Page 296 - Contributed Paper Session (CPS) - Volume 2

CPS1863 La Gubu et al.
2.4 Portfolio selection using robust estimation FMCD

In this study, the weight of the selected stocks that formed the optimum
portfolio is determined using the robust FMCD estimation method. To see the
advantages of this method, the results will be compared with the classic MV
method. The following will be briefly presented the procedure for determining
portfolio weights using the robust FMCD estimation method.
The minimum covariance determinant (MCD) estimation aims to find robust
estimates based on the observations of total observations (n), where the
covariance matrix has the smallest determinant. The MCD estimation is a pair
of ∈ ℝ and Σ is a symmetric positive definite matrix with a dimension of
̂

from a sample of h observation, where (++1) ≤ ℎ ≤ with
2
1 ℎ
= ∑
ℎ =1
(5)
The estimation of the covariance matrix can be obtained by solving the
following equation:
1
̂
Σ = ∑ ℎ ( − )( − )′
ℎ =1
(6)
MCD calculations can be very complicated if the data dimensions are
getting bigger, this is because this method must examine all possible subsets
of h from a number of n data. Therefore, Roesseeuw and Van Driessen (1999)
found a faster calculation algorithm for calculating MCD called Fast MCD
(FMCD). The FMCD method is based on the C-Step theorem described below.

Theorem 1 (Rousseeuw and Driessen,1999)
If is the set of size h taken from data of size n, the sample statistics are:
1
1
1
= ∑

ℎ ∈ 1
(7)
1
1
̂
1
1
Σ = ∑ ( − )( − ) ′
ℎ ∈ 1
(8)
If |Σ | > 0 than distance = ( ; , Σ ). Next, specify is subset consist of
1
1
1
̂
̂
2

the observation with the smallest distance , namely { ()| ∈ } =

1
2
{( ) , … , ( ) } where ( ) ≤ ( ) ≤ ⋯ ≤ ( ) is a sequential distance.
1 1
1 2
1 ℎ
1
1 1
Based on , using equations (7) and (8), we obtained
2
2
̂
|Σ | ≤ |Σ |
̂
1
(9)
̂
1
̂
1
2
2
Equation (9) will be the same if = dan Σ = Σ
̂
̂
̂
C-Step theorem is done repeatedly until | Σ | > 0 or | Σ | = | Σ |.

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