Portfolio performance evaluation with loss aversion

In this paper we consider a loss-averse investor equipped with a specific, but still quite general, utility function motivated by behavioral finance. We show that, under certain concrete assumptions concerning the form of this utility, one can derive closed-form solutions for the investor's portfolio performance measure. We investigate the effects of loss aversion and demonstrate its important role in performance measurement. The framework presented in this paper also provides a sound theoretical foundation for all known performance measures based on partial moments of the distribution.     

Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model

We develop a two-factor general equilibrium model of the term structure. The factors are the short-term interest rate and the volatility of the short-term interest rate. We derive closed-form expressions for discount bonds and study the properties of the term structure implied by the model. The dependence of yields on volatility allows the model to capture many observed properties of the term structure. We also derive closed-form expressions for discount bond options. We use Hansen's generalized method of moments framework to test the cross-sectional restrictions imposed by the model. The tests support the two-factor model.     

Portfolio performance evaluation with generalized Sharpe ratios: Beyond the mean and variance

This paper presents a theoretically sound portfolio performance measure that takes into account higher moments of distribution. This measure is motivated by a study of the investor’s preferences to higher moments of distribution within Expected Utility Theory and an approximation analysis of the optimal capital allocation problem. We show that this performance measure justifies the notion of the Generalized Sharpe Ratio (GSR) introduced by Hodges (1998). We present two methods of practical estimation of the GSR: nonparametric and parametric. For the implementation of the parametric method we derive a closed-form solution for the GSR where the higher moments are calibrated to the normal inverse Gaussian distribution. We illustrate how the GSR can mitigate the shortcomings of the Sharpe ratio in resolution of Sharpe ratio paradoxes and reveal the real performance of portfolios with manipulated Sharpe ratios. We also demonstrate the use of this measure in the performance evaluation of hedge funds.     

Gini-type measures of risk and variability: Gini shortfall,capital allocations,and heavy-tailed risks

We introduce and explore Gini-type measures of risk and variability, and develop the corresponding economic capital allocation rules. The new measures are coherent, additive for co-monotonic risks, convenient computationally, and require only finiteness of the mean. To elucidate our theoretical considerations, we derive closed-form expressions for several parametric families of distributions that are of interest in insurance and finance, and further apply our findings to a risk portfolio of a bancassurance company.     

Versicherungsnachfrage und Ausfallerwartung

Recently, several undesirable feature of classical risk measure like variance or value-at-risk have been mentioned in the literature. Due to this drawbacks, alternative risk measures as the lower partial moments receive increasing attention. The present paper analyzes the lower partial moments with respect to their compatibility to expected utility theory as well as their psychological plausibility. With respect to the latter point, shortfall expectation seems to be more realistic than the other lower partial moments. We proceed by analyzing demand for coinsurance in the context of risk measurement by shortfall expectation.     

Analytic bond pricing for short rate dynamics evolving on matrix Lie groups

We provide closed-form expressions for bond prices in interest rate models based on compact Lie groups. Our approach uses a Doob transform technique and PDE solutions by the Mathieu periodic functions. As a by-product, we derive formulas for bond option prices as well as new identities for the Laplace transform of periodic functionals of Brownian motion and Brownian diffusion processes.     

Singular risk-neutral valuation equations

Many risk-neutral pricing problems proposed in the finance literature do not admit closed-form expressions and have to be dealt with by solving the corresponding partial integro-differential equation. Often, these PIDEs have singular diffusion matrices and coefficients that are not Lipschitz-continuous up to the boundary. In addition, in general, boundary conditions are not specified. In this paper, we prove existence and uniqueness of (continuous) viscosity solutions for linear PIDEs with all the above features, under a Lyapunov-type condition. Our results apply to European and Asian option pricing, in jump-diffusion stochastic volatility and path-dependent volatility models. We verify our Lyapunov-type condition in several examples, including the arithmetic Asian option in the Heston model.     

Optimal investment and risk control for an insurer with partial information in an anticipating environment

Abstract

This paper considers an optimal investment and risk control problem under the criterion of logarithm utility maximization. The risky asset process and the insurance risk process are described by stochastic differential equations with jumps and anticipating coefficients. The insurer invests in the financial assets and controls the number of policies based on some partial information about the financial market and the insurance claims. The forward integral and Malliavin calculus for Lévy processes are used to obtain a characterization of the optimal strategy. Some special cases are discussed and the closed-form expressions for the optimal strategies are derived.     

Approximate option pricing and hedging in the CEV model via path-wise comparison of stochastic processes

This paper presents a methodology of finding explicit boundaries for some financial quantities via comparison of stochastic processes. The path-wise comparison theorem is used to establish domination of the stock price process by a process with a known distribution that is relatively simple. We demonstrate how the comparison theorem can be applied in the constant elasticity of variance model to derive closed-form expressions for option price bounds, an approximate hedging strategy and a conditional value-at-risk estimate. We also provide numerical examples and compare precision of our method with the distribution-free approach.     

Polynomial goal programming and the implicit higher moment preferences of US institutional investors in hedge funds

Polynomial goal programming (PGP) is a flexible method that allows investor preferences for different moments of the return distribution of financial assets to be included in the portfolio optimization. The method is intuitive and particularly suitable for incorporating investor preferences in higher moments of the return distribution. However, until now, PGP has not been able to meet its full potential because it requires quantification of “real” preference parameters towards those moments. To date, the chosen preference parameters have been selected somewhat “arbitrarily”. Our goal is to calculate implied sets of preference parameters using investors’ choices of and the importance they attribute to risk and performance measures. We use three groups of institutional investors—pension funds, insurance companies, and endowments—and derive implied sets of preference parameters in the context of a hedge fund portfolio optimization. To determine “real” preferences for the higher moments of the portfolio return distribution, we first fit implied preference parameters so that the PGP optimal portfolio is identical to the desired hedge fund portfolio. With the obtained economically justified sets of preference parameters, the well-established PGP framework can be employed more efficiently to derive allocations that satisfy institutional investor expectations for hedge fund investments. Furthermore, the implied preference parameters enable fund of hedge fund managers and other investment managers to derive optimal portfolio allocations based on specific investor expectations. Moreover, the importance of individual moments, as well as their marginal rates of substitution, can be assessed.     

Pricing Options with Extendible Maturities: Analysis and Applications

Many common types of financial contracts incorporate options with extendible maturities. This paper derives closed-form expressions for options that can be extended by the optionholder and presents a number of applications including the valuation of American options with stochastic dividends, junk bonds, and shared-equity mortgages. We also derive closed-form expressions for writer-extendible options and discuss the writer's economic incentives for extending an out-of-the-money option. We apply these results to show that corporate debtholders have a strong incentive to extend the maturity of defaulting debt if there are liquidation costs. We model and solve the debtholders' optimal extension problem and show that the possibility of an extension can induce shareholders in highly levered firms to accept negative NPV projects.     

Strategic Debt Service

When firms experience financial distress, equity holders may act strategically, forcing concessions from debtholders and paying less than the originally-contracted interest payments. This article incorporates strategic debt service in a standard, continuous time asset pricing model, developing simple closed-form expressions for debt and equity values. We find that strategic debt service can account for a substantial proportion of the premium on risky corporate debt. We analyze the efficiency implications of strategic debt service, showing that it can eliminate both direct bankruptcy costs and agency costs of debt.     

Ruin probabilities in classical risk models with gamma claims

In this paper, we provide three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag–Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.     

Asymmetry,tail risk and time series momentum

In this paper, we investigate how to improve the time series momentum strategy by using partial moments. We find that reversals of time series momentum can be partly predicted by tail-distributed upper and lower partial moments derived from daily returns of commodity futures. Based on such information, we propose rule-based approaches to improve the trading signals suggested by the time series momentum strategy. The empirical results based on Chinese commodity futures document statistically significant improvements of the Sharpe ratio in the out-of-sample period. These improvements are robust to different look-back windows.     

On the Asymmetric U‐Shaped Relationship between Inflation,Inflation Uncertainty,and Relative Price Skewness in the UK

We investigate the asymmetric relationships between aggregate inflation and the second and third moments of the cross‐sectional distribution of relative prices using a modified Calvo pricing model with regime‐dependent price rigidities. Calibration experiments reveal that the inflation‐standard deviation and inflation‐skewness relationships exhibit U‐shaped asymmetries around the historical mean inflation rate. UK sectoral data support our results. We conclude that monetary policy should target an inflation rate proximate to the (common) minima of these nonlinear relationships and that core inflation measures should not be used for policy purposes as they exclude much of the information contained in the higher moments.     

Orthogonal expansions for VIX options under affine jump diffusions

In this work we derive new closed-form pricing formulas for VIX options in the jump-diffusion SVJJ model proposed by Duffie et al. [Econometrica, 2000, 68, 1343–1376]. Our approach is based on the classic methodology of approximating a density function with an orthogonal expansion of polynomials weighted by a kernel. Orthogonal expansions based on the Gaussian distribution, such as Edgeworth or Gram–Charlier expansions, have been successfully employed by a number of authors in the context of equity options. However, these expansions are not quite suitable for volatility or variance densities as they inherently assign positive mass to the negative real line. Here we approximate option prices via expansions that instead are based on kernels defined on the positive real line. Specifically, we consider a flexible family of distributions, which generalizes the gamma kernel associated with the classic Laguerre expansions. The method can be employed whenever the moments of the underlying variance distribution are known. It provides fast and accurate price computations, and therefore it represents a valid and possibly more robust alternative to pricing techniques based on Fourier transform inversions.     

Measuring systemic risk in the European banking sector: a copula CoVaR approach

We propose a new methodology based on copula functions to estimate CoVaR, the Value-at-Risk (VaR) of the financial system conditional on an institution being under financial distress. Our Copula CoVaR approach provides simple, closed-form expressions for various definitions of CoVaR for a broad range of copula families and allows the CoVaR of an institution to have time-varying exposure to its VaR. We extend this approach to estimate other ‘co-risk’ measures such as Conditional Expected Shortfall (CoES). We focus on a portfolio of large European banks and examine the existence of common market factors triggering systemic risk episodes. Further, we analyse the extent to which bank-specific characteristics such as size, leverage, and equity beta are associated with institutions' contribution to systemic risk and highlight the importance of liquidity risk at the outset of the financial crisis in summer 2007. Finally, we investigate the link between macroeconomy and systemic risk and find that changes in major macroeconomic variables can contribute significantly to systemic risk.     

An approximate multi-period Vasicek credit risk model

Financial institutions and regulators usually measure credit risk only over a one-year time horizon. Hence, current statistical models can generate closed-form expressions for the one-year loss distribution. Losses over longer horizons are considered using scenario analysis or Monte Carlo simulation. This paper proposes a simple multi-period credit risk model and uses Taylor expansion approximations to estimate the multi-period loss distribution. In this paper we extend the currently available second-order Taylor expansion approximations to credit risk with a third-order term and we use this new approximation to obtain the loss distribution in the multi-period framework. Our results show that the approximation is more accurate under recessions or for portfolios with high probability of default. We also show that, in general, the effect of this third-order adjustment is quite small.