A key limitation of using the log-utility CAPM as the basis for PME's risk adjustment is that:

Question

BlackTom AI · Accepted Answer

Log-utility investors maximize expected logarithmic growth (the Kelly criterion), which produces a closed-form link between expected excess return and variance that underpins the stated limitation.

1) Relevant concepts
- Log utility: u(W) = ln(W). The investor chooses portfolio weight w to maximize E[ln(1 + wR)] for a risky asset with excess return R having mean μ and variance σ^2.
- Equilibrium CAPM for a representative log-utility investor implies the market portfolio is the growth-optimal one, tying the market risk premium to its variance.
- PME (public market equivalent) uses a public-market benchmark; if the benchmark’s required premium is underestimated, PME’s risk adjustment will be biased.

2) Step-by-step reasoning and calculations
Start with the expected log-return expansion (sufficient for typical small-period returns or exact in continuous-time):
E[ln(1 + wR)] ≈ wμ − 0.5 w^2 σ^2.
Take derivative with respect to w and set to zero (first-order condition):
d/dw = μ − w σ^2 = 0 ⇒ w* = μ / σ^2.
In a representative-agent equilibrium where the market portfolio is the chosen portfolio, the aggregate weight on the risky market asset is w* = 1 (i.e., the market asset comprises the full risky allocation). Setting w* = 1 gives:
μ = σ^2.
Thus the implied equilibrium market risk premium (μ, or μ − r_f if r_f ≈ 0) equals the market variance σ^2.

3) Numerical interpretation
If annual volatility σ is 15%–20%, then σ^2 = 0.0225–0.04, i.e., 2.25%–4.0%. That is substantially lower than commonly reported historical equity premia (often 4–6% or higher).

4) Why this is a limitation
Using the log-utility CAPM as the basis for PME’s risk adjustment forces an implied market premium equal to variance — a low number (≈2–4%). That understates the compensation investors historically require for bearing market risk, causing PME adjustments to be too small and potentially biasing conclusions about private-equity outperformance.

5) Why other plausible criticisms are not the central one (brief)
- “Log-utility is unrealistic” is a valid concern but not the core numerical limitation asked here.
- “It overstates the premium” is incorrect — it actually implies a much smaller premium.
- “It ignores higher moments” is true but secondary to the quantitative implication that premium ≈ variance.

Conclusion: Therefore, the key limitation is that the log-utility CAPM implies a market risk premium equal to market variance (roughly 2–4%), which is much lower than historically observed premia, matching the provided answer.

A key limitation of using the log-utility CAPM as the basis for PME's risk adjustment is that:单项选择题

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