37 research outputs found
A note on conditional covariance matrices for elliptical distributions
In this short note we provide an analytical formula for the conditional
covariance matrices of the elliptically distributed random vectors, when the
conditioning is based on the values of any linear combination of the marginal
random variables. We show that one could introduce the univariate invariant
depending solely on the conditioning set, which greatly simplifies the
calculations. As an application, we show that one could define uniquely defined
quantile-based sets on which conditional covariance matrices must be equal to
each other if only the vector is multivariate normal. The similar results are
obtained for conditional correlation matrices of the general elliptic case
The 20-60-20 Rule
In this paper we discuss an empirical phenomena known as the 20-60-20 rule.
It says that if we split the population into three groups, according to some
arbitrary benchmark criterion, then this particular ratio implies some sort of
balance. From practical point of view, this feature often leads to efficient
management or control. We provide a mathematical illustration, justifying the
occurrence of this rule in many real world situations. We show that for any
population, which could be described using multivariate normal vector, this
fixed ratio leads to a global equilibrium state, when dispersion and linear
dependance measurement is considered
Backtesting Expected Shortfall: a simple recipe?
We propose a new backtesting framework for Expected Shortfall that could be
used by the regulator. Instead of looking at the estimated capital reserve and
the realised cash-flow separately, one could bind them into the secured
position, for which risk measurement is much easier. Using this simple concept
combined with monotonicity of Expected Shortfall with respect to its target
confidence level we introduce a natural and efficient backtesting framework.
Our test statistics is given by the biggest number of worst realisations for
the secured position that add up to a negative total. Surprisingly, this simple
quantity could be used to construct an efficient backtesting framework for
unconditional coverage of Expected Shortfall in a natural extension of the
regulatory traffic-light approach for Value-at-Risk. While being easy to
calculate, the test statistic is based on the underlying duality between
coherent risk measures and scale-invariant performance measures
New fat-tail normality test based on conditional second moments with applications to finance
In this paper we introduce an efficient fat-tail measurement framework that
is based on the conditional second moments. We construct a goodness-of-fit
statistic that has a direct interpretation and can be used to assess the impact
of fat-tails on central data conditional dispersion. Next, we show how to use
this framework to construct a powerful normality test. In particular, we
compare our methodology to various popular normality tests, including the
Jarque--Bera test that is based on third and fourth moments, and show that in
many cases our framework outperforms all others, both on simulated and market
stock data. Finally, we derive asymptotic distributions for conditional mean
and variance estimators, and use this to show asymptotic normality of the
proposed test statistic
Unbiased estimation of risk
The estimation of risk measures recently gained a lot of attention, partly
because of the backtesting issues of expected shortfall related to
elicitability. In this work we shed a new and fundamental light on optimal
estimation procedures of risk measures in terms of bias. We show that once the
parameters of a model need to be estimated, one has to take additional care
when estimating risks. The typical plug-in approach, for example, introduces a
bias which leads to a systematic underestimation of risk. In this regard, we
introduce a novel notion of unbiasedness to the estimation of risk which is
motivated by economic principles. In general, the proposed concept does not
coincide with the well-known statistical notion of unbiasedness. We show that
an appropriate bias correction is available for many well-known estimators. In
particular, we consider value-at-risk and expected shortfall (tail
value-at-risk). In the special case of normal distributions, closed-formed
solutions for unbiased estimators can be obtained. We present a number of
motivating examples which show the outperformance of unbiased estimators in
many circumstances. The unbiasedness has a direct impact on backtesting and
therefore adds a further viewpoint to established statistical properties
The least squares method for option pricing revisited
It is shown that the the popular least squares method of option pricing
converges even under very general assumptions. This substantially increases the
freedom of creating different implementations of the method, with varying
levels of computational complexity and flexible approach to regression. It is
also argued that in many practical applications even modest non-linear
extensions of standard regression may produce satisfactory results. This claim
is illustrated with examples
Dynamic Limit Growth Indices in Discrete Time
We propose a new class of mappings, called Dynamic Limit Growth Indices, that
are designed to measure the long-run performance of a financial portfolio in
discrete time setup. We study various important properties for this new class
of measures, and in particular, we provide necessary and sufficient condition
for a Dynamic Limit Growth Index to be a dynamic assessment index. We also
establish their connection with classical dynamic acceptability indices, and we
show how to construct examples of Dynamic Limit Growth Indices using dynamic
risk measures and dynamic certainty equivalents. Finally, we propose a new
definition of time consistency, suitable for these indices, and we study time
consistency for the most notable representative of this class -- the dynamic
analog of risk sensitive criterion
A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time
In this paper we provide a flexible framework allowing for a unified study of
time consistency of risk measures and performance measures (also known as
acceptability indices). The proposed framework not only integrates existing
forms of time consistency, but also provides a comprehensive toolbox for
analysis and synthesis of the concept of time consistency in decision making.
In particular, it allows for in depth comparative analysis of (most of) the
existing types of time consistency -- a feat that has not be possible before
and which is done in the companion paper [BCP2016] to this one. In our approach
the time consistency is studied for a large class of maps that are postulated
to satisfy only two properties -- monotonicity and locality. The time
consistency is defined in terms of an update rule. The form of the update rule
introduced here is novel, and is perfectly suited for developing the unifying
framework that is worked out in this paper. As an illustration of the
applicability of our approach, we show how to recover almost all concepts of
weak time consistency by means of constructing appropriate update rules
A note on Multiplicative Poisson Equation: developments in the span-contraction approach
In this paper we study the problem of Multiplicative Poisson Equation (MPE)
bounded solution existence in the generic discrete-time setting. Assuming
mixing and boundedness of the risk-reward function, we investigate what
conditions should be imposed on the underlying non-controlled probability
kernel or the reward function in order for the MPE bounded solution to always
exists. In particular, we consolidate span-norm framework based results and
derive an explicit sharp bound that needs to be imposed on the cost function to
guarantee the bounded solution existence under mixing. Also, we study the
properties which the probability kernel must satisfy to ensure existence of
bounded MPE for any generic risk-reward function and characterise process
behaviour in the complement of the invariant measure support. Finally, we
present numerous examples and stochastic-dominance based arguments that help to
better understand the intricacies that emerge when the ergodic risk-neutral
mean operator is replaced with ergodic risk-sensitive entropy
Unbiased estimation and backtesting of risk in the context of heavy tails
While the estimation of risk is an important question in the daily business
of banks and insurances, many existing plug-in estimation procedures suffer
from an unnecessary bias. This often leads to the underestimation of risk and
negatively impacts backtesting results, especially in small sample cases. In
this article we show that the link between estimation bias and backtesting can
be traced back to the dual relationship between risk measures and the
corresponding performance measures, and discuss this in reference to
value-at-risk and expected shortfall frameworks. Motivated by this finding, we
propose a new algorithm for bias correction and show how to apply it for
generalized Pareto distributions. In particular, we consider value-at-risk and
expected shortfall plug-in estimators, and show that the application of our
algorithm leads to gain in efficiency when heavy tails exist in the data