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Special Semester on Stochastics with Emphasis on Finance
Linz, September 2008 - December 2008
Presentation: On a subjective risk measurement and capital allocation

Workshop: Concluding Workshop

Time: Thu, December 04, 2008, 15:30-16:00

Speaker: Piotr Jaworski

Abstract

There are several approaches to the risk measurement. The most popular can be summarized in the following way:
The risk measure of a position (investment) is equal to the minimal amount of capital
necessary to allocate to make it save (solvable).


Basing on the above, the several systems of axioms were provided to be fulfilled by
"good" risk measures like coherent or convex risk measures. Note that in all these approaches
the cost of capital is neglected.

In my talk I would like to present a slightly different point of view on risk measurement. The one that
takes the cost of capital into account.

If a financial institution has a risky position then it has
to prepare some reserves (i.e. allocate some capital) to secure it. It is natural to regard
the optimal level of reserves as a measure of risk.
Furthermore we assume that the "risk-bearers" are "rational".
Their decisions depend on their subjective preferences.
In presented model these preferences are described by two non-decreasing, weakly convex functions,
$L_1$ and $L_2$,
which are measuring the cost of keeping reserves (or the cost of allocated capital),
and the loss when the reserves
are not sufficient. Let the random variable $X$ describe the liabilities,
and the function $L(X,r)=L_1(r)+L_2(X-r)$ costs.
The rational risk-bearer should choose the optimal level of reserves $r$,
the one which minimizes the expected value of costs.
In my talk I am going to show the existence of such $r$ and discuss the properties of associated risk measures.

When the cost of capital is linear we get a monetary risk measure.
For example, if the disadvantage functions
are proportional to the allocated amount and to the excess of the loss,
respectively, then the optimal level of reserves $r$ equals
Value at Risk with the confidence level depending on the slopes of $L_1$ and $L_2$.

Furthermore when left-side derivative of $L_2$ is convex then the associated risk measure
becomes a convex risk measure
induced by the utility function $-D^-L_2(-t)$.

Presentation slides (pdf, 91 KB)

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