Risk bounds for factor models

Abstract

Recent literature has investigated the risk aggregation of a portfolio $X=(X_{i})_{1\leq i\leq n}$ under the sole assumption that the marginal distributions of the risks $X_{i}$ are specified, but not their dependence structure. There exists a range of possible values for any risk measure of $S=\sum_{i=1}^{n}X_{i}$, and the dependence uncertainty spread, as measured by the difference between the upper and the lower bound on these values, is typically very wide. Obtaining bounds that are more practically useful requires additional information on dependence.

Here, we study a partially specified factor model in which each risk $X_{i}$ has a known joint distribution with the common risk factor $Z$, but we dispense with the conditional independence assumption that is typically made in fully specified factor models. We derive easy-to-compute bounds on risk measures such as Value-at-Risk ($\mathrm{VaR}$) and law-invariant convex risk measures (e.g. Tail Value-at-Risk ($\mathrm{TVaR}$)) and demonstrate their asymptotic sharpness. We show that the dependence uncertainty spread is typically reduced substantially and that, contrary to the case in which only marginal information is used, it is not necessarily larger for $\mathrm{VaR}$ than for $\mathrm{TVaR}$.

Appendices

Appendix A: Proof of Proposition 2.1

For any admissible risk vector ${X}\in A(H)$, the conditional distribution of $X_{i}$ given $Z=z$ is given by $F_{i| z}$. Therefore, conditionally under $Z=z$, the random vector ${X}$ has marginal distributions $F_{i| z}$, $1\le i\le n$. As a consequence, we obtain by conditioning

$$ P\bigg[\sum_{i=1}^{n} X_{i}\ge t\bigg] = \int P\bigg[\sum _{i=1}^{n} X_{i}\ge t\bigg| Z=z\bigg] dG(z) \le\int\overline{M} _{z} (t) dG(z), $$

and thus $\overline{M}{}^{f}(t)\le\int\overline{M_{z}} (t) dG(t)$.

Conversely, let ${X}_{z}=(X_{i,z})$ be random vectors with marginal distributions $F_{i| z}$ such that for given $\varepsilon>0$,

$$ P\bigg[\sum_{i=1}^{n} X_{i,z}\ge t\bigg] \ge\overline{M_{z}}(t)-\varepsilon. $$

(A.1)

The risk vector ${X}$ has a representation as a mixture model ${X}= {X}_{Z}$, where $Z$ is a random variable with distribution $G$, independent of $(X_{i,z})$. Then by conditioning, we obtain that $({X},Z)$ is admissible, i.e., ${X}\in A(H)$ and

$$ P\bigg[\sum_{i=1}^{n} X_{i} \ge t\bigg] \ge\int\overline{M_{z}} (t) dG(z)-\varepsilon. $$

(A.2)

As a result, (A.1) and (A.2) establish equality in (2.4). The lower bound is proved in a similar way.  □

Remark A.1

By a measurable selection result as in [39], a worst case distribution for $\overline{M}{}^{f}$ exists, and thus the $\varepsilon$-argument in the proof of Proposition 2.1 could be avoided in the case of the upper bound. However, the lower bounds $\underline{M}^{f}$ and $\underline{M_{z}} (t)$ are only attainable when we modify the definition of the Value-at-Risk slightly; see [3, 4].

Appendix B: Proof of Proposition 3.2

a) Consider the vector $X_{Z}^{c}$ having components $F_{i| Z}^{-1}(U)$, and observe that their conditional distribution functions are $F_{i| z}$ and their marginal distribution functions are $F_{i}$. Hence, $X_{Z}^{c}\in A(H)$ and $S_{Z}^{c}\in\mathcal{S}(H)$. Furthermore, for any $X\in A(H)$, we can use the mixture representation $X_{Z}$ for $X$ with $X_{i,z}=F_{i| z}^{-1}(U_{i,z})$ as in Sect. 2. From the convex ordering result in (3.2), it follows that

$$ S_{z}=\sum_{i=1}^{n} X_{i| z}\le_{\mathrm{cx}} \sum_{i=1}^{n} F_{i| z}^{-1}(U). $$

This implies by conditioning that $S_{Z}\preceq_{\mathrm{cx}} \sum_{i=1}^{n} F_{i| Z}^{-1}(U)=S_{Z}^{c}$.

b) Since $\varrho$ is consistent with the convex order, the result follows from a).

Appendix C: Proof of Proposition 4.9

For any $X_{Z}\in A(H)$, it holds that

$$\begin{aligned} \mathrm{VaR}_{\alpha}(S_{Z}) =& \mathrm{VaR}_{\alpha}\,\big(\,\mathrm {VaR}_{U}(S_{Z})\big)\\ \leq& \mathrm{VaR}_{\alpha}\,\bigg(\min \bigg\{ \,\mathrm{TVaR}_{U}(S_{Z}^{c}),\mu_{Z}+v_{Z}\sqrt{\frac {U}{1-U}}\bigg\} \bigg), \end{aligned}$$

where we have used that for all $z\in D$ and $u\in(0,1)$, ${\mathrm {VaR}}_{u}(S_{z}) \leq\mathrm{TVaR}_{u}(S_{z}^{c})$ and ${\mathrm {VaR}}_{u}(S_{z}) \leq\mu_{z}+v_{z}\sqrt{\frac{u}{1-u}}$ (Cantelli bound). This shows the desired result for $\overline{\mathrm{VaR}}_{\alpha }^{f}$. The case of $\underline{\mathrm{VaR}}_{\alpha}^{f}$ is similar.  □

Appendix D: Proof of Proposition 4.10

For any $X_{Z}\in A(H)$, it holds that $\mathrm{TVaR}_{\alpha}(S_{Z})\leq\mathrm{TVaR}_{\alpha}(S^{c}_{Z})= \mathrm{TVaR}_{\alpha}(\mathrm{VaR}_{U}(S^{c}_{Z}))$. Furthermore, $\mathrm{TVaR}_{\alpha}(S_{Z})= \mathrm{TVaR}_{\alpha}(\mathrm {VaR}_{U}(S_{Z}))$ and for all $z\in D$ and $u\in(0,1)$, ${\mathrm {VaR}}_{u}(S_{z}) \leq\mu_{z}+v_{z}\sqrt{\frac{u}{1-u}}$ (Cantelli bound). Hence, by combining, we obtain the desired result.  □