Skewness of Returns, Capital Adequacy, and Mortgage Lending

We calibrate a simulation model of credit value-at-risk for mortgage lending to UK experience. Simulations to capture the skewness of returns that might arise in the context of a financial crisis suggest that the IRB calculations of the new Basel Accord can substantially understate prudential capital adequacy. The same model shows that raising capital requirements has only a small impact on bank funding costs. We conclude that Pillar 2 supervisory review should increase capital requirements above IRB levels for secured bank assets—those whose returns can potentially fall furthest, relative to other, normally “riskier” assets, in extreme outcomes. JEL classification: G21, G28, R31. Presented at the December 2003 conference at the University of Tor Vegata, Rome. We are grateful for comments from William Lang, Mario Onarato, Larry Wall, and from an anonymous referee. All errors and omissions are our own responsibility. “The lady doth protest too much, methinks. The Queen's response to the players in Hamlet, Act 3, scene 2.     

A simple measure of skewness

Groeneveld (1986) in discussing the skewness for the Weibull family has pointed out the shortcomings of the classical measures of asymmetry—the standardized third moment and the Pearson measure of skewness. He has shown that a modified form of the Pearson measure b₃= (μ-m)/E|X-m| portrays the skewness of Weibull family quite well. We give another competitive measure of skewness T that is easy to interpret and is based on conditional expectations. The proposed measure satisfies the desirable properties of a skewness measure.     

Portfolio selection with skewness: A multiple-objective approach

In the presence of skewness, the portfolio selection entails considering competing and conflicting objectives, such as maximizing both its expected returns and skewness, and minimizing its risk for decreasing absolute risk-aversion investors. Since it is unlikely that a portfolio can solve the multiple-objectives problem simultaneously, a portfolio selection must depend on the investor's preference among objectives. This article shows that investor preference can be incorporated into a polynomial goal programming problem from which a portfolio selection with skewness is determined. An inefficient mean-variance portfolio may be optimal in the mean-variance-skewness content. The features of applying polynomial goal programming in portfolio selection are 1) the existence of an optimal solution, 2) the flexibility of the incorporation of investor preference, and 3) the relative simplicity of computational requirements.     

Shortfall minimization and the Naive (1/N) portfolio: an out-of-sample comparison

Naive portfolio selection, wherein an investor allocates an equal portion of their wealth to the field of candidate assets, is a simple ad-hoc way to create a portfolio. Naive portfolio selection contrasts to the many sophisticated portfolio selection rules that are optimal with respect to a specific portfolio allocation objective and which often perform well in sample. However, some recent research finds that many of these ‘optimal’ portfolio allocation mechanisms perform no better than naive diversification in out-of-sample data. This paper extends this line of inquiry by comparing the out-of-sample performance of naive portfolio selection to several recently developed shortfall-minimizing portfolio selection methods. The results corroborate the prior findings that optimal portfolio methods struggle to beat the naive portfolio in out-of-sample environments.     

The negative news threshold—An explanation for negative skewness in stock returns

Abstract

A vast literature documents negative skewness in stock index return distributions on several markets. In this paper the issue of negative skewness is approached from a different angle to previous studies by combining the Trueman's 1997 model of management disclosure practices with symmetric market responses in order to explain negative skewness in stock returns. Empirical tests reveal that returns for days when non-scheduled news items are disclosed are the source of negative skewness in stock returns, as predicted. These findings suggest that negative skewness in stock returns is induced by asymmetries in the news disclosure policies of firm management. Furthermore, it is found that the returns are negatively skewed only for non-scheduled firm-specific news disclosures for firms where the management is compensated with stock options.     

Skewed background risks and higher-order risk preferences: prudent versus temperate behavior

In recent economic experiments, lotteries with left-skewed background risks elicit more prudent choices than lotteries with right-skewed background risks. In this letter, we use an expected utility framework to show that a skewed zero-mean background risk may induce not only prudent but also temperate behaviour. We suggest that the experimental findings could also be due to temperance, rather than merely to prudence.     

Karl Pearson in Russian Contexts

The confluence of statistics and probability into mathematical statistics in the Russian Empire through the interaction, 1910–1917, of A.A. Chuprov and A.A. Markov was influenced by the writings of the English Biometric School, especially those of Karl Pearson. The appearance of the Russian-language exposition of Pearsonian ideas by E. E. Slutsky in 1912 was instrumental in this confluence. Slutsky's predecessors in such writings (Lakhtin, Orzhentskii, and Leontovich) were variously of mathematical, political economy, and biological backgrounds. Work emanating from the interpolational nature of Pearson's system of frequency curves was continued subsequently through the work of Markov, Bernstein, Romanovsky, and Kravchuk (Krawtchouk), who laid a solid probabilistic foundation. The correlational nature in the interpolational early work of Chebyshev, and work of the English Biometric School in the guise of linear least-squares fitting exposited as the main component of Slutsky's book, was developed in population as well as sample context by Chuprov. He also championed the expectation operation in providing exact relations between sample and population moments, in direct interaction with Karl Pearson. Romanovsky emerges as the most adaptive and modern mathematical statistician.     

A one-factor volatility smile model with closed-form solutions for European options

The common practice of using different volatilities for options of different strikes in the Black-Scholes (1973) model imposes inconsistent assumptions on underlying securities. The phenomenon is referred to as the volatility smile. This paper addresses this problem by replacing the Brownian motion or, alternatively, the Geometric Brownian motion in the Black-Scholes model with a two-piece quadratic or linear function of the Brownian motion. By selecting appropriate parameters of this function we obtain a wide range of shapes of implied volatility curves with respect to option strikes. The model has closed-form solutions for European options, which enables fast calibration of the model to market option prices. The model can also be efficiently implemented in discrete time for pricing complex options.
G1     

L‐moments skewness and kurtosis as measures of regional convergence and cohesion

The paper deals with the statistical modeling of convergence and cohesion over time with the use of kurtosis, skewness and L‐moments. Changes in the shape of the distribution related to the spatial allocation of socio‐economic phenomena are considered as an evidence of global shift, divergence or convergence. Cross‐sectional time‐series statistical modeling of variables of interest is to overpass the minors of econometric theoretical models of convergence and cohesion determinants. L‐moments perform much more stable and interpretable than classical measures. Empirical evidence of panel data proves that one pure pattern (global shift, polarization or cohesion) rarely exists and joint analysis is required.