We here detail how to build through variance optimization a range of portfolio over a multi risk framework factor. This methodology is much used by practitioner as the factors covariance is non singular even with few observations and more stable.

1. 1
Multi risk factor model
Minimum variance optimization
under constraints
Market neutrality
130/30 long short portfolio
DUPREY Stefan

2. 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

3. 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 = ξ2
ijδij (2)
χi, fA = 0 (3)
fA, fB = ΦAB (4)
Ri, Rj = Θij (5)

4. Covariance matrix 4
2. Covariance matrix
The Ri covariance matrix can be written as :
Θ = Ξ + ΩΦΩ⊺
(6)
where Ξij = ξ2
ijδij 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
ξ2
ij + f⊺
Φf (8)
where f = Ωω is the factor values for our portfolio with weights ω

5. 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

6. Minimum variance optimization rebalancement at each date 6
4. Minimum variance optimization rebalancement at each date
argmin



ω ∈ RN
, ωi > 0, ωi < 1
N
i=1 ωi = 1
n
i=1 ωiβi = n
i=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
N
i=1 ωi = 1
n
i=1 ωiβi = n
i=1 ωi
Cov(Ri,Rmarket)
Var(Ri) = 1
Ωω = f
{xQx⊺
− q × µ⊺
x} (12)

7. Minimum variance optimization rebalancement at each date 7
where :
Q =














Φ 0 · · · 0
0
ξ2
11
0...
...
0 ξ2
nn
...
0














, µ =


0
µ

 (13)

8. 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 ωi
Cov(Ri,Rmarket)
Var(Ri) = 1
Ωω = f
−v < ω < v
v > 0
N
i=1 vi = 1.6
{xQx⊺
− q × µ⊺
x} (14)

9. Long/short 130/30 portfolio 9
where :
Q =














Φ 0 · · · 0
0
ξ2
11
0...
...
0 ξ2
nn
...
0














, µ =


0
µ

 (15)

10. 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)

11. Long/Short 100/100 Market Neutral portfolio 11
where :
Q =




Φ 0 · · · 0
0 Qidio −Qidio
0 −Qidio Qidio



 , Qidio =








ξ2
11
0...
...
0 ξ2
nn








(17)

12. R code : computing factors loadings 12
7. R code : computing factors loadings

13. R code : estimating factors covariance matrix and idiosyncratic risk 13
8. R code : estimating factors covariance matrix and idiosyncratic risk

14. R code : quadratic programming, constraints and multi risk factor models14
9. R code : quadratic programming, constraints and multi risk factor models

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