##### From Quantitative Finance And Risk Management: A Physicist's Approach

In this chapter, we discuss begin a discussion of VAR, an acronym for Value at Risk ^{[i]}. The "Plain-Vanilla VAR" *(PV-VAR)* along with its incarnation as a quadratic form ( *QPV-VAR)* is a standard risk measure that we discuss first. PV-VAR is rather blunt and unrefined ^{[1]}. In the next chapter, we will discuss refinements. The first refinement stage defines the "Improved Plain-Vanilla VAR" ( *IPV-VAR*). Further refinements produce the "Enhanced/Stressed VAR" ( *ES-VAR*) ^{[2]}. We give a few previews in the footnotes, which also contain other important points.

The various components of the VAR, called CVARs, will also be discussed. The CVARs are useful because they give a consistent picture of the composition of the risk. We show that the CVARs have uncertainties (i.e. there is a CVAR volatility) and we show how to calculate these uncertainties ^{[3]}. The CVAR volatility is useful because it shows the uncertainty in different possible compositions for a given total risk.

### Plain-Vanilla VAR (PV-VAR)

We first describe "Plain-Vanilla VAR" ( *PV-VAR*). Basically, PV-VAR is a one-step simulator in time that measures risks at a given confidence level of a portfolio ^{[4]} *C* using a variety of simplifying assumptions. The portfolio *C* is a function of underlying variables { *x* _{ ?}}. ? = 1 *n*. The variable *x* _{ ?} can be a physical variable (e.g. interest rate, stock price, etc.), or *x* _{ ?} can be a function,...

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In this chapter, we discuss various stages of refinements of the plain-vanilla VAR discussed in Ch. 26. Increasingly realistic aspects will be included, with the final aim to obtain a risk measure...

The CVAR Volatility with Two Variables Here, we restrict our attention to two variables. We begin with the CVAR volatility. Here is a picture of the geometry: The CVAR volatility [1] turns out to be...

Chapter List Chapter 21: Fat Tail Volatility (Tech. Index 5/10) Chapter 22: Correlation Matrix Formalism; the N-Sphere (Tech. Index 8/10) Chapter 23: Stressed Correlations and Random Matrices...

In this chapter, we present a formal functional derivation of the VAR and CVAR equations for the linear case. We pay particular attention to the CVAR volatility. The derivation is done for in the...

10.1. MOTIVATION Risk comprises uncertainty and exposure. A VaR measure represents uncertainty with a characterization of the conditional distribution of 1 R. It represents exposure with a portfolio...