34 Session 25: Split Track — PE Buyout Workshop / Full HJB Derivation
| Unit | 5 — Split Track |
| Track 1 source | Class 21 (PE Buyout Full Valuation) + new case material |
| Track 2 source | Classes 7 (HJB) + 14 (HJB numerical) + new derivation material |
| Assessment milestone | PS4 drops at end of class |
Track 1 (Practitioner) and Track 2 (Researcher) students meet in different rooms (or different breakout sessions in synchronous online format). Both sessions run the same 75 minutes. Confirm your track location before class — see course site for room assignment.
Lecture recordings from both tracks are posted to the course site afterward. Cross-track audit is encouraged.
34.1 Track 1: PE Buyout Case Workshop
34.1.1 Track 1 Learning Objectives
By the end of this session, Track 1 students will be able to:
- Apply the GE-LAV framework end-to-end to a real PE buyout fund scenario.
- Calibrate OU parameters using provided secondary market data for the case.
- Compute DCF, LAV, and GE-LAV values for the case fund and explain the differences.
- Construct an IC-style recommendation based on GE-LAV outputs.
- Defend the recommendation orally to a peer reviewer.
34.1.2 Track 1 Pre-Class Assignment
- Read: Bain Capital Asia IX case (distributed via course site)
- Bring: Laptop with platform access
- Optional: Pre-class spreadsheet exercise (link on course site)
34.1.3 Track 1 In-Class Outline (75 minutes)
| Time | Segment | Format |
|---|---|---|
| 0:00–0:10 | Case introduction and team formation (groups of 3) | Lecture |
| 0:10–0:30 | Hands-on: calibration and DCF baseline | Group work |
| 0:30–0:50 | Hands-on: LAV and GE-LAV computation | Group work |
| 0:50–1:05 | Group presentations (3 minutes each) | Presentations |
| 1:05–1:15 | Synthesis and PS4 introduction | Lecture |
34.1.4 Track 1 Case Setup
The Case: Bain Capital Asia Fund IX
- Vintage: 2018
- Strategy: Asian mid-market buyout
- NAV as of Q4 2024: $1.2B (LP’s share)
- Remaining hold: ~4 years
- Reason for valuation review: LP is considering secondary sale offer
- Secondary bid: $980M (18% discount to NAV)
Provided data: - Quarterly secondary market discounts 2018–2024 for Asian PE funds - Fund cash flow history (capital calls and distributions) - Public benchmark: MSCI Asia ex-Japan - Asset class calibration: PE buyout, calibrated parameters per Table 7.2
34.1.5 Track 1 Discussion Questions
- The secondary bid is $980M against a $1.2B NAV. Is the secondary buyer offering fair value, exploiting distress, or pricing in something the LP doesn’t see? How would GE-LAV decompose the gap?
- Asian PE has had different liquidity dynamics than US PE over 2018-2024 (less institutional secondary infrastructure, more strategic buyers). How does this affect the calibration?
- The LP’s IC will likely prefer a “hold” recommendation since it preserves the carry track record. How do you advocate for a “sell” if the GE-LAV analysis supports it?
34.2 Track 2: Full HJB Derivation
34.2.1 Track 2 Learning Objectives
By the end of this session, Track 2 students will be able to:
- Derive the Hamilton-Jacobi-Bellman equation from the Bellman optimality principle and Itô’s lemma.
- Apply the HJB equation to the GE-LAV optimal stopping problem and identify the variational inequality structure.
- State and verify the smooth pasting condition as the optimality condition at the free boundary.
- Implement a finite-difference solver for the GE-LAV exit boundary.
- Reproduce the \(L^*(t)\) chart numerically from first principles.
34.2.2 Track 2 Pre-Class Assignment
- Read: Book Chapter 11 in full (with proofs)
- Optional pre-reading: Karatzas & Shreve, Brownian Motion and Stochastic Calculus, Chapter 5; or Pham, Continuous-time Stochastic Control and Optimization, Chapters 3-4
34.2.3 Track 2 In-Class Outline (75 minutes)
| Time | Segment | Format |
|---|---|---|
| 0:00–0:10 | Recap measure-theoretic foundations (assumed prereq) | Lecture |
| 0:10–0:30 | Derivation: from Bellman to HJB | Lecture + board work |
| 0:30–0:50 | The variational inequality and smooth pasting | Lecture |
| 0:50–1:05 | Numerical implementation: finite differences | Lecture + code walkthrough |
| 1:05–1:15 | Verification theorem | Lecture |
34.2.4 Track 2 Discussion Questions
- The verification theorem requires the candidate value function \(V^*\) to be smooth (e.g., \(C^{1,2}\)). What if smooth pasting failed and \(V^*\) had a kink at the boundary? What would that imply economically?
- The finite-difference solver uses Crank-Nicolson for stability. What’s the alternative? When does explicit time-stepping suffice vs. require implicit treatment?
- The HJB derivation assumes Markov dynamics. If \(L_t\) had memory (fractional Brownian motion), how would the derivation change? Hint: research on rough volatility extensions.
34.3 Common to Both Tracks: PS4 Drops
PS4 drops at the end of today’s session. Track-specific content:
- Track 1 PS4: Apply GE-LAV to a multi-asset private market portfolio. Use the platform. Produce an IC memo.
- Track 2 PS4: Mathematical work on advanced topics (TBD per track-specific assignment).
Due: Session 31.
← Session 24 | Schedule | Next: Session 26 →
Try this case in the engine: The RJR Nabisco GE-LAV® analysis is available as a worked template at liquidityillusion.com.