### 8b.2 Euler decomposition

We so far have considered the exposures in the ex-ante performance decomposition (8b.6) as a fixed vector, computed once for all in the Ex-ante attribution Step 8a.

Here, we let the exposures vary. More precisely, we define the random variable

 Yb≡bZβ (8b.40)

where the factors are considered fixed and static, and the exposure vary in such a way that the variable (8b.40) becomes the actual ex-ante performance (8b.6) when . The subscript “” in (8b.40) stresses out that the factors depend on the exposures , because the factor (8a.61) depends on the residual of the attribution model (8a.12).

Then, for a general satisfaction/risk measure (7.2) we define the function

 b↦Satis{bZβ}, (8b.41)

which associates to an vector of exposures the ensuing satisfaction , is not necessarily the exposures from the attribution model (8a.12). We will omit the subscript from now on for brevity.

If the satisfaction/risk measure (7.2) is positive homogeneous of degree (7.27), then the function (8b.41) is positive homogenous [W]. If the function (8b.41) is further differentiable (15.28), then the Euler decomposition of the ex-ante performance (8b.6) into additive marginal contributions holds

 Satis{Y}=∑¯kk=0βk×1m∂Satis{βZ}∂βk, (8b.42)

where is a shorthand for . The contribution associated with the -th risk factor (8b.8) is called the Euler marginal contribution

 [Satis{Y}]Eulerk≡βk×1m∂Satis{βZ}∂βk. (8b.43)

We highlight in the equation above that the contribution from the -th factor can be further expressed as a per-unit contribution times the “amount” of that factor (exposure) .

The marginal contributions (8b.43) are the Aumann-Shapley marginal contributions (8b.33), see , and .

In the following we show the decomposition (8b.42)-(8b.43) in practice for the most important satisfaction measures encountered in the Evaluation Step 7 which are positive homogenous (7.27).

#### 8b.2.1 Standard deviation and variance

The standard deviation risk measure (7.60) is positive homogeneous of degree , see (7.64). In this case, we can apply the Euler decomposition (8b.42) where the Euler marginal contributions (8b.43) read E.8b.1

 [Sd{Y}]Eulerk=βk×[Cv{Z}×β']k√β×Cv{Z}×β', (8b.44)

and where the term on the right of is the partial derivative .

The variance risk measure (7.55) is positive homogeneous of degree , see Table 7.4. In this case, we can apply the Euler decomposition (8b.42). The Euler marginal contributions (8b.43) read E.8b.2

 [V{Y}]Eulerk=βk×[Cv{Z}×β']k, (8b.45)

where the term on the right of is half the partial derivative .

##### Scenario probability

If the joint distribution of the factors is a scenario-probability distribution (18.312), as in (8a.78)

 Z∼{z(j),p(j)}¯ȷj=1, (8b.46)

we can compute the Euler contributions (8b.43) for the standard deviation (8b.44) and the variance (8b.45) by replacing the covariance of the factors with the corresponding scenario-probability covariance (18.368).

Example 8b.10. Let us continue from Example 8a.13, where we derived ex-ante performance decomposition (8b.6) of the return of the stocks portfolio (Example 8b.2). The factors have the scenario-probability distribution (8a.81). Consider as risk measure the standard deviation (7.60). Then, the Euler marginal contributions (8b.43) can be computed as in (8b.44) and read S.8b.1

 [Sd{R}]Euler0=0.0045,[Sd{R}]Euler1=0.0071,[Sd{R}]Euler2=0.0019, (8b.47)

which sum to the total (7.62), as purported by the general decomposition (8b.8).

Example 8b.11. Let us consider the same scenario-probability setting as of Example 8b.10. Consider as risk measure the variance (7.55). Then, the Euler marginal contributions (8b.43) can be computed as in (8b.45) and read S.8b.1

 [V{R}]Euler0=6.03×10−5,[V{R}]Euler1=9.58×10−5,[V{R}]Euler2=2.59×10−5, (8b.48)

which sum to the total (7.57), as purported by the general decomposition (8b.8).

##### Elliptical

If the joint distribution of the factors is elliptical (18.241), as in (8a.69)

 Z∼El(μZ,σ2Z,g¯k+1), (8b.49)

we can compute the Euler contributions (8b.43) for the standard deviation (8b.44) and the variance (8b.45) by replacing expectation and covariance of the factors with the corresponding elliptical expectation and covariance (18.259) as

 [Sd{Y}]Eulerk=βk×√γ×[σ2Z×β']k√β×σ2Z×β', (8b.50)

and

 [V{Y}]Eulerk=βk×γ×[σ2Z×β']k, (8b.51)

where is determined by (18.260).

In particular, if the factors are normally distributed, i.e. , the Euler contributions for the standard deviation (8b.44) become

 [Sd{Y}]Eulerk=βk×[σ2Z×β']k√β×σ2Z×β', (8b.52)

and the Euler contributions for the variance (8b.45) become

 [V{Y}]Eulerk=βk×[σ2Z×β']k. (8b.53)

Example 8b.12. Let us continue from Example 8b.2, where we derived ex-ante performance decomposition (8b.6) of the ex-ante P&L of the portfolio of zero coupon bonds obtained in Example 8a.11. The factors are normally distributed (8a.73). Consider as risk measure the standard deviation (7.60). Then, the marginal contributions (8b.43) can be computed as in (8b.52) and read S.8b.2

 [Sd{Π}]Euler0=56,[Sd{Π}]Euler1=2,089, (8b.54)

which sum to (7.61), as purported by the general decomposition (8b.8).

Example 8b.13. Let us consider the same elliptical setting of Example 8b.12. Consider as risk measure the variance  (7.55). Then the marginal contributions (8b.45) can be computed as in (8b.53) and read S.8b.2

 [V{Π}]Euler0=0.12×106,[−V{Π}]Euler1=4.48×106, (8b.55)

which sum to (7.56), as purported by the general decomposition (8b.8).

##### Relative marginal contributions for the negative variance

Traditional risk budgeting relies on quantifying the Euler contributions to the overall variance due to each single factor . To this purpose, we introduce the relative marginal contributions

 rEulerk≡[V{Y}]EulerkV{Y}, (8b.56)

or in vector notation

 rEuler≡β'⊙(Cv{Z}β')βCv{Z}β'. (8b.57)

The relative contributions sum to one, but they are not necessarily positive due to negative correlations or the presence of negative exposures to factors

 ∑¯kk=0rEulerk=1,rEulerk≶0. (8b.58)

A portfolio can be considered diversified if the relative contributions (8b.57) are uniformly distributed.

Example 8b.14. Let us consider the same setting as of Example 8b.2. The relative marginal contributions to risk (8b.57) corresponding to are S.8b.4

 rEuler=⎛⎜⎝0.230.670.10⎞⎟⎠. (8b.59)

Due to the correlation between the risk factors, from (8b.45), the covariance (21.45) can be decomposed into pure and spurious components E.8b.17

 V{Y}=∑¯kk=0(β2kV{Zk}+βkSd{Zk}∑j≠kCr{Zj,Zk}Sd{Zj}βj). (8b.60)

Example 8b.15. Let us consider the same setting as of Example 8b.2. The risk contribution associated with the risk factor can be decomposed as in (8b.60S.8b.4

 [−V{Y}]Euler1=−(324.072β1V{Z1}+23.52β1Sd{Z1}∑j≠1Cr{Zj,Z1}Sd{Zj}βj). (8b.61)

From (8b.60), we observe that if the factors were uncorrelated, then the marginal contributions (8b.45), and thus the relative contributions  (8b.57), would be affected by the -th factors only, and thus they would be always all positive. We will exploit this observation in Section 8b.4.

#### 8b.2.2 Certainty-equivalent

The certainty-equivalent (7.95) is positive homogeneous of degree if and only if the utility function (7.90) is the power utility function defined by where , see Table 7.11 (other than the trivial case of linear utility, which leads to the expectation measure). In this case, we can apply the Euler decomposition (8b.42). The Euler marginal contributions (8b.43) read E.8b.3

 [Ceq{Y}]Eulerk=βk×E{Zk(βZ)λ−1}(E{(βZ)λ})λ−1λ, (8b.62)

where the term on the right of is the partial derivative .

##### Scenario probability

If the joint distribution of the factors is a scenario-probability distribution (8b.46), we can compute the Euler contributions for the certainty equivalent with power utility (8b.62) via the expectation rule (18.337), provided that all the scenarios are positive.

Example 8b.16. Let us consider the same scenario-probability setting as of Example 8b.10. Consider as satisfaction measure the equivalent (7.106) associated to the exponential utility function . We cannot perform the Euler decomposition of the certainty equivalent (7.106) computed in Example 7.37 because it is not positive homogeneous.
On the other hand, we cannot apply a power utility function, which gives rise to a positive homogeneous certainty-equivalent (7.95), to measures of performance with negative scenarios, such as the return or P&L.

#### 8b.2.3 Quantile

The quantile satisfaction measure (7.117) with confidence level is positive homogenous of degree , see Table 7.13. In this case, we can apply the Euler decomposition (8b.42). The Euler marginal contributions (8b.43) read E.8b.4

 [qY(α)]Eulerk=βk×E{Zk|βZ=qβZ(α)}, (8b.63)

where the term on the right of is the partial derivative and reads (18.68)

 E{Zk|βZ=qβZ(α)}=∫zkfZk|βZ(zk|qβZ(α))dzk. (8b.64)

##### Scenario probability

If the joint distribution of the factors is a scenario-probability distribution (8b.46), the quantile is no longer differentiable (15.28) in  E.8b.5 . However, if the effective number of scenarios (37.21) for (8b.46) is large, then is approximately differentiable and we can compute its approximated Euler marginal contributions (8b.43) as the scenario-probability counterpart of (8b.63), as follows.

First, we observe that the joint distribution of the ex-ante performance (6.97) and of the factors can be written equivalently in terms of the sorted scenario-probability pairs

 (Y,Z)∼{(y(j)sort,z(j)sort),p(j)sort}¯ȷj=1, (8b.65)

where the sorted scenarios and probabilities are obtained according to the order induced by the portfolio performance sorting (7.119) in the Evaluation Step 7, as in the below Figure 8b.1.

Example 8b.17. Let us consider the same scenario-probability setting as of Example 8b.10. The sorted scenario-probability (7.119) of the ex-ante return reads S.8b.1

 R∼{−0.0241↑r(1)sort,10%↑p(1)sort,−0.0106↑r(2)sort,30%↑p(2)sort,0.0066↑r(3)sort,40%↑p(3)sort,0.0193↑r(4)sort,20%↑p(4)sort}. (8b.66)

Thus, the joint sorted scenario-probability distribution (8b.65) reads S.8b.1

 ⎛⎜ ⎜ ⎜⎝RZ0Z1Z2⎞⎟ ⎟ ⎟⎠∼{⎛⎜ ⎜ ⎜⎝−0.0241−0.0215−0.00980.0145⎞⎟ ⎟ ⎟⎠↑(r(1)sort,z(1)sort)',10%↑p(1)so