Russian-Doll Risk Models

Zura Kakushadze

Quantigicr Solutions LLC, Stamford, CT, USABusiness School & School of Physics, Free University of Tbilisi, Georgia

zura@quantigic.com

(Talk presented at Bloomberg L.P., New York)

April 30, 2015

Zura Kakushadze (Quantigic & FreeUni) Russian-Doll Risk Models April 30, 2015 1 / 7

Russian-Doll Risk Models

Motivation: Portfolio Optimization

Stock expected returns: Ri , i = 1, . . . ,N (∼ 1000− 2500)

E.g.: Sharpe → max

Dollar holdings: Hi = const.×∑

j C−1ij Rj

Sample cov.mat Cij : singular if #(observations) < N + 1

Short-holding/ephemeral strats: long lookbacks not desirable/avail

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Russian-Doll Risk Models

Factor Model

Replace sample cov.mat → factor model (positive-definite):

Γij = ξ2i δij +K∑

A,B=1

ΩiA ΦAB ΩjB

Specific (idiosyncratic) risk ξi : diagonal (var, not cov)

Factor risk cov.mat ΦAB : much smaller, A = 1, . . . ,K N

Style factors ΩiA: size, value, growth, mom, vol, liq, etc. (∼< 10)

Industry factors ΩiA: ind.class (GICS, ICB, BICS, etc.):

BICS : Sector→ Industry→ Sub-Industry

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Russian-Doll Risk Models

ssrn.com/abstract=2538123

Too many industry factors: ∼ few hundred

Calc factor cov.mat: problematic, e.g., short-lookback strats

Simple idea: model factor cov.mat via a factor model

Repeat until: remaining factor cov.mat can/need not be computed

# of remaining factors: dramatically reduced, even to 1 (var only)

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Russian-Doll Risk Models

Math

Γij = ξ2i δij +K∑

A,B=1

ΩiA ΦAB ΩiB

ΦAB = ζ2A δAB +F∑

a,b=1

ΛAa Ψab ΛBb

Γij = ξ2i δij +K∑

A=1

ζ2A ΩiA ΩjA +F∑

a,b=1

Ωia Ψab Ωjb

Ω ≡ Ω Λ

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Russian-Doll Risk Models

Examples

Binary ind.class, e.g., BICS:

Sub-Industries→ Industries→ Sectors (10)→ “Market” (1)

Add few style factors (e.g., 4F Model, ssrn.com/abstract=2511874):

4F + Sub-Ind→ 4F + Ind→ 4F + Sec (14)→ 4F + “Mkt” (5)

Need to calc: spec.risks (var) & small factor cov.mat (“secret sauce”)

Powerful tool: organic custom risk model building

“Secret sauce”: Forthcoming!. . .

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Russian-Doll Risk Models

Thank you!

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