THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

A new approach to optimizing or hedging a portfolio of nancial instruments to reduce risk is presented and tested on applications. It focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value-at-Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is anyway considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage rms, mutual funds, and any business that evaluates risks. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can be applied also to the optimization of percentiles in contexts outside of nance.

/pdf/optimization-of-conditional-value-at-risk-338oy3ddeu.pdf

Optimization of conditional value-at-risk

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Fundamental properties of conditional value-at-risk (CVaR), as a measure of risk with significant advantages over value-at-risk (VaR), are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. CVaR is able to quantify dangers beyond VaR and moreover it is coherent. It provides optimization short-cuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking.

/pdf/conditional-value-at-risk-for-general-loss-distributions-4789km634z.pdf

Conditional value-at-risk for general loss distributions

Fundamental properties of conditional value-at-risk, as a measure of risk with significant advantages over value-at-risk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. Conditional value-at-risk is able to quantify dangers beyond value-at-risk, and moreover it is coherent. It provides optimization shortcuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID267256_code010418130.pdf?abstractid=267256&type=2

Conditional Value-at-Risk for General Loss Distributions

Standard market risk optimization tools, based on assumptions of normality, are ineffective for evaluating credit risk. In this article, the authors develop three scenario optimization models for portfolio credit risk. They first create the trading risk profile and find the best hedge position for a single asset or obligor. The second model adjusts all positions simultaneously to minimize the regret of the portfolio subject to general linear restrictions. Finally, a credit risk‐return efficient frontier is constructed using parametric programming. While scenario optimization of quantile‐based credit risk measures leads to problems that are not generally tractable, regret is a relevant and tractable measure that can be optimized using linear programming. The three models are applied to optimizing the risk‐return profile of a portfolio of emerging market bonds.

/pdf/applying-scenario-optimization-to-portfolio-credit-risk-1q69alcs0q.pdf

Applying Scenario Optimization to Portfolio Credit Risk

The risk/return trade‐off has been a central tenet of portfolio management since the seminal work of Markowitz [1952]. The basic premise, that higher (expected) returns can only be achieved at the expense of greater risk, leads naturally to the concept of an efficient frontier. The efficient frontier defines the maximum return that can be achieved for a given level of risk or, alternatively, the minimum risk that must be incurred to earn a given return. Traditionally, market risk has been measured by the variance (or standard deviation) of portfolio returns, and this measure is now widely used for credit risk management as well. For example, in the popular Credit‐Metrics methodology (J.P. Morgan [1997]), the standard deviation of credit losses is used to compute the marginal risk and risk contribution of an obligor. Kealhofer [1998] also uses standard deviation to measure the marginal risk and, further, discusses the application of mean‐variance optimization to compute efficient portfolios. While this is reasonable when the distribution of gains and losses is normal, variance is an inappropriate measure of risk for the highly skewed, fat‐tailed distributions characteristic of portfolios that incur credit risk. In this case, quantile‐based measures that focus on the tail of the loss distribution more accurately capture the risk of the portfolio. In this article, we construct credit risk efficient frontiers for a portfolio of bonds issued in emerging markets, using not only the variance but also quantile‐based risk measures such as expected shortfall, maximum (percentile) losses, and unexpected (percentile) losses.

Efficient Risk/Return Frontiers for Credit Risk

This paper evaluates several alternative formulations for minimizing the credit risk of a portfolio of financial contracts with different counterparties. Credit risk optimization is challenging because the portfolio loss distribution is typically unavailable in closed form. This makes it difficult to accurately compute Value-at-Risk (VaR) and expected shortfall (ES) at the extreme quantiles that are of practical interest to financial institutions. Our formulations all exploit the conditional independence of counterparties under a structural credit risk model. We consider various approximations to the conditional portfolio loss distribution and formulate VaR and ES minimization problems for each case. We use two realistic credit portfolios to assess the in- and out-of-sample performance for the resulting VaR- and ES-optimized portfolios, as well as for those which we obtain by minimizing the variance or the second moment of the portfolio losses. We find that a Normal approximation to the conditional loss distribution performs best from a practical standpoint.

http://optlab.mcmaster.ca/index2.php?option=com_docman&task=doc_view&gid=150&Itemid=51

Portfolio credit-risk optimization.

The minimax relative regret solution to a linear programme with interval objective function coefficients can be found using an algorithm that, at each iteration, solves a linear programme to generate a candidate solution and a mixed integer programme (MIP) to find the corresponding maximum regret. This paper first shows that there exists a regret-maximising solution in which all uncertain costs are at a bound, and then uses this to derive a MIP formulation that maximises the regret of a candidate solution. Computational experiments demonstrate that this approach is effective for problems with up to 50 uncertain objective function coefficients, significantly improving upon the existing enumerative method.

Minimising the maximum relative regret for linear programmes with interval objective function coefficients

Quantile‐based measures of risk, e.g., value at risk (VaR), are widely used in portfolio risk applications. Increasing attention is being directed toward managing risk, which involves identifying sources of risk and assessing the economic impact of potential trades. This article compares the performance of two quantile‐based VaR estimators commonly applied to assess the market risk of option portfolios and the credit risk of bond portfolios.

Calculating Quantile‐Based Risk Analytics with L‐Estimators

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