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Characterise and use the different approaches for modeling asset liquidity, among them one for bonds that trade rarely and the t-cost model for working out transaction costs on corporate bonds.

FRM® Part II · Risk Management and Investment Management · Liquidity Risk Management · FRM2.IM.16.c

The idea

Asset liquidity is modelled in several ways. The quoted or effective bid-ask spread gives the cost of a sale as half the spread times the amount sold. Bid-ask bounce makes successive price changes negatively correlated, and cov denotes their first-order serial covariance. Roll's estimator then infers the effective spread with Spread = 2 × √(−cov). It fails when the covariance is positive or the sample is thin. The Amihud ratio is the average, over days with trades, of the absolute return divided by trading volume, and a higher value means less liquid. Zero-trading-day measures count days without trades. For bonds that rarely trade, or for transaction costs on corporate bonds, a t-cost regression relates cost to trade size, rating, maturity and age. It is fitted on bonds that do trade and then applied to the bond in question. The usual mistake is dropping the factor 2 in Roll or averaging over days with no volume.

Formulas

Roll estimator of the effective bid-ask spread

Spread = 2 × √(−Cov)

Cov
the first-order serial covariance of price changes

Works only when the covariance is negative, from bid-ask bounce. The square root gives the half-spread, so dropping the factor 2 halves the answer.

Amihud illiquidity ratio

Amihud = Σ(|rₜ| ÷ Vₜ) ÷ N

rₜ
the return on trading day t
Vₜ
the trading volume on day t
N
the number of days with trades

Compute the ratio day by day, then average over days with trades only. A higher value means less liquid.

A practice question

From the question bank, with the reason each wrong option is wrong.

The Amihud illiquidity measure for a bond is computed as:

  1. the quoted bid-ask spread divided by the mid price, averaged over the quoting dealersWhy not: Describes a relative quoted spread, not the Amihud price-impact ratio.
  2. the average of absolute return over dollar volume; higher means less liquid - correct
  3. the fraction of days in a period on which the bond did not tradeWhy not: Describes a zero-trading-day measure, not the Amihud ratio.
  4. the negative of the first-order serial covariance of price changesWhy not: Describes an input to the Roll estimator, which needs a square root and a factor 2 to become a spread.

Amihud's ratio measures price impact per unit of volume: average of |return| / dollar volume. A higher ratio means a given volume moves the price more, i.e. lower liquidity.

A harder one

A bond's observed trade-price changes, measured in percent of price, have a first-order serial covariance of -0.16 (percent squared). Using the Roll estimator, the implied effective spread is closest to:

  1. 0.40%Why not: Omits the factor 2: the square root of the negative covariance is only the half-spread.
  2. 0.32%Why not: Doubles the covariance instead of taking its square root.
  3. 0.80% - correct
  4. 0.16%Why not: Reports the covariance itself, without the square root or the factor 2.

Spread = 2 x sqrt(-cov) = 2 x sqrt(0.16) = 2 x 0.40 = 0.80%. The square root gives the half-spread, which must be doubled.

The hardest tier

Over three days on which a bond traded (it did not trade on other days), the absolute returns in percent were 0.30, 0.10 and 0.50, and the trading volumes were USD 5 million, 2 million and 20 million. The Amihud illiquidity ratio, the average over trading days of absolute return divided by volume (percent per USD million), is closest to:

  1. 0.033Why not: Divides total absolute return by total volume (0.90 / 27) instead of averaging the daily ratios.
  2. 0.135Why not: Sums the three daily ratios instead of averaging them.
  3. 0.045 - correct
  4. 0.011Why not: Divides the average absolute return (0.30) by the total volume of 27 instead of averaging daily ratios.

Daily ratios are 0.30/5 = 0.060, 0.10/2 = 0.050 and 0.50/20 = 0.025. Their average is 0.135 / 3 = 0.045. The ratio is computed day by day and then averaged.

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