effect at different levels of the price distribution. A difficulty with the conditional quantile framework proposed by [START_REF] Koenker | Regression quantiles[END_REF] is that these regressions do not provide estimates of the impact of changes in the distribution of covariates on the marginal quantiles of the outcome variable.

To deal with this issue, [START_REF] Firpo | Unconditional quantile regressions[END_REF] have recently proposed a modified regression framework called unconditional quantile regressions. The method consists of using a recentered influence function (456 hereafter), which represents the influence of an individual observation on a given distributional statistic. Let 56(7; 1 2 , 6 9 ) be the influence function of the 3th quantile. Then, the recentered influence function is given by : 456(7; 1 2 , 6 9 ) = 1 2 + 56(7; 1 2 , 6 9 ) (5) with 56(7; 1 2 , 6 9 ) = :3 -1 9;< = >/? 9 (1 2 ). Formally, 456(7; 1 2 , 6 9 ) is computed by estimating the sample quantile 1 2 , the density ? 9 (1 2 ) at the point 1 2 (using a kernel method for instance), and by forming a dummy variable equal to 1 when 7 ≤ 1 2 and 0 otherwise. The unconditional quantile regressions are then obtained by estimating a simple OLS regression of the new dependent variable 456(7; 1 2 , 6 9 ) on the set of explanatory factors .13

Table 5 reports the RIF-OLS estimated coefficients of the log price model. We first focus on the impact of the diamond's characteristics other than weight. They strongly depend on the location in the price distribution. The marginal effect of cut shape is for instance much lower on the bottom part of the price distribution, except for princess and marquise. Compared to brilliant stones, other shapes are significantly cheaper and the difference in price strongly increases as one considers more and more expensive stones. The coefficient of almost all shapes is more than 3 times higher at the 90th percentile than at the 10th percentile.

The impact of poor colors (from K-L to M-Z) is highly detrimental both for low and high diamond prices. Conversely, with respect to the best grade D, there is a fall in price for E and F colors that significantly increases at the 90th percentile. A similar profile is found for clarity. Poor clarity strongly reduces price whatever the stone's value, but the negative effect of very slight inclusions is much higher at the top of the distribution. Finally, the unconditional estimates show that excellent 13 More complex estimation techniques are further described in [START_REF] Firpo | Unconditional quantile regressions[END_REF]. Their results suggest that turning to a non-parametric method, for instance, has very little effect on the magnitude of the unconditional quantile estimates.

polish, excellent symmetry and lack of fluorescence have a higher payoff in the upper part of the price distribution (75th percentile), but not really at the top (90th percentile). All these findings suggest that there is no place for small imperfections among the most expensive stones.

We next considered the weight coefficient. As previously highlighted with the row conditional estimates of Figure 4, the marginal impact of carat weight does not remain constant along the price distribution. The effect of carat weight is much smaller at the bottom of the distribution (with a coefficient of 0.908 at the 10th percentile) than at the top (with a coefficient of 2.598 at the 90th percentile). So, the price elasticity of diamond with respect to weight is nearly three times higher for the largest stones. It is interesting here to compare the RIF-OLS unconditional quantile estimates with the conditional quantile estimates.

Insert Table 5 We find much less variation in the weight coefficient along the distribution since the conditional estimates are respectively 1.906 at the 10th percentile, 1.951 at the 50th percentile, and 2.008 at the 90th percentile.14 On the one hand, the conditional estimates show that the carat weight increases the price dispersion within groups, defined as diamonds characterized by similar values of the covariates (other than carat). On the other hand, there is a premium for the largest stones and thus there is an inequality-enhancing effect between groups of diamonds. So the between and within group effects go in the same direction. This explains why the unconditional price-weight estimates are much larger than the conditional ones.

Discontinuity in the carat-price relationship

So far, we have assumed in our empirical analysis that the weight-price relationship is continuous. Interestingly, the pattern described by [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF] suggests that there are significant discontinuities in both price and quantities when considering the market of diamonds. For instance, a diamond that is one-half carat is on average 18% more expensive than a diamond that is slightly less than a half carat.

As a preliminary step, we decided to look closely at the weight distribution of the selected diamonds. As shown in Figure 4, we can observe the existence of strong discontinuities in the number of diamonds by size. Indeed, the frequency of diamonds whose weight is round appears to 14 Estimates from the conditional quantile regressions are available from the authors upon request.

be much higher. For example, there are 4422 diamonds whose weight is 0.4 carat, but only 1590 whose weight is 0.41 carat. When considering weights around 0.5 carat, the sample includes 5985 diamonds of 0.5 carat against 215 of 0.49 carat and 2539 of 0.51 carat. While the number of diamonds weighting 0.01 carat less is much smaller compared to diamonds with a round weight, we also note that sometimes the number of diamonds with +0.01 additional carat is substantially higher.

Insert Figure 4 This pattern is essentially found for diamonds exceeding 1 carat. For instance, there are 5555 diamonds of 1 carat in our sample, a figure to compare to 7460 diamonds weighting 1.01 carat.

Similarly, there are 20 diamonds whose weight is 1.99 carat, 984 with 2 carats, 1961 with 2.01 carats and 918 with 2.02 carats. A simple explanation of these discontinuities in weight could be a certain risk aversion on the part of suppliers. If diamond producers expect their gemstones to be less sought after if they fall just below certain threshold (round) values, then they would be tempted to produce diamonds just equal to or slightly above the thresholds when cutting the gems.

In their study, [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF] report large discontinuities in prices and interpret them as price anomalies. As a preliminary step, without controlling for any features of the diamonds, we decided to compare the row prices of the gems by considering successive intervals of 0.2 carat.

From the Least Absolute Deviations estimates reported in column 1 of Table 5, we find significant increases in prices for diamonds featured with round weights. For example, the price of a diamond of 0.5 carat is +57.7% higher than that of 0.49-carat diamond. The rise is respectively +27.9% between 0.59 and 0.6 carat, +33.6% between 0.69 and 0.7 carat, +20.2% between 0.79 and 0.8 carat, +40.3% between 0.89 and 0.9 carat, and +39.6% between 0.99 and 1 carat. We find less convincing evidence when considering diamonds above 1 carat, the discontinuity in price being much lower.15

Insert Table 5 We report the change in price as a function of weight over the various intervals in Figure 6.

Interestingly, we find that the curve of the row prices is rather flat and even sometimes negative before the mid values of the interval. If we consider for instance the case of diamonds ranging from 0.4 to 0.59 carats, the row price of a gem is 9.2% lower for a diamond of 0.45 carat compared to a diamond of 0.4 carat. These figures are 15.8% for 0.46 carat, 15.7% for 0.47 carat, 18.4% for 0.48 carat and 12.4% for 0.49 carat. The phenomenon is clearly observed up to 1.1 carats, while the shape is less clear beyond. The gross price curve tends to grow from 1.1 to 1.3 carats, while there are many more fluctuations above (this is presumably related to the much smaller sample sizes).

Insert Figure 6 At first sight, this pattern is very surprising. A consumer interested in buying a diamond will probably not understand why buying a gem with 0.05 additional carat would be cheaper compared to the corresponding round weight. A simple explanation could be that the cut of diamonds leads to a selection effect so that gems far from round prices have less attractive characteristics. In column 2

of Table 5, we control for the composition effect of the diamonds. Once the objective characteristics of the diamonds are taken into account, we find a different relationship between the price of diamonds and their weights.

As shown in Figure 6, we now observe an increasing profile between price and weight, with a jump at exactly the round quantity. For instance, the transition from 0.49 to 0.50 carats leads to an increase of +26.2% in the diamond price against 57.7% without controlling for the composition effect. The increase in price is 13.3% from 0.99 to 1 carat (the row jump was equal to +39.6%) and 20.1% from 1.49 to 1.5 carats (the row increase was 35.4%). Thus our results highlight the importance of taking the other diamond features into account when explaining prices. On average, the quality of diamonds whose weight is just below the round weight tends to be lower. This explains why net of the composition effect we never observe any decrease in price for larger gems.

Discussion and concluding comments

In this paper, we have used a unique dataset on more than 100,000 diamonds to bring evidence on the relationship between the price setting of diamonds and their weight. Drawing on decomposition techniques and unconditional quantile regressions, our framework based on hedonic price functions à la Rosen leads to the three following results. First, the weight in carat is the main predictor of the diamond value as it explains about 95% of variation in price. Secondly, the weightprice elasticity, which is around 2 at the mean values of the sample, is rising along the price distribution of diamonds. Thirdly, we find substantial increases in price for gems with round weights compared to gems just below these round weights. These discontinuities were recently described in [START_REF] Scott | Pricing anomalies in the market for diamonds: Evidence of conformist behavior[END_REF], who consider them price anomalies.

Attempting to rationalize these jumps in price is a challenging issue. From our own results, we suggest the following explanation. On the demand side, consumers are looking for diamonds of high quality (remembering that diamonds are eternal), which means that price is presumably not the sole purchase criterion. If diamonds just below round weights are perceived as diamonds of lower quality, then consumers would have a strong preference for round weight diamonds. This would lead to a discontinuous demand, higher on focal points because diamonds are better at these points. On the supply side, producers would cut their stones to meet this increased demand. The fact that the weight in carats is a round number becomes in itself an indicator of quality of the diamond.

Unfortunately, with the supply data at hand (nothing is known about purchases), it is not possible to better assess the causality in the relationship between supply and demand. It may be that producers deliberately keep their best stones for carving in round weight, but it is also possible that consumers have shifted their preferences by discovering that the quality of diamonds whose weight is not round was lower. Whether increased demand at focal points shapes the cut of gems or whether consumers expect gems with round weights to be of higher quality remains moot. Whatever the case, the good news from our empirical analysis is that consumers wishing to buy a diamond can make substantial savings by purchasing a gem with a weight just slightly lower (0.01 carat less is most often enough) than the one originally intended. Note : The sample is restricted to diamonds whose weight is comprised between 0.4 and 1.25 carats. Source: authors' calculations using data from Info Diamond.

Note: estimates from OLS models, with standard errors in brackets. Significance levels are respectively 1% ( *** ), 5% ( ** ) and 10%.

Figure

  Figure 1. Shapes of diamondAsscherBrilliant round Cushion Emerald

Figure 2 .Figure 3 .

 23 Figure 2. Distribution of diamond prices

Figure 4 .

 4 Figure 4. Conditional quantile estimates of the diamond price-weight relationship

Figure 5 .

 5 Figure 5. Number of diamonds by carat weight

Table 1 . Descriptive statistics of the sample

 1 Note: Price increases are obtained from Least-Absolute Deviation estimates.

		8,000
		6,000
	Number of diamonds	4,000
		2,000
		0
		.4	.41	.42	.43	.44	.45	.46	.47	.48	.49	.5	.51	.52	.53	.54	.55	.56	.57	.58	.59	.6	.61	.62	.63	.64	.65	.66	.67	.68	.69	.7	.71	.72	.73	.74	.75	.76	.77	.78	.79	.8	.81	.82	.83	.84	.85	.86	.87	.88	.89	.9	.91	.92	.93	.94	.95	.96	.97	.98	.99	1	1.01	1.02	1.03	1.04	1.05	1.06	1.07	1.08	1.09	1.1	1.11	1.12	1.13	1.14	1.15	1.16	1.17	1.18	1.19	1.2	1.21	1.22	1.23	1.24	1.25

Table 2 . Hedonic diamond price functions (OLS estimates)

 2

	Variables		(1) All diamonds	(2) Brilliant diamonds (3) Brilliant diamonds
			coef		s.e.	coef	s.e.	coef	s.e.
	Constant		9.311 ***	(0.004)	9.294 ***	(0.005)	9.254 ***	(0.007)
	Weight (in carats)		1.940 ***	(0.001)	1.960 ***	(0.001)	1.958 ***	(0.001)
	Cut shape	Asscher	-0.468	***	(0.004)
		Brilliant round	Ref
		Heart	-0.422	***	(0.004)
		Cushion	-0.461	***	(0.002)
		Marquise	-0.425	***	(0.003)
		Oval	-0.453	***	(0.003)
		Pear	-0.449	***	(0.003)
		Princess	-0.457	***	(0.002)
		Radiant	-0.479	***	(0.003)
		Triangular	-0.509	***	(0.009)
		Emerald	-0.497	***	(0.002)
	Colour	Exceptional white + (D)	Ref			Ref		Ref
		Exceptional white (E)	-0.080	***	(0.002)	-0.091 ***	(0.003)	-0.090 ***	(0.003)
		Rare white + (F)	-0.128	***	(0.002)	-0.145 ***	(0.003)	-0.144 ***	(0.003)
		Rare white (G)	-0.208	***	(0.002)	-0.225 ***	(0.003)	-0.225 ***	(0.003)
		White (H)	-0.317	***	(0.002)	-0.317 ***	(0.003)	-0.317 ***	(0.003)
		Slightly tinted white (I-J)	-0.505	***	(0.002)	-0.502 ***	(0.003)	-0.502 ***	(0.003)
		Tinted white (K-L)	-0.804	***	(0.003)	-0.809 ***	(0.003)	-0.808 ***	(0.003)
		Tinted color (M-Z)	-1.101	***	(0.004)	-1.135 ***	(0.005)	-1.133 ***	(0.005)
	Clarity	Flawless -Internally Flawless (FL -IFL)	Ref			Ref		Ref
		Very Very Slightly Included (VVS1)	-0.108	***	(0.003)	-0.080 ***	(0.004)	-0.079 ***	(0.004)
		Very Very Slightly Included (VVS2)	-0.200	***	(0.003)	-0.173 ***	(0.003)	-0.170 ***	(0.003)
		Very Slightly Included (VS1)	-0.282	***	(0.003)	-0.261 ***	(0.003)	-0.257 ***	(0.003)
		Very Slightly Included (VS2)	-0.378	***	(0.003)	-0.376 ***	(0.003)	-0.372 ***	(0.003)
		Slightly Included (S1)	-0.529	***	(0.003)	-0.534 ***	(0.003)	-0.529 ***	(0.003)

Table 3 . Fields decomposition of diamond's price

 3

	Variables	All cut shapes			Brilliant diamonds
		0.5 carat	1 carat	1.5 carats	All	0.5 carat	1 carat	1.5 carats	All
	Weight (in carats)	-	-	-	94.7%	-	-	-	95.2%
	Cut shape	28.2%	30.9%	26.5%	0.9%	-	-	-	-
	Colour	21.7%	22.1%	26.2%	-0.3%	30.4%	31.6%	38.3%	-0.2%
	Clarity	36.4%	28.7%	27.9%	2.5%	54.2%	44.2%	36.7%	3.1%
	Certificate	3.6%	5.7%	10.0%	-0.4%	5.5%	10.7%	16.9%	-0.7%
	Polish	1.2%	1.5%	0.7%	0.1%	1.1%	1.3%	0.2%	0.1%
	Symmetry	2.1%	2.5%	1.1%	0.2%	1.4%	2.0%	0.6%	0.1%
	Fluorescence	-0.1%	0.1%	0.3%	0.0%	0.2%	0.4%	0.5%	0.0%
	Residual	6.8%	8.5%	7.3%	2.4%	7.2%	9.9%	6.9%	2.3%
	Total	100.0%	100.0%	100.0%	100.0%	100.0%	100.0%	100.0%	100.0%
	Number of observations	5,985	5,555	1,856	112,080	4,515	3,034	1,143	63,717
	R²	0.932	0.914	0.927	0.976	0.928	0.902	0.931	0.977
	Source: authors' calculations using data from Info Diamond.

Table 4 . Hedonic diamond price functions (unconditional quantile estimates)

 4

	Variables		Q10		Q25		Q50		Q75		Q90
			coef	s.e.	coef	s.e.	coef	s.e.	coef	s.e.	coef	s.e.
	Constant		7.836 ***	(0.021)	8.323 ***	(0.028)	8.922 ***	(0.021)	10.031 ***	(0.030)	11.235 ***	(0.044)
	Weight (in carats)		0.908 ***	(0.007)	1.719 ***	(0.008)	1.787 ***	(0.005)	2.663 ***	(0.007)	2.598 ***	(0.013)
	Cut shape	Asscher	-0.280 ***	(0.017)	-0.267 ***	(0.022)	-0.357 ***	(0.020)	-0.809 ***	(0.028)	-0.719 ***	(0.041)
		Brilliant round	Ref		Ref		Ref		Ref		Ref
		Heart	-0.188 ***	(0.024)	-0.384 ***	(0.028)	-0.475 ***	(0.019)	-0.640 ***	(0.027)	-0.572 ***	(0.032)
		Cushion	-0.181 ***	(0.010)	-0.151 ***	(0.012)	-0.426 ***	(0.010)	-0.907 ***	(0.014)	-0.771 ***	(0.019)
		Marquise	-0.337 ***	(0.020)	-0.334 ***	(0.020)	-0.439 ***	(0.015)	-0.601 ***	(0.021)	-0.524 ***	(0.026)
		Oval	-0.112 ***	(0.015)	-0.166 ***	(0.020)	-0.480 ***	(0.016)	-0.800 ***	(0.021)	-0.718 ***	(0.028)
		Pear	-0.147 ***	(0.013)	-0.221 ***	(0.017)	-0.484 ***	(0.013)	-0.779 ***	(0.017)	-0.706 ***	(0.021)
		Princess	-0.483 ***	(0.011)	-0.513 ***	(0.011)	-0.373 ***	(0.007)	-0.527 ***	(0.011)	-0.501 ***	(0.014)
		Radiant	-0.189 ***	(0.012)	-0.211 ***	(0.016)	-0.514 ***	(0.013)	-0.833 ***	(0.017)	-0.701 ***	(0.025)
		Triangular	-0.212 ***	(0.050)	-0.343 ***	(0.057)	-0.498 ***	(0.041)	-0.718 ***	(0.068)	-0.821 ***	(0.083)
		Emerald	-0.260 ***	(0.011)	-0.257 ***	(0.014)	-0.414 ***	(0.012)	-0.871 ***	(0.018)	-0.868 ***	(0.024)
	Colour	D	Ref		Ref		Ref		Ref		Ref
		E	-0.067 ***	(0.010)	-0.066 ***	(0.013)	-0.052 ***	(0.009)	-0.083 ***	(0.012)	-0.119 ***	(0.016)
		F	-0.132 ***	(0.010)	-0.030 **	(0.012)	-0.043 ***	(0.009)	-0.124 ***	(0.012)	-0.276 ***	(0.017)
		G	-0.173 ***	(0.010)	-0.062 ***	(0.012)	-0.055 ***	(0.009)	-0.265 ***	(0.013)	-0.454 ***	(0.018)
		H	-0.272 ***	(0.011)	-0.189 ***	(0.013)	-0.219 ***	(0.009)	-0.390 ***	(0.013)	-0.529 ***	(0.019)
		I-J	-0.357 ***	(0.011)	-0.336 ***	(0.013)	-0.421 ***	(0.009)	-0.685 ***	(0.013)	-0.780 ***	(0.018)
		K-L	-0.559 ***	(0.016)	-0.761 ***	(0.018)	-0.658 ***	(0.012)	-1.057 ***	(0.018)	-0.978 ***	(0.025)
		M-Z	-0.852 ***	(0.029)	-1.103 ***	(0.028)	-0.931 ***	(0.019)	-1.291 ***	(0.029)	-1.242 ***	(0.037)
	Clarity	FL -IFL	Ref		Ref		Ref		Ref		Ref
		VVS1	0.009	(0.012)	-0.064 ***	(0.020)	-0.115 ***	(0.014)	-0.130 ***	(0.021)	-0.155 ***	(0.031)
		VVS2	-0.079 ***	(0.012)	-0.134 ***	(0.019)	-0.164 ***	(0.014)	-0.241 ***	(0.021)	-0.263 ***	(0.031)
		VS1	-0.225 ***	(0.012)	-0.241 ***	(0.018)	-0.187 ***	(0.013)	-0.325 ***	(0.020)	-0.299 ***	(0.029)
		VS2	-0.424 ***	(0.012)	-0.345 ***	(0.018)	-0.238 ***	(0.013)	-0.391 ***	(0.020)	-0.448 ***	(0.030)
		S1	-0.468 ***	(0.012)	-0.413 ***	(0.018)	-0.333 ***	(0.013)	-0.601 ***	(0.020)	-0.780 ***	(0.029)
		S2	-0.458 ***	(0.012)	-0.422 ***	(0.018)	-0.498 ***	(0.014)	-0.951 ***	(0.020)	-0.995 ***	(0.030)

Table 5 . Effect of a 0.01-carat increase on price at focal weights

 5

	Focal carat weight	Effect of a 0.01-carat increase on price	Number of	Number of
		Row increase	Net increase	Diamonds	variety
	0.5	57.7	26.2	5985	366

For an overall description of the diamonds market, see for instance http://www.safariminerals.com/en/overallmarket.html or http://www.diamineexplorations.com/web/index.php?id=147. It is estimated that approximately 80% of the world's annual production of rough diamonds is under the control of the De Beers Group and other major diamond producers including Rio Tinto Group, BHP Billiton Group and Alrosa Group. Antwerp is the world's largest diamond trading center with other key centers including Mumbai, Johannesburg and Tel Aviv.

2 Following[START_REF] Rauch | Networks versus markets in international trade[END_REF], differentiated products are goods not having a reference price or goods whose price is not quoted on organized exchanges.

According to Info Diamond, a slight fluorescence should depreciate diamonds D, E, and F, while it should give an increase in value to diamonds equal or below G.

7A depth too high or too low is likely to impact how a diamond reflects light. Ideal round diamonds should have a depth of between 59% and 62.5%.8 Both for depth and table, the range considered the best for round diamonds may slightly differ from company to company.

The exact sampling distribution of this unbiased estimator is further investigated in[START_REF] Giles | Interpreting dummy variables in semi-logarithmic regression models: Exact distributional results[END_REF].

Interestingly, the visible effects of faint (+7.1%) or even medium (+3.5%) fluorescence are mainly perceptible to gemologists using a special UV light source.

For instance, a basic search of the expression "diamond price calculator" on Google yields 63500 hits. A calculator is available on Info Diamond at http://www.info-diamond.com/others/diamond-prices.html.

The shape of the price-weight curve is slightly different when the sample is restricted to brilliant stones. The profile increases strongly till the 10th percentile, then remains rather flat until it increases again sharply after the 70th percentile.

The increase is equal to +1.2% between 1.09 and 1.1 carat, -0.7% between 1.19 and 1.2 carat and +4.7% between 1.29 and 1.3 carat. Nevertheless, the discontinuity in price is again higher for heavier diamonds (+25.9% between 1.39 and 1.4 carat, +35.4% between 1.49 and 1.5 carat).

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Pierre Cariou
email: pierre.cariou@euromed-management.com

François-Charles Wolff

Chartering practices in liner shipping

Keywords: Liner shipping, chartering, profitability Chartering practices in liner shipping

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Asset management is a key component of the operational and commercial success of container shipping lines. Confronted with over-capacity resulting from the recent financial and economic crisis over the last two years, container lines have implemented many measures to restore their profitability. The most cited actions are: postponing or cancelling the delivery of vessels, scrapping or laying-up vessels and slow steaming services [START_REF] Notteboom | Container Shipping[END_REF].

When considering this list of actions, it turns out that little attention has been devoted to chartering. Liner operators always own a certain number of vessels to meet the minimum requirements of their contract engagements, but they also charter extra tonnage to meet specific market requirements [START_REF] Gorton | Shipbroking and chartering practice[END_REF]. For the sake of illustration, out of the 4,916 containerships operated by the top-100 liner operators in June 2011, 57.8% in number and 52.7% in capacity were chartered. These figures are fairly similar to those observed in March 2007 since the proportions were 57.1% and 51.8% respectively [START_REF] Alphaliner | Alphaliner Top 100[END_REF].

The extent of these charter practices and its potential impact on profitability of liner shipping companies is the topic under investigation in this paper. Our contribution is to shed light on these practices using a large dataset of the containership fleet. For a subsample of the largest operators, we also study the relationship between chartering practices and profitability.

To do so, our paper is organized as follows. The first section presents a literature review on asset management in liner shipping and provides some background information on chartering practices. In Section 3, we gather some descriptive statistics on the extent of this decision by liner operators in 2009. Section 4 provides an econometric analysis describing how vessel and firm characteristics influence chartering practices. We further study chartering strategies using a sub-sample of 17 large liner operators. Section 5 presents some preliminary estimations of the relationship between operating profit margins and chartering practices and size of these firms in 2007, 2008 and 2009. Finally, Section 6 provides concluding comments and outlooks for further research.

Literature review

There are many operational strategies to absorb over-capacity. According to Notteboom [START_REF] Notteboom | Container Shipping[END_REF], lay-up and slow steaming/re-routing services have been the most effective in recent years [START_REF] Vernimmen B | The effect of high fuel costs on liner service configuration in container shipping[END_REF][START_REF] Cariou | Is slow steaming a sustainable means of reducing CO2 emissions from container shipping[END_REF]. These strategies combined with strategic alliances and mergers and acquisitions [START_REF] Ryoo | Liner alliances in the globalisation era: a strategic tool for Asian container carriers[END_REF][START_REF] Slack | Strategic alliances in the container shipping industry: a global perspective[END_REF][8] contribute to restoring profitability in a proportion that is difficult to assess in markets that have recently subjected to large fluctuations in freight rates.

Fleet management in chartering instead of owning the vessel is another option largely used by the main operators. The proportion of chartered vessels was equal to 57.8% among the top-100 operators in June 2011 [START_REF] Alphaliner | Alphaliner Top 100[END_REF]. In 2010, 2,406 transactions were recorded: this number has substantially increased over the last few years since it was respectively 1,485 and 1,717 in 2008 and 2009 [8][9][10]. This increase in the number of transactions is a likely consequence of the financial crisis that has led most operators to charter-in vessels.

Compared to the owing of a vessel, chartering presents a number of advantages. Firstly, it reduces the debt leverage ratio [START_REF] Stopford | Maritime Economics[END_REF]. Secondly, chartering offers the possibility of allocating resources to other activities such as door-to-door services [START_REF]Empirical evidence for integration and disintegration of maritime shipping, port and logistics activities[END_REF]. Thirdly, it secures future slot costs that are pre-established within the charter party [START_REF] Drewry | Containerships charter market: A positive course for the new millennium[END_REF]. Fourthly, chartering provides an indirect access to ship funds, leasing or partnership structures [START_REF] Oldershausen | German KG Financing, a well proven ship finance alternative: overview and future challenge[END_REF]. Finally, it is a way for a company to benchmark its owned vessel operating performances with those of chartered vessels [START_REF] Wolff F-C | Ship-owners' decisions to outsource vessel management[END_REF].

Most background information on chartering practices is found in the "grey" literature and can be summarized in the following way. Chartered container vessels are predominantly controlled by German interests with a specific partnership structure known as "Kommandit Gesellschaft" or KG funds which has existed since 1969 [START_REF] Stopford | Maritime Economics[END_REF]. KGs are closed end funds, not limited to shipping, that provide tax concessions to investors. They combine the advantages of a partnership with those of a limited liability corporation. Initially implemented to stimulate investment in German flag registered ships, this agreement led the country to own approximatively 58% of all chartered containerships in 1998 [START_REF] Drewry | Containerships charter market: A positive course for the new millennium[END_REF]. While the tax advantages expected from KGs have since been considerably reduced, 15% of KG investments were still allocated to containerships in 2005, amounting to 10 billion Euros [START_REF] Oldershausen | German KG Financing, a well proven ship finance alternative: overview and future challenge[END_REF]. Today, German owners control around 35% of all containerships (in teu) on the chartering market [START_REF] Deutsche | Container shipping: Successful turnaround[END_REF].

The vast majority of chartered vessels are mid-sized vessels: only 18 out of the 404 transactions recorded during the fourth-quarter of 2010 were more than 5,000 teu vessels, compared to 221 vessels that were less than 2,000 teu [10]. The mean charter durations and rates are subject to significant fluctuations. For a 4,000 teu vessel, the average duration was 24 months in 2007, but it was three times lower (8.1 months) for transactions recorded during the last trimester of 2010. Charter rate fluctuations are also paramount, moving from 26,100 USD per day in 2008 to 5,200 USD per day in 2009 for 4,000 teu vessels. Finally, some differences among the various vessel types are explained by changes in operating costs, which vary respectively according to size and age [START_REF] Hsh Nordbank | A study on the operating costs of German ships[END_REF].

Companies' chartering practices are expected to impact their financial performance. The latter has been greatly affected during recent years, as highlighted by the case of Maersk Line (-1,938 USD Million in operating profits in 2009), CGA-CGM (-1,148 USD Million) and COSCON (-1,144 USD Million). Interestingly, to the best of our knowledge, there is no study to date that has attempted to investigate the relationship between chartering decisions and financial performance. The main difficulty is that the relationship between chartering practices and liner profitability is ambiguous on a priori grounds.

On the one hand, operators who were chartering significantly more with long duration contracts were probably in a worse situation when the crisis hit the liner shipping markets in mid-2008. Once the market collapsed, they were unable to quickly reduce their overcapacity. On the other hand, operators who were chartering more with short term duration contracts were able to reduce over-capacity and to take advantage of the recent decrease in chartering rates in 2009.

Using data on a sub-sample of 17 large liner operators, this last issue will be subject to further investigation in the final section of this article. Meanwhile, we in the next section provide an econometric analysis of chartering practices in 2009.

Chartering practices: data selection and identification

Our empirical analysis is based on exhaustive data from Lloyd's Register Fairplay collected in 2009 [START_REF] Fairplay | IHS Fairplay World Shipping Encyclopaedia[END_REF]. This data set includes information on all merchant vessels in the world fleet. In what follows, we focus our attention on containerships. This leaves us with a sample comprising 4,220 vessels in 2007, 4,616 in 2008 and 5,005 vessels in 2009. For each vessel, information was retrieved on its owner and on its operator with the view that when the company's name is different, the vessel is chartered by the operator.

One difficulty when analysing the extent of chartering practices in liner shipping is that a prior identification of potential shareholding ties between legal proprietors and operators in charge of the commercial decisions on vessel deployment is required. Due to shipping practices where companies often finance the acquisition of vessels through subsidiaries, this identification is far from being straightforward.

On the basis of their names, we identified 890 different ship-owners and 580 different operators in our original data. However, these numbers undoubtedly over-estimate the exact numbers of owners and operators as they include subsidiaries. To limit this bias, we use owners' and operators' websites as well as professional publications (Alphaliner and Drewry publications) to reconstitute shareholding ties between operators and owners.

For the sake of illustration, vessels operated by Safmarine Container Lines, Moller AP, Moller-Maersk A/S, Mercosul Line, Ocean Africa Container Lines were all considered as part of the same operating group (renamed Maersk Line). At the owner level, Moller AP, Moller W, Nedlloyd Genoa Ltd, Nedlloyd Marseille Ltd, Maersk Line, Royal P&O Nedlloyd NV were also considered as part of the Maersk Line's owner group. It follows that a vessel belonging to these two groups was not considered as chartered, but as a Maersk Line's vessel 1 .

Following this screening of our initial dataset, our sample was downsized to 760 owners and 510 operators. For 2,767 of the 5,005 vessels operated in 2009 (55.2% of vessels), the name of the operator differs from the name of the owner. The proportion was equal to 57.6% in 2007 and 57.2% in 2008 respectively. It is interesting to note that these chartering rates are very similar to the proportion of 57.1% reported in March 2009 by Alphaliner for the top-100 liner operators [START_REF] Alphaliner | [END_REF].

For each vessel, we also gathered information from the LRF database on its age (in years) and size (in teu). We exploit this information to calculate a proxy for the chartering rate in the following way. We first use representative one-year time charter rates contracted in 2008 [START_REF] Drewry | Shipping Insights[END_REF]. These rates were 5,000 USD/day for a 500 teu, 9,000 USD/day for a 1,000 teu, 15,000 USD/day for a 1,700 teu, 25,000 USD/day for a 2,500 teu and 35,000 USD/day for a 4,000 teu. As a rule of thumb, we decided to use a 10 USD/day/teu rate for less than 1,000 teu vessels, and then reduced by 1 USD every 2,000 teu increase.

Keeping the same assumption f, the method leads to 4 USD/day/teu, or for a 14,000 teu vessel 56,000 USD/day2 . The few figures reported by ship brokers in 2008 show that 13,100-14,000 teu vessels are chartered at a rate of 58,700-60,650 USD/day, a value in line with our estimates. Then, a conservative assumption of a 3% increase in charter rates for every year of age was finally considered to account for the impact of age on operating costs [START_REF] Hsh Nordbank | A study on the operating costs of German ships[END_REF].

Table 1 presents some descriptive statistics on vessels and operators in 2009. A first insight is the similarity in the rate of chartering between small (less than 5 vessels) and large (more than 100 vessels) operators, respectively 56.7% and 57.5%. As expected, Germany is the first country of ownership in terms of number of vessels operated, with 13.8% of all vessels in number, equivalent to 17% of the 13.7 million teu world capacity.

A detailed analysis for chartered vessels stresses that 52% of the 2,767 vessels in number and 26% in teu capacity are controlled by owners registered in Germany -a result lower than the 35% reported by Deutsche Bank Research for 2010 [START_REF] Deutsche | Container shipping: Successful turnaround[END_REF]. In 2009, Doehle owned 77 vessels on the chartering market, ER Shiffahrt GmBH (72 vessels), MPC (67), Hansa Shipping (61) and Buss H (61). In the total rankings, the Chinese government with 75 vessels is the only non-German entity company within the top-5 chartering companies. Finally, vessels with higher chartering rates due to either their size and/or their age are, as expected, less chartered (43.6%) than other vessels. They are however less frequent, representing 7% of all containerships.

Insert table 1 around here

To further identify differences in chartering practices among the largest operators, we chose to focus on a subset of 17 operators. These companies were selected because of the availability of information on their financial performances. These companies are Maersk Line, CMA-CGM, Hapag-Lloyd AG, APL Ltd, NYK, MOL, Hanjin, COSCON, OOCL, Evergreen, CSCL, CSAV, Hyundai, Yang Ming, Zim, Wan Hai and RCL. In 2009, they operated 2,382 vessels (corresponding to 47.6% of all vessels) equivalent to a carrying capacity of 8,697,831 teu (63.5% of the world teu capacity). The major liner operator missing from our sample is MSC. This company operates 393 vessels (it is ranked second after Maersk Line and before CMA-CGM), but it wasimpossible for us to retrieve any financial information. Figure 1 presents the number of owned and chartered vessels in 2009 within the 17 operators' respective fleets.

Insert figure 1 around here

The 17 companies have a mean chartering rate of 55.3%, a proportion similar to the one observed earlier for the entire sample of 510 operators (55.2%). Maersk Line is clearly ahead of all firms in terms of operated fleet, with about 500 vessels. The second operator in our sample is CMA-CGM (ranked third in the world), with about 300 vessels. There are large differences in the rate of chartering by operators. CSAV (98% of its operated fleet), Hyundai (83.6%) and MOL (71.5%) charter much more on average, while the reverse pattern is found for RCL (15.9%), COSCON (25.9%) and Wan Hai (29.1%).

There are also significant differences in the characteristics of vessels operated. As shown in table 2, companies like COSCON, Evergeen or RCL operate vessels of approximatively 11 years old on average. This is more than 5 years older than the mean age of vessels operated, for instance, by CSAV (5.9) or Hyundai (5.7). The largest fleet in size is operated by APL Ltd, Hanjin, OOCL and Hyundai with a mean size exceeding 4000 teu, while Wan Hai and RCL operate the smallest fleet.

Insert table 2 around here

Another interesting finding is that there are significant differences between the characteristics of chartered and owned vessels at the operator level. Consider for instance Maersk Line and CMA-CGM. On average, the vessels they own are much larger (155% for Maersk Line, 180% for CMA-CGM) and slightly younger than the vessels they charter.

Conversely, the mean size of chartered vessels is larger than that of owned vessels in the case of operators like COSCON, CSAV, or APL Ltd. The next section provides further insight by developing an econometric analysis applied to the entire sample and then to the 17 operators.

Econometric analysis of chartering practices

We turn to an econometric analysis to further study the role played by the characteristics of operators and vessels on chartering rates. Let * be a latent variable measuring the propensity that a company decides to charter a vessel . We rely on the following linear expression for * :

* = + + + (1)
where and are sets of characteristics respectively specific to the operators and to the vessels, and are vectors of coefficients to estimate, and and are random perturbations. On the one hand, is an operator-specific unobserved heterogeneity term that will pick up preferences for chartering. On the other hand, is a pure random term specific to each vessel. We assume that both perturbations are normally distributed such that ~ (0; ) and ~ (0; 1). By definition, * is unobserved, but we know that the decision to charter a vessel is such that = 1 when * > 0 and = 0 when * ≤ 0. It thus follows that the probability of a vessel being chartered is:

Pr( = 1) = Φ( + + ) (2)
with Φ the cumulative distribution function of the univariate standard normal distribution.

We note that the probability Pr( = 1) depends on the unobserved operator component . Let Γ ! = + + . For a given operator, it follows that:

Pr( " , … , % ) = & ∏ (Φ)Γ ! *+ , -. (1 -Φ)Γ ! *+ "0, -. % 1" 23 03 4) *d( ) (3)
with 4 the density function of the multivariate normal distribution. This specification defines a random effect Probit model that is estimated using quadrature techniques [START_REF] Butler | A computationally efficient quadrature procedure for the one factor multinomial Probit model[END_REF]. The underlying assumption behind the random effect framework is that the operator-specific unobserved heterogeneity terms are uncorrelated with the characteristics of the vessels.

The corresponding estimates, applied to the sample of 510 operators and 5,005 vessels, are presented in table 3.

Insert table 3 around here

As shown in column (1), at conventional levels, the size of the operator does not significantly influence the decision of chartering a vessel. In column (2), we introduced the growth rate in the fleet operated by an operator over the 2007-2009 period. As a reminder, the growth rate in the operated fleet over this period amounts to 18.6%, from 4,220 vessels in 2007 to 5,005 in 2009. We find that this covariate is positively correlated with the rate of chartering. This suggests that operators that have increased their fleet have done so by slightly increasing the proportion of vessels chartered. An explanation of this result is that it is obviously easier and faster to charter than to buy.

According to table 3, chartering is less frequently observed for older vessels. The negative correlation between chartering and age of vessel is significant for vessels over 10 years old and is much higher for vessels over 20 years old. A similar pattern is found for vessel size.

The probability of chartering is significantly lower for larger vessels than for 4,000 teu. Given the high correlation between vessel characteristics (age/size) and chartering rates, we only include the latter covariate in column [START_REF] Alphaliner | Alphaliner Top 100[END_REF]. Finally, we find that vessels with high chartering costs are less likely to be chartered.

In table 4, we enrich the analysis on the impact yielded by operator size by providing some descriptive statistics on the fleet management strategy. Since the chartering decision is binary, this implies that a given operator may face the following three configurations: either all vessels are chartered, all vessels are owned, or some vessels are owned and some are chartered. The strategy is strongly affected by the size of an operator. The likelihood of having all vessels chartered strongly decreases with an operator's size, from 41.4% for companies with two vessels to 17.8% for companies with 11-50 vessels. The proportion of companies with owned vessels also decreases with size, but essentially for operators with more than 5 vessels. Finally, the intermediate option or mixed strategy strongly increases with size: 33.3% for companies with 6-10 vessels, 57.7% with 11-50 vessels to reach 100%

when an operator has a fleet of more than 100 vessels.

Insert table 4 around here

We then re-estimated the random effect Probit regressions for the sub-sample of 17 operators. Estimates in columns 1 and 3 of table 5 confirm earlier findings. We again observe the positive impact of a company's past two years growth on chartering rate, an effect higher than the earlier one (+0.022 against +0.003). This result is not explained by the higher growth rates for these 17 companies, since the increase from 2007 to 2009 is equal to 14.1% versus 18.6% for the 510 operators. It mainly stems from the presence in our sample of companies that have a relative tendency to charter more often 3 . Finally, in line with our former findings, chartering for these companies is also less frequent when vessels are older, larger (above 6000 teu) or have higher chartering costs (column 3).

Insert table 5 around here

Since there are many vessels per operator in the data, we decided to further investigate how the characteristics of chartered vessels differ from those of the vessels he owns (column 2).

To analyze the issue for the 17 selected operators, we estimated a fixed effect Logit model using a conditional likelihood maximization [START_REF] Chamberlain | Analysis of covariance with qualitative data[END_REF]. Since the characteristics of an operator are by definition the same for all the vessels of its fleet, they are picked up by a fixed effect and they are thus excluded from the list of covariates. The probability that a vessel is chartered within an operator's fleet can therefore only be a function of its age and size.

As shown in column 2, the fixed effect framework had little influence on our results.

Estimates confirmed the existence of a negative correlation between the likelihood of chartering and both the age and size of the vessel. On average, operators own older and larger vessels while they charter-in younger and smaller vessels. This pattern is still observed when using the chartering cost proxy (column 4). This last result is interesting in light of the impressive growth in the number of large vessels during the period. For instance, the number of more than 6,000 teu vessels increased by 51% over the period to reach 478 vessels in 2009 compared to 317 in 2007, while less than 6,000 teu containerships grew by 16% in number.

The impact of chartering practices on profitability

In this last section, we attempt to measure to what extent the chartering practices identified earlier for the 17 selected operators may have impacted their financial performance in 2007, 2008 and 2009. Information from the companies' financial statements was retrieved from [10]. It included their operating profit margins and revenues used to determine an operating profit margin to revenue ratio (OPM hereafter). These results only concern the container activity of such companies. 11% at the global level and in particular by more than 24% for the Europe/Far East trade [9].

It is therefore not surprising to find that the two Chinese companies in our sample, CSCL and COSCON, recorded the highest drop in 2008 with around a 30% decrease in OPM. The high exposure of these two companies to Asian markets is an obvious reason. COSCON reported a drop of 30% in its average volume transported in 2008 compared to a 14% drop for Maersk Line.

When turning to results for 2009 compared to 2008, we observe that some operators experienced a faster recovery than others. This is the case for instance for RCL, Hyundai and OOCL. Conversely, four companies still reported negative figures: Zim, NYK, MOL and CSAV.

For Zim, this result is due to their fleet deployment with an average drop in freight rate estimated at 3% combined with more than a 30% decrease in volume [9]. For NYK and MOL, this is mainly explained by their exposure to Europe/Far East trade where average freight rates dropped by 50%. Finally, for the Chilean based shipping company CSAV, this is related to the collapse of South American exports and the company's choice in 2008 to opt for long term chartered contracts.

In figure 3, we examined whether these observed differences in OPMs may be attributed to operators' chartering practices, although we are fully aware of the fact that this can only be seen as one factor among others. Furthermore, given the huge changes in the economic context over the 2007-2009 period which hampered the potential relationship, we chose to represent year-specific correlations.

As illustrated in the case of CSAV, greater chartering using long term contracts may negatively impact profitability when markets suddenly plummet such as in 2009. At the same time, greater chartering using short term contracts can positively impact profitability as the operator's fleet can be quickly reduced in the event of over-capacity. Figure 3 illustrates the ambiguous role played by the chartering rate. We observed a negative relationship in 2007. Companies that were chartering less were relatively more profitable and showed a slight upturn to a positive relationship in 2009. During this period, as mentioned earlier, the average charter duration dropped from 24 months in 2007 to 4 months in 2009.

Insert figure 3 around here

To further investigate this relationship, we first estimate an OLS model of OPM as a function of the operator's size in number of vessels and of the operator's chartering rate. Operator size was used as a proxy of company exposure on the market, with the idea that larger companies share similar positioning in terms of fleet deployment. According to the results reported in panel A of table 6, we find that operator size has no influence, leading to the conclusion that small and large companies have been similarly impacted.

A negative correlation is found between OPM and a company's chartering rate, but the corresponding coefficient is not significant in 2007 (at the 1% level). At the same time, one likely reason for such a weak relationship might be due to our sample size which remains limited since we consider only 17 operators. In such a context, the presence of outliers in the data is likely to distort the classical ordinary least square estimates and to lead to unreliable results [START_REF] Rousseeuw | Robust Regression and Outlier Detection[END_REF].

the dispersion of the data confirmed in figure 3, we decided to rely on a linear regression estimator, robust to outliers, and to apply the MM-estimator described in [START_REF] Verardi | Robust regression in Stata[END_REF].

This estimator is robust to both vertical outliers (observations with outlying values for the ydimension), good leverage points (observations with outlying values in the space of covariates, but located close to the regression line) and bad leverage points (observations outlying in the space of covariates and located far from the regression line). For instance, Figure 3 suggests that in 2009 a company like Zim is a vertical outlier, RCL and CSAV are good leverage points, but COSCON or Wan Hai tend to be bad leverage points.

Insert table 6 around here

In Panel B, the MM-estimates suggest that it is indeed important to take into account the influence of outliers. The operator's size now plays a role and the increase in its coefficient over time suggests that larger operators had a tendency to recover faster from the crisis. For the three years under consideration, the chartering rate of a company has a statistically significant role when understanding profitability, albeit only a small one. In 2007 and 2008, when for instance the typical 12 month chartering rates for a 3,500 teu vessel were 29,350-26,450 USD/day [9], a 1% increase in chartering rate slightly reduced the OPM of companies by -0.174% in 2007 and -0.071% in 2008 respectively.

This confirms the negative relationship that was suggested in figure 3 for these two years.

The situation is then reversed in 2009, when chartering rates soared to 7,600 USD/day: operators that chartered more generated more profits (+0.101%) during this period. Our results therefore show the difficulty in linking chartering practices and profitability due to the substantial fluctuation in chartering rates over the period.

Conclusions

This article is a preliminary attempt to shed light on the extent and similarity of chartering practices by liner operators and to explore their impact on profitability. Far from being complete, it raises many questions both to the quality of information used and the interpretations of the results.

On the first issue, the shipping world has always been an intricate industry where direct and indirect relationships, strategic alliances and partnerships co-exist. Our investigation attempts to clarify certain relationships by establishing some shareholding ties between owners and operators. Nevertheless, this still leaves many open questions as to when these two entities should be considered separately.

To provide an example, when dealing with vessels operated by MSC and analyzing the case of 510 operators, we decided to consider that each time a vessel was owned by a single ship company, registered only with the name of the vessel as owner, this was indeed a subsidiary of MSC and not a chartered vessel. To what extent this reflects the true ownership is an issue that probably requires more in-depth analysis. Another issue is the limited size of our dataset since we only consider 17 operators. This constraint is due to the availability of financial results, which reduces the possibility of implementing a more comprehensive analysis on factors influencing profitability.

Interpretation of our results is also subject to caution. Our empirical analysis suggests that the relationship between chartering practices and profitability evolves over time as both the rates of chartering per vessel and the contract duration change over time. Furthermore, many factors were not considered in this article [START_REF] Hoffmann | Determinants of Vessel Flag[END_REF]. For instance, what is the ship-owner's nationality and does he have stakes in a shipyard? Is the company private or government owned? Does past performance influence a company's decision to charter in a given year?

Nevertheless, this article provides a preliminary framework to tackle these issues. Further investigation would for instance incorporate additional information on charter contract durations and on company market-positioning.

Figure 2

 2 provides some primary results on the changes in OPM in 2008 compared to 2007 and 2009 compared to 2008. Insert figure 2 around here Due to the financial crisis that severely hit the shipping markets in mid-2008, all firms experienced negative figures in their OPMs in 2008 compared to 2007. This is the result of collapsing markets in the last quarter of 2008, when the average container revenues fell by

Figure 1 .Figure 2 .

 12 Figure 1. Number of owned and chartered vessels in 2009, for selected operators

Figure 3 .

 3 Figure 3. Chartering rates and operating profit margin A. 2007

Table 1 . Descriptive statistics of the sample for 2009

 1

	Variable (N=5,005 vessels)		Chartering rate (in %)	Distribution of vessels (in %)
	Operator's size	0-4 vessels	56.7	12.1
		5-9 vessels	48.5	8.1
		10-49 vessels	47.7	18.6
		50-99 vessels	62.0	9.7
		≥100 vessels	57.5	51.5
	Operator's nationality	Germany	43.5	13.8
		Denmark	59.3	10.0
		China	46.5	7.9
		Switzerland	71.3	7.9
		Japan	60.0	7.2
		Taiwan	35.9	6.0
		France	69.6	5.9
		Korea (South)	60.6	5.1
		Singapore	40.2	4.7
		USA	67.0	4.4
		Others	57.3	27.2
	Chartering rate	<10,000	54.4	41.5
	in USD/year	10,000 -19,999	59.6	23.8
		20,000 -29,999	58.7	14.3
		30,000 -39,999	52.7	13.4
		≥40,000	43.6	7.0
	Total		55.2	100.0

Source: Author's calculation from Lloyd's Register Fairplay -World Shipping Encyclopaedia (June 2009)

Similar groupings were done for all operators and owners. The coding is available upon request from the authors.

For this category of vessels, a representative rate does not exist due to the limited number of transactions on the chartering market.

For instance, broker information reported in[10] stresses that during the last trimester of 2009, CMA-CGM (46 vessels), Maersk Line (38) and CSAV[START_REF] Verardi | Robust regression in Stata[END_REF] were the most active companies on the chartering market, these three companies being amongst our 17 selected operators.