Regensburg 2007 – wissenschaftliches Programm
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AKSOE: Arbeitskreis Physik sozio-ökonomischer Systeme
AKSOE 2: Financial Markets and Risk Management I
AKSOE 2.2: Vortrag
Montag, 26. März 2007, 10:45–11:15, H8
Bounds for Value at Risk of currency portfolios — •Piotr Jaworski — Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland
In my talk I shall deal with the following simple case. An investor operating on an emerging market, has in his portfolio several foreign currencies which are highly dependent. Let si be the quotients of the currency rates at the end and at the beginning of the investment. Let wi be the part of the capital invested in the i-th currency, ∑wi=1, wi ≥ 0. So the final value of the investment equals
| W1(w)=(w1 s1 + … + wdsd)· W0. |
The crucial point is to estimate the risk of keeping the portfolio. As a measure of risk I shall consider "Value at Risk" (VaR), which last years became one of the most popular measures of risk in the "practical" quantitative finance. Roughly speaking the idea is to determine the biggest amount one can lose on certain confidence level α
| VaR1− α(w)=sup{V:P(W0−W1(w) ≤ V) < 1−α }. |
The main result, I would like to present, is the following estimate of the Value at Risk of a given portfolio w in terms of Value at Risk of one-currency portfolios ei:
| ∑wi VaR1−α(ei) ≥ VaR1−α(w) ≥ ∑wi VaR1−α′(ei), |
where α′= α/L(1, … ,1). The above estimate is valid for sufficiently small α under the mild assumption that the lower tail part of the copula C of si’s is homogeneous of degree 1, i.e. for sufficiently small q
| C(q)=L(q), ∀ t>0 L(tq)=tL(q). |