47b.2 Euler decomposition

Key points

• The Euler decomposition (47b.42) allows us to compute additive contributions for positive homogeneous risk/satisfaction measures.
• We illustrate the Euler decomposition computations in the elliptical and scenario-probability case for several measures, including the variance (47b.45), the quantile (47b.63) and sub-quantile (47b.73).

We so far have considered the exposures $\beta$ in the ex-ante performance decomposition $Y=\beta Z$ (47b.6) as a fixed vector, computed once for all in the Ex-ante attribution step (Chapter 47a).

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

$$Y_b\equiv b{Z}_{\beta }$$

(47b.40)

where the factors $Z_{\beta }$ are considered fixed and static, and the exposure vary in such a way that the variable $Y_b$ (47b.40) becomes the actual ex-ante performance $Y$ (47b.6) when $b=\beta$. The subscript “$\beta$” in $Z_{\beta }$ (47b.40) stresses out that the factors $Z\equiv {(Z_0,\dots ,Z_{\overline{k}})}^{\text{'}}$ depend on the exposures $\beta$, because the factor $Z_0$ (47a.61) depends on the residual $U$ of the attribution model (47a.12).

Then, for a general satisfaction/risk measure $Satis\{\cdot \}$ (46.2) we define the function

$$b\mapsto Satis\left\{b{Z}_{\beta }\right\}\text{,}$$

(47b.41)

which associates to an $1\times (\overline{k}+1)$ vector of exposures $b$ the ensuing satisfaction $Satis\left\{b{Z}_{\beta }\right\}$, $b$ is not necessarily the exposures $\beta$ from the attribution model (47a.12). We will omit the subscript $\beta$ from now on for brevity.

If the satisfaction/risk measure $Satis\{\cdot \}$ (46.2) is positive homogeneous of degree $m$ (46.27), then the function (47b.41) is positive homogenous [W]. If the function (47b.41) is further differentiable (32.28), then the Euler decomposition of the ex-ante performance (47b.6) into additive marginal contributions holds

$$Satis\left\{Y\right\}={\sum }_{k=0}^{\overline{k}}{\beta }_k\times \frac{1}{m}\frac{\partial Satis\left\{\beta Z\right\}}{\partial {\beta }_k}\text{,}$$

(47b.42)

where $\frac{\partial Satis\left\{\beta Z\right\}}{\partial {\beta }_k}$ is a shorthand for $\frac{\partial Satis\left\{bZ\right\}}{\partial b_k}{|}_{b=\beta }$. The contribution associated with the $k$-th risk factor (47b.8) is called the Euler marginal contribution

$${\left[Satis\right\{Y\left\}\right]}_k^{Euler}\equiv {\beta }_k\times \frac{1}{m}\frac{\partial Satis\left\{\beta Z\right\}}{\partial {\beta }_k}\text{.}$$

(47b.43)

We highlight in the equation above that the contribution from the $k$-th factor can be further expressed as a per-unit contribution $\partial Satis∕\partial {\beta }_k$ times the “amount” of that factor (exposure) ${\beta }_k$.

The marginal contributions ${\left[Satis\right\{Y\left\}\right]}_k^{Euler}$ (47b.43) are the Aumann-Shapley marginal contributions (47b.33), see [Aumann and Shapley, 2015], and [Denault, 2001].

In the following we show the decomposition (47b.42)-(47b.43) in practice for the most important satisfaction measures encountered in the Evaluation step (Chapter 46) which are positive homogenous (46.27).

47b.2.1 Standard deviation and variance

The standard deviation risk measure $Sd\{\cdot \}$ (46.60) is positive homogeneous of degree $m=1$, see (46.64). In this case, we can apply the Euler decomposition (47b.42) where the Euler marginal contributions (47b.43) read 58.1 

$${\left[Sd\right\{Y\left\}\right]}_k^{Euler}={\beta }_k\times \frac{{\left[Cv\right\{Z\}\times {\beta }^{\text{'}}]}_k}{\sqrt{\beta \times Cv\left\{Z\right\}\times {\beta }^{\text{'}}}}\text{,}$$

(47b.44)

and where the term on the right of $\times$ is the partial derivative $\partial Sd\left\{\beta Z\right\}∕\partial {\beta }_k$.

The variance risk measure $V\{\cdot \}$ (46.55) is positive homogeneous of degree $m=2$, see Table 46.3. In this case, we can apply the Euler decomposition (47b.42). The Euler marginal contributions (47b.43) read 58.2 

$${\left[V\right\{Y\left\}\right]}_k^{Euler}={\beta }_k\times {\left[Cv\right\{Z\}\times {\beta }^{\text{'}}]}_k\text{,}$$

(47b.45)

where the term on the right of $\times$ is half the partial derivative $\partial V\left\{\beta Z\right\}∕\partial {\beta }_k$.

Scenario probability

If the joint distribution of the factors $Z\equiv {(Z_0,\dots ,Z_{\overline{k}})}^{\text{'}}$ is a scenario-probability distribution (37.239), as in (47a.78)

$$Z\sim {\{z^{\left(j\right)},p^{\left(j\right)}\}}_{j=1}^{\overline{ȷ}}\text{,}$$

(47b.46)

we can compute the Euler contributions (47b.43) for the standard deviation (47b.44) and the variance (47b.45) by replacing the covariance of the factors $Z$ with the corresponding scenario-probability covariance (37.302).

Example 47b.10. Let us continue from Example 47a.13, where we derived ex-ante performance decomposition (47b.6) of the return $R$ of the stocks portfolio (Example 47b.2). The factors $Z={(Z_0,Z_1,Z_2)}^{\text{'}}$ have the scenario-probability distribution (47a.81). Consider as risk measure the standard deviation $Sd\{\cdot \}$ (46.60). Then, the Euler marginal contributions (47b.43) can be computed as in (47b.44) and read S.54b.1 

$${\left[Sd\right\{R\left\}\right]}_0^{Euler}=0.0045,{\left[Sd\right\{R\left\}\right]}_1^{Euler}=0.0071,{\left[Sd\right\{R\left\}\right]}_2^{Euler}=0.0019\text{,}$$

(47b.47)

which sum to the total $Sd\left\{R\right\}=0.0135$ (46.62), as purported by the general decomposition (47b.8).

Example 47b.11. Let us consider the same scenario-probability setting as of Example 47b.10. Consider as risk measure the variance $V\{\cdot \}$ (46.55). Then, the Euler marginal contributions (47b.43) can be computed as in (47b.45) and read S.54b.1 

$${\left[V\right\{R\left\}\right]}_0^{Euler}=6.03\times 10^{-5},{\left[V\right\{R\left\}\right]}_1^{Euler}=9.58\times 10^{-5},{\left[V\right\{R\left\}\right]}_2^{Euler}=2.59\times 10^{-5}\text{,}$$

(47b.48)

which sum to the total $V\left\{R\right\}=1.82\times 10^{-4}={(0.0135)}^2$ (46.57), as purported by the general decomposition (47b.8).

Elliptical

If the joint distribution of the factors $Z\equiv {(Z_0,\dots ,Z_{\overline{k}})}^{\text{'}}$ is elliptical (37.110), as in (47a.69)

$$Z\sim El({\mu }_Z,{\sigma }_Z^2,g_{\overline{k}+1})\text{,}$$

(47b.49)

we can compute the Euler contributions (47b.43) for the standard deviation (47b.44) and the variance (47b.45) by replacing expectation and covariance of the factors $Z$ with the corresponding elliptical expectation and covariance (37.121) as

$${\left[Sd\right\{Y\left\}\right]}_k^{Euler}={\beta }_k\times \sqrt{\gamma }\times \frac{{[{\sigma }_Z^2\times {\beta }^{\text{'}}]}_k}{\sqrt{\beta \times {\sigma }_Z^2\times {\beta }^{\text{'}}}}\text{,}$$

(47b.50)

and

$${\left[V\right\{Y\left\}\right]}_k^{Euler}={\beta }_k\times \gamma \times {[{\sigma }_Z^2\times {\beta }^{\text{'}}]}_k\text{,}$$

(47b.51)

where $\gamma$ is determined by $g_{\overline{k}+1}$ (37.122).

In particular, if the factors $Z$ are normally distributed, i.e. $Z\sim N({\mu }_Z,{\sigma }_Z^2)$, the Euler contributions for the standard deviation (47b.44) become

$${\left[Sd\right\{Y\left\}\right]}_k^{Euler}={\beta }_k\times \frac{{[{\sigma }_Z^2\times {\beta }^{\text{'}}]}_k}{\sqrt{\beta \times {\sigma }_Z^2\times {\beta }^{\text{'}}}}\text{,}$$

(47b.52)

and the Euler contributions for the variance (47b.45) become

$${\left[V\right\{Y\left\}\right]}_k^{Euler}={\beta }_k\times {[{\sigma }_Z^2\times {\beta }^{\text{'}}]}_k\text{.}$$

(47b.53)

Example 47b.12. Let us continue from Example 47b.2, where we derived ex-ante performance decomposition (47b.6) of the ex-ante P&L $\Pi$ of the portfolio of zero coupon bonds obtained in Example 47a.11. The factors $Z\equiv {(Z_0,Z_1)}^{\text{'}}$ are normally distributed (47a.73). Consider as risk measure the standard deviation $Sd\{\cdot \}$ (46.60). Then, the marginal contributions (47b.43) can be computed as in (47b.52) and read S.54b.2 

$${\left[Sd\right\{\Pi \left\}\right]}_0^{Euler}=56,{\left[Sd\right\{\Pi \left\}\right]}_1^{Euler}=2,089\text{,}$$

(47b.54)

which sum to $Sd\left\{\Pi \right\}=2,145$ (46.61), as purported by the general decomposition (47b.8).

Example 47b.13. Let us consider the same elliptical setting of Example 47b.12. Consider as risk measure the variance $V\{\cdot \}$ (46.55). Then the marginal contributions (47b.45) can be computed as in (47b.53) and read S.54b.2 

$${\left[V\right\{\Pi \left\}\right]}_0^{Euler}=0.12\times 10^6,{[-V\{\Pi \left\}\right]}_1^{Euler}=4.48\times 10^6\text{,}$$

(47b.55)

which sum to $V\left\{\Pi \right\}=4.60\times 10^6={(2,145)}^2$ (46.56), as purported by the general decomposition (47b.8).