PROVINCIA DE SANTO DOMINGO DE LOS TSÁCHILAS

We now want to apply the concept of PCA to the Risk Parity problem. The PCA is used here as a blind source separation method as uncorrelated risk factors are later needed to determine the solution set. The idea of factor risk parity that is used is thereby similar to the concept described in Roncalli and Weisang (2012) whereby the specic risk factors in this work are not, as mentioned above, further economically motivated. A core advantage of using a risk parity approach is that return expectations are not needed for the model, especially as setting those expectations are often a point of criticism. Also, historical returns are not put in any relationship to risk or used as an approximation for future returns.

We therefore take the price changes x of the assets described in Section 3.5 and calculate the matrixC∈_Rn×n_{as described in the subsection above. We assume the matrix to be fully ranked.}

As mentioned above, we can transform asset price changes into changes along the directions of the principal components and vice versa by multiplying withCorCT _{respectively. We therefore}

e

x=x C, (37)

withxe∈R

k×n_{being the changes in the principal components. In the same way we can transform}

the weightswinto w_e by

e

w=CT w, (38)

with we ∈ R

n _{interpreted as weights in a principal component space. The portfolio return of}

principal componentsPeis then calculated as

e

P :=_exw,_e (39)

with Pe ∈_Rk. Given the asset space portfolio returns P as described in Section 3.3, the daily portfolio returns inP are equal to the daily portfolio returns inPe.

With a covariance matrixΣe∈_Rn×nof the principal componentsx_e, the marginal risk contribution for each component can be calculated by

^ M RCi:= ∂σ(w_e) ∂w_ei = (Σewe)i p e wT e Σw_e . (40)

Through the orthogonal transformation used by the PCA, each component is orthogonal to the preceding components. Therefore, by usingfσ_i2as the variance of thei-th principal component, the covariance matrix is given by:

e

Σ =diag(_eσ₁2, . . . ,_eσ_n2) (41) Hence, the correlation matrix is:

ρx_e=diag[1, 1, 1, . . . , 1]. (42)

The orthogonality in the principal components and equation (40) imply that changing the weight wei only changes thei-th marginal risk contributionM RC^i and leavesM RC^j withj6=i unchanged.

As mentioned above, orthogonality considerably simplies the calculation of the risk contribution:

g RCi= e w2_{i e}σ_i2 f σP ∀i∈ {1, . . . , n}. (43)

We now try to nd weightswe ∈R

n _{which lead to an equal risk contribution of the rst m}

principal components in the factor space of solutions. Those rstmprincipal components should

explain most of the variation. In practice,mwill be 2.2

e

w_i2_eσ_i2=w_e2_j σ_e2_j ∀i, j∈ {1,2, .., m} (44) Considering an equality in squared variables, βi ∈ {−1,1} serves as a long-short indicator

related to these factors:

β1w_e1_eσ1=β2w_e2_eσ2,

β1we1eσ1=β3we3eσ3, · · ·

β1w_e1_eσ1=βmw_em_eσm.

. (45)

βiequates to the principal components direction so that we receive2m−1 possible solutions. For

each permutation ofβ, the vectorw_eis a solution in the factor space. The directions of the other n−mprincipal components are irrelevant. Due to the transformationw=Cw_ewe do not need

to consider the direction of the other principal components.

The concept presented here is similar to the approach of Meucci (2009) who does not use the term of risk contributions but volatility concentration curves. As the varianceσe

2

i of thei-th

principal component equals thei-th eigenvalue of the PCA, those terms though largely overlap.

Additionally, Meucci (2009) focuses on setting up portfolios by analyzing diversication distri- butions represented by the (exponential of the) entropy (see equation (73)). There are major dierences to this model, however: rst, risk parity is independent of any historical returns or return distributions which in general is seen as a key advantage of this approach. Meucci (2009) on the other hand relates the eective number of bets to the expected return. For the case that solely the entropy is maximized, all principal components, excluding those determined through the constraints, are taken into consideration. That is a core dierence to the approach discussed in this paper, which lets the residual risk components oat, thereby generalizing the classical fac- tor risk parity solution set. As the residual components in a PCA are more unstable and often considered to be noisy, letting the weights oat to some extend is in our view a more intuitive approach.

As PCA is not free of criticism, it is sometimes replaced by other methods as for example done by Meucci et al. (2014), who present a model that is based on minimum torsion bets to

determine uncorrelated factors to track the original ones. They argue, among others, that prin- cipal components are statistically unstable, especially those regarding to the lower eigenvalues. They further mention the problem of uniqueness as well as interpretation issues of the principal components. The model described here, by focussing only on the rst principal components, bypasses some of those issues: the focus is placed on the more stable, important principal com- ponents and the residual ones are allowed to oat to some extend. As the number of equal risk contributions is set to only a few components, the amount of solutions due to dierent principal component directions does not increase signicantly.