  Backtesting risk measure forecasts requires identifiability (for model
calibration and validation) and elicitability (for model comparison). We show
that the three widely-used systemic risk measures conditional value-at-risk
(CoVaR), conditional expected shortfall (CoES) and marginal expected shortfall
(MES), which measure the risk of a position $Y$ given that a reference position
$X$ is in distress, fail to be identifiable and elicitable on their own. As a
remedy, we establish the joint identifiability of CoVaR, MES and (CoVaR, CoES)
together with the value-at-risk (VaR) of the reference position $X$. While this
resembles the situation of the classical risk measures expected shortfall (ES)
and VaR concerning identifiability, a joint elicitability result fails.
Therefore, we introduce a completely novel notion of multivariate scoring
functions equipped with some order, which are therefore called multi-objective
scores. We introduce and investigate corresponding notions of multi-objective
elicitability, which may prove beneficial in various applications beyond
finance. In particular, we prove that conditional elicitability of two
functionals implies joint multi-objective elicitability with respect to the
lexicographic order on $\mathbb{R}^2$, which makes it applicable in the context
of CoVaR, MES or (CoVaR, CoES), together with VaR. We describe corresponding
comparative backtests of Diebold-Mariano type, for two-sided and 'one and a
half'-sided hypotheses, which respect the particularities of the lexicographic
order and which can be used in a regulatory setting. We demonstrate the
viability of these backtesting approaches in simulations and in an empirical
application to DAX 30 and S&P 500 returns.
