Rosella Giacometti

Stable distributions in the Black-Litterman approach to asset allocation

The integration of quantitative asset allocation models and the judgment of portfolio managers and analysts (i.e. qualitative view) dates back to a series of papers by Black and Litterman in the early 1990s. In this paper we improve the classical Black–Litterman model by applying more realistic models for asset returns (the normal, the t-student, and the stable distributions) and by using alternative risk measures (dispersion-based risk measures, value at risk, conditional value at risk). Results are reported for monthly data and goodness of the models are tested through a rolling window of fixed size along a fixed horizon. Finally, we find that incorporation of the views of investors into the model provides information as to how the different distributional hypotheses can impact the optimal composition of the portfolio.

Measuring financial risk and portfolio optimization with a non-Gaussian multivariate model

In this paper, we propose a multivariate market model with returns assumed to follow a multivariate normal tempered stable distribution. This distribution, defined by a mixture of the multivariate normal distribution and the tempered stable subordinator, is consistent with two stylized facts that have been observed for asset distributions: fat-tails and an asymmetric dependence structure. Assuming infinitely divisible distributions, we derive closed-form solutions for two important measures used by portfolio managers in portfolio construction: the marginal VaR and the marginal AVaR. We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average, first statistically validating the model based on goodness-of-fit tests and then demonstrating how the marginal VaR and marginal AVaR can be used for portfolio optimization using the model. Based on the empirical evidence presented in this paper, our framework offers more realistic portfolio risk measures and a more tractable method for portfolio optimization.

Aggregation issues in operational risk

In this paper we study copula-based models for aggregation of operational risk capital across business lines in a bank. A commonly used method of summation of the value-at-risk (VaR) measures, which relies on a hypothesis of full correlation of losses, becomes inappropriate in the presence of dependence between business lines and may lead to overestimation of the capital charge. The problem can be further aggravated by the persistence of heavy tails in operational loss data; in some cases, the subadditivity property of VaR may fail and the capital charge becomes underestimated. We use α-stable heavy-tailed distributions to model the loss data and then apply the copula approach in which the marginal distributions are consolidated in the symmetric and skewed Student t-copula framework. In our empirical study, we compare VaR and conditional VaR estimates with those obtained under the full correlation assumption. Our results demonstrate a significant reduction in capital when a t-copula is employed. However, the capital reduction is significantly smaller than in cases where a moderately heavy-tailed or thin-tailed distribution is calibrated to loss data. We also show that, when historical weekly data is used, VaR exhibits the superadditivity property for confidence levels below 94% and that, when the loss distribution approach is used, the superadditivity of VaR is observed at a higher confidence level (98%).

Risk factor analysis and portfolio immunization in the corporate bond market

In this paper we develop a multi-factor model for the yields of corporate bonds. The model allows the analysis of factors which influence the changes in the term structure of corporate bonds. More than 98% of the variability in the corporate bond market is captured by the model, which is then used to develop credit risk immunization strategies. Empirical results are given for the U.S. market using data for the period 1992-1999.

On pricing of Credit spread options

Abstract This paper describes and analyses different pricing models for credit spread options such as Longstaff–Schwartz, Black, Das–Sundaram and Duan (GARCH-based) models. The first two models, Longstaff–Schwartz and Black, assume respectively a mean-reverting dynamic and a lognormal distribution for the spread and are representative of the so-called “spread models”. Such models consider the spread as a unique variable and provide closed form solutions for option pricing. On the contrary Das–Sundaram propose a recursive backward induction procedure to price credit spread options on a bivariate tree, which describes the dynamic of the term structure of forward risk-neutral spread and risk-free rate. This model belongs to the class of structural models, which can be used to price a wider range of credit risk derivatives. Finally, we consider the pricing of credit spread options assuming a discrete time GARCH model for the spread.

Estimating the Joint Probability of Default Using CreditDefault Swap and Bond Data

Systemic default risk—that is, the risk of the simultaneous default of multiple institutions—has caused great concern in the recent past. The aim of this article is to estimate the joint probability of default for multiple financial institutions. Both bond and credit derivative markets convey information on the default process: The former provides information on the marginal, the latter, on the joint default probabilities. We consider the corporate bond and the credit default swap (CDS) markets. The over-the-counter nature of the CDS market implies the presence of counterparty risk, or the risk that the protection seller will fail to fulfill its obligations. The counterparty risk is reflected in the CDS price through the joint default probability of the reference entity and the protection seller. Applying a no-arbitrage argument, we extract from market data forward-looking joint default probabilities of financial institutions operating in the CDS market from January 3, 2005–Mar 15, 2010.

Credit default swaps: implied ratings versus official ones

Recently, in line with the progressive development of the credit derivatives market, the academic research has begun to explore the relationship between Credit Default Swap market and rating events. In this paper, following a market based approach, we calibrate an Implied Rating model on Credit Default Swap market spreads. The non parametric mapping of Implied Ratings is calibrated on a large data set of Credit Default Swap quotes that includes the years of financial turmoils. This allows also to investigate the existence of possible differences between normal and abnormal market conditions. Unlike other models, the one proposed considers a linear penalty function which allows to evaluate market quotes in a neutral way and to formalize a more computationally efficient programming model. We compare the behaviors of credit rating agencies in different markets (EU and USA) and in different sub-periods, in order to analyze whether Implied Rating changes anticipate or follow the effective rating changes supplied by Fitch Ratings, Moody’s and Standard and Poor’s.

A Stochastic Model for Mortality Rate on Italian Data

A new stochastic model for mortality rate is proposed and analyzed on Italian mortality data. The model is based on a stochastic differential equation derived from a generalization of the Milevesky and Promislow model (Milevesky, M.A., Promislow, S.D.: Insur. Math. Econ. 29, 299–318 (2001)). We discuss and present a methodology, based on the discretisation approach by Wymer (Wymer, C.R.: Econometrica 40(3), 565–577 (1972)) to evaluate the parameters of our model. The comparison with the Milevesky and Promislow model shows the relevance of our proposal along an horizon, which includes periods of time with a different volatility of mortality rates. The estimate of the parameters turns out to be stable over time with the exception of the mean reverting parameter, which shows, for a person of a fixed age, an increase over time.

Heavy-tailed distributional model for operational losses

We examine the statistical properties of operational losses obtained from a large European bank using an actuarial-type framework. The simplistic assumption of a Poisson frequency distribution fails and we show that the frequency process follows closely a non-homogeneous Poisson process with a deterministic intensity of the form of a continuous cdf-like function. Further, operational losses are modeled using a variety of distributions. We address the problems of (1) reporting bias; (2) supplementing internal data with external data; (3) tail estimation; and (4) mixing the distributions of the body and the tail, and propose practical solutions to such problems. Finally, our empirical findings are consistent with other studies reporting very heavy-tailed loss distributions with the tail index below unity.

Estimating the probability of multiple EU sovereign defaults using CDS and bond data

The ongoing EU sovereign debt crisis is causing great concern about the sustainability of national debt issued by the member states. In this paper, we propose a methodology to estimate the likelihood of the default of one or more countries in the Euro Area by extending the approach in Pianeti et al. [J. Fixed Income, 2012, 21, 44–58] to the case of multiple defaults. We provide an assessment of the marginal, the joint and the conditional default probabilities within the Euro Zone. The adopted measure of systemic risk is the probability of a joint default of the EU countries over a 5 year time horizon. We find evidence of increasing systemic risk and danger of contagion from early 2007 and more significantly from late 2011 onwards. We show that our measure has predictive ability with respect to the equity market.