39  Session 30: Project Draft Workshop · Valuation Hierarchy Proof (T2)

Unit 5 — Split Track
Track 1 Project draft workshop (no new content)
Track 2 Valuation hierarchy proof (Theorem 20.1) + project draft workshop
Track 2 source Class 20 (Pigouvian Tax remainder, hierarchy proof)
Assessment milestone Project draft due (start of class)
ImportantProject draft due today

Both tracks submit their 15-page project draft + supporting materials at the start of class. Peer review activity uses these drafts.

39.1 Both Tracks: Project Draft Workshop (60 minutes)

39.1.1 Workshop Learning Objectives

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

  1. Provide structured peer review on a classmate’s GE-LAV project draft.
  2. Identify common weaknesses in their own draft based on peer feedback.
  3. Plan revisions for the final paper and presentation.
  4. Practice delivering the project’s “elevator pitch” — the IC summary in 2 minutes.

39.1.2 Workshop Structure (Both Tracks, Same Activity)

Time Activity Detail
0:00–0:05 Logistics: pair assignments, rubric distribution Lecture
0:05–0:25 Reading time: read partner’s draft + take notes Individual
0:25–0:50 Peer feedback session (12.5 min each direction) Pair work
0:50–1:00 Wrap-up: list 3 priority revisions for your draft Individual

39.1.3 Peer Review Rubric

Each student provides written feedback to their partner addressing:

1. Clarity (5 questions, 1-5 scale) - Is the asset/question stated precisely? - Is the methodology clear? - Are the calibrations documented? - Are the results easy to follow? - Is the recommendation defensible?

2. Rigor (5 questions) - Is the GE-LAV implementation correct? - Are the comparisons (DCF vs. LAV vs. GE-LAV) accurate? - Are sensitivity analyses adequate? - Is the data appropriate? - For Track 2: are mathematical derivations correct?

3. Communication (3 questions) - Could you explain this paper to a non-finance friend? Try in 2 sentences. - Is the IC-relevance clear? - Is the writing professional quality?

4. Improvement priorities (open) - “The single most important revision before the final paper would be…” - “The presentation should emphasize…” - “One thing that would delete if I had to cut to 12 pages…”

39.2 Track 2: Valuation Hierarchy Proof (15 additional minutes)

39.2.1 Track 2 Learning Objectives

By the end of the Track 2 portion, students will additionally be able to:

  1. State and prove Theorem 20.1 (strict valuation hierarchy DCF ⊊ LAV ⊊ GE-LAV).
  2. Identify Theorem 20.2 (conditions for approximate equality among the hierarchies).
  3. Compute the gap between hierarchy levels for arbitrary OU calibrations.
  4. Discuss why the strict nesting matters for regulatory and pedagogical purposes.

39.2.2 Track 2 Slide Briefs

Slide T2-30.1 — Theorem 20.1: Strict Valuation Hierarchy

  • Title: “Theorem 20.1: DCF ⊊ LAV ⊊ GE-LAV”
  • Statement: Under positive liquidity volatility (\(\sigma_L > 0\)) and positive general-equilibrium coupling (\(\gamma > 0\)):
    • \(V^{DCF}(L, t) \neq V^{LAV}(L, t)\) (strict inequality somewhere on the state space)
    • \(V^{LAV}(L, t) \neq V^{GE-LAV}(L, t)\) (strict inequality)
  • In words: The three operators give genuinely different answers when liquidity is stochastic and the externality is present.

Slide T2-30.2 — Theorem 20.1 Proof Sketch

  • Title: “Proof: strict nesting”
  • Part 1 (DCF ≠ LAV):
    • Jensen’s inequality applied to \(e^{-r(L)T}\): \(E[e^{-r(L)T}] > e^{-E[r(L)]T}\)
    • Strict whenever \(L\) is non-degenerate (i.e., \(\sigma_L > 0\))
  • Part 2 (LAV ≠ GE-LAV):
    • GE-LAV adds the equilibrium uplift to \(r(L)\): \(r_{GE}(L, \mu) = r(L) + \text{externality}(L, \mu)\)
    • Strict whenever the externality is positive (i.e., \(\gamma > 0\))
  • Combination: Both inequalities strict → strict nesting holds.

Slide T2-30.3 — Theorem 20.2: Conditions for Equality

  • Title: “When the hierarchies converge”
  • Statement: \(V^{DCF} = V^{LAV} = V^{GE-LAV}\) in the limit:
    • As \(\sigma_L \to 0\): liquidity is deterministic, no path dependence
    • As \(\gamma \to 0\): no externality, partial equilibrium
  • Result: The framework reduces to DCF in the “no-uncertainty, no-coupling” limit.

39.2.3 Track 2 Discussion Questions

  1. The strict nesting theorem requires \(\sigma_L > 0\) and \(\gamma > 0\). What if \(\sigma_L > 0\) but \(\gamma = 0\)? Is LAV strictly necessary then, or does DCF + simple Jensen suffice?
  2. The proof of strict inequality relies on Jensen’s inequality applied pointwise. What if some pricing kernel is approximately linear (e.g., very short horizons)? Does the inequality “vanish” in the limit?
  3. Theorem 20.2 (equality conditions) is the “consistency limit.” Is there a third limit where GE-LAV reduces to something other than DCF? Perhaps when the externality dominates?

39.3 Both Tracks: Final Reminders

39.3.1 What’s Due When

Deliverable Due Date
Project draft + peer review notes Today (Session 30)
PS4 Session 31
Final paper (with revisions) Session 32
Project presentation Session 32

39.3.2 Session 31 Preview

  • Both tracks: Final research frontiers + course synthesis
  • PS4 due (track-specific)
  • Final preparation for project presentations

39.3.3 Session 32 Preview

  • Both tracks reunified
  • Project presentations (20 min + 5 min Q&A each)
  • Final wrap-up

← Session 29 | Schedule | Next: Session 31 →

For your capstone: Students often use liquidityillusion.com to compute final-project results. The production engine accepts your own cash-flow inputs.