Multi risk factor model
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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