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# Multi-dimensional liquidity measures

## 3.3 Liquidity measures

### 3.3.2 Multi-dimensional liquidity measures

Quote slope

The quote slope , as used by Hasbrouck and Seppi (2001), is calculated by dividing the absolute spread by the logarithm to the best bid and ask depth.

The following figure is a visualization of the quoted spread, and shows the relation between a deeper order book at the best bid and ask and the liquidity. A low quote slope logically denotes high liquidity.

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From the first to the second graph, the spread is constant but the order depth at the best bid and ask price increases. From the second to the third graph, the spread additionally narrows. Both steps could be said to enhance liquidity.

The quote slope is a comparatively intuitive measure, yet it faces some challenges when compared across stocks, because of its use of the absolute spread.

Log quote slope

If multiple stocks are compared the log quote slope solves the problem of comparing absolute spreads across stocks by using the logarithmized relative spread in the numerator.

Hasbrouck and Seppi see the two aforementioned measures “…as summary measures of the liquidity supply curve” (Hasbrouck and Seppi, 2001, p9), since they combine both price and quantity information. However, looking into the log quote slope, one might question the intuition if there is a significant difference in the best bid and ask depth. This is typically the case in a strongly upward or downward moving market.

If the best bid and ask depth is substantially asymmetric the adjusted log quote slope may be preferred, as it accounts for this asymmetry by adding a correction term.

Increasing liquidity

Figure 3

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The equation is more intuitive when rewritten as follows:

The intuition behind the log quote slope is shown in the figure below. The red hatched area relates to the parts of the equation with the red font. The correction term basically scale the degree of upward- or downward movement by the aggregated bid and ask depth and multiply it by the spread.

Figure 4

The extra term, therefore, increases the liquidity measure (implying reduced liquidity) if there is asymmetry between the best bid and ask depth. Note that if the best bid and ask depths are the same, the correction term will be zero. Even though this correction intuitively makes sense, one should remember that when comparing securities across different periods, the correction term will be affected by general upward or downward moving markets.

While these measures are essentially ex-ante measures of liquidity (i.e. anticipating what can happen) similar effects could be captured analyzing realized transactions.

28 Liquidity ratio 1

Liquidity ratio 1 is unlike the quote slope measure based on ex-post data, thus giving another dimension to liquidity.

Liquidity ratio 1 divides the turnover by the absolute price change for a certain period. is also known as the Amivest liquidity ratio. With the absolute price change placed in the denominator, the measure describes how much turnover that corresponds to a one unit change in the price. A high thus denotes high liquidity as it means that is has historically been possible to trade large quantities without significantly affecting the price. When comparing liquidity measures over time, one may argue that could be biased by variations in the market return, since the market return is expected to have a higher correlation with the return of an individual stock than with the volume of that particular stock. A variant of is the return per turnover , which is used by Amihud (2002), the measure is shown below.

The intuition behind the reciprocal of liquidity ratio 1 is to calculate what price impact one unit of turnover has on the price when analyzing a specific time interval. Clearly, as opposed to a low Amihud measure denote high liquidity. As with liquidity ratio 1, the Amihud measure is a rather ingenious way of estimating the cost per volume traded.

Liquidity ratio 2

Building on , liquidity ratio 2 divides by the term which denotes the difference between the total number of shares and the number of shares owned by the company , hereby applying an ‘easy to calculate’ estimation of the free float.

Liquidity ratio 2 expresses how much turnover is needed to change the value of the free floated shares by one unit, over a given time interval. Hence, a high measure again indicates high liquidity.

29 Liquidity ratio 3

Along the idea of liquidity ratio 1 and 2, liquidity ratio 3 measures the average price impact per trade .

This measure assumes that each trade has the same impact on the price, hereby implying that each trade is equally sized, which is presumably a rough approximation. Similar criticism could however be stated in relation to the other measures.

Flow ratio

The flow ratio measures the ratio between the turnover and average waiting time between trades, over a given time interval. The average waiting time is calculated by adding all time intervals between trades and averaging it over the number of trades .

The higher frequency of trades the lower the denominator, and hence, a high flow ratio indicates high liquidity. As Wyss (2004) notes, the flow ratio can, for most practical purposes (when analyzing the same time interval across different securities), be restated as follows (Wyss, 2004, p20).

However, the term should be replaced with in order to get the same result as returned by the ‘original’ flow ratio.

Order ratio

In relation to the depth measures described earlier, the order ratio focuses on the likely asymmetry in the depths at the best bid and ask quotes, measured as the absolute difference between the best bid and ask depth scaled by the turnover .

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Note that a low order ratio denotes high liquidity, as for example, a substantial difference in the best bid and ask depth, accompanied with a low turnover, would be a sign of an imbalance in the supply and demand of the stock.

Market impact

At this point, the former measures have focused on either the best bid and ask prices, the best bid and ask depth, various time related measures or the combinations of these. While the Amivest and Amihud measure are both impact measures, the market impact , looks beyond the actual number of shares traded, and measures the price impact incurred when trading a certain amount of money e.g. one million Euros. This is done by subtracting the average price paid (if buying for a certain amount of money) by the average price achieved (if selling for a certain amount of money). The market impact measure, thereby takes into account if one has to trade over several ticks by enlarging the quoted spread.

A high market impact denotes low liquidity, as it essentially tries to describe the round trip cost of trading a certain amount of money.

Depth for price impact

Associated with the market impact is the depth for price impact . It measures how many shares have to be traded in order to move the price ticks away from the quote mid-price.

The measure can be calculated from the bid-side of the market too.

The depth for price impact measure is faced with a lack of comparability across stocks as it does not account for the relative ownership differences across shares, i.e. it does not account for the aggregated turnover that has to be traded. A high measure denotes high liquidity.

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