Measuring Bond Market Liquidity

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Measuring Bond Market Liquidity: Devising A Composite Aggregate Liquidity Score

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fications as required due to the diverse nature of the constituent components that we use. We also use a method based on that used to measure “diversity” in structured financial or securitization products such as asset-backed securities and collateralized debt obligations. We applied additional techniques in calculating our liquidity measure; these included an Index method using Laspeyres and Paasche indices, and a matrix-style linear programming method used to calculate maximum and minimum index values. Using our composite score, we found that the level of liquidity in the gilt market had increased during the period under study, with the increase in score being marked following full implementation of the reforms. For the empirical study we consider the UK gilt market during the period 1993–2002. This time period is dictated by the timing of the structural reforms undertaken by the BoE, implemented during 1996–1998, and takes into account the period immediately before and after the implementation of the reforms. The market reforms, which were designed to improve liquidity and accessibility of the U.K. gilt market (see Bank of England [1995]), included the introduction of new products as well as technical changes to operating processes. The following reforms were introduced:

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inancial institutions as well as the central monetary authorities are concerned that financial markets operate in an orderly manner and offer sufficient liquidity. The importance of having adequate levels of liquidity is emphasized strongly in the academic literature. It is accepted that liquidity is one of the most important characteristics of an orderly market. However, there is more than one definition of liquidity. This can lead to confusion when attempting to measure liquidity levels. The issue of measuring liquidity level in any market is a problematic one, and one that is often solved through the use of various proxy indicators, such as the extent of an asset’s bid-offer spread, level of market turnover, or asset issuance volume. No single measure is a completely satisfactory indicator, however, and the use of proxy measures renders comparison across different markets difficult. The objective of this article is to devise a composite aggregate measure of liquidity that makes use of a range of proxy measures and infrastructure factors, and that can be applied to any financial market. We attempt to measure the overall liquidity of the market as an index-type score, which enables comparison to be made between different years in the observation period. The calculation of the liquidity score is based on techniques used to calculate the value of an equity or bond index, with modi-

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is visiting professor at London Metropolitan Business School in England, U.K. [email protected]

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Moorad Choudhry

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Moorad Choudhry

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LITERATURE REVIEW

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Our research makes two contributions to the financial economics literature. First, we devise a quantitative indicator that allows us to assess whether the stated objectives of the BoE, at the time the reforms of the gilt market were undertaken, were actually achieved. Secondly, we devise a measure of liquidity that can be applied to any debt or equity market, thus enabling a comparison of liquidity levels across markets. The article is set out as follows: first, we present a review of the relevant literature. We then describe the methodologies used in calculating our different liquidity scores and the rationale behind them. We then apply these methods to calculate the level of liquidity in the gilt market during the observation period. We calculate 1) a diversitybased liquidity score and 2) Paasche and Laspeyeres index scores, which are tested using linear programming. Either method can be applied to any capital market. We conclude that U.K. gilt market liquidity did increase during the period under observation, and that the introduction of the structural reforms contributed in part to this increase, even during times of market correction. We also suggest that the new liquidity measure has applicability for other markets as a standard measure of liquidity. Institutional investors can apply the methodology we describe here to other financial markets to test for relative liquidity levels, prior to making the investment decision.

found that the difference in bid-offer spread between U.S. Treasury bills and Treasury securities had an impact of yield-to-maturity. Amihud, Mendelson, and Lauterbach [1997] observed that asset values on the Israeli stock exchange underwent changes when the equities began to be traded on a more liquid electronic system. Researchers have identified a number of factors that are determinants of liquidity. Alexander, Edwards, and Ferri [2002] and Sarig and Warga [1989] found that corporate bonds that were issued more recently were more actively traded, implying age of bond as a liquidity factor. Among numerous studies that make this observation, Babbel et al. [2001] showed that benchmark or “on-the-run” U.S. Treasury securities were more actively traded than older Treasuries. Bollerslev, Cai, and Song [1999] and Fleming and Remolona [1999] found that macroeconomic announcements had a significant impact on the bid-offer spread. Another factor is the outstanding amount in issue for a bond; Fisher [1959] observed this to be of some inf luence in an early study. Garman [1976], Stoll [1978], Amihud and Mendelsohn (1980), and Ho and Stoll [1981] found that the bid-offer spread increases with the bond price and the credit risk of the bond, and also decreased with higher levels of trading activity. For corporate bond markets, credit ratings have an impact on bond liquidity, as shown by Fridson and Garman [1998]. They analyzed a sample of lower-creditquality bonds and found that the credit ratings of the bonds had the biggest impact on bid-offer spread at time of issue. The primary concern with looking at corporate bonds when measuring liquidity is that there are other issues, such as credit risk, that will inf luence the results and that make separation from liquidity more problematic. This is why we limit our investigation into market liquidity to government bonds first. This viewpoint is supported by Kamara [1994], who concluded that looking solely at Treasury bonds removed liquidity issues arising from credit risk, because all the bonds in the sample are essentially identical and credit-risk-free. The same applies to U.K. gilts; all bonds in a sample of gilts have uniform tax, trading, and settlement issues, with zero credit risk.

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• a market in gilt zero-coupon bonds (strips); • a regular auction issuance and benchmarking program; • changes to operating methods including a different accrued interest day-count basis, bringing gilts into line with practice in the U.S. Treasury and euroarea sovereign bond markets.

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The importance of liquidity to the smooth functioning of financial markets, and hence the global economic system, is emphasized frequently in the literature. Datar, Naik, and Radcliffe [1998] suggest that liquidity has an impact on asset returns. Amihud and Mendelson [1986], in a study of equities traded on the New York stock exchange, concluded that investors allow for lower liquidity by demanding a higher return premium, which is the trade-off required for bearing the higher cost of trading in illiquid markets. The same authors [1991a] also

Literature: Liquidity as a Market Concept

Gravelle [1998] defines liquidity as being the ease with which large-size bond transactions can be effected

2 M easuring Bond M arket Liquidity: Devising A Composite Aggregate Liquidity Score Fall 2009

• return spreads; measured in Crabbe and Turner [1995]; • the bid-offer spread; in Schultz [1998], Chakravarty and Sarkar [1999 and 2003], and Hong and Warga [2000]; • volume, or issue size; cited in Kamara [1994] and Alexander, Edwards, and Ferri [2000].

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Nunn, Hill and Schneeweis [1986] use three proxies for individual bond liquidity: 1) the age of the bond, 2) the bid-offer price spread, and 3) the nominal outstanding value of the bond. Their results indicate that of these proxies, the bond’s age is the most reliable indicator; their sample showed that a bond would tend to exhibit greater illiquidity the older it became. Bid-offer spreads differ with types of bonds even for credit-risk-free bonds. Amihud and Mendelson [1991a] observe a difference in bid-offer spread between Treasury bills, which are three-month assets, and Treasury bonds of 10 years maturity or greater. Their study shows that this has an impact on yield-to-maturity. Other proxy measures include yield or return spreads, which were considered by Sarig and Warga [1989], Blume, Keim, and Patel [1991], Warga [1992], and Crabbe and Turner [1995]. Volume as a proxy measure was studied by Kamara [1994] as well as Alexander, Edwards, and Ferri [2000]. A number of empirical studies have considered size of issue as a measure of liquidity. Support for this hypothesis has been observed in Hong and Warga [2000], who considered bid-offer spreads but noted that larger issue sizes are usually associated with smaller quote spreads. While looking at the determinants of trading volumes, Alexander, Edwards, and Ferri [2000] concluded that the larger bond issues among corporate issuers were more liquid. However Warga [1992], Crabbe and Turner [1995], and Fridson and Garman [1998], when studying yield spreads or return spreads as proxies for liquidity, found no backing for this hypothesis. Some studies have looked at the impact of volatility on liquidity. In an observation of U.S. corporate bonds, Kalimipalli and Warga [2002] observed that level of volatility was positively correlated to the bid-offer spread. They also found that issue size was not significantly positively correlated to liquidity. Kalimipalli and Warga [2002] conducted an analysis of the time series relationship between bidoffer price spreads and volatility in the U.S. corporate

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• trading cycle: the level of trading activity for new bond issues; • persistence of turnover: where a bond that is consistently among the most actively traded; • smoothness of yield curve: where the curve truly ref lects market expectations, and so becomes smoother, may indicate (among other things) an improved liquidity.

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without market prices being impacted. He also states that a liquid market is desired by central authorities. Borio [2000] describes a liquid market as one where “… transactions can take place rapidly and with little impact on price” (Borio [2000], p. 38). The author gives indicators of liquidity as bid-offer spread, average quote size, and daily turnover. McCauley and Remolona [2000] note that a number of OECD governments have recorded an interest in fostering market liquidity. They emphasize the importance of a liquid market in government bonds. A study comparing country liquidity was presented by the BIS Study Group on fixed-income markets [2001]. It looked at changes in quote size, turnover. and price impact in the U.S. Treasury and other markets during the 1997–2000 period of market turbulence. This work also considers other measures of liquidity, including:

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O’Hara [1995] defines liquidity as “the ability to trade a security quickly and with little cost.” The O’Hara definition refers to the concept in terms of a market. Practitioners also define an individual bond issue as being liquid if there is a two-way market available in it to investors in a “normal” market size,1 which size can be dealt at any time during market opening hours without there being any impact on the rest of the market or any market disruption. Literature: Liquidity Measurement by Proxy

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A range of proxy measures of liquidity have been used in previous research, and which have enabled the concept to be measured quantitatively. Noteworthy proxies among the recent literature include the following: • yield spreads; this is cited and measured in Sarig and Warga [1989], Blume, Keim, and Patel [1991], and Warga [1992]; Fall 2009

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Liquidity Measurement Methodology

There is precedent for calculating weighted scores in the financial markets, for instance in indices and as part of market returns, for which see Blake [1990, 2000]. A standard indexation technique is the Laspeyres index, described for instance in Jacques [2006]. Certain equity and bond indices use a method to weight the inf luence of selected stocks; for example, see Fabozzi [1993], while bond traders use a weighted average calculation when determining the hedge ratio for bond portfolios (see Garbade [1996] and Grieves and Mann [2007]). Another approach is that used by the securitization markets when determining the weighted average credit rating and diversity score of collateralized debt obligations (Fabozzi and Goodman [2001]). Operations research techniques such as linear programming (see Gass [1970]) are also used where a matrix-based approach may be preferred, for example because of a large number of variables. We look at each method in turn. We consider first the selection of contributory factors to liquidity.

by Fleming, such as bid-offer spread, average quote size, and turnover were also cited as key indicators by Borio [2000] and Bernanke and Blinder [1992]. We consider these proxy components, and also add components not considered in the previous literature but which merit inclusion for a number of reasons, discussed below. The components of liquidity that we believe are significant, and so merit inclusion in our proposed liquidity measurement, are summarized below. Size of issue. This is a key measure of market liquidity and is cited in a number of previous studies, such as O’Hara [1995], Engle and Lange [1997], the DMO [1998], and Fleming [2001 and 2002]. Standard practice in the U.S. Treasury market for many years was to build up large-size individual benchmark issues. This has two benefits, the first being sufficient availability for investors; it also makes it difficult for one practitioner to manipulate the issue by buying and holding a large quantity of stock. We set at £5 billion the size at which we consider a benchmark issue to be. This figure is used because it is the DMO definition of what is the minimum outstanding volume of stock for it to be eligible for benchmark status.2 Number of clearing systems. A bond that can be “cleared” (settlement of cash proceeds against stock) in only one clearing system is of lower attraction, all else being equal, to an investor outside the domestic market. We assign a higher score for each year where settlement has been possible in more than solely the domestic system. Number of market makers. Previous literature including Fleming and Sarkar [1999], Borio [2000], and Fleming [2001] has highlighted how the presence of an adequate number of dealers has contributed to greater liquidity. This earlier research suggested that a sufficient number of dealers ranged from 4 to 10 firms. With such an adequate number, it becomes difficult for dealers to act in concert or operate a cartel arrangement. This contributes to greater price transparency. Bid-offer spread. As a signif icant and widely observed proxy for liquidity, the average bid-offer spread for each year is therefore given a high weighting in the aggregate score. Overseas holdings. The DMO [2000, 2004] has previously stated an objective of having a higher percentage of gilts holders who are based overseas. Such an

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bond market. They look specifically at bonds traded on the New York Stock Exchange automated bond quote system, which does not report prices at regular intervals, but only at times of customer order. To compensate for data collected at uneven intervals (a problem that does not exist with developed government bond markets such as the gilt market), they aggregate the data and use a GARCH model. The authors state that market microstructure theory suggests that the bid-offer spread is a function of two things: the operating and carrying costs of holding positions, and an element representing adverse selection costs.

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Components of Liquidity

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Our choice of components of liquidity is driven by market practice as well as the academic literature precedent. We draw on the basic elements considered by Mackintosh [1995] and the United Kingdom Debt Management Office [2000, 2004]. We note the importance of trading systems and clearing systems, cited in the work by Amihud, Mendelson, and Lauterbach [1997]. We also note the components cited by Fleming [2000] in his review of U.S. Treasury market liquidity, such as turnover and volume. Some of the proxy measures used


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market liquidity, as the gilt return becomes a benchmark against which equity risk-return profiles are assessed. Average normal market size (NMS). The NMS is the basic size of order that a market maker commits to trading in when quoting a two-way price. Just as the bid-offer spread is a reasonable proxy for liquidity because its spread indicates the level of cost for a market maker when hedging its inventories, similarly the NMS level is an indicator of the size of trade which a market maker believes can be laid off in the market or otherwise hedged at reasonable cost. A higher NMS is therefore indicative of higher liquidity in the market. Benchmark yield curve. This is the spread of the benchmark bond below the yield curve. It is a standard liquidity measure in the U.S. Treasury market. This ref lects that, as liquidity is of value to investors, more liquid securities will have higher prices (hence lower yields) than less liquid securities of similar maturity. Studies that have reported this include Amihud and Mendelson [1991b] and Kamara [1994]. Fleming [2001] includes this measure in his study on U.S. Treasury market liquidity. Swap spreads. Swap spreads are a measure of interbank credit risk relative to the sovereign bond market. Generally they widen in times of recession, and narrow during periods of economic boom as business confidence increases. Conversely, widening spreads result in greater trading volumes in the sovereign market as investors embark on a “f light to quality.” Hence this can be expected to be a period of greater activity, and as a result greater liquidity, in government bond markets. We therefore mark down a period of narrower swap spreads, with a spread above the average for the period being seen to contribute to liquidity. We strip out this inf luence by assigning such periods a zero score. Note that the swap spread component is used only in our second liquidity calculation method. We have not included name recognition among the liquidity constituent components, although this was included by Mackintosh [1995]. This is because the U.K. gilt market, which is a credit-risk-free and well-developed market, is assumed to have a very high level of name recognition among the investor community. For similar reasons, we have replaced Mackintosh’s use of an “initial placement geography” indicator with our measure of percentage of gilt portfolio held overseas. This is because gilts, unlike corporate bonds (the instruments that Mackintosh considered), are offered to investors worldwide

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increase is viewed by the DMO as implying an increased level of liquidity. Turnover. The level of turnover is another widely used proxy for liquidity. Although turnover is used frequently as an indicator of liquidity, it can be misleading because it is inf luenced by volatility. For this reason we also include the turnover ratio as a component inf luencing liquidity. Electronic trading. Amihud, Mendelson, and Lauterbach [1997] cited an increase in liquidity for equities following the introduction of an electronic trading ability. We include the availability of this, and live screen prices from the DMO, as a component indicating increased liquidity. Additional components, which have not been included in previous studies, but which we include for the first time in this study, are listed below. Average bargain size. Higher average bargain size would suggest a liquid market, or at the least a relatively higher level of liquidity. We assign a low weighting to this component due to the inf luence of macroeconomic conditions on the values. The value is also closely related to turnover. Average number of bargains per day (benchmark stocks). This indicator is related to turnover. A higher value indicates a liquid market. This is also assigned a low weighting due to the inf luence of overall economic conditions on the value. Level of business confidence. In their study of corporate yield curves, Diaz and Skinner [2001] incorporated one hitherto-unused explanatory variable in their econometric model. This was the level of consumer confidence. Their proxy for consumer confidence was the Consumer Board’s monthly consumer confidence index. Their reasoning was that as the index rose, they expected to observe rising liquidity; this would be expected to lead to a lower average spread error in the fitted corporate bond yield curve. On the other hand, if consumer confidence dropped, the authors expected to see a “f light-to-quality” to the U.S. Treasury market. This would tend to have an opposite effect, as higher investment in Treasuries would be expected to depress yields and hence increase the corporate bond yield spread.3 We therefore include a proxy for market confidence, except that we use the level of the FTSE-100 equity index and not a general consumer confidence measure for the U.K. economy. We view an increase in equity market confidence and activity as contributing to increasing gilt Fall 2009


Database Construction

i =1

(1)

where Wi is the weighting for each constituent stock i in the Index. If stocks in the index are bonds, then the weighting is given by the market value of the bond divided by the market value of the portfolio. In our case, because the components themselves are not homogeneous, we are not able to use a summation method to assign the weights. Hence we use only three categories of weighting—low, medium, and high, with the highest weighting having a value of 1. This assigns a score to each component based on the deviation of the score away from the average, the average being given a 0 or 1 score while above-average scores are given a higher score. The scores are then weighted. Weighted scores are then aggregated to give a composite liquidity score. Therefore TRi is the individual score for a component, weighted according to its value relative to its average value for the whole period. This value is then weighted according to its component level of importance, whether low, medium, or high. The second approach is based on the “diversity score” methodology used by credit rating agencies in the structured finance bond market. This is described in Charpentier and Quipildor [2004]; we have modified the approach as shown in Exhibit 1. This is the Excel spreadsheet model used to calculate the liquidity score. The Excel spreadsheet formulae may be obtained from the author on request. The calculation is undertaken as follows:

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The data required to construct the database of the above components is obtained from the market sources Bloomberg, Reuters, and the DMO.

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Diversity Score Method: Weighting and Calculation Methodology

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We consider it prudent to weight components in order of relative importance. The weighting decision is driven by the previous literature; the intent is to score the most widely accepted components and aggregate them into a liquidity rating. Those components cited in the previous literature, and/or used by central authorities such as the DMO to measure liquidity, are given higher weightings. This would include proxy measures such as size of bid-offer spread. Previously unused components— that is, ones that are being considered for the first time in this study—are weighted according to the opinion of gilt primary dealers, recorded in a specially commissioned survey. The weights we assign to each component are noted in Exhibit 3. Because the components are widely varying in nature, and also reasons of accessibility, we assign only three weights: low, medium, and high.

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The weighting of each component is described in the next section.

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TRIndex = ∑ TRi × Wi

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• Market structural factors: issue size, clearing systems, electronic trading ability. • Trading factors: market makers, bid-offer spread, turnover, turnover ratio, FTSE 100 level, normal market size, benchmark yield spreads, swap spreads. • Investor factors: overseas holdings, average bargain size, average bargains per day.

We calculate the liquidity score in two ways. The first method is a simple formula that assigns a score to each component and then aggregates the score. Each score is weighted. The second approach is based on the weighting methodology used by credit rating agencies to calculate the “diversity score” of collateralized structured finance bonds. Both methods are described below. For the first method, we use a simple formula that is related to that used to calculate equity index values such as the FTSE100, to score the aggregate liquidity. For instance, the total return of an index is given by

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in an identical manner. Geographical placement does not affect their initial liquidity. However, we do wish to incorporate the level of overseas holdings, because this has an impact on secondary market liquidity; also, it was a stated target indicator of the Bank of England (BoE).4 The first method we use to calculate the composite liquidity measure is the “diversity score” method. In this, each component is categorized as a structural factor, trading factor, or investor factor. This categorization facilitates the model calculation (see Exhibit 1). The split is as follows:

• Each component is assigned a score for each year of the period under study, based on the level for that year compared to the average value for the entire period. In other words, the average value


Exhibit 1

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Extract from Liquidity Score Spreadsheet

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of the component (whether it is number of market makers, or issue size, or so on) over the whole observation period is given the median score. The value for each year is then compared to this median and scored relative to the median score. • The scores are aggregated and an average score obtained. • Each component is scored based on its value relative to the average value, with a maximum value of 1. • Each component is then weighted in accordance with low, medium, or high weighting. The liquidity score for that year is then the aggregate of these two summed values. This calculation is carried out for each year in the observation period.

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Components that are higher-weighted are stated in Exhibit 2. As we note above, the decision on which weighting to assign is based on previous literature, which suggests which of the components are most inf luential or the most accurate proxies for liquidity, and survey-obtained practitioner views on liquidity measures We also note that the methodology may be sensitive to a change in weighting for any one or more of the components. As such we also conduct a test with changed weightings, to determine if this has a signif icant inf luence on the conclusions we derive from the results. These test results are shown later in the article.


Exhibit 2

Diversity Score Method: Results of the Weighting Approach

The liquidity score based on an un-weighted aggregate of the components is shown in Exhibit 3.5 The results using the first (simple) method are shown in Exhibit 4. These results are presented in graphical format in Exhibit 5. We acknowledge that the data are fixed points and not continuous lines; they are joined into a line purely for illustration purposes. The scores for the second approach are shown in Exhibit 6. This shows a steady increase in liquidity score for the period overall, if less emphatically than the scores calculated using the simpler method. Exhibit 7 is a graphical representation of the scores obtained from the second method. Finally, we change the weights for comparison purposes and to assess the robustness of this specific simple approach. Each weight was adjusted by one level, upwards if it was a low weighting and downwards if it

Exhibit 3 Unweighted Aggregate Liquidity Score

Exhibit 4 Final Liquidity Score, First Method

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A common approach to calculating indices and weighted returns in financial economics is to apply actual values; that is, unlike the practice in industrial economics, where an index may range from 0 to 1, there is no theoretical maximum score. The raw quantitative value is used in the calculation. This can be seen for example in the calculation for the FTSE100 equity index (Blake [2000]), as well as the return calculation for bond indices (Brown [1994]). The calculation of the diversity score for structured finance securities such as collateralised debt obligations (CDOs) also uses actual values in its calculation; there is no maximum value (see Charpentier and Quipildor [2004]). This approach is used in our composite score methodology. The scores are then weighted before being aggregated. The weighting is that shown in Exhibit 2; the calculation formulae are obtainable from the author on request. Note that as the weighting of some of the components retains an element of subjectivity, prior to calculating the liquidity scores we carried out a measurement assuming equal weighting of all components. We also calculated a score based on changed weights. We recorded the former for later comparison with the two weighted calculation methods we describe above.

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Diversity Score Method: Weighting Assigned to Components


Exhibit 5

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Liquidity Score First Method, Graphical Illustration

Exhibit 6

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Liquidity Scores, Second Approach

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was a high weight. The results are reported in Exhibit 8 and show the same pattern. Using either method, we observe that by the end of the observation period, the liquidity score had increased from that recorded for the start of the period. This suggests that the level of market liquidity, measured using our selected components, had increased during the observation period. It does not tell us how much of this increase was as a direct result of the introduction of structural reforms; indeed, this would be difficult to determine. Nevertheless, we do note that liquidity has improved during the period being studied.

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We also note that using these measures, it appears that liquidity was increasing before the period of the implementation of the structural reforms. There is more than one possible explanation for this. With respect to the impact of the structural reforms themselves, it must be remembered that the BoE gave advance notice of the implementation of the various new measures; for example, it issued a proposal paper on the new planned repo market in December 1994, which is over 12 months before its actual introduction. The expected introduction of the new measures may have resulted in an immediate perception of increased liquidity, such that this became a self-fulfilling prophecy. Liquidity Score Using Index Method

The liquidity score calculation method described and reported in the previous section is based on the “diversity score” method used in debt capital structured finance markets. In this section we report the results of a second method used to calculate the liquidity score, one based on standard index series techniques. Three different measures are calculated and reported. Laspeyres Index-Based Liquidity Score

We use the raw data for each of the 14 different components to calculate an aggregate score based on the The Journal of T rading 9

Exhibit 7

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Liquidity Score Second Approach, Graphical Illustration

We list the following aggregate index scores:

Exhibit 8

• The absolute aggregate index value. • The unweighted index score. This does not weight the components in the aggregate calculations, so in effect it assumes that each determinant of liquidity is of equal importance. • Components weighted in accordance to their citation in the previous financial economics literature. If the component has been used as a proxy measure of liquidity, or otherwise cited as a determinant of liquidity, in previous trade or academic literature, then it is given a double weighting. This is “Weighting [1]” in Exhibit 11. • Components weighted in accordance with results arising from a survey of U.K. gilt primary dealers, carried out by the author specifically for the purpose of assigning weightings to this index calculation. The results of this survey are shown in Exhibit 12. The weights are shown as Weighting [2] in Exhibit 11. Incidentally, note that the survey indicated that primary dealers viewed the most important determinants of liquidity to be 1) normal market size, 2) size of bid-offer spread, and 3) number and size of benchmark issues.

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Liquidity Scores, Weightings Changed for Comparison Purposes

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Laspeyres base-weighted index. Exhibit 9 shows the raw data over the time period in question. The line underneath each component value is the index value for that component. Exhibit 10 is a graphical illustration of this raw data (scales have been adjusted by order of 10’s where necessary so that all component values can fit on the same scale; this graph is shown purely for illustration purposes). Indices are calculated from the basket of individual index scores shown in Exhibit 9. That is, we convert each of the individual component statistics into a Laspeyres index. Then we aggregate these into a basket and calculate an overall index value from the individual component index values.

The weightings used in the three methods for each of the 14 components of liquidity are shown in Exhibit 11,


Exhibit 9

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Individual Liquidity Component Statistical Time Series Data and Individual Base-Weighted Index Values

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Exhibit 10

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Graphical Illustration of the Absolute Trend During Observation Period for Each of the Components Used in the Liquidity Score Calculation

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Observation of results. For all three basket (aggregate) index scores, we observe a steady rise in value during the observation period. We consider the most significant measure to be Weighting [2], where the individual component index scores are weighted in accordance with importance as suggested by gilt market primary dealers. This aggregate index shows a steady increase in liquidity score, and by implication market liquidity, each year. This contrasts with the unweighted score, which implies a decline in liquidity in the second year under observation. The implied increase in liquidity cannot be ascribed with any certainty to the structural reforms; that liquidity was rising before the completion of the reforms suggests that other factors were also at work. Nevertheless, we can posit that the reforms were associated with a period of rising liquidity, and we suggest that they contributed in part to this perceived rise.

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labeled as Weighting [1] and Weighting [2]. Weighting [1] ref lects the incidence of the component in the literature precedent, with a greater weighting for those components that occur more frequently. Weighting 2 is what was suggested by gilt primary dealers following a survey of the market undertaken by the author (see Exhibit 12). The index results are shown in Exhibit 13. A graphical illustration is shown in Exhibit 14.

Exhibit 11

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Weightings Used in Index Calculations

Paasche Index-Based Liquidity Score

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Calculating a base-weighted index series follows an established and straightforward logic and is computationally simple. For this reason the procedure is widely used by central banks and other sovereign

Exhibit 12

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The Gilt Primary Dealer (GEMM) Liquidity Weighting Survey: Results

Weighting: 1-lowest, 5-highest Survey sample: 17 primary dealers (100% of population. Firm names obtainable on request). Survey response: 15 responses. Date undertaken: September–December 2006.


Exhibit 13

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Laspeyres-Based Aggregate and Basket Index: Results

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Exhibit 14

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Laspeyres-Based Basket Index Results: Graphical Illustration

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authorities; for example, see Curwin and Slater [2004]. The implied assumption behind the calculation is that the pattern of change in the indicator being measured remains essentially identical over time. Over a long-run period, such as the period of time over which a typical business cycle runs, this may become an unrealistic assumption. Therefore we also calculate a measure based on the current-weighted index series, known as the Paasche index. We adapt the standard Paasche index calculation so that it can be applied to our data series. We calculate index scores based on the final-year value; that is, the series is converted into an index with the “current” year

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used as the base value (the last year of observations is taken to be the current year). Exhibit 15 shows the individual index values for each of the 14 components calculated using this adapted current-weighting approach. Exhibit 16 shows the aggregate (basket) index scores. Again we show an unweighted score, and then two series weighted with the weights shown previously in Exhibit 11. A graphical illustration of the three index series is shown at Exhibit 17. Observation of results. The general conclusion is similar to that made earlier for the base-weighted basket index series: a steady increase in the basket score, implying an increase in liquidity. An interesting obser-


Exhibit 15

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Individual Liquidity Component Statistical Time Series Data and Individual Current-Weighted Index Values

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Exhibit 16

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Paasche-Based Aggregate and Basket Index: Results

Exhibit 17

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Paasche-Based Basket Index Results: Graphical Illustration


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Fisher Ideal Index Scores

As a final index measure we calculate the Fisher Ideal index score from the aggregate basket index series we calculated earlier. The series is calculated, as before, as an unweighted series of the basket (aggregate) of the individual index scores, and weighted in the two further series (see Exhibit 11 for the weightings). As it is a geometric mean of the two earlier measures, we wish to use it for further illustration of the trend. Results are shown in Exhibit 18. The conclusions we draw from this series results are unchanged from those drawn from observation of the earlier index series results.

measures used a weighting approach when calculating the composite score. To confirm that our approach in aggregating and weighting the individual components is a rigorous one we apply a technique commonly used in operations research and industrial economics, that of linear programming and optimisation. This is a well-established technique, for example see Dorfman et al. [1958] and Gass [1970]. By undertaking this additional method of linear programming, we choose the combination of weights that maximizes the aggregate liquidity score and minimizes the score. That is, we calculate an upper and lower bound to the index value, which constrains the aggregate score. These maximum and minimum values are subject to each of the variables (liquidity components) being in the index. The upper and lower limits are compared to the actual aggregate scores, and if the actual scores lie within these constraints then this is a confirmation of the reasonableness of the weights used in the earlier liquidity score calculations. Additionally, if the scores given by the upper and lower bounds indicate the same increase in score, this strengthens our earlier conclusions.

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vation is that compared to the Laspeyres-based score, the unweighted index here shows a fall in liquidity score in the year before 2002, the final year in our dataset. This is observed in the unweighted score only. However, we view this as statistically insignificant because both weighted series, significantly the one marked Weighting [2], demonstrate the same steady increase in liquidity score, and hence by implication market liquidity, for the entire period under observation.

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Matrix Linear Programming Approach To Maximize And Minimize Index Value

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In the underlying methodology to the liquidity score calculation methods used above, the raw statistical data for each of the 14 determinants of market liquidity is aggregated to produce a composite score. This score is then used as a measure of the liquidity level, and its value over the observed time period used to determine if liquidity as been rising or not. We accept that an accurate measure could not rely solely on aggregating the time series values of each component, because 1) they are measures of different units of value and 2) their significance as a determinant of liquidity is not equal. Hence both the diversity score approach and the index

Formulating the Linear Program Model

In conventional linear programming we are presented with an assignment or maximization/ minimization problem involving a specified number of variables, for which we formulate constraints under which each of the variables must operate.6 Where the number of variables is more than two or three, a matrix approach is used, as the number of equations required to formulate the problem would be unwieldy. The solution to the problem is also expedited if one uses a unit matrix and the “simplex method”, which is applicable in virtually all cases; see Gass [1970]. Hence our solution is obtained using the simplex method and a unit matrix. For the composite liquidity

Exhibit 18

Fisher Ideal Aggregate Basket Index: Results

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subject to the constraints X11 + X12 + X13 + … + X114 = 1 X11 + X 21 + X 31 + … + X141 = 1 

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and so on (we do not list all the equations above). Additional equations result from the addition of the artificial variables; these are the equations from X11 to X14, as additional rows are added to the tableau. Solution. The solution to this problem has an explicit constraint that variables take on integer values of either 1 or 0. The first (immediate) feasible solution is where the coefficient variables are assigned in a diagonal line across the matrix. This is presented in Exhibit 21, which shows the coefficients assigned to the variables and the resulting matrix value. Where a variable is indicated for a component value that lies in the range X11 to X14 we use the last recorded value for the component. This “immediate solution” indicated an aggregate value of 42,204. To obtain the maximum and minimum values, we use an Excel VBA program to generate the matrix that produces the optimum solution. The matrix tableau that maximizes the value is given in Exhibit 22; the matrix tableau for the minimum value is given in Exhibit 23. The matrix solutions shown in Exhibits 22 and 23 indicate maximum and minimum index values of 52,799 and 23,597, respectively. Using the timespan of

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score measure, we wish to maximize/minimize the value of the index. The variables are the coefficients to the actual time series scores for each of the liquidity components, and the years in which the values were recorded. The variables are restricted to a value of 1 or 0. We assume a linear relationship between the values of the individual measures. The matrix we obtain is shown at Exhibit 19. Note that the matrix is not a unit matrix. Therefore, following Dorfman et al. [1958] and Gass [1970] we create an artificial unit basis and hence create a unit matrix. This results in a set of artificial non-negative variables that are attached to the problem, creating one new variable for each equation. The precedent for this is seen in Dorfman [1958] and Gass [1970]. So although a unit basis is not initially available, we add four artificial variables, which are x11, x12, x13, and x14. These correspond to the additional years 2003–2006 seen in Exhibit 19. In this way we arrive at the required unit matrix. Each element in the matrix is the relevant coefficient xij. The elements in Exhibit 20 are the coefficient identifiers. We wish to find two sets of non-negative values of the variables xij ≥ 0, which maximize and minimize the following function:

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4X11 + 1X12 + 20X13 + 6X14 + 15X15 + 22541X16 + 18X17 + 7X18 + 203X19 + 2900X110 + 15X111 + 0X112 + 8X113 + 32X114 + 4X 21 + 1X 22 + · · · + 35X1014

Exhibit 19

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Linear Programming Formulation: Simplex Tableau


Exhibit 20

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Corresponding Matrix Coefficient Variables

Exhibit 22

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Exhibit 21

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scoring system. This adopts a composite methodology that attempts to use the infrastructure and procedural components in a market and assign a measure of value to them. These components are selected using criteria based on their relative importance to promoting liquidity; this is indicated by precedent in the academic literature and statements made by the central treasury authorities and primary dealers. The components include structural, trading, and investor factors. They include proxy measures such as bid-offer spread as well as market factors such as the number of settlement (clearing) systems. Our first liquidity measurement methodology assigns a value to each of the components and then a weighting to each of these values, based on a technique used in the structured finance market known as the “diversity score.” The second method is based on a Laspeyres index approach and uses both an unweighted index and a weighted base index; the weighted method assigns greater weight to those proxy values that are cited in previous and existing literature as being market-standard proxies. The second index weighting approach uses weights indicated from a survey of gilt primary dealers, conducted specifically to obtain market inputs as to what weights we should use in our index. We also calculated Paasche current-weighted index series for the same data. The third method applied a linear programming matrix approach as employed in industrial economics and operations research, and was used to obtain a maximum and minimum index value. That is, we used this method to obtain upper and lower bounds for the aggregate index score; when the unweighted aggregate score was found to lie within these bounds we concluded that our approach was logical and consistent. With all three of the methods used, we observed a steady increase in the liquidity score during the time period under observation. This became marked in the period following the conclusion of the implementation of the structural reforms in the market. That liquidity was suggested as rising in the pre-reform period was as a result, we believe, of increasing trading volume in global sovereign bond markets, increased trading volume in the gilt market itself during this time, and the anticipation effect of the BoE’s reform announcements themselves. Under the “diversity score” measurement method we observed an initial dip in implied liquidity from 1994, which was then followed by a consistent increase in liquidity. Note that in February 1994 the U.S. Federal

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Exhibit 23

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the observations as one set of variables, we observe the minimum and maximum values of the index, with an upper and lower bound for the weights. Referring to the aggregate index values (unweighted) shown in Exhibit 13, we see that these lie within the upper and lower bounds given in the matrix solution. Of course, by definition any index value would have to lie within the upper and lower bounds (and indeed all the values, irrespective of the weighting or approach, do so); however, we have shown the actual limits within which the index can lie. We conclude that the methodology used for the earlier index calculations is logical and consistent. Conclusions

We have attempted to devise a transparent, accessible, and straightforward yet robust method to measure the level of liquidity in the gilt market, using an aggregate


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required minimum amount outstanding will be in existence in short order. 3 Diaz and Skinner [2001] reported a positive coefficient for consumer confidence as an explanatory variable for the average spread error. 4 The DMO continues to monitor this statistic for the same reason; for instance see its Quarterly Review, December 2005, June 2007 and September 2007. 5 The unweighted score is an input to the main model. For instance, the score for 2002 is 38.23, which is shown in Exhibit 1. 6 For background on implementing the technique and the rationale behind it, see for instance Gass [1970] and Jacques [2006], although the subject is covered in a large number of finance and mathematics texts.

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ENDNOTES

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Reserve began a process of raising interest rates that precipitated a major bear market in bonds worldwide that continued for the rest of the year. Such a bear market might be expected to fuel negative investor sentiment, and this is ref lected in lower liquidity indicators for that year. From 1995 onwards, however, we observe the possibility of increasing liquidity, irrespective of the calculation method we adopt. We conclude the following: our observations suggest that market liquidity may have improved in our study period. Our measure does not suggest to what extent any increase in liquidity was as a direct result of the structural reforms themselves that were undertaken by the BoE. Nevertheless, we suggest that liquidity had increased by the end of the observation period. This has positive implications for our hypothesis, as we had expected the introduction of the BoE structural reforms to lead to an improvement in liquidity. Without being able to prove causality, we conclude that the reforms are associated with an improvement in liquidity. Finally, we suggest that our liquidity measurement methodology may be employed as a practical tool by institutional investors who wish to obtain a measure of relative liquidity in any stock or bond market. It would enable institutions to effect comparisons across different markets, prior to making the investment decision. Such an approach would be of value to investors seeking to invest in one out of a number of (otherwise similar) markets. We have developed a methodology using a theoretical approach to calculating index scores. Thus we have now a liquidity score measure that combines a rigorous theoretical and academic background with a sound practical approach. It may be used in practical investment analysis by institutional investors worldwide, and in any financial market.

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To order reprints of this article, please contact Dewey Palmieri at dpalmieri@ iijournals.com or 212-224-3675.

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