Real-life decisions are usually made in the state of uncertainty or risk. In this article we present the risk measuring techniques value at risk (VaR) and tail value at risk (TVaR) under uncertainty. Firstly, we introduce the VaR concept of uncertain variable based on uncertainty theory and examine its fundamental properties. Then, the TVaR concept is evolved and some fundamental properties of ...

Constant proportion portfolio insurance (CPPI) allows an investor to limit downside risk while retaining some upside potential by maintaining an exposure to risky assets equal to a constant multiple m > 1 of the cushion, the difference between the current portfolio value and the guaranteed amount. In diffusion models with continuous trading, this strategy has no downside risk, whereas in real m...

This paper examines the intertemporal relation between downside risk and expected stock returns. Value at risk (VaR), expected shortfall, and tail risk are used as measures of downside risk to determine the existence and significance of a risk-return tradeoff for several stock market indices. We find a positive and significant relation between downside risk and the portfolio returns on the NYSE...

There are more to a risk-measure than being coherent. Both the popular VaR and the coherent Tail-VaR ignore useful information in a large part of the loss distribution; As a result they lack incentive for risk-management. I propose a new coherent risk-measure that utilizes information in the whole loss distribution and provides incentive for risk-management.

This paper proposes differentiability properties for positively homogeneous risk measures which ensure that the gradient can be applied for reasonable risk capital allocation on non-trivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the one-sided moments of a risky payoff. In contrast to quantile-based risk measur...

Risk measurement involves estimating some functional of a loss distribution. This calls for nested simulation, in which risk factors are sampled at an outer level of simulation, while the inner level of simulation provides estimates of loss given each realization of the risk factors. We present a general method for providing a confidence interval for the risk measurement. It involves combining ...

At present, all value at risk (VaR) implementations ± i.e., all risk measures of thè`maximum loss at a given level of con®dence'' type ± are based on the assumption that the portfolio mix will not change before the VaR horizon. This hypothesis may be unrealistic, especially when the VaR horizon is established by the regulators (BIS). At the opposite, we measure VaR dynamically, i.e., taking int...

Recent experience in capital markets has highlighted the need for risk-sensitive portfolio strategies in both domestic investment grade as well as opportunistic high-yield and emerging market portfolios. We have previously developed a fixed-income sector optimization methodology to facilitate tradeoffs between various sectors based on their contribution to the total portfolio return and risk. W...

K e y w o r d s: Family of Sign RCA Models, Value at Risk, backtesting, loss function.

Steven Morrison Steven [email protected] A common definition of an insurer’s economic capital requirements is based around a 1-year Value at Risk (VaR) metric. This defines capital requirements in terms of some tail percentile (typically the 99.5th percentile) of the market-consistent value of the insurer’s balance sheet in 1 year’s time. The problem of estimating such a metric naturally le...