This article will show how recent advances in value-at-risk methodology can be effectively applied for risk measurement and management. The second article will focus on the application of the methodology for international equity portfolios.
One of the main criticisms of VaR is it is a unidirectional analysis, as the final result is just one number. In the process of aggregating and simplifying the portfolio risk, essential information that could be useful to manage a portfolio actively is lost. Once the VaR numbers are calculated, the "drill-down" capabilities are an essential part of any VaR system, as they offer crucial insights to determine the main sources of risk of the portfolio--hot spots--and which components of the portfolio act as a natural hedge.
Following the standard analytical methodology of VaR, we know that, VaR = (check) p´Qp , where p is the amounts of cashflow assigned to each vertex, after "mapping" the portfolio of trades stated in present value of the currency of reference; p is a column vector where p0 = f(P0) and f is the cash flow mapping function, and Q is the variance-covariance matrix adjusted for a certain time horizon and confidence level.
CASHFLOW MAP, DIVERSIFIED VaR AND UNDIVERSIFIED VaR.
Raw cashflow maps can be thought of as a "pre-covariance analysis," which employs neither volatilities nor correlations. The cashflow map shows the exposure to fluctuations in each of the sectors and currencies involved in the analysis.
The undiversified VaR can be obtained by multiplying each cashflow allocated to each vertex by the volatility of that risk factor adjusted by the horizon and confidence interval chosen, i.e. without considering the correlations among them. This type of risk analysis is, however, very narrow and limited, as it does not take into account the interaction among the different sectors and the possible benefits of diversification within the portfolio. The underlying theme behind modern portfolio theory is that 2 + 2 does not equal 4 in risk terms. With the introduction of the correlation information, it is possible to introduce the diversification effects between the different components of the portfolio to obtain its diversified VaR.
VaRDELTA AND COMPONENT VaR.
With the VaRdelta and component VaR technology, originally developed by Garman, an entire portfolio's diversified VaR can be taken and additively allocate it to the individual components comprising the portfolio.
VaRdelta tells us the marginal exposure of our current portfolio, posterior to correlation and volatility information. In other words, it answers the question of how VaR will change if another (numeraire) unit of a vertex cashflow is injected, given the covariance matrix (market) information. It is always measured in percentage terms, usually in basis points for short horizons.
Each element in the VaRdelta vector is a measure of the sensitivity of the VaR of the portfolio to an additional unit in cash flow in each risk factor, an increase or reduction in one of the portfolio exposures.
Before reaching the conclusion that a certain position is considerably increasing portfolio risk, and exposure to that particular risk factor should be reduced, it is important to introduce the analysis of diversified VaR, which can be accomplished through VaRdelta and component VaR.
An application of VaRdelta is "component VaR", which allows us to break down the diversified VaR into its main sources or components as well as to identify the trades that act as a hedge with respect to total portfolio risk.
The component VaR vector can be calculated through the inner product of the cash flow map vector of component I times the VaRdelta vector of the reference portfolio P, (see flow diagram).
With component VaR and VaR-Beta (component VaR divided by diversified VaR), it is possible to create risk management reports drilling down into portfolio VaR multiple ways, in terms of traders, trades, market sectors, countries, counterparty, etc.
This week's Learning Curve was written by Carlos Blanco, head of global support services at Financial Engineering Associatesin Berkeley, CA, and Jose Ramón Aragonés, professor of finance at Universidad Computensein Madrid, Spain.
