9  Problem Sets Overview

This course assigns four problem sets, each worth 6.25% of your grade (25% total). Problem sets are track-differentiated — Track 1 students submit applied/empirical work; Track 2 students submit mathematical work. Same due dates, same topical coverage; different question content.

9.1 PSet Schedule

PSet Drops Due Sessions Covered Topics
PS1 Session 1 Session 8 1–7 Diagnosis · OU · Term structure · IRR/PME
PS2 Session 9 Session 16 11–15 Exit timing · Portfolio · Regulatory
PS3 Session 17 Session 24 17–23 Platform · Math intuition · Jensen · Pigouvian
PS4 Session 25 Session 31 25–30 Track-specific advanced topics

9.2 Submission

  • Submitted via course site as a single PDF
  • Supporting files (Excel, Python notebooks, R scripts) submitted as a zip
  • Cover page must include: name, track, collaborators (if any, max 3), AI use disclosure
  • Late penalty: 10% per 24-hour period, up to 72 hours; no credit thereafter

9.3 Collaboration Policy

  • Permitted: groups of up to 3 students discussing approach and method
  • Each student must independently write and submit their own solutions — no shared documents, no copy-paste between submissions
  • Disclose collaborators on cover page
  • Track 1 and Track 2 students may discuss general concepts but not specific problem solutions (since the questions differ)

9.4 PSet Structure Standards

Every PSet contains:

  • Conceptual question (15%) — Test understanding of recent lecture material with no math required
  • Applied / mathematical core (60%) — The substantive work for the PSet, track-differentiated
  • Stretch question (15%) — Designed to be genuinely challenging; partial credit available
  • Reflection (10%) — 200-word reflection: What did you learn? Where did you get stuck? How does this connect to your project?

9.5 Sample PSet (Brief Sketches)

9.5.1 PS1 — Track 1 (Applied)

  1. Conceptual (15 pts): Explain in 200 words why the IRR’s two biases (reinvestment assumption + sign-pattern dependence) make it unsuitable for assessing vintage-year liquidity effects.

  2. Applied core (60 pts):

      1. Using the provided dataset of Preqin secondary market discounts (2003–2024), compute the implied liquidity state L_t for each quarter using the linear discount-to-L mapping from Ch. 2.
      1. Fit the OU process (κ, σ, L̄) to your computed L_t series using MLE. Report your point estimates with 95% confidence intervals. Compare to the book’s calibration.
      1. Compute the Jensen bias B(T) for T = 5 and T = 10 using your calibration. Comment on the difference vs. the book’s calibration.

  3. Stretch (15 pts): Identify and characterize the four “regimes” of the secondary market discount series (2003–2024). Are they consistent with the book’s regime classification? Where do they differ?

  4. Reflection (10 pts): As specified above.

9.5.2 PS1 — Track 2 (Mathematical)

  1. Conceptual (15 pts): Explain in 200 words why the Jensen convexity bias must be strictly positive whenever the discount factor is a strictly convex function of the liquidity state.

  2. Mathematical core (60 pts):

      1. Derive the stationary distribution of the OU process dL_t = κ(L̄ − L_t)dt + σ dW_t from first principles using the Fokker-Planck equation. Show your work.
      1. Prove that E[L_t] → L̄ as t → ∞ and Var[L_t] → σ²/(2κ) regardless of L_0.
      1. Derive the conditional moments E[L_t | L_0] and Var[L_t | L_0]. State the result and prove it.
      1. Using your derivation, prove that the Jensen bias B(T) under a quadratic discount rate r(L) = r̄ + α(1−L) + β(1−L)² is affine in T: B(T) = A·T + C. Identify A and C explicitly.

  3. Stretch (15 pts): Extend the OU process to a jump-diffusion with compensated Poisson jumps. Derive the new stationary distribution. Under what conditions does it remain Gaussian?

  4. Reflection (10 pts): As specified above.

9.6 PS2, PS3, PS4 — Topic Outlines

Full question sets will be released at PSet drop. Outlines below.

9.6.1 PS2 (Sessions 11–15)

Track 1 topics:

  • Exit boundary computation for a real PE fund position
  • Portfolio construction with hedge demand — applied to a $1B endowment allocation
  • Regulatory case: compute the Solvency II SCR for a real insurance company’s private market portfolio under both standard and LA approaches

Track 2 topics:

  • Numerical solution of the HJB equation for h(L) using finite differences
  • Mean-field equilibrium fixed-point iteration — implement and verify convergence
  • Welfare gap derivation under heterogeneous beta

9.6.2 PS3 (Sessions 17–23)

Track 1 topics:

  • Full GE-LAV platform pipeline: take a real fund, calibrate, value, stress-test
  • Conceptual: explain the Pigouvian tax to a non-economist (writing exercise)
  • Apply LA-IRR and LA-PME to a real fund and compare to standard metrics

Track 2 topics:

  • Prove a property of the LAV operator (e.g., path-dependence)
  • McKean-Vlasov mean-field game — finite-N approximation and rate of convergence
  • Derive the Pigouvian tax formula explicitly from the welfare-gap analysis

9.6.3 PS4 (Sessions 25–30)

Track 1 topics:

  • Multi-asset GE-LAV applied to a real diversified private market portfolio
  • Stress scenario analysis: 2008 crisis, 2020 COVID, hypothetical 2030 climate event
  • Communication exercise: 1-page IC memo summarizing GE-LAV vs. DCF for a specific deal

Track 2 topics:

  • Network topology with heterogeneous β — analytical and numerical
  • Neural SDE calibration to GE-LAV — implement and benchmark
  • Extension proposal: a 5-page research note on a novel GE-LAV extension

9.7 Grading Rubric (Common Across PSets)

Score range Meaning
90–100% Excellent work — correct, well-explained, with insight beyond the minimum
80–89% Solid work — correct conclusions; some minor errors or missing detail
70–79% Acceptable work — main results obtained; some confusion in derivation or interpretation
60–69% Weak work — partial credit for effort and partial correctness
Below 60% Insufficient — significant misunderstanding or incomplete submission

For mathematical work (Track 2): Show your steps. Final answer only earns ~40% credit even if correct.

For applied work (Track 1): Document your assumptions, your data, and your calculations. Final number only earns ~40% credit.

9.8 AI Use Policy for PSets

See the syllabus AI policy. For PSets specifically:

  • Permitted: Using AI to explain concepts, debug code, suggest derivation strategies, check intermediate steps
  • Required: You must understand and be able to defend every step of your submission
  • Disclosure: Brief note on cover page describing how AI was used (e.g., “Used Claude to debug Python code in Q2; verified all derivations independently”)

Forbidden: Submitting AI-generated solutions verbatim. The PSets are designed so that AI alone cannot produce a correct submission — most questions require your specific dataset, your specific calibration, or your specific interpretation.

9.9 Solutions Release

  • Solutions are released 48 hours after the deadline on the course site
  • Both tracks’ solutions are released to all students (encouraging cross-track learning)
  • Solutions explain not just the answer but also common errors and good vs. bad approaches

Next: Session 1 →