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Suppose that we want to minimize the expected shortfall of a portfolio. The key contribution of Rockafellar and Uryasev in their 2000 paper is to introduce the auxiliary function (,) for the expected shortfall: (,) = + (,) [(,)] + Where = and (,) is a loss function for a set of portfolio weights to be applied to the returns.
The FRTB revisions address deficiencies relating to the existing [8] Standardised approach and Internal models approach [9] and particularly revisit the following: . The boundary between the "trading book" and the "banking book": [10] i.e. assets intended for active trading; as opposed to assets expected to be held to maturity, usually customer loans, and deposits from retail and corporate ...
Basel III requires banks to have a minimum CET1 ratio (Common Tier 1 capital divided by risk-weighted assets (RWAs)) at all times of: . 4.5%; Plus: A mandatory "capital conservation buffer" or "stress capital buffer requirement", equivalent to at least 2.5% of risk-weighted assets, but could be higher based on results from stress tests, as determined by national regulators.
Here, α is a "multiplier" of 1.4, acting as a buffer to ensure sufficient coverage; and: RC is the "Replacement Cost" were the counterparty to default today: the current exposure, i.e. mark-to-market of all trades, is aggregated by counterparty, and then netted-off with haircutted-collateral.
1. Subadditivity. A risk measure is subadditive if for any portfolios A and B, the risk of A+B is never greater than the risk of A plus the risk of B. In other words, the risk of the sum of subportfolios is smaller than or equal to the sum of their individual risks. Standard deviation and expected shortfall are subadditive, while VaR is not.
In this approach, banks calculate their own risk parameters subject to meeting some minimum guidelines. However, the foundation approach is not available for Retail exposures. For equity exposures, calculation of risk-weighted assets not held in the trading book can be calculated using two different ways: a PD/LGD approach or a market-based ...
Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at (), the value at risk of level . [2] Under some other settings, TVaR is the conditional expectation of loss above a given value, whereas the expected shortfall is the product of this value with the probability of ...
That is, if portfolio always has better values than portfolio under almost all scenarios then the risk of should be less than the risk of . [2] E.g. If is an in the money call option (or otherwise) on a stock, and is also an in the money call option with a lower strike price.