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This chapter presents a multi-criteria portfolio model with the expected return as a performance measure and the expected worst-case return as a risk measure. The problems are formulated as a single-objective linear program, as a bi-objective linear program, and as a triple-objective mixed integer program. The problem objective is to allocate the wealth on different securities to optimize the portfolio return. The portfolio approach has allowed the two popular financial engineering percentile measures of risk, value-at-risk (VaR) and conditional value-at-risk (CVaR) to be applied. The decision-maker can assess the value of portfolio return, the risk level, and the number of assets, and can decide how to invest in a real-life situation comparing with ideal (optimal) portfolio solutions. The concave efficient frontiers illustrate the trade-off between the conditional value-at-risk and the expected return of the portfolio. Numerical examples based on historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided. The computational experiments prove that both proposed linear and mixed integer programming approaches provide the decision-maker with a simple tool for evaluating the relationship between the expected and the worst-case portfolio return.

Regulators require banks to employ value‐at‐risk (VaR) to estimate the exposure of their trading portfolios to market risk, in order to establish capital requirements. However, portfolio‐level VaR analysis is a high‐dimensional problem and hence computationally intensive. This article presents two new portfolio‐based approaches to reducing the dimensionality of the VaR analysis.

Spectacular bankruptcies of the Orange County Investment Pool in December 1994 and Barings Bank in February 1995 mounted a pressure on the U.S. Financial Accounting Standards Board (FASB) to issue Statement No. 133, Accounting for Derivatives Instruments and Hedging Activities (FAS 133). Although measuring derivatives at fair value is a major improvement in accounting for derivatives, such type of accounting falls short of quantifying and reporting the risk of losses associated with derivative instruments. The purpose of this paper is to suggest an alternative approach to market valuation by integrating quantitative market risk estimation into the valuation method. The paper will use the Barings Bank experience to demonstrate how FAS no. 133 disclosure falls short of disclosing the magnitude of the market risk held by the bank at the end of 1994. It will also demonstrate how using a risk‐impacted value would have improved the disclosure of how much the bank stood to lose from their open positions.

This study examines problems that may occur when conventional Value‐at‐Risk (VaR) estimators are used to quantify market risks in an agricultural context. For example, standard VaR methods, such as the variance‐covariance method or historical simulation, can fail when the return distribution is fat tailed. This problem is aggravated when long‐term VaR forecasts are desired. Extreme Value Theory (EVT) is proposed to overcome these problems. The application of EVT is illustrated by an example from the German hog market. Multi‐period VaR forecasts derived by EVT are found to deviate considerably from standard forecasts. We conclude that EVT is a useful complement to traditional VaR methods.

The purpose of this article is to introduce a new method of estimating risk as an alternative to value at risk (VaR), drawing on the risk assessment literature in environmental science.

A commonly used and accepted measure of market risk is VaR, defined as the difference between initial portfolio value and a probabilistic lower bound B on the portfolio value at time T. To take account of situations where the portfolio value may fall below B prior to time T, an an alternative to VaR is proposed based on first passage time distributions.

It is argued that the resulting expected minimum portfolio value over the time frame T provides a clear alternative measure of market risk. Analytical expressions are obtained and numerical comparisons given when the distribution of portfolio returns is lognormal.

Analytical results are presented for lognormal distributions for returns. Results for other models can be easily obtained from simulation.

The new measure of risk recognizes that investors might be sensitive to risks of decline in the value of a portfolio at any time within a given time horizon, not just at the end of the anticipated period of investment. The expected minimum portfolio value measures the largest loss that is expected at some stage over that period.

A new measure of risk is presented that arises from literature on risk assessment in environmental science. It is complementary to VaR for assessing risk.

Maximum Loss was one of the risk measures proposed as alternatives to Value at Risk following criticism that Value at Risk is not coherent. Although the power of Maximum Loss is recognised for non‐linear portfolios, there are arguments that for linear portfolios Maximum Loss does not convey more information than Value at Risk. This paper argues for the usefulness of Maximum Loss for both linear and non‐linear portfolios.

This is a synthesis of existing theorems. Results are established by means of counterexamples. The worst case based risk‐return management strategy is presented as a case study.

For linear portfolios under elliptic distributions Maximum Loss is proportional to Value at Risk, and to Expected Shortfall, with the proportionality constant not depending on the portfolio composition. The paper gives a new example of Value at Risk violating subadditivity, using a portfolio of simple European options. For non‐linear portfolios, Maximum Loss need not even approximately be explained by the sum of Maximum Loss contributions of the individual risk factors. Finally, is proposed a strategy of risk‐return management with Maximum Loss.

The paper is restricted to elliptically distributed risk factors. Although Maximum Loss can be defined for more general continuous and even discrete distributions of risk factor changes, the paper does not address this matter.

The paper proposes an intuitive, computationally easy way how to improve average returns of linear portfolios while reducing worst case losses.

One is a synthesis of existing theorems. The counterexample establishing result is the first example of a portfolio of plain vanilla options violating Value at Risk subadditivity, an effect hitherto only known for portfolios of exotic options. Furthermore, the strategy of risk‐return management with Maximum Loss is original.

This paper aims to show that the accuracy of real estate portfolio valuations and of real estate risk management can be improved through the simultaneous use of Monte Carlo simulations and options theory.

The authors' method considers the options embedded in Continental European lease contracts drawn up with tenants who may move before the end of the contract. The authors combine Monte Carlo simulations for both market prices and rental values with an optional model that takes into account a rational tenant's behaviour. They analyze how the options significantly affect the owner's income.

The authors' main findings are that simulated cash flows which take account of such options are more reliable that those usually computed by the traditional method of discounted cash flow.

Some limitations are inherent to the authors' model: these include the assumption of the rationality of tenant's decisions and the difficulty of calibrating the model given the lack of data in many markets.

The main contribution of the paper is both by accounting for market risk (Monte Carlo simulations for the prices and market rental values) and for accounting for the idiosyncratic risk (the leasing risk).

Risk capital is an important input for management functions. Capital structure decisions, capital budgeting, and ex post performance measurement require different measures of risk capital. While it has become common to estimate risk capital using VaR models, it is not clear that VaR‐based capital estimates are optimal for applications to management functions (e.g. risk management, capital budgeting, performance measurement, or regulation). This article considers three typical problems that require an estimate of credit risk capital: an optimal equity capital allocation; an optimal capital allocation for capital budgeting decisions; and an optimal capital allocation to remove moral hazard incentives from a compensation contract based on ex post performance. The optimal credit risk capital allocation is different for each problem and is never consistent with a credit VaR estimate of unexpected loss. The results demonstrate that the optimal risk capital allocation depends on the objective.

It is widely accepted that equity return volatility increases more following negative shocks rather than positive shocks. However, much of value‐at‐risk (VaR) analysis relies on the assumption that returns are normally distributed (a symmetric distribution). This article considers the effect of asymmetries on the evaluation and accuracy of VaR by comparing estimates based on various models.