This chapter uses the case of long-short hedge fund manager to demonstrate how a quantitative portfolio manager can use value at risk to measure, manage and optimize portfolio risk. The example portfolio includes 12 long and short positions in international stock index futures contracts, futures contracts on a benchmark ... click here for more details.

For large portfolios with many positions, risk decomposition by individual positions generally provides little insight as the risk contribution of each position is small. For these cases, effective risk decomposition must be done by groups of positions and/or factors. This chapter explains the use of factor models in ... click here for more details.

This chapter explains the four properties of a coherent risk measure: subadditivity, homogeneity, monotonicity and the risk-free condition. Further, the chapter shows how value at risk is not a coherent risk measure since it fails the subadditivity property. The chapter also briefly discusses scenario based measures and the ... click here for more details.

This chapter provides a summary of the mathematics of risk decomposition, showing how how to calculate the risk contribution of each position in a portfolio and the important role of covariance in that calculation. The chapter also discusses how risk decomposition can be done with historical simulation and Monte ... click here for more details.

One limitation of the parametric VaR method and Monte Carlo simulation VaR method is the assumption that market returns are normally distributed. Even the historical simulation VaR method often fails to capture extreme events because of insufficient data. Extreme value theory models extreme events using the generalized Pareto ... click here for more details.

By knowing the assumptions and limitations of VaR methods, a portfolio manager could game the value at risk by exploiting estimation errors that cause the value at risk calculation to underestimate actual risk. This chapter explains how estimation errors in the covariance matrix can be exploited, allowing a portfolio ... click here for more details.

This chapter demonstrates how the concepts of value-at-risk and risk budgeting can be applied to the problem of managing active risk at the total fund level. The chapter explores both the analysis of the existing manager roster and the determination of the optimal manager roster, and demonstrates both by ... click here for more details.

Shortcomings of value at risk include the lack of information about the magnitude of potential losses beyond the value at risk and the lack of information about the direction of the risk exposure. Stress testing helps reduce these shortcomings. Although there is no standard way to stress test ... click here for more details.

Although more complicated, the application of the delta-normal (or parametric) method to fixed-income portfolios is similar to its application to equity portfolios. This chapter explains the steps in applying the delta-normal VaR method to fixed-income portfolios: identifying market factors, mapping positions to the factors, estimating the distribution of changes ... click here for more details.

For portfolios with a large number of positions, factor models offer an intuitive way to think about its risks. This chapter provides an effective explanation of how factor models can be used for calculating a portfolio`s value at risk using the parametric VaR method and using Monte Carlo simulation. ... click here for more details.