1

Multi risk factor modelMinimum variance optimization

under constraintsMarket neutrality

130/30 long short portfolioDUPREY Stefan

Plan 2

Plan

1. Factor model and stochastic processes

2. Covariance matrix

3. Regressing returns over factor loading/exposure

4. Minimum variance optimization rebalancement at each date

5. Long/short 130/30 portfolio

6. Long/Short 100/100 Market Neutral portfolio

7. R code : computing factors loadings

8. R code : estimating factors covariance matrix and idiosyncratic risk

9. R code : quadratic programming, constraints and multi risk factor

models

Factor model and stochastic processes 3

1. Factor model and stochastic processes

We have N ∈ N stocks. Their returns Ri, ∀i ∈ 1, · · · , N are modeled by

N random processes χi corresponding to idiosyncratic risk together with K

random processes fA :

Ri = χi +K∑

A=1

ΩiAfA (1)

〈χi, χj〉 = ξ2ijδij (2)

〈χi, fA〉 = 0 (3)

〈fA, fB〉 = ΦAB (4)

〈Ri, Rj〉 = Θij (5)

Covariance matrix 4

2. Covariance matrix

The Ri covariance matrix can be written as :

Θ = Ξ+ ΩΦΩ⊺ (6)

where Ξij = ξ2ijδij is the diagonal idiosyncratic part of the variance and Φ

the K ×K factor covariance matrix, Ω is the K ×K factor loading matrix.

Note that K << N gives more out of samples stability. Through Cholesky :

Θ = Ξ+ ΩΩ⊺ (7)

where Ω = ΩΦ and ΦΦ⊺ = Φ

Note that the standard variance for a portfolio with weights ω decomposes

as :

ω⊺Ωω =N∑

i=1

ξ2ij + f⊺Φf (8)

where f = Ωω is the factor values for our portfolio with weights ω

Regressing returns over factor loading/exposure 5

3. Regressing returns over factor loading/exposure

At each observation time we have to regress our stock returns to our stock

factors :

R(t) =K∑

k=1

βk(t)× Fk(t) + ǫ(t) (9)

R(t) =K∑

k=1

fk(t)× Ωk(t) + ǫ(t) (10)

Ωi, Fi is the factor exposure or loading for the stock i and βi, fi is the

factor return or factor itself.

The variance of ǫ(t) is the idiosyncratic risk for stock i

The covariance of (fk(t))k∈1,··· ,K is the factor covariance matrix.

Here a proprietary strategy is to be defined to compute the factor matrix : a

rolling standard deviation or garch volatility estimation for idiosyncratic

variance and a rolling Ledoit-Wolf covariance matrix estimation

Minimum variance optimization rebalancement at each date 6

4. Minimum variance optimization rebalancement at each date

argmin

ω ∈ RN , ωi > 0, ωi < 1

∑Ni=1 ωi = 1

∑ni=1 ωiβi =

∑ni=1 ωi

Cov(Ri,Rmarket)Var(Ri)

= 1

ωΣω⊺ − q × µ⊺ω , becomes

(11)

argmin

x = (f, ω) ∈ R(N+K), ωi > 0, ωi < 1

∑Ni=1 ωi = 1

∑ni=1 ωiβi =

∑ni=1 ωi

Cov(Ri,Rmarket)Var(Ri)

= 1

Ωω = f

xQx⊺ − q × µ⊺x (12)

Minimum variance optimization rebalancement at each date 7

where :

Q =

Φ 0 · · · 0

0

ξ211 0. . .

. . .

0 ξ2nn...

0

, µ =

0

µ

(13)

Long/short 130/30 portfolio 8

5. Long/short 130/30 portfolio

argmin

x = (f, ω) ∈ R(N+K)

v ∈ R(N)

−0.1 < ωi < 0.1∑N

i=1 ωi = 1∑n

i=1 ωiβi =∑n

i=1 ωiCov(Ri,Rmarket)

Var(Ri)= 1

Ωω = f

−v < ω < v

v > 0∑N

i=1 vi = 1.6

xQx⊺ − q × µ⊺x (14)

Long/short 130/30 portfolio 9

where :

Q =

Φ 0 · · · 0

0

ξ211 0. . .

. . .

0 ξ2nn...

0

, µ =

0

µ

(15)

Long/Short 100/100 Market Neutral portfolio 10

6. Long/Short 100/100 Market Neutral portfolio

argmin

x = (f, lω, sω) ∈ R(N+3∗K)

b ∈ R(N)

0 < lωi < 0.1, 0 < sωi < 0.1∑N

i=1 lωi =∑N

i=1 sωi∑N

i=1 lωi +∑N

i=1 sωi = 2∑n

i=1 lωiβi =∑n

i=1 sωiβi

Ωlω − Ωsω = f

lω < b

sω < 1− b

xQx⊺ − q × µ⊺x (16)

Long/Short 100/100 Market Neutral portfolio 11

where :

Q =

Φ 0 · · · 0

0 Qidio −Qidio

0 −Qidio Qidio

, Qidio =

ξ211 0. . .

. . .

0 ξ2nn

(17)

R code : computing factors loadings 12

7. R code : computing factors loadings

R code : estimating factors covariance matrix and idiosyncratic risk 13

8. R code : estimating factors covariance matrix and idiosyncratic risk

R code : quadratic programming, constraints and multi risk factor models14

9. R code : quadratic programming, constraints and multi risk factor models

R code : quadratic programming, constraints and multi risk factor models15