Chapter 5 · Illiquidity and Non-Tradability

Chapter at a glance

Illiquidity ceases to be a premium bolted onto a discount rate and becomes a constraint on the action set. The chapter models liquidity as a finite-state process, classifies transaction-cost structures, derives the secondary-market discount from a search model, characterizes the optimal-sale policy as a threshold in urgency, and defines the illiquidity valuation operator L. It previews the option value of waiting that Chapters 13–14 make endogenous.

Learning Outcome Statements

LOS 5.1 — Model liquidity as a finite-state process that constrains the investor’s admissible actions, and distinguish this friction-side treatment from the priced-factor treatment of Chapter 4.

LOS 5.2 — Classify transaction-cost structures (proportional, fixed, impact) and explain why the episodic, block-sized trading of private markets makes state-contingent availability, not the spread, the binding friction.

LOS 5.3 — Derive the distribution of holding periods and of discounted realizable value under Markov liquidity states, in closed form for the two-state chain.

LOS 5.4 — Derive the secondary-market discount as the outcome of a search-andreservation problem, and prove its comparative statics in urgency and market depth.

LOS 5.5 — Formulate forced and voluntary sale timing as a stopping problem over observation-filtration stopping times, anticipating the full theory of Chapter 14.

LOS 5.6 — Define the liquidity-adjusted valuation operator, prove its monotonicity in liquidity transition intensities, and explain in what precise sense it is nonlinear.

LOS 5.7 — Decompose the illiquidity component of required returns into clientele amortization, priced liquidity risk, and state-dependent shadow costs, and assign each to its correct model address.

Laboratory (book §5.9)

Module: Liquidity Shock Simulator — open in the Laboratory

Manage an illiquid position through simulated liquidity cycles and funding shocks; watch realizable value diverge from fundamental value and price the divergence with L. E1 replay the fading bid (decompose the discount into urgency and state-transition effects); E2 option value of waiting (sweep the threshold policy, locate the interior optimum, watch precautionary behavior emerge); E3 depth versus urgency (iso-discount curves); E4 tail pricing (the variance-finiteness boundary of the holding-period proposition).

Downloads: Python notebook · Excel workbook · Slides

Exercises

Solutions are distributed to instructors with the Instructor’s Solutions Manual; they are not posted here.

Conceptual Problems

5.1 Classify the following frictions by cost structure (proportional, fixed, impact, availability) and by correct model address (operator constraint, operator discount, SDF factor): a 1% secondary-agent fee; legal costs of an LP transfer; the GP’s right to refuse a transferee; a market-wide freeze of secondary activity; deeper discounts for larger blocks.

5.2 The opening problem’s buyers bid lower in November partly because they inferred urgency. Distinguish the decision-theoretic channel (the seller’s reservation value fell) from the informational channel (buyers updated on the seller’s constraint), and explain which one Proposition 5.5 captures and which Exercise 5.13 adds.

5.3 A consultant proposes reporting “liquidity-adjusted NAV” as (1 − 𝛿(ℓ, 𝑢, ¯ 𝑞))×NAV with a fixed average urgency 𝑢. ¯ Using Sections 5.6 and 5.8, explain what this number is (a realizable value under a hypothetical squeeze) and what it is not (the holder’s value), and when each is the right report.

5.4 Explain why L ’s nonlinearity (Proposition 5.7(iii)) implies that the illiquidity discount of a portfolio of LP stakes is generally smaller than the sum of stand-alone discounts, and connect this to the pooling logic of secondary funds.

Mathematical Problems

5.5 Solve (5.1) in closed form for uniform 𝐹 on [1 − 𝑑, ¯ under which the used in Figure 5.2, and derive the exact condition on (𝑢, 𝜈, 𝜌, 𝑑) seller accepts every offer (the reservation constraint binds at the support’s bottom).

5.6 Extend Proposition 5.4 to the three-state chain via the linear-system method of Proposition 4.6: compute e[𝑒 −𝜌 𝜎 ] and e[𝑒 −𝜌 𝜎 𝑉 𝜎 ] starting from each state, where 𝜎 is the first Liquid time, and verify the two-state formulas as a merging limit.

5.7 Derive the variance-finiteness condition of Proposition 5.4(iv) and compute the full Mellin transform e[(𝑒 −𝜌 𝜎 𝑉 𝜎 ) 𝜃 ], identifying the power-law tail index of realizable value induced by exponential waiting against lognormal growth.

5.8 In the search model, let urgency follow a two-state chain (calm/squeeze) rather than a constant. Write the coupled reservation equations, prove existence and uniqueness by the monotone system argument of Proposition 4.6, and show 𝑆 ∗ (calm) > 𝑆 ∗ (squeeze).

5.9 Construct the two-claim counterexample promised in the proof of Proposition 5.7(iii): claims whose optimal sale states differ, with L (𝐷 1 + 𝐷 2 ) <L (𝐷 1 ) +L (𝐷 2 ) under the single-policy constraint, and compute the subadditivity gap.

Computational Problems

5.10 Implement the search-model panel and reproduce Figure 5.2 (left) from the printed parameters; add beta-distributed offers and report how offer-dispersion changes (Proposition 5.5(iii)) move the curves.

5.11 Implement the holding-period dashboard: verify Proposition 5.4(ii) by Monte Carlo across a (𝜈, 𝜌, 𝑔) grid, then map the empirical variance-explosion boundary and compare with (iv).

5.12 Run experiment E2 programmatically: for call intensities {0.5, 1, 2, 4} per year, compute the optimal urgency threshold per state and the value function; plot the precautionary shift and quantify the value of doubling the liquid reserve at each intensity.

5.13 Replace the exogenous offer distribution with Nash bargaining in which the buyer observes a noisy signal of the seller’s urgency. Characterize the equilibrium mapping from the seller’s funding state to the accepted discount, and design an empirical test separating the reservation-value channel from the inference channel using deadline-driven (quarter-end) variation in seller urgency.