33 Session 24: Jensen Bias · Pigouvian Tax · Welfare (Synthesis)

Unit	4 — Math Intuition Bridges (final synthesis)
Book Chapters	18, 19 (concepts only — full derivations in Session 29 Track 2)
Track	Common core (both tracks)
Assessment milestone	PS3 due (start of class)

33.1 Learning Objectives

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

State the closed-form Jensen bias formula $B(T) = (\pi_2/2) \cdot \Pi_{liq}(T)$ and compute it for any asset class.
Articulate the welfare gap ($\Delta W \approx 2.3\%$/yr) as the cost of the McKean-Vlasov externality.
Explain the Pigouvian exit tax intuitively: tax on secondary exits during stress, designed to align private and social incentives.
Compute the optimal tax rate $\tau^*(L)$ at different liquidity states using the calibrated formula.
Articulate “GE-LAV in one slide” — the complete framework synthesis.

Time	Segment	Format
0:00–0:05	PS3 collection · Today: synthesis	Lecture
0:05–0:25	The Jensen convexity bias (full formula)	Lecture
0:25–0:45	The welfare gap and externality cost	Lecture
0:45–1:05	The Pigouvian exit tax	Lecture + chart
1:05–1:15	“GE-LAV in one slide” + Q&A	Synthesis

The Jensen bias formula $B(T) = (\pi_2/2) \cdot \Pi_{liq}(T)$ depends on five calibrated parameters. If any of those parameters are wrong, the bias estimate is wrong. Which parameter is most fragile? How would you stress-test it?
The Pigouvian tax raises ~$14B at GFC depth. Distributing this back to remaining LPs is the textbook answer. Politically, would governments distribute, or absorb? How does that change the framework’s appeal?
The valuation hierarchy DCF ⊊ LAV ⊊ GE-LAV implies GE-LAV is the “right” framework. Why isn’t DCF abandoned today, given the framework exists?

Setup: $\$100M$ infrastructure asset, 15-year horizon, $L_0 = -0.5$ (mild stress).

Step 1: Jensen bias

$\Pi_{liq}(15) = 0.1024 \cdot [15 - (1 - e^{-13.5})/0.9] = 0.1024 \cdot [15 - 1.111] = 0.1024 \cdot 13.889 = 1.422$

Wait — let me recompute: $\Pi_{liq}(T) = \sigma^2 \cdot [T - (1 - e^{-2\kappa T})/(2\kappa)]$

$\Pi_{liq}(15) = 0.1024 \cdot [15 - (1 - e^{-13.5})/0.9] \approx 0.1024 \cdot [15 - 1.111] \approx 1.422$

$B(15) = (\pi_2/2) \cdot \Pi_{liq}(15) = (0.021/2) \cdot 1.422 = 0.0149$ ≈ 1.5%

Step 2: LAV value

DCF value (at constant 7.5%): $V^{DCF} \approx \$100M$ (assuming normalization)

LAV value (Jensen-adjusted): $V^{LAV} = V^{DCF} \cdot (1 + B(15)) = \$101.5M$

Step 3: GE-LAV value at current stress

At $L_0 = -0.5$, equilibrium premium ≈ 5.5% (vs. DCF assumption 3.5%) → effective rate ~9.5%

$V^{GE-LAV} \approx V^{DCF} \cdot e^{-(0.095 - 0.075) \cdot 15} \approx \$100M \cdot 0.74 = \$74M$

(Account for ongoing cash flow generation roughly cancels the discount rate effect over the term)

Step 4: Welfare loss attributable to this position

Per-year welfare loss for this asset: ~$\$2.3M$ (2.3% of $\$100M)$

Cumulative over 15 years: ~$\$25M$ (with discount)

Step 5: Optimal Pigouvian tax if held in stress

If LP exits at $L = -0.5$: optimal $\tau^* = 1.5\%$. On $\$100M$ sale: $\$1.5M$ tax.

If LP exits at GFC depth: optimal $\tau^* = 7.0\%$. On (deeply discounted) sale: substantial.

Sessions 25 onward are split track:

Track 1 students report to the applied case workshop. Sessions 25 covers PE buyout valuation.
Track 2 students report to the mathematical session. Session 25 covers the full HJB derivation.

Both tracks: Session 25 is your first split-track session. Make sure you know which track you’re in and where it meets.

Reading for Track 1: Bain Capital case study or similar (will be distributed).

Reading for Track 2: Book Chapter 11 (HJB, with full derivations).

PS4 drops at end of Session 25. Track-specific.