20  Session 11: Exit Timing · The Trapped Investor Problem

Unit 3 — Decision and Application
Book Chapter 6 (sections 6.1–6.4)
Track Common core (both tracks)

20.1 Learning Objectives

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

  1. Frame the LP’s exit decision as an optimal stopping problem with stochastic state.
  2. Distinguish between forced exit (regulatory, operational) and discretionary exit (optimization).
  3. Identify the four conditions under which an LP becomes “trapped” — unable to exit at fair value despite wanting to.
  4. State the smooth pasting condition at the optimal exit boundary, at the intuition level (Track 2 will derive this in Session 25).
  5. Compute the value of holding vs. exiting for a representative position using the LAV operator.

20.2 Pre-Class Assignment

  • Read: Book Chapter 6, sections 6.1–6.4 (~10 pages)
  • Bring back: Any midterm reflections or questions

20.3 In-Class Outline (75 minutes)

Time Segment Format
0:00–0:10 Midterm debrief · Set up Unit 3 Lecture
0:10–0:25 The LP exit decision in practice Lecture
0:25–0:45 The optimal stopping formulation Lecture
0:45–1:00 Smooth pasting (intuition only) Lecture
1:00–1:15 Trapped investor scenarios — four cases Discussion + cases

20.4 Discussion Questions

  1. CalPERS publicly sold PE secondaries at significant discounts in 2009. Were they “trapped” by scenario 1, 2, 3, or 4 from today’s slide? Could they have avoided the discount through better planning?
  2. The IPEV valuation guidelines suggest GPs should “hold to liquidation when feasible.” Does this guideline implicitly assume scenario 1 doesn’t exist? What would the guideline say if it acknowledged GE-LAV?
  3. A pension fund’s actuarial assumption is 7% expected return on private markets. If GE-LAV-optimal exit timing reduces realized returns by ~50 bps relative to “hold to liquidation,” is the actuarial assumption violated? Or is the realized return improvement on a risk-adjusted basis still positive?

20.5 Worked Numerical Example: Hold vs. Exit Decision

Setup: An LP holds a 5-year remaining PE position. Reported NAV: $100M. Current liquidity state: \(L_t = -0.8\) (moderately stressed). OU parameters: κ = 0.45, σ = 0.32, \(\bar{L} = 1.0\).

Step 1: Expected hold value

Under OU dynamics, \(E[L_{t+5}] = \bar{L} + (L_t - \bar{L}) e^{-5\kappa} = 1.0 + (-0.8 - 1.0) \times e^{-2.25} = 1.0 + (-1.8)(0.105) = 0.81\)

The expected liquidity state recovers from −0.8 toward \(\bar{L} = 1.0\) over 5 years, ending at +0.81.

Using \(\pi(L) = 0.045 - 0.025L + 0.021L^2\): - Average premium over 5-year horizon ≈ \(\pi(0.0) = 0.045 - 0 + 0 = 4.5\%\) (rough average of current and future) - More carefully: integrate path; result ~ 4.0% effective

Hold value (Jensen-adjusted LAV): \(V_\text{hold} \approx \$100M \times (1 + \text{Jensen bias at 5 years})\) With \(A = 0.16\%\)/yr (buyout): \(B(5) = 0.8\%\) \(V_\text{hold} \approx \$100.8M\)

Step 2: Current secondary price

At \(L_t = -0.8\), secondary discount calibration suggests a 20% discount to NAV: \(P_\text{secondary} = \$100M \times (1 - 0.20) = \$80M\)

Step 3: Compare

\(V_\text{hold} = \$100.8M\) vs. \(P_\text{secondary} = \$80M\)

Decision: Hold. The mean-reversion benefit (≈$20M expected over 5 years) exceeds any liquidity insurance value at this stress level.

Step 4: Where would exit make sense?

If \(L_t\) dropped to \(-1.5\) (GFC depth) and secondary discount widened to 40%: - \(P_\text{secondary} = \$60M\) - But \(V_\text{hold}\) also drops (higher expected premium during recovery period) — maybe to ~\(\$70M\) - At \(L_t = -1.5\), hold still slightly dominates exit for a 5-year position - For shorter remaining horizons (e.g., 2 years), the trade flips: exit becomes optimal because there’s less time for mean reversion

20.6 What to Expect Next Session

Session 12 computes the exit boundary \(L^*(t)\) numerically using the calibrated GE-LAV parameters. We’ll cover:

  • Numerical solution of the value function (intuition for both tracks)
  • The \(L^*(t)\) curve across remaining horizons
  • Sensitivity to OU parameters and asset class
  • Historical episodes where the exit rule would have changed outcomes

Reading: Book Chapter 6, sections 6.5–6.7 (~8 pages).

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