AbbottModelDLOM

Liquidity, Block Size, Total Delay

From Market Liquidity and Transfer Restrictions to the Effective DLOM Horizon

DLOM liquidity analysis measures the economic disadvantage that arises when an owner cannot convert an interest into cash as readily as an owner of an otherwise comparable freely traded interest. The magnitude of that disadvantage depends not simply on whether a restriction exists, but also on how long liquidity is delayed, how rapidly the market can absorb the interest once sale becomes possible, the size of the block relative to the relevant share base, and the volatility of value during the resulting period of exposure.

The AbbottModelDLOM framework therefore treats the time associated with lack of marketability as an economic quantity that should be estimated from the facts of the valuation rather than assigned mechanically from a historical average.

DLOM Liquidity:Marketability and Liquidity

Marketability and liquidity are related, but they are not identical.

Marketability concerns whether an ownership interest may be transferred and the conditions under which a transfer can occur. Legal, contractual, regulatory, or practical restrictions may delay a sale even when an active market exists for otherwise comparable securities.

Liquidity concerns the speed, ease, cost, and price certainty with which an interest can actually be converted into cash. An interest may be transferable yet remain relatively illiquid if its size is large compared with normal trading activity or if the relevant market has limited capacity to absorb it.

Blockage is a specific manifestation of this liquidity problem. A sufficiently large position may require an extended execution period, unusual selling costs, or price concessions before the market can absorb it at or near the quoted price.

These distinctions are important because a restriction period and a post-restriction liquidation period represent different economic impediments. The analyst should estimate these periods separately before combining them into the valuation horizon.

Notation and Definitions

The principal terms used in the liquidity and blockage calculations are summarized below. Time-based quantities are expressed in trading days unless otherwise stated.

SymbolDefinition
bBlock size expressed as a fraction of the relevant share base, with 0 < b ≤ 1.
λDaily exponential liquidation-rate parameter derived from average daily trading volume V and the relevant shares outstanding S: λ = −ln(1 − V/S).
tElapsed trading time.
Q₀Quantity represented at the beginning of the modeled liquidation process.
Q(t)Quantity remaining after elapsed time t under the exponential process.
F(t)Cumulative fraction liquidated by trading day t.
f(t)Instantaneous liquidation density; the volume weight applied at time t.
TbTerminal time required for cumulative liquidation to reach block fraction b; the endpoint of the modeled liquidation interval.
H or t1/2Firm-level half-life in trading days: the estimated time required for one-half of the modeled quantity to be absorbed.
VWLP(b)Volume-Weighted Liquidation Period: the average trading day on which the shares comprising block b are liquidated.
TrestrictionApplicable statutory, contractual, regulatory, or other no-sale period.
TDLOMTotal valuation horizon used for the DLOM analysis: restriction period plus block-specific VWLP.
σApplicable volatility measure used with the estimated horizon in the valuation model.

From Trading Volume to the Daily Liquidation Rate

The base liquidity input is the relationship between the relevant share base and observed daily trading volume. Let S denote the relevant shares outstanding and V denote average daily trading volume over the applicable measurement period. The observed daily turnover fraction is V/S.

τ=VS\tau=\frac{V}{S}

To express that observed turnover as the constant daily rate of the exponential liquidation process, the framework converts the turnover fraction into the continuously compounded rate λ:

λ=−ln⁡(1−VS)\lambda=-\ln\left(1-\frac{V}{S}\right)

Thus λ is derived from observable market trading capacity rather than assumed independently. A larger daily trading volume relative to shares outstanding produces a larger λ, a shorter half-life, and shorter block-specific liquidation horizons. When V/S is small, λ is numerically close to V/S, but the logarithmic expression is the exact rate consistent with the exponential model.

Worked Example: From Trading Volume to VWLP

Consider a company with 10,000,000 shares outstanding, average daily trading volume of 100,000 shares, and a subject block of 2,500,000 shares.

InputValue
Shares outstanding, S10,000,000
Average daily trading volume, V100,000
Subject block shares, B2,500,000

First, daily turnover is:

VS=100,00010,000,000=0.01\frac{V}{S}=\frac{100{,}000}{10{,}000{,}000}=0.01

Next, convert that turnover fraction into the daily exponential liquidation rate:

λ=−ln⁡(1−0.01)=0.0100503\lambda=-\ln(1-0.01)=0.0100503

This means the exponential process has a daily rate parameter of approximately 1.005%.

Using that rate, calculate the firm’s liquidity half-life:

H=ln⁡(2)0.0100503≈68.97 trading daysH=\frac{\ln(2)}{0.0100503}\approx68.97\ \text{trading days}

Next, the block is expressed as a fraction of shares outstanding:

b=BS=2,500,00010,000,000=0.25b=\frac{B}{S}=\frac{2{,}500{,}000}{10{,}000{,}000}=0.25

For this 25% block, the terminal liquidation time is:

Tb=−ln⁡(1−0.25)0.0100503≈28.62 trading daysT_b=-\frac{\ln(1-0.25)}{0.0100503}\approx28.62\ \text{trading days}

That 28.62-day figure is the modeled endpoint for the block. It is not the average time at which the shares in the block are sold.

Finally, calculate the Volume-Weighted Liquidation Period for the same 25% block:

VWLP(0.25)=10.0100503[1+0.750.25ln⁡(0.75)]≈13.63 trading daysVWLP(0.25)=\frac{1}{0.0100503}\left[1+\frac{0.75}{0.25}\ln(0.75)\right]\approx13.63\ \text{trading days}

Accordingly, the same observed market data lead sequentially from trading volume to λ, from λ to firm-level half-life, and from half-life and block size to the transaction-specific liquidation measures.

Derived measureExample result
Daily turnover, V/S1.000%
Daily exponential rate, λ0.010050
Firm-level half-life, H68.97 trading days
Block fraction, b25.0%
Terminal liquidation time, Tᵦ28.62 trading days
Volume-Weighted Liquidation Period, VWLP13.63 trading days

Measuring the Firm’s Underlying DLOM Liquidity

The liquidity analysis starts with the rate at which the relevant market can absorb shares. The framework represents this process as an exponential liquidation or transfer process with rate parameter λ.

If Q₀ is the quantity initially remaining, the model represents the quantity remaining after time t as:

Q(t)=Q0e−λtQ(t)=Q_0e^{-\lambda t}

A convenient way to express this underlying rate of market absorption is through the firm-level half-life. Half-life is the estimated time required for the market to absorb one-half of the quantity represented by the exponential liquidity process. It is a measure of the trading liquidity of the security rather than the liquidation period of the particular subject block. A shorter half-life indicates that shares are absorbed more rapidly and the security is more liquid; a longer half-life indicates slower market absorption and lower liquidity. Under the exponential process, firm-level half-life is:

t1/2=ln⁡(2)λt_{1/2}=\frac{\ln(2)}{\lambda}

Equivalently:

λ=ln⁡(2)t1/2\lambda=\frac{\ln(2)}{t_{1/2}}

A shorter half-life corresponds to greater liquidity and a faster rate of market absorption. A longer half-life indicates that the market requires more time to absorb shares and therefore imposes a longer liquidity exposure.

The firm-level half-life describes the security’s underlying liquidity. By itself, however, it does not determine the liquidation period for the subject interest. Block size also matters.

DLOM Liquidity and Block Size

Let b denote the subject block as a fraction of the relevant share base, where:

0<b≤10<b\leq1

As block size increases, disposing of the position uses a larger proportion of the market’s available liquidity capacity.

Under the exponential framework, the time at which cumulative liquidation reaches fraction b is:

Tb=−ln⁡(1−b)λT_b=-\frac{\ln(1-b)}{\lambda}

or, using the firm-level half-life:

Tb=t1/2−ln⁡(1−b)ln⁡(2)T_b=t_{1/2}\frac{-\ln(1-b)}{\ln(2)}

This quantity identifies the endpoint of the liquidation process: the time required for cumulative liquidation to reach the specified block size.

However, it does not measure the average holding period of the shares in the block.

That distinction is economically important. When the owner sells a block progressively, some shares convert to cash early, others later, and only the final portion remains until Tb. If the analyst treats every share as illiquid until the terminal date, the calculation overstates the average duration of the liquidity exposure.

Volume-Weighted Liquidation Period

The AbbottModelDLOM framework therefore uses the Volume-Weighted Liquidation Period (VWLP) to measure the average realization time of the shares comprising the block.

Under the exponential liquidation process, cumulative liquidation through time t is:

F(t)=1−e−λtF(t)=1-e^{-\lambda t}

and the corresponding liquidation density is:

f(t)=λe−λtf(t)=\lambda e^{-\lambda t}

For a block representing fraction b, liquidation extends from time zero through Tb. VWLP averages the liquidation times over that interval and weights each time by the volume realized at that point:

VWLP(b)=1b∫0Tbtλe−λtdtVWLP(b)=\frac{1}{b}\int_0^{T_b}t\lambda e^{-\lambda t}\,dt

Evaluating the integral gives:

VWLP(b)=1λ[1+1−bbln⁡(1−b)]VWLP(b)=\frac{1}{\lambda}\left[1+\frac{1-b}{b}\ln(1-b)\right]

Because:

1λ=t1/2ln⁡(2)\frac{1}{\lambda}=\frac{t_{1/2}}{\ln(2)}

the VWLP can also be calculated directly from the firm-level half-life:

VWLP(b)=t1/2ln⁡(2)[1+1−bbln⁡(1−b)]VWLP(b)=\frac{t_{1/2}}{\ln(2)}\left[1+\frac{1-b}{b}\ln(1-b)\right]

or equivalently:

VWLP(b)=t1/2[1+1−bbln⁡(1−b)ln⁡(2)]VWLP(b)=t_{1/2}\left[\frac{1+\frac{1-b}{b}\ln(1-b)}{\ln(2)}\right]

This form directly links the company’s measured liquidity to the block-specific liquidation horizon used in the valuation analysis.

How Block Size Changes the Liquidation Horizon

The table below expresses the block-specific VWLP in normalized form. VWLP / H shows the average liquidation period as a multiple of the firm’s half-life. λ × VWLP shows the same average period relative to the full-block limiting horizon 1/λ. The terminal-time column is included to distinguish the endpoint of the modeled liquidation interval from the average realization horizon used in valuation.

Block sizeVWLP / Hλ × VWLPTerminal time / H
5%0.0366840.0254270.074001
25%0.1975830.1369540.415037
50%0.4426950.3068531.000000
75%0.7760280.5379022.000000
100%1.4426951.000000∞

For example, a 50% block has a VWLP of approximately 0.4427 times the firm’s half-life, while the terminal time required to reach 50% cumulative liquidation is exactly one half-life. At a 100% block, VWLP approaches 1/λ, or approximately 1.4427 half-lives, whereas literal terminal liquidation time is unbounded.

Why VWLP Differs from Terminal Liquidation Time

The distinction between VWLP and terminal liquidation time becomes increasingly important as block size grows.

Terminal liquidation time is:

Tb=−ln⁡(1−b)λT_b=-\frac{\ln(1-b)}{\lambda}

As b approaches 100 percent:

Tb→∞T_b\rightarrow\infty

because an exponential process approaches complete liquidation asymptotically.

VWLP behaves differently. Taking the limit of the VWLP equation as the block approaches 100 percent gives:

limb→1⁡VWLP(b)=1λ\lim_{b\to1}VWLP(b)=\frac{1}{\lambda}

and therefore:

limb→1⁡VWLP(b)=t1/2ln⁡(2)\lim_{b\to1}VWLP(b)=\frac{t_{1/2}}{\ln(2)}

Thus the terminal time required to reach literal 100 percent liquidation is unbounded, while the average realization time of the shares remains finite.

The model does not impose this result as an arbitrary cap. The value follows from the mathematical limit of the same continuous liquidation process used for every smaller block.

The result also preserves the expected economic relationships. Very small blocks have relatively short VWLPs; VWLP increases monotonically with block size; less liquid securities generate longer VWLPs; and the measure approaches 1/λ as block size approaches the entire relevant share base.

Restriction Period and Total Delay

The analyst must distinguish liquidity delay from any period during which legal or contractual terms prohibit a sale.

A statutory, contractual, regulatory, or other restriction may impose a no-sale period on the interest. During that interval, the holder cannot begin liquidation regardless of the market’s trading capacity.

After the restriction expires and transfer becomes possible, the market may still need additional time to absorb the block.

The framework therefore treats the two periods separately:

Trestriction=applicable no-sale periodT_{\mathrm{restriction}}=\text{applicable no-sale period}
Tliquidity=VWLP(b)T_{\mathrm{liquidity}}=VWLP(b)

The total DLOM horizon is then:

TDLOM=Trestriction+VWLP(b)T_{\mathrm{DLOM}}=T_{\mathrm{restriction}}+VWLP(b)

This formulation recognizes two distinct sources of delayed liquidity:

  • a period during which sale cannot occur; and
  • a subsequent period during which the market progressively absorbs the position.

If no applicable restriction exists, then:

Trestriction=0T_{\mathrm{restriction}}=0

and the relevant horizon is determined solely by the block-specific VWLP.

Conversely, a restriction does not eliminate the need to evaluate liquidity after it expires. Legal transferability does not necessarily make a security instantly liquid.

From the DLOM Horizon to the Valuation Models

The resulting total horizon is an input to the DLOM analysis rather than a DLOM by itself.

The economic cost of delayed liquidity also depends on how much value may vary while the holder waits to realize the investment. The platform therefore combines the estimated horizon with market-based volatility and the assumptions of the applicable valuation model.

Conceptually:

DLOM=f(σ,TDLOM,model assumptions,information setting)DLOM=f\left(\sigma,T_{\mathrm{DLOM}},\text{model assumptions},\text{information setting}\right)

where:

  • σ represents the applicable volatility measure;
  • TDLOM represents the estimated total delay;
  • the selected model determines how the economic exposure is valued; and
  • the information setting determines which model assumptions are appropriate to the facts.

This structure allows the analyst to distinguish the separate economic effects of transfer restrictions, market liquidity, block size, time, volatility, and information conditions rather than embedding them in a single historical discount percentage.

Professional Judgment and Appropriate Use

These calculations provide analytical anchors rather than an automatic valuation conclusion.

The analyst should evaluate the estimated half-life, VWLP, restriction period, total DLOM horizon, volatility measures, and model indications in light of the actual valuation facts. The analyst should consider whether the selected market evidence, reference cohort, share base, restriction assumptions, liquidity measures, and model structure are appropriate for the interest being valued.

Relevant evidence may also exist outside the platform. Depending on the assignment, the analyst may need to consider shareholder agreements, contractual transfer provisions, registration rights, expected liquidity events, ownership concentration, information access, financial condition, management considerations, capital requirements, litigation, or other facts affecting the realistic ability to dispose of the interest.

The analyst should evaluate and explain differences among model indications rather than conceal them within an unexplained mechanical average.

AbbottModelDLOM organizes market evidence, makes the calculation process transparent, and provides quantitative indications that support professional analysis. It does not replace investigation, due diligence, assignment-specific research, verification of inputs, selection of appropriate assumptions, consideration of contrary evidence, documentation, or the analyst’s professional responsibility for the final valuation conclusion.