Reference 0016

Factor Exposure Recipe Card

Caps bound names; walls bound themes. Exposure is e = Fᵀw — a weighted sum no per-name cap can see. One wall per factor; one dual per wall; one tax per name.

The Four Wall Forms

tilt cap:      βᵀw ≤ L                     one tilted wall
neutrality:    βᵀw  = 0                     equality: wall + floor welded
band:          |βᵀw| ≤ τ                    -τ ≤ βᵀw ≤ τ
                                             (Ref 0015 pattern 1: NO auxiliaries)
many factors:  lower ≤ Fᵀw ≤ upper         F is n × k; one wall per column
benchmark-relative: constrain Fᵀ(w - w_bench), not Fᵀw

The Dual Is a Factor Tax

wall binds with rent λ  ⇒  fund name i while  α_i > λβ_i
                             rank by after-toll alpha:  α_i - λβ_i

one dual, n taxes: each name pays λ × (its loading)
zero-beta alpha rides free; high-beta alpha must clear a higher bar

two hats (Ref 0013): λ = tax rate that replaces the wall
                     λ = alpha gained per unit of exposure headroom
slack wall ⇒ λ = 0   (no rent on a wall nobody touches)
at the dual: the marginal names TIE after tax (solver indifference)

CVXPY Snippets

e = F.T @ w                          # exposures, one per factor
constraints = [e <= upper, e >= lower]   # bands
constraints += [beta @ w == 0]           # beta-neutral equality

prob.solve()
lam = constraints[0].dual_value      # rent per unit exposure, per factor
taxed_alpha = alpha - F @ lam        # what the optimizer really ranked

Whiteboard Traps

tightening weight caps to control a theme:  blind (caps don't see loadings)
                                            and blunt (rations low-beta alpha too)
|βᵀw| ≤ τ with auxiliaries:               legal but wasteful — two walls suffice
reading raw exposure on a benchmark fund:   the bet is Fᵀ(w - w_bench)
equality walls:                             dual is sign-free (push from either side)

Use this page with Lesson 0016 (the hidden tilt, the toll-hunting lab), Reference 0014 (toll replaces wall), Reference 0013 (two hats), and Reference 0015 (the band rewrite). Sources: MOSEK Portfolio Cookbook (factor models); Boyd & Vandenberghe §4.2, §5.6; Boyd et al. multi-period trading paper (holding constraints).