Resultados de la investigación.

To recap the conditions in the factor risk parity model, using the notation from the second study above, and to introduce unfunded positions, the following restrictions which are formulated here as a set of linear equalities and inequalities have to hold:

1. Risk parity condition

The risk contributions of the rst two principal components have to be equal. This condi- tion can be written as

(_fσ1 −fσ2 0 · · · 0)   f w1 ... e wn  =   0 ... 0  . (74)

2. Positive weights in the asset space

Prohibiting short selling means that the conditionw=Cw_e≥0 must be fullled. 3. Funded asset weights sum up to 1

ExpressionP_w

ini=1 = 1only applies to funded assets. Unfunded assets, such as volatility

constraint dened in the part above has to be modied. We redeneci,jbeing the elements

of the transformation matrixC from the PCA for funded positions and

ˆ ci,j=     

ci,j, if assetiis funded

0, if assetiis unfunded

, (75)

The sum for funded positions are constituted byCi = P n

j=1(ˆci,j) and exclude unfunded

positions from the sum of all elements in thei-th column in the matrix C. One can then

write the constraints as:

(C1 C2 · · · Cn)   f w1 ... e wn  = 1. (76)

In the second study above, the marginal risk contribution has been calculated with the stan- dard deviation as the risk measure. The standard deviation, however, may not be perfectly suit- able as historical return distributions often exhibit fat tails or are not symmetrically distributed. This is even more distinct if asset classes such as volatility are included in the multi-asset port- folio. The expected shortfall as a non-parametric tail-risk measure therefore appears suitable to capture special distributional properties.

Using the quantile function V aRα(z) =Fz−1(α)for the value at risk to the quantile α, the

expected shortfall for a continuously distributed random variablezto the same probability level αis dened as ESα(z) := 1 1−α Z 1 α V aRu(z)du. (77)

Given the portfolioP :=x w∈_Rk_{consisting of empirical time series, the empirical expected}

shortfall of portfolio P can be written in the form of

ESα(P) =µ[P|P ≤V aRα(P)], (78)

with µ[P|x] as the conditional expectation of P under x. Rewriting the portfolio using P = P

ixiwi, the empirical portfolio expected shortfall can be decomposed into

ESα(P) =

X

i

wiµ[xi|P ≤V aRα(P)]. (79)

As it can be seen below, this decomposition is important for calculating the marginal expected shortfall (M ESi). Following Tasche (2002), the marginal expected shortfall is equal to the rst

order partial derivative of expression (79). The component expected shortfall (CESi), which can

be interpreted as the risk contribution of the empirical time series xi to the empirical portfolio

expected shortfallESα(P), is then the marginal expected shortfall multiplied by the weights of

the time seriesxi in the portfolio:

M ESi= ∂ESα ∂wi =µ[xi|P≤V aRα(P)], (80) CESi=wi ∂ESα ∂wi =wiµ[xi|P ≤V aRα(P)]. (81)

The marginal expected shortfall is equal to the the average value of asset i if the portfolio loss

is bigger than the historical value at risk to the quantileα.

For historical scenarios, this value can be calculated by rst taking the change in the portfolio value for each scenario, sorting them to take the cuto point for determining the value at risk and nally taking the average value of price change for asset ifor all those scenarios where the

portfolio loss is larger than V aRα(P).

Given the portfolioPe:=_exw_eand the quantile functionV aRα(Pe) =F−_e1

P (α)for the principal

component time series, one can formulate an equation similar to the expression in the standard deviation case for the expected shortfall as the risk measure:

e

w1M ES^1(we) =we2M ES^2(we). (82) So far, the solutions forwe1andwe2to this equation is not trivial asM ES^1andM ES^2depend onwe. In the case of the standard deviation, thei-th marginal risk contribution only depends on weight fwi which simplies the equations signicantly. The orthogonal attribute of the principal components is not sucient to guarantee thatM ES^idoes not also depend on a dierent weight

j. Rewriting the equation (82) may clarify those dependencies:

e

w1µ[ex1|(exwe)≤V aRα(xewe)] =we2µ[xe2|(exwe)≤V aRα(exwe)]. (83) The solution to the equation depends heavily on the form of the assets' distributions and thereby on the principal components' distributions. Before proceeding with the FRP model, some general statements are rst made. Later, closed form solutions for the marginal expected shortfall under dierent distribution assumptions will be presented.

Nonetheless, a numerical process for computing the FRP model weightsweis needed and will be described in the last part of this section.

General statements

As mentioned above in the case of the standard deviation as the risk measure, the ratio of the weights wf1

f

w2 is equal for all risk parity solutions in the principal component space. This is a

direct consequence of the attribute that only the i-th weight w_ei changes thei-th marginal risk

contributionM RC^i.

When using the expected shortfall we allow returns to also be non-normally distributed. As nancial data are analyzed, we expect distributions e.g. to be fat tailed and skewed. Regardless of the asset or principal component return distributions, the component expected shortfall is homogeneous, i.e.

CESi(c·w) =c·CESi(w), (84)

with c >0. Particularly, it follows that if a portfolio has equal component expected shortfalls, so does a portfolio with weightsc·wforc >0. The proof is straightforward as for givenwand c >0, due to positive homogeneity of the value at risk,

CESi(c·w) =c·wiE[xi|c·x≤V aRα(c·x)]

=c·wiE[xi|c·x≤c·V aRα(x)]

=c·CESi(w).

(85)

So far, we know that scaling the weights of a factor risk parity portfolio with a factorc >0 means that the new scaled portfolio stays a factor risk parity portfolio when the expected short- fall as the risk measure is used. This, in general, is only true if the scaling factor is positive. It will be shown later that, in the case of the normal or elliptical distribution assumptions when the distributions are symmetrical with mean zero, the solution is still a solution to the problem even if all the optimal portfolio weights are multiplied by minus one, meaning that scaling by a factor of c <0 is a valid operation. Therefore, the characteristic of scaling the weights of a factor risk parity portfolio with the expected shortfall for the normal and elliptical distributions is the same as in the general case with the standard deviation as the risk measure.

With skewed distributions the remarks above are incorrect. The reason for this is that by using skewed distributions and the expected shortfall as a risk measure, a left skewed distribution

becomes a right skewed distribution and vice versa when the sign of the weight changes.

Closed form solution - normal case

Let X denote asset return, whereX is a normally distributed random variable with meanµand

varianceσ2_{,X ∼ N(µ, σ}2_{). Following Boudt et al. (2008), the component expected shortfall in}

this case can be written as

CESi=wi (Σw)i √ wT_Σw 1 αφ[Φ −1_{[α]], (86)}

withαbeing the corresponding quantile,φ(),Φ() andΦ−1()being the standard Gaussian den- sity, distribution and quantile functions. As the time series are normally distributed, it is not surprising that the rst part of equation (86) is identical to the equation in the standard de- viation case. As the term 1

αφ[Φ

−1_{[α]] is independent of i, the term crosses out when we set}

CESi =CESj. This leads to the same solution set when we compare the standard deviation

versus the expected shortfall results with normal distributed returns.

Closed form solution - elliptical case

Landsman and Valdez (2003) extend the calculation of the component expected shortfall to the larger class of elliptical distributions which includes the case of the normal distribution above. Multivariate elliptical distributions are those where the characteristic function can be expressed as ϕX(t) =exp itTµψ ₁ 2t T_Σt , (87)

for some column vectorµ∈_Rn_{, complex numberi, a positive-denite matrixΣ∈}

Rn×nand some functionψ(t) : [0,∞)→Rsuch thatψ(Pni=1(t

2

i))is ann-dimensional characteristic function. In

this case the solution can be given as

CESi=wiµi+λPwiσiσPρXi,P (88)

for someλP as given in Landsman and Valdez (2003) (be aware thatCESi=wi·M ESi). Note

that the paper discusses the caseP =X1+· · ·+Xn. However, without loss of generality,Xican

be replaced bywiXi as linear combinations of elliptical distributions are elliptical distributions

CESi=wiµi+λPwiσiσP Cov(wiXi, P) wiσiσP =λPCov wiXi, n X k=1 wkXk ! =λPwi2σ 2 i. (89)

ForCESi to be equal to CESj,

λPw2iσ 2

i =λPw2jσ 2

j. (90)

Therefore, the solution is the same as in the case using the standard deviation as the risk measure.

For the sake of completeness, the (marginal) modied expected shortfall (see appendix 4.A) is additionally described. The idea is to approximate the tail by using a Cornish-Fisher expansion. This approach will be used for validating the numerical results determined in Section 4.6.1.

Numerical solutions

As discussed above, a numerical process for computing FRP portfolios in the principal component space of solutions under the expected shortfall as the risk measure is required for non-normally distributed returns. The computations now consist of 2 steps: the calculation of the component expected shortfall followed by nding the optimal weights so that the component expected short- fall of the rst two components are equal.

Given the weightsw, the calculation of the component expected shortfall is straightforward

as given directly by equations (81) and (82). We rst determine the portfolio in the factor space, determine the tail scenarios for the portfolio, take a componentiand average the scenario

changes for exactly those tail scenarios.

Being able to determine theCES and M ES this way, the optimal portfolio can be found

numerically by minimizing the squared dierence between theCESof the rst 2 principal compo-

nents. The general optimization problem to achieve equality in the rstmprincipal components

can be written as:6

f∗=argmin   m X i=1 X j<i wi ∂ESα ∂wi −wj ∂ESα ∂wj 2  , (91)

or equivalently f∗=argmin   m X i=1 X j<i (wiM ESi−wjM ESj) 2  . (92)

The constraints we dene for the optimization problem above are directly taken from the standard deviation case. These include:

• Asset weights sum up to 1

• Non-negative asset weights - short sale prohibited

• Minimum explanation level of the rst two risk contributions of at least 66%

As mentioned above, an additional objective function, similar to the case where the standard deviation is used, is needed to narrow the choices down to one portfolio. The allocations we want to analyze are the

• Minimum Variance allocation and • Maximum Diversication allocation.

Both allocations and the constraints have already been explained in the second study above.