120 research outputs found
The Law of Total Odds
The law of total probability may be deployed in binary classification
exercises to estimate the unconditional class probabilities if the class
proportions in the training set are not representative of the population class
proportions. We argue that this is not a conceptually sound approach and
suggest an alternative based on the new law of total odds. We quantify the bias
of the total probability estimator of the unconditional class probabilities and
show that the total odds estimator is unbiased. The sample version of the total
odds estimator is shown to coincide with a maximum-likelihood estimator known
from the literature. The law of total odds can also be used for transforming
the conditional class probabilities if independent estimates of the
unconditional class probabilities of the population are available.
Keywords: Total probability, likelihood ratio, Bayes' formula, binary
classification, relative odds, unbiased estimator, supervised learning, dataset
shift.Comment: 12 pages, 1 figure, new reference
Expected Shortfall and Beyond
Financial institutions have to allocate so-called "economic capital" in order
to guarantee solvency to their clients and counter parties. Mathematically
speaking, any methodology of allocating capital is a "risk measure", i.e. a
function mapping random variables to the real numbers. Nowadays
"value-at-risk", which is defined as a fixed level quantile of the random
variable under consideration, is the most popular risk measure. Unfortunately,
it fails to reward diversification, as it is not "subadditive". In the search
for a suitable alternative to value-at-risk, "Expected Shortfall" (or
"conditional value-at-risk" or "tail value-at-risk") has been characterized as
the smallest "coherent" and "law invariant" risk measure to dominate
value-at-risk. We discuss these and some other properties of Expected Shortfall
as well as its generalization to a class of coherent risk measures which can
incorporate higher moment effects. Moreover, we suggest a general method on how
to attribute Expected Shortfall "risk contributions" to portfolio components.
Key words: Expected Shortfall; Value-at-Risk; Spectral Risk Measure;
coherence; risk contribution.Comment: 18 pages, LaTeX with hyperref package, Remark 3.8 and references
update
Incorporating exchange rate risk into PDs and asset correlations
Intuitively, the default risk of a single borrower is higher when her or his
assets and debt are denominated in different currencies. Additionally, the
default dependence of borrowers with assets and debt in different currencies
should be stronger than in the one-currency case. By combining well-known
models by Merton (1974), Garman and Kohlhagen (1983), and Vasicek (2002) we
develop simple representations of PDs and asset correlations that take into
account exchange rate risk. From these results, consistency conditions can be
derived that link the changes in PD and asset correlation and do not require
knowledge of hard-to-estimate parameters like asset value volatility.Comment: 7 pages, 1 figur
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