On measuring nonlinear risk with scarce observations

We propose a methodology for estimating the risk of portfolios that exhibit nonlinear dependence on the risk driving factors and have scarce observations, which is typical for portfolios of investments in hedge funds. The methodology consists of two steps: first, regressing the portfolio return on nonlinear functions of each single risk driving factor and second, merging together the obtained estimates taking into account the dependence between different factors. Performing the second step leads us to a certain probabilistic problem, for which we propose an analytic and computationally feasible solution for the case where the joint law of the factors is a Gaussian copula. A typical practical application can be to estimate the risk of a hedge fund or a portfolio of hedge funds. As a theoretical consequence of our results, we propose a new definition of the factor risk, i.e., the risk of a portfolio brought by a given factor.     

Exploring risk premium factors for country equity returns

In this paper, we study a comprehensive set of risk premia of country equity returns for 45 countries over the sample period 2002 - 2018 in both a single and a multiple factor setting. Using a new three-pass estimation method for factor risk premia by Giglio and Xiu (2021), we find that several factors, including default risk, are also priced in country equity excess returns, controlled by the Fama–French 5-factor and Carhart models. Moreover, we apply a novel approach to investigate the multi-factor impact on country equity returns. We find that the multi-factor information, constructed from the first principal component of the statistically significant single factors, provides a consistent and stronger prediction of anomalies in country equity returns.     

How do global investors differentiate between sovereign risks? The new normal versus the old

When global investors go into emerging markets or get out of them, how do they differentiate between economies? Has this behavior changed since the crisis of 2008 to reflect a “new normal”? We consider these questions by focusing on sovereign risk as reflected in monthly returns on credit default swaps (CDS) for 18 emerging markets and 10 developed countries. Tests for breaks in the time series of such returns suggest a new normal that ensued around October 2008 or soon afterwards. Dividing the sample into two periods and extracting risk factors from CDS returns, we find an old normal in which a single global risk factor drives half of the variation in returns and a new normal in which that risk factor becomes even more dominant. Surprisingly, in both the old and new normal, the way countries load on this factor depends not so much on economic fundamentals as on whether they are designated an emerging market.     

Equilibrium asset pricing: with non-Gaussian factors and exponential utilities

We analyse the equilibrium asset pricing implications for an economy with single period return exposures to explicit non-Gaussian systematic factors, that may be both skewed and long-tailed, and Gaussian idiosyncratic components. Investors maximize expected exponential utility and equilibrium factor prices are shown to reflect exponentially tilted prices for non-Gaussian factor risk exposures. It is shown that these prices may be directly estimated from the univariate probability law of the factor exposure, given an estimate of average risk aversion in the economy. In addition, a residual form of the capital asset pricing model continues to hold and prices the idiosyncratic or Gaussian risks. The theory is illustrated on data for the US economy using independent components analysis to identify the factors and the variance gamma model to describe the probability law of the non-Gaussian factors. It is shown that the residual CAPM accounts for no more than 1% of the pricing of risky assets, while the exponentially tilted systematic factor risk exposures account for the bulk of risky asset pricing.     

债券信用溢价的共同因子与结构变化:基于机器学习方法

共同因子是刻画风险溢价的重要基础,将共同因子模型应用于公司债券市场有助于合理估计信用风险溢价.本文利用机器学习算法探究债券信用溢价因子的存在性以及结构变化后发现:规模、下行风险、价值、波动率以及流动性等五个公司债券共同因子对单个债券信用溢价有较好的解释能力,动量因子对信用溢价的解释能力较差,流动性因子具有较强的逆周期防...     

Quantile regression analysis of hedge fund strategies

Extending previous work on hedge fund pricing, this paper introduces the idea of modelling the conditional quantiles of hedge fund returns using a set of risk factors. Quantile regression analysis provides a way of understanding how the relationship between hedge fund returns and risk factors changes across the distribution of conditional returns. We propose a Bayesian approach to model comparison which provides posterior probabilities for different risk factor models that can be used for model averaging. The most relevant risk factors are identified for different quantiles and compared with those obtained for the conditional expectation model. We find differences in factor effects across quantiles of returns, which suggest that the standard conditional mean regression method may not be adequate for uncovering the risk-return characteristics of hedge funds. We explore potential economic impacts of our approach by analysing hedge fund single strategy return series and by constructing style portfolios.     

A comparison of single factor Markov-functional and multi factor market models

We compare single factor Markov-functional and multi factor market models and the impact of their correlation structures on the hedging performance of Bermudan swaptions. We show that hedging performance of both models is comparable, thereby supporting the claim that Bermudan swaptions can be adequately risk-managed with single factor models. Moreover, we show that the impact of smile can be much larger than the impact of correlation. We use the constant exercise method for calculating risk sensitivities of callable products in market models, which is a modification of the least-squares Monte Carlo method. The hedge results show the constant exercise method enables proper functioning of market models as risk-management tools.     

The Simple Analytics of a Pooled Annuity Fund

This article provides a formal analysis of payout adjustments from a longevity risk‐pooling fund, an arrangement we refer to as group self‐annuitization (GSA). The distinguishing risk diffusion characteristic of GSAs in the family of longevity insurance instruments is that the annuitants bear their systematic risk, but the pool shares idiosyncratic risk. This obviates the need for an insurance company, although such instruments could be sold through a corporate insurer. We begin by deriving the payout adjustment for a single entry group with a single annuity factor and constant expectations. We then show that under weak requirements a unique solution to payout paths exists when multiple cohorts combine into a single pool. This relies on the harmonic mean of the ratio of realized to expected survivorship rates across cohorts. The case of evolving expectations is also analyzed. In all cases, we demonstrate that the periodic‐benefit payment in a pooled annuity fund is determined based on the previous payment adjusted for any deviations in mortality and interest from expectations. GSA may have considerable appeal in countries which have adopted national defined contribution schemes and/or in which the life insurance industry is noncompetitive or poorly developed.     

When Will Mean-Variance Efficient Portfolios Be Well Diversified?

We characterize the conditions under which efficient portfolios put small weights on individual assets. These conditions bound mean returns with measures of average absolute covariability between assets. The bounds clarify the relationship between linear asset pricing models and well-diversified efficient portfolios. We argue that the extreme weightings in sample efficient portfolios are due to the dominance of a single factor in equity returns. This makes it easy to diversify on subsets to reduce residual risk, while weighting the subsets to reduce factor risk simultaneously. The latter involves taking extreme positions. This behavior seems unlikely to be attributable to sampling error.     

Capital Share Risk in U.S. Asset Pricing

A single macroeconomic factor based on growth in the capital share of aggregate income exhibits significant explanatory power for expected returns across a range of equity characteristic portfolios and nonequity asset classes, with risk price estimates that are of the same sign and similar in magnitude. Positive exposure to capital share risk earns a positive risk premium, commensurate with recent asset pricing models in which redistributive shocks shift the share of income between the wealthy, who finance consumption primarily out of asset ownership, and workers, who finance consumption primarily out of wages and salaries.     

A Cautionary Note on Natural Hedging of Longevity Risk

In this article, we examine the so-called natural hedging approach for life insurers to internally manage their longevity risk exposure by adjusting their insurance portfolio. In particular, unlike the existing literature, we also consider a nonparametric mortality forecasting model that avoids the assumption that all mortality rates are driven by the same factor(s).

Our primary finding is that higher order variations in mortality rates may considerably affect the performance of natural hedging. More precisely, although results based on a parametric single factor model—in line with the existing literature—imply that almost all longevity risk can be hedged, results are far less encouraging for the nonparametric mortality model. Our finding is supported by robustness tests based on alternative mortality models.     

Economic capital for nonperforming loans

A portfolio of nonperforming loans requires economic capital. We present two models for forecasting the portfolio loss and its probability distribution. In the first model, the loss for each nonperforming loan entails a change in provision over the risk horizon. The risk determinants are the single-name concentration, measured by the Herfindahl–Hirschmann index, as well as a systematic factor and the idiosyncratic risk. Our second model allows for interportfolio diversification with a portfolio of performing loans because banks typically own both performing and nonperforming loans. In this model, the nonperforming loan is identified with its systematic risk. Both models allow for closed-form expressions of economic capital and for the capital charge of the single loan. We calibrate the macroeconomic model parameters statistically with a loss panel; the microeconomic parameters depend on the portfolio. The portfolio risk for nonperforming loans mainly depends on the volatility of the systematic economic factor. The dependence becomes more pronounced when interportfolio diversification is taken into account. The magnitude of interportfolio diversification is also marked. Finally, we calculate regulatory capital charges according to Basel II for past-due loans. The regulatory charges are on average smaller than our economic charges and, additionally, take the volatility of economic activity into account only implicitly.     

Factor decomposition of the Eurozone sovereign CDS spreads

In the present study, we examine the factors driving Eurozone sovereign credit default swap (CDS) spreads during the Eurozone sovereign debt crisis. For identifying factors we utilize independent component analysis (ICA), a technique similar to principal component analysis (PCA). We identify three factors that impact spreads and capture the features specific to the crisis such as the breakup risk of the Eurozone: peripheral factor, global factor, and Eurozone common factor. In contrast, when PCA is applied, only a single factor is identified. Moreover, using ICA with a GARCH model, we show that the source of volatility for CDS spreads shifted from the global factor in 2009 and the peripheral factor in 2010 to the Eurozone common factor in 2012, and that the dynamic correlation reflects the decoupling between low credit risk countries such as Germany and high credit risk countries such as Greece. We also show that the goodness-of-fit of the ICA-based model is better than other models used such as the Student's t copula model.     

Asset liquidity,business risk,and beta

Extant literature posits that because of leverage, equity beta estimates from a single factor capital asset pricing model based on an equity-only market index are biased. We show analytically that this leverage bias is intimately related to the firm's asset structure per se, the firm's asset liquidity (i.e., cash holdings) and business risk. This is mainly because riskless cash holdings and risky real assets jointly determine the relevant risk for asset pricing. We empirically confirm that asset liquidity and business risk can marginally explain the leverage bias in the cross-section of stocks returns.     

Measuring Basis Risk in Longevity Hedges

Abstract

In examining basis risk in index longevity hedges, it is important not to ignore the dependence between the population underlying the hedging instrument and the population being hedged. We consider four extensions to the Lee-Carter model that incorporate such dependence: Both populations are jointly driven by the same single time-varying index, the two populations are cointegrated, the populations depend on a common age factor, and there is an augmented common factor model in which a population-specific time-varying index is added to the common factor model with the property that it will tend toward a certain constant level over time. Using data from the female populations of Canada and the United States, we show the augmented common factor model is preferred in terms of both goodness-of-fit and ex post forecasting performance. This model is then used to quantify the basis risk in a longevity hedge of 65-year old Canadian females structured using a portfolio of q-forward contracts predicated on U.S. female population mortality. The hedge effectiveness is estimated at 56% on the basis of longevity value-at-risk and 81.61% on the basis of longevity risk reduction.     

A new family of equity style indices and mutual fund performance: Do liquidity and idiosyncratic risk matter?

We propose and test novel multifactor models of daily mutual fund performance. To this aim, we set up equity style indices and derive risk factors, which nest the established Fama and French (1992) and Carhart (1997) factors. We add two additional risk factors, namely idiosyncratic risk and Amihud (2002) liquidity. Our sample contains 528 actively managed mutual funds with European stock market focus during 2002 to 2009. Model estimation reveals that—while market excess return and size appear significant for the cross-section of all funds—the remainder factors explain the performance of subsets of funds. About one third of the funds exhibit significant factor sensitivities not only with respect to valuation or momentum, but also with respect to liquidity or idiosyncratic risk. No single risk factor is dominated and hence our six factor model may serve as a valid performance benchmark. In a four factor model setting, the Carhart model and a model with valuation replaced by liquidity perform best. Our results remain stable under various robustness checks. We further document that managers on average prefer liquid stocks, show no aggregate idiosyncratic risk preference and deliver results that are consistent with equilibrium models of fund performance.     

The multi-layer network nature of systemic risk and its implications for the costs of financial crises

The inability to see and quantify systemic financial risk comes at an immense social cost. Systemic risk in the financial system arises to a large extent as a consequence of the interconnectedness of its institutions, which are linked through networks of different types of financial contracts, such as credit, derivatives, foreign exchange, and securities. The interplay of the various exposure networks can be represented as layers in a financial multi-layer network. In this work we quantify the daily contributions to systemic risk from four layers of the Mexican banking system from 2007 to 2013. We show that focusing on a single layer underestimates the total systemic risk by up to 90%. By assigning systemic risk levels to individual banks we study the systemic risk profile of the Mexican banking system on all market layers. This profile can be used to quantify systemic risk on a national level in terms of nation-wide expected systemic losses. We show that market-based systemic risk indicators systematically underestimate expected systemic losses. We find that expected systemic losses are up to a factor of four higher now than before the financial crisis of 2007–2008. We find that systemic risk contributions of individual transactions can be up to a factor of one thousand higher than the corresponding credit risk, which creates huge risks for the public. We find an intriguing non-linear effect whereby the sum of systemic risk of all layers underestimates the total risk. The method presented here is the first objective data-driven quantification of systemic risk on national scales that reveal its true levels.     

Capital Allocation for Portfolio Credit Risk

Capital allocation rules are derived that maximize leverage while maintaining a target solvency rate for credit portfolios where risk is driven by a single common factor and idiosyncratic risk is fully diversified. Equilibrium conditions ensure that capital allocations depend on interest earnings as well as credits’ probability of default, endogenous loss given default, and asset correlation. Capitalization rates exceed those estimated using Gaussian credit loss models. Results demonstrate that credit risk is undercapitalized by the Basel II AIRB approach in part because of ambiguities regarding the definition of loss given default. An alternative proposed capital rule removes this bias.     

Expected Returns,Time-varying Risk,and Risk Premia

A new empirical model for intertemporal capital asset pricing is presented that allows both time-varying risk premia and betas where the latter are identified from the dynamics of the conditional covariance of returns. The model is more successful in explaining the predictable variations in excess returns when the returns on the stock market and corporate bonds are included as risk factors than when the stock market is the single factor. Although changes in the covariance of returns induce variations in the betas, most of the predictable movements in returns are attributed to changes in the risk premia.     

Smart defaults: Determining the number of default funds in a pension scheme

We propose a new methodology for the smart design of the default investment fund(s) in occupational defined contribution pension schemes based on the observable characteristics of scheme members. Using a unique dataset of member risk attitudes and characteristics from a survey of a large UK pension scheme, we apply factor analysis to identify single factors for risk aversion, risk capacity and ethical investment preferences, and then apply cluster analysis to these factors to identify two distinct groups of members across age cohorts. We find membership of these clusters depends on a number of personal characteristics, with the principal differentiating feature being that one group had previously engaged with the pension scheme, while the other had not. These identified characteristics can be utilised in the design of smart default funds, including appropriate engagement strategies.