Existence of underestimation bias on risk for optimized portfolio is well known to quantitative fund managers who construct their portfolio from optimizer (mathematical software) by using multi factor risk model. There are some reasons for underestimation of portfolio risk. One of the reasons of...

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a...

Empirical evidence suggests that asset returns correlate more strongly in bear markets than conventional correlation estimates imply. We propose a method for determining complete tail-correlation matrices based on Value-at-Risk (VaR) estimates. We demonstrate how to obtain more effi cient...

We characterize the investor’s optimal portfolio allocation subject to a budget constraint and a probabilistic VaR constraint in complete markets environments with a finite number of states. The set of feasible portfolios might no longer be connected or convex, while the number of local optima...

Monetary risk measures classify a financial position by the minimal amount of external capital that must be added to the position to make it acceptable.We propose a new concept: intrinsic risk measures. The definition via external capital is avoided and only internal resources appear. An...

The plain market-beta was a good predictor of stock returns not only during bull and ordinary markets, but also during bear markets and crashes. Thus, it was indeed a good measure of the hedge against market risk. This plain beta also predicted the subsequent down-beta (i.e., measured only on...