Lisa R. Goldberg

Affine Point Processes and Portfolio Credit Risk

This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting and facilitate the description of complex event dependence structures. ODEs characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

A Top-Down Approach to Multiname Credit

A multiname credit derivative is a security that is tied to an underlying portfolio of corporate bonds and has payoffs that depend on the loss due to default in the portfolio. The value of a multiname derivative depends on the distribution of portfolio loss at multiple horizons. Intensity-based models of the loss point process that are specified without reference to the portfolio constituents determine this distribution in terms of few economically meaningful parameters and lead to computationally tractable derivatives valuation problems. However, these models are silent about the portfolio constituent risks. They cannot be used to address applications that are based on the relationship between portfolio and component risks, for example, constituent risk hedging. This paper develops a method that extends these models to the constituents. We use random thinning to decompose the portfolio intensity into a sum of constituent intensities. We show that a thinning process, which allocates the portfolio intensity to constituents, uniquely exists, and is a probabilistic model for the next-to-default. We derive a formula for the constituent default probability in terms of the thinning process and the portfolio intensity, and develop a semi-analytical transform approach to evaluate it. The formula leads to a calibration scheme for the thinning processes and an estimation scheme for constituent hedge sensitivities. An empirical analysis for September 2008 shows that the constituent hedges generated by our method outperform the hedges prescribed by the Gaussian copula model, which is widely used in practice.

Beyond Value at Risk: Forecasting Portfolio Loss at Multiple Horizons

We develop a portfolio risk model that uses high-frequency data to forecast the loss surface, which is the set of loss distributions at future time horizons. Our model uses a fully automated, semi-parametric fitting procedure that has its basis in extreme value statistics. We take account of distributional asymmetry, heavy tails, heteroscedasticity and serial correlation. Loss distributions are time aggregated by taking products of characteristic functions. We test loss-surface-implied forecasts of value at risk and expected shortfall out of sample on a diverse set of portfolios and we compare our forecasts to industry-standard risk forecasts that are based on asset and factor covariance matrices. The empirical results make a compelling case for the application and further development of our approach.

Is There a Green Factor

Climate change has far-reaching implications for the global economy and is increasingly being recognized by investors as a long-term investment theme. As more investors take note of companies that are well positioned to handle climate change, a common factor may account, in part, for the share prices of these companies. The authors search for the existence of this common factor, which they call the green equity factor. During the three-year period from May 2005 through May 2008, the authors find that a sample portfolio of renewable energy stocks outperformed the broad, global MSCI All Country World Index (ACWI), as well as a subindex consisting of traditional energy stocks. Their findings persist after adjusting for size, value, and country biases in the sample portfolio. Furthermore, a systematic analysis based on the Barra Global Equity Model supports the existence of a renewable energy risk factor.

Modeling Credit Risk

Spreads for credit instruments denominated in euros, sterling, and U.S. dollars over their local swap curves are examined here. The findings indicate that monthly spread changes were strongly currency-dependent during the period May 1999–May 2001. Sector-by-rating factor returns are at best weakly correlated across currencies, and U.S. dollar spread returns are generally more volatile than the other two by a factor of two or three. This is contrary to what would be expected from covered interest arbitrage. The conclusion is that analysts should estimate credit factor risk models separately in each market, as risk forecasting models using a single set of spread factors for different markets will not be accurate.

Risk Without Return

Risk-only investment strategies have been growing in popularity as traditional in- vestment strategies have fallen short of return targets over the last decade. However, risk-based investors should be aware of four things. First, theoretical considerations and empirical studies show that apparently dictinct risk-based investment strategies are manifestations of a single effect. Second, turnover and associated transaction costs can be a substantial drag on return. Third, capital diversification benefits may be reduced. Fourth, there is an apparent connection between performance and risk diversification. To analyze risk diversification benefits in a consistent way, we introduce the Risk Diversification Index (RDI) which measures risk concentrations and complements the Herfindahl-Herschman Index (HHI) for capital concentrations.

t-Statistics for Weighted Means in Credit Risk Modelling

Purpose – The purpose of this paper is to describe a generalization of the familiar two-sample t-test for equality of means to the case where the sample values are to be given unequal weights. This is a natural situation in financial risk modeling when some samples are considered more reliable than others in predicting a common mean. We also describe an example with real credit data showing that ignoring this modification of the two-sample test can lead to the wrong statistical conclusion. Design/methodology/approach – We follow the analysis of the classical two-sample tests in the more general situation of weighted means. We also test our methods against some market data to assess the importance of the findings. Findings – We formulate some explicit test statistics that should be used when the sample values are to be assigned differing known weights. Different cases are presented depending on how much is known about the variances. In the most typical case (the unpooled two-sample test), we approximate the test statistic with a t-distribution. Proofs are given where possible. Research limitations/implications – In the unpooled case, we still only have an approximate t-distribution. This is related to the classical Behrens-Fisher problem, which is still not fully solved. We also focus on the case where the sample values are normally distributed. It would be valuable to see how far the discussion can be extended to non-normal distributions. Practical implications – Researchers should use the two-sample test statistics given in this paper instead of the standard ones when testing for equality of weighted means. Originality/value – Weighted means occur frequently in situations when the credibility or reliability of data vary. However, standard tests for equality of means do not take weights into account. These results will be of value to any researchers studying statistical means of data of varying reliability, such as corporate bond spreads.

Determinants of Levered Portfolio Performance

Working Paper # 2013- 01 The Decision to Lever Robert M. Anderson, University of California at Berkeley Stephen W. Bianchi, University of California at Berkeley Lisa R. Goldberg, University of California at Berkeley July 1, 2013 University of California BerkeleyThe cumulative return to a levered strategy is determined by ve elements that t together in a simple, useful formula. A previously undocumented element is the covariance between leverage and excess return to the fully invested source portfolio underlying the strategy. In an empirical study of volatility-targeting strategies over the 84-year period 1929{2013, this covariance accounted for a reduction in return that substantially diminished the Sharpe ratio in all cases.

Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA

A financial institution’s counterparty credit exposures may be correlated with the credit quality of a counterparty; wrong way risk refers to the case where this correlation is negative. Hull and White [9] are the first to model wrong way risk in Credit Value Adjustment (CVA) calculations by expressing the counterparty’s default intensity in terms of the financial institution’s credit exposure to the counterparty. We derive a formula for CVA for a class of models that includes the formulation of Hull and White [9], and we show that wrong way risk does not affect the credit quality of the counterparty. We provide numerical examples based on the Hull and White [9] formulation to estimate CVA for forward contracts and European options. These examples demonstrate that independent CVA can exceed wrong way CVA. This is inconsistent with the scalar multiples of independent CVA that have been adopted by regulators as a proxy for wrong way CVA.

Central Limits and Financial Risk

Systematic model bias has been implicated in the global recession that began in 2007, and this bias can be traced back to assumptions about the normality of data. Nonetheless, the normal distribution continues to play a foundational role in quantitative finance. One reason for this is that the normal often emerges, without prompting, as the distribution of sums or averages of large collections of random variables. Precise statements of this miracle are known as Central Limit Theorems, and they appear throughout the physical and social sciences. In this note, we review some of the most widely-used Central Limit Theorems. Subsequently, we explore the gap between the normal distribution and financial risk. This can be traced to a failure of the financial data to satisfy the assumptions of even the most liberal versions of the Central Limit Theorem.