courant nyu 04142015

Factor Models for Stocks & Alphas

Zura Kakushadze

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

[email protected]

(Talk presented at Courant Institute of Mathematical Sciences, New York University)

April 14, 2015

Zura Kakushadze (Quantigic & FreeUni) Factor Models for Stocks & Alphas April 14, 2015 1 / 25

Outline

1 Factor Models for StocksMotivation: Portfolio OptimizationStyle & Industry Factors

Short v. Long HorizonsAlpha NeutralizationIndustry Granularity & Custom Universe

4-Factor Model for Overnight ReturnsRussian-Doll Risk Models“Secret Sauce”

2 Factor Models for AlphasMotivation: Alpha CombosStyle FactorsCluster FactorsUnderlying Tradables as Risk Factors

3 References


Factor Models For Stocks

Motivation: Portfolio Optimization

N stocks: i = 1, . . . ,N

Expected returns: Ri

Optimization: e.g., maximize Sharpe ratio

Vanilla: no constraints, costs, etc.

Dollar holdings: Hi = const.×∑

j C−1ij Rj

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

Off-diag Cij : not out-of-sample stable unless M N (diag rel. stable)

Liquid portfolios: N ∼ 1000− 2500

5 years: ∼ 1260 daily observations

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

Need: replace sample cov.mat



Factor Model

Factor risk & specific (idiosyncratic) risk:

Ri = χi +∑

A

ΩiAfA

Risk factors: fA, A = 1, . . . ,K N

Specific risk cov.mat: Cov(χi , χj ) ≡ Ξij = ξ2i δij

Factor risk cov.mat: Cov(fA, fB) ≡ ΦAB

Uncorrelated: Cov(χi , fA) = 0

Model cov.mat Γij ≡ Cov(Ri ,Rj ):

Γ = Ξ + ΩΦΩT

Φ positive-definite: Γ positive-definite, invertible



Style & Industry Risk Factors

Style factors: stocks’ estimated/measured properties

E.g.: size, value, growth, momentum, volatility, liquidity, etc.

Industry factors: similarity criterion, stocks’ membership in industries

Industry classification: GICS, ICB, BICS, etc.

Hierarchy, e.g., BICS (others use diff names):

Sector→ Industry→ Sub-Industry



Style Factors: Short v. Long Horizons [ZK & Liew, 2014]

Consider: short-holding strategies

Relevant returns: short horizons, e.g., overnight, intraday, etc.

Returns: not highly correlated w/ longer-horizon quantities

E.g.: value (book value) and growth (earnings) update quarterly

No predictive power: add noise, cause overtrading, cost money



Inadvertent Alpha Neutralization

Suppose: by design, alpha is long/short some risk factor(s)

E.g.: alpha skewed toward small cap value

Standardized risk model: optimization/regression neutralizes alpha

Need: remove undesirable risk factors

Need: recompute factor covariance matrix, specific risk

Standardized RM: cannot drop factors from factor cov.mat calc

Need: custom risk model



Industry Granularity & Custom Universe

Standardized RM: fixed # of “cookie-cutter” industries

Reduced ind granularity: adversely affects hedging ind risk

Empty ind: e.g., trading univ Utr w/ no telecom stocks

Standardized RM: cannot omit ∅ ind from factor cov.mat calc

#(true ind): typically larger, depends on Utr (GICS, ICB, BICS,. . . )

Standardized RM: Ucoverage → Utr spoils style factor normalization

Need: custom risk model



4-Factor Model for Overnight Returns [ZK, 2014b]

Factors: size, momentum, volatility, liquidity

Size: log(mkt.cap) = log(prc) + log(shares.outstanding)

shares.outstanding: low correlation

“Size”: log(prc)

“prc”: adjusted or unadjusted prev.close

Momentum: log(prev.close/prev.open)

Volatility: 12 log

[Mean

((high− low)2/close2

)]Mean: 21 days, smooth out noise

Liquidity: log(ADDV) ≈ log(prc) + log(Mean(volume))

“Liquidity”: log(Mean(volume))

4-factor model: prc, mom, hlv, vol



4-Factor Model: Tests

Overnight returns: Ri (ts) = log(Padj .openi (ts)/Padj .close

i (ts+1))

ts : trading days

Loadings βiA(ts): int, prc, mom, hlv, vol

X-sectional regressions: Ri (ts) ∼∑

A βiA(ts)fA(ts) + εi (ts)

Universe: top-2000-by-ADDV, rebalanced every 21 days

Annualized serial t-statistic [Fama & McBeth, 1973]:

int prc mom hlv vol2.28 -4.66 -3.78 2.95 2.91

“Sanity” check: intraday open-to-close mean-reversion alphas

αi (ts) ∼ −εi (ts) (normalized)


Table: Intraday Alphas for Testing 4-Factor Model

Model ROC SR CPS

int only 26.94% 7.79 0.82int+prc+mom+hlv+vol 28.79% 11.22 0.87BICS sectors only 33.27% 11.55 1.02BICS sectors+prc+mom+hlv+vol 34.30% 14.94 1.04

Univ: top-2000-by-ADDV, rebalanced every 21 days. No trading costs. “Delay-0” executions:

established at the open, liquidated at the same day’s close. More details: [ZK, 2014b].


Figure: Intraday Alphas for Testing 4-Factor Model

0 200 400 600 800 1000 1200

0.0e

+00

1.0e

+07

2.0e

+07

3.0e

+07

Trading Days

P&

L

Bottom-to-top-performing: i) intercept only, ii) 4-factors plus intercept, iii) 10 BICS sectors

only, and iv) 4-factors plus 10 BICS sectors. Investment level: $10M × $10M.



Russian-Doll Risk Models [ZK, 2014c]

Too many industry factors: a few hundred or more

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

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 or 0



Russian-Doll Risk Models: Math

Γij = ξ2i δij +K∑

A,B=1

ΩiAΦABΩiB

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



Russian-Doll Risk Models: Examples

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

Sub-Industries→ Industries→ Sectors (10)→ Market (1)→ ∅

Binary ind.class + few style factors (e.g., BICS + 4-Factor Model):

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

Need specific risks and remaining factor cov.mat, e.g., binary BICS:

Γij = ξ2i δij + ζ2G(i)δG(i),G(j) + η2G(i)

δG(i),G(j)

+ σ2G(i)

δG(i),G(j)

+ λ2

Specific risks (rel. stable): Sub-Ind ζG(i); Ind ηG(i)

; Sec σG(i)

; Mkt λ

G (i) : ticker 7→ Sub-Ind; G (i) : ticker 7→ Ind; G (i) : ticker 7→ Sec



Russian-Doll Risk Models: Specific Risks, Remaining Factor Cov.Mat

Must reproduce in-sample total variance:

Γii = ξ2i + ζ2G(i) + η2G(i)

+ σ2G(i)

+ λ2 = Cii

Must have: nonnegative ξ2i , ζ2A, η2a , σ2α, λ2

Then: all ζ2A, η2a , σ

2α, λ

2 ≤ min(Cii )

Cii : skewed (“log-normal”) distribution

Result: small effect on off-diag Cij for larger Cii , i.e., small correlations

Observation: if Cii were more uniform, correlations would not be small

Correlation matrix: Cij =√Cii

√Cjj Ψij , Ψii ≡ 1

Simple “hack”: model corr.mat Ψij via a Russian-doll model Γij

Toy Ansatz (only for binary), e.g.: ξ2i = ζ2A = η2a = σ2α = λ2 = 1/5



“Secret Sauce”

Specific risks and factor cov.mat: how to calc?

Formal ∼ w/ X-sec regression: Ri (ts) = εi (ts) +∑

A βiA(ts)fA(ts)

“Lore”: βiA(ts) ≡ ΩiA; ΦAB ≡ Cov(fA, fB); ξ2i ≡ Var(εi )

Cov(ε, εT ) = [1− Q]C [1− Q] 6= diag

Γii = ([1− Q]C [1− Q] + QCQ)ii 6= Cii

Tr(Γ) = Tr(C )

Q2 = Q ≡ Ω(

ΩT Ω)−1

ΩT

Define ξ2i ≡ Cii −∑

A,B ΩiA ΦAB ΩiB? No: ξ2i 6> 0

Prop algos: commercial RM, Quantigic Risk Model, some prop shops


Factor Models For Alphas

Motivation: Alpha Combos

Alpha: instr. to take predefined pos. in tradables at specified times

Olden days: built “by hand”, not many alphas

Nowadays: datamined, automated, thousands of αi

Challenge: too many (ephemeral) alphas, too few observations

Challenge: out-of-sample stability

Challenge: sample cov.mat Cij ≡ Cov(αi , αj ) is badly singular

Why care? Trade weighted alpha combos∑

i wiαi , save on costs

Automatic internal crossing: turnover reduction, increased capacity

Alpha weights wi : optimization, e.g., Sharpe → max, etc.

Need: invertible alpha cov.mat

Idea: Factor Models for Alpha Streams [ZK, 2014a]



How Is It Usually Done in Practice?

Sample cov.mat Cij : singular

Deformation: Γ = C + εZ , e.g., Zij = Ciiδij

Prin.comp: C = UΛUT , ΛAB = diag(λA)

K -factor model (A = 1, . . . ,K = #(observations)− 1):

Γ = εZ + UΛUT

ε→ 0 : Γ−1 = ε−1Θ +O(1)

Θ ≡ Z−1 − Z−1U(UTZ−1U)−1UTZ−1

Weights: wi ∼∑

i Θijαj = εi/Cii (factor-neutral:∑

i wiUiA = 0)

εi : residuals of 1/Cii weighted regression of αi over prin.comp UiA

Finite deformations: not expected to be out-of-sample stable



Beyond Regression: Motivation

Increase #(factors) #(observations): improve α portfolio perf

Full risk model: specific risk & factor cov.mat

Optimization w/ linear costs: P&L =∑

i wiαi − Li |wi |Module: too many combos (∼ 3N) for general (invertible) cov.mat

Factor model: solvable [ZK, 2015a], [ZK, 2015b] (≈ for impact)

Anything correlations based: limited by #(observations)

Factor model: style + cluster (∼ ind) factors?



Style Factors

Style factors: few

E.g.: volatility, momentum, turnover, capacity

Turnover: roughly ∼ ADDV/Mkt.Cap ratio

Capacity: roughly ∼ size (Mkt.Cap); not easy (nonlinear)

Effect: not too substantial compared w/ many more prin.comp

Cluster Factors

Clusters: based on alpha classification, how alphas are constructed

Number: a priori can be many

Issue: often, only positions are known, not how alphas are constructed

Problem: w/o cluster factors, hard to compete w/ prin.comp

Need: novel approach



Underlying Tradables as Risk Factors

Alphas: αi , i = 1, . . . ,N

Stocks: RA, A = 1, . . . ,F (e.g., F ∼ 2500)

Data: dollar positions HiA(ts) at times ts

Normalization:∑

A |HiA(ts)| = 1

Idea: construct factor loadings ΩiA from HiA(ts)

Factor cov.mat ΦAB : use avail (or build) factor model for stocks

Zeroth approx: ΦAB ≈ vAδAB

vA = Var(RA): relatively stable, do not require long lookbacks

Evident choice: ΩiA =∑

s HiA(ts) (e.g., monthly)

Unstable: no better than Cor(αi ,RA) (HiA(ts) flips sign frequently)

Need: unsigned quantity



Underlying Tradables as Risk Factors

Another choice: ΩiA =∑

s |HiA(ts)|, no longer unstable

Issue: if mostly αi trade stocks at uniform capacity, Feff F

Solution: ΩiA =√

Var(HiA(ts)), ΩiA =√

Var(|HiA(ts)|), etc.

Normalization(ΩiA): fixed when calc specific risk (“secret sauce”)

Fewer factors: usually HiA(ts) already neutral w.r.t. style+ind factors

Underlyings as Risk Factors: Generalization

Processes: Xi (ts), i = 1, . . . ,N

Processes: YA(ts), A = 1, . . . ,K < N

Previsible ZiA(ts): Xi (ts) =∑

A ZiA(ts)YA(ts)

YA(ts) as risk factors: so long as ZiA(ts) data is available

General: not just for trading or finance


Factor Models for Alphas & Stocks

Good place to stop.Thank you!


References

Fama, E.F. & MacBeth, J.D. (1973) Risk, Return and Equilibrium: EmpiricalTests. Journal of Political Economy 81(3): 607-636.

ZK (2014a) Factor Models for Alpha Streams. The Journal of InvestmentStrategies 4(1): 83-109; http://ssrn.com/abstract=2449927.

ZK (2014b) 4-Factor Model for Overnight Returns;http://ssrn.com/abstract=2511874.

ZK (2014c) Russian-Doll Risk Models; http://ssrn.com/abstract=2538123.

ZK (2015a) Combining Alpha Streams with Costs. The Journal of Risk 17(3):57-78; http://ssrn.com/abstract=2438687.

ZK (2015b) Notes on Alpha Stream Optimization. The Journal of InvestmentStrategies 4(3): June 2015; http://ssrn.com/abstract=2446328.

ZK & Jim Liew (2014) Custom v. Standardized Risk Models;http://ssrn.com/abstract=2493379.


courant nyu 04142015

Documents