32  Session 23: The LAV Operator · GE Equilibrium · Market Clearing

Unit 4 — Math Intuition Bridges
Book Chapters 16, 17 (concepts only — full derivation in Session 28 Track 2)
Track Common core (both tracks)

32.1 Learning Objectives

By the end of this session, students will be able to:

  1. State the LAV operator in plain language — how a single asset’s value depends on the entire \(L_t\) path.
  2. Distinguish the LAV operator from a standard DCF operator (constant rate) and from a simple Jensen-corrected operator.
  3. Articulate what “GE equilibrium” means in the GE-LAV framework: the simultaneous determination of \(\lambda^*\) and individual decisions.
  4. Describe market clearing in the secondary market: how the equilibrium price reflects the marginal investor’s required premium.
  5. Identify the valuation hierarchy mathematically: DCF ⊊ LAV ⊊ GE-LAV as nested operators.

32.2 Pre-Class Assignment

  • Read: Book Chapters 16, 17 (concepts only)

32.3 In-Class Outline (75 minutes)

Time Segment Format
0:00–0:05 Synthesis: 19-22 → today’s payoff Lecture
0:05–0:25 The LAV operator Lecture
0:25–0:45 GE equilibrium and market clearing Lecture
0:45–1:00 The valuation hierarchy DCF ⊊ LAV ⊊ GE-LAV Lecture
1:00–1:15 What this means for practice + Q&A Discussion

32.4 Discussion Questions

  1. The valuation hierarchy DCF ⊊ LAV ⊊ GE-LAV implies that DCF is a special case of GE-LAV (vanishing liquidity volatility) and LAV is a special case of GE-LAV (vanishing MFG coupling). Does this nesting make the framework “obviously correct,” or does it just mean GE-LAV has more parameters to fit?
  2. The 4.31× amplification factor decomposes into ~5% LAV path correction + ~20% GE externality uplift. Which decomposition piece is more important for regulators? For practitioners managing portfolios?
  3. The GE equilibrium requires solving a fixed-point problem (HJB + Fokker-Planck + market clearing). What’s the practical implication if the fixed-point has multiple solutions (e.g., in extreme stress when stability condition fails)?

32.5 Worked Example: Computing LAV vs. GE-LAV vs. DCF for One Asset

Setup: 10-year PE buyout fund. Annual cash flow stream: \(\$10M\)/year. Current liquidity state \(L_0 = -0.5\).

Step 1: DCF

Using constant rate \(r_{DCF} = 12\%\):

\(V^{DCF} = \sum_{t=1}^{10} \dfrac{10}{1.12^t} = 56.5M\)

Step 2: LAV (path-dependent, partial equilibrium)

Need to integrate \(E[\exp(-\int_0^t r(L_s) ds)]\) over the OU path distribution.

At calibrated parameters and starting state \(L_0 = -0.5\):

  • Expected path of \(L_s\): mean reverts from -0.5 toward 1.0 over time
  • Expected \(\pi(L_s)\): starts at \(\pi(-0.5) = 0.058\), decreases to \(\pi(1.0) = 0.041\) at long horizon
  • Effective average \(r(L)\) over 10 years: approximately 7.5% + 0.045 = \(11.5%\)
  • Including Jensen correction: approximately 0.7%

\(V^{LAV} \approx \sum_{t=1}^{10} \dfrac{10}{1.115^t} \cdot (1 + 0.7\%) \approx 58.2M\)

Step 3: GE-LAV (full equilibrium)

At normal liquidity, equilibrium premium ≈ partial equilibrium premium. So GE-LAV ≈ LAV at \(L_0 = -0.5\).

\(V^{GE-LAV} \approx 58.0M\)

Step 4: Now stress the scenario

Suppose we evaluate at \(L_0 = -1.5\) (GFC conditions):

  • DCF (using same fixed rate): still $56.5M (DCF is regime-blind)
  • LAV (partial eq): premium higher, \(V^{LAV} \approx 38M\)
  • GE-LAV (full eq): equilibrium premium amplifies, \(V^{GE-LAV} \approx 22M\)

Comparison at GFC depth:

Method Value Gap from DCF
DCF $56.5M —
LAV $38M -33%
GE-LAV $22M -61%

Interpretation: “DCF says $56.5M regardless. LAV says $38M because the path-corrected discount rate is higher. GE-LAV says $22M because the equilibrium premium amplifies under collective stress. The 4.31× amplification factor shows up in this comparison.”

32.6 What to Expect Next Session

Session 24 is the synthesis session of Unit 4. We cover:

  • The Jensen convexity bias as a closed-form result (Theorem 18.1)
  • The Pigouvian exit tax as a welfare-maximizing instrument (Theorem 19.1)
  • Quantitative welfare analysis (~2.3%/yr)
  • The complete “GE-LAV in one slide” picture

PS3 due Session 24. Both tracks. Track 1 includes intuition questions on Sessions 19–23; Track 2 includes mathematical work.

Reading: Book Chapters 18, 19 (concepts).

← Session 22 | Schedule | Next: Session 24 →