portfolio-theoryoptimal-weightingmean-variancetheory

Ferson–Siegel Optimal Weighting (FS 权重)

The theoretical foundation for nonlinear signal-to-position mapping in trend-following. The "S-curve" that maximizes unconditional Sharpe ratio for a single risky asset with a noisy expected-return signal.

Setting

Following Ferson & Siegel (2001), consider one riskless asset (return r_f) and one risky asset with excess return:

R_{t+1} = \mu(s_t) + \varepsilon_{t+1}, \quad \mathbb{E}[\varepsilon_{t+1} | s_t] = 0, \quad \mathbb{E}[\varepsilon_{t+1}^2] = \sigma_\varepsilon^2

where s_t is a public signal (e.g., past returns) believed to predict \mu(s_t) = \mathbb{E}[R_{t+1} | s_t]. An investor with mean–variance preferences over the unconditional distribution chooses weight w^*(s_t) on the risky asset to minimize unconditional variance for a target unconditional mean.

The optimal weight

For a quadratic-utility / mean-variance investor, the optimal weight on the risky asset is:

w^*(s_t) = \frac{\mu(s_t)}{\sigma_\varepsilon^2 + \lambda \,\mathbb{E}[\mu(s_t)^2]}

Wait — the cleaner closed form (from the unconditional formulation in Moskowitz et al., 2025, citing FS 2001) when the signal perfectly predicts the next-period risk-adjusted return (\mu(s_t) = s_t):

f_{FS}(s_t) = \frac{\lambda \, s_t}{\sigma_\varepsilon^2}

subject to variance normalization \text{var}(f_{FS}(s_t)) \approx 1 for s_t \sim \mathcal{N}(0,1), which pins down \lambda \approx 0.394 (when \sigma_\varepsilon^2 = 1).

The S-shape

flowchart LR
    A["s_t = -3"] -->|"convex region: weight rolls back toward zero"| B["f(s_t) ≈ -1"]
    C["s_t = -1"] -->|"linear region"| D["f(s_t) ≈ -1"]
    E["s_t = 0"] -->|"neutral"| F["f(s_t) = 0"]
    G["s_t = 1"] -->|"linear region"| H["f(s_t) ≈ 1"]
    I["s_t = 3"] -->|"concave region: weight rolls back toward zero"| J["f(s_t) ≈ 1"]

In block form:

f_{FS}(s) \approx \begin{cases} \lambda \, s, & |s| \leq 1 \quad \text{(linear)} \\ \lambda \, \text{sign}(s), & |s| \to \infty \quad \text{(saturating / rolling back)} \end{cases}

The shape is linear in the middle, concave for large positive s (weight grows slower than signal), convex for large negative s (weight rolls back toward zero from below). Visually: an S-curve.

Intuition

For a fixed signal \mu(s) with variance \sigma_\varepsilon^2, the unconditional-variance penalty of placing weight w scales as w^2 \sigma_\varepsilon^2, while the expected return scales linearly in w. For very large |s|, increasing w further raises variance faster than it raises expected return, so the optimum flattens. This is the same logic that produces Kelly-style fractional betting under noisy return estimates.

Why it matters in practice

  • Moskowitz et al. (2025) show that a neural network trained to maximize out-of-sample Sharpe ratio recovers a function f very close to f_{FS}. Theory and ML agree.
  • The "rollback" region is most active during extreme market environments — exactly when TSMOM is supposed to deliver tail-hedge returns. The empirical paper shows that most of NLTSMOM's outperformance vs. linear TSMOM is concentrated in bad states (down markets), consistent with FS intuition.
  • In smooth regions (|s| \lesssim 1), FS is essentially linear; the binary rule is a poor approximation of FS even there because FS doesn't fully saturate.

Implementation notes

  • The exact functional form derived by FS is more elaborate than the linear-in-middle S-curve summary above. See Moskowitz et al. (2025) §3.2 for the full derivation.
  • In practice, the F&S specification used as a benchmark is the linear-with-saturation rule above with \lambda \approx 0.394.
  • Sensitivity: if signal noise increases, the optimal weight should pull back from the extremes even more (Bayesian shrinkage intuition). FS captures this via the \sigma_\varepsilon^2 in the denominator.

References

  • Ferson, W.E. & Siegel, A.F. (2001). "The Efficient Use of Conditioning Information in Portfolios." Journal of Finance, 56(3), 19–38.
  • Moskowitz, T.J., Sabbatucci, R., Tamoni, A., & Uhl, B. (2025). "Nonlinear Time Series Momentum." Working paper, §3.2.