A financial institution's most important management problem is: given the uncertain future value of its balance sheet, how much capital must it hold today to remain solvent with high probability? Answering this operationally requires three things: a mathematical framework for converting a portfolio into a random loss variable (the mapping); a choice of risk measure to summarise that loss distribution (VaR, ES, or something better); and a framework for deciding what makes a risk measure a good one (coherence, convexity, dual representation).
This topic provides all three. It forms the conceptual engine room of the entire unit: every subsequent topic — empirical return series, EVT, copula models, market risk, credit risk — is ultimately about getting the loss distribution right so that the measures introduced here are computed accurately.
By the end of this topic you should be able to:
- write down the loss operator and explain the role of the mapping function ;
- apply the linearized (delta) and delta–gamma approximations to a stock portfolio and an options position;
- describe the three computation methods for loss distributions (variance–covariance, historical simulation, Monte Carlo) and state when each is appropriate;
- define VaR and ES formally, compute them for normal and Student loss distributions, and translate between the two;
- state the four coherence axioms and prove that VaR is not coherent (it can be superadditive) while ES is;
- explain what a spectral risk measure is and why the weighting function must be non-decreasing;
- set up the Euler allocation formula for portfolio capital decomposition and interpret each component as a marginal risk contribution.
In this lesson
- Contents
- The big picture
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