MEASURING DISTRIBUTION MODEL RISK

We propose to interpret distribution model risk as sensitivity of expected loss to changes in the risk factor distribution, and to measure the distribution model risk of a portfolio by the maximum expected loss over a set of plausible distributions defined in terms of some divergence from an estimated distribution. The divergence may be relative entropy or another f‐divergence or Bregman distance. We use the theory of minimizing convex integral functionals under moment constraints to give formulae for the calculation of distribution model risk and to explicitly determine the worst case distribution from the set of plausible distributions. We also evaluate related risk measures describing divergence preferences.     

UNIVERSAL SEMICONSTANT REBALANCED PORTFOLIOS

In this paper, we investigate investment strategies that can rebalance their target portfolio vectors at arbitrary investment periods. These strategies are called semiconstant rebalanced portfolios in Blum and Kalai and Helmbold et al. Unlike a constant rebalanced portfolio, which must rebalance at every investment interval, a semiconstant rebalanced portfolio rebalances its portfolio only on selected instants. Hence, a semiconstant rebalanced portfolio may avoid rebalancing if the transaction costs outweigh the benefits of rebalancing. In a competitive algorithm framework, we compete against all such semiconstant portfolios with an arbitrary number of rebalancings and corresponding rebalancing instants. We investigate this framework with and without transaction costs and demonstrate sequential portfolios that asymptotically achieve the wealth of the best semiconstant rebalanced portfolios whose number of rebalancings and instants of rebalancings are tuned to the individual sequence of price relatives.     

Cover's universal portfolio,stochastic portfolio theory,and the numéraire portfolio

Cover's celebrated theorem states that the long‐run yield of a properly chosen “universal” portfolio is almost as good as that of the best retrospectively chosen constant rebalanced portfolio. The “universality” refers to the fact that this result is model‐free, that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numéraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model‐free result is complemented by a comparison with the numéraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.     

A MODEL OF OPTIMAL CONSUMPTION UNDER LIQUIDITY RISK WITH RANDOM TRADING TIMES

We consider a portfolio/consumption choice problem in a market model with liquidity risk. The main feature is that the investor can trade and observe stock prices only at exogenous Poisson arrival times. He may also consume continuously from his cash holdings, and his goal is to maximize his expected utility from consumption. This is a mixed discrete/continuous stochastic control problem, non‐standard in the literature. The dynamic programming principle leads to a coupled system of Integro‐Differential Equations (IDE), and we provide a convergent numerical algorithm for the resolution to this coupled system of IDE. Several numerical experiments illustrate the impact of the restricted liquidity trading opportunities, and we measure in particular the utility loss with respect to the classical Merton consumption problem.     

MEAN‐VARIANCE POLICY FOR DISCRETE‐TIME CONE‐CONSTRAINED MARKETS: TIME CONSISTENCY IN EFFICIENCY AND THE MINIMUM‐VARIANCE SIGNED SUPERMARTINGALE MEASURE

The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.     

BENCHMARKED RISK MINIMIZATION

This paper discusses the problem of hedging not perfectly replicable contingent claims using the numéraire portfolio. The proposed concept of benchmarked risk minimization leads beyond the classical no‐arbitrage paradigm. It provides in incomplete markets a generalization of the pricing under classical risk minimization, pioneered by Föllmer, Sondermann, and Schweizer. The latter relies on a quadratic criterion, requests square integrability of claims and gains processes, and relies on the existence of an equivalent risk‐neutral probability measure. Benchmarked risk minimization avoids these restrictive assumptions and provides symmetry with respect to all primary securities. It employs the real‐world probability measure and the numéraire portfolio to identify the minimal possible price for a contingent claim. Furthermore, the resulting benchmarked (i.e., numéraire portfolio denominated) profit and loss is only driven by uncertainty that is orthogonal to benchmarked‐traded uncertainty, and forms a local martingale that starts at zero. Consequently, sufficiently different benchmarked profits and losses, when pooled, become asymptotically negligible through diversification. This property makes benchmarked risk minimization the least expensive method for pricing and hedging diversified pools of not fully replicable benchmarked contingent claims. In addition, when hedging it incorporates evolving information about nonhedgeable uncertainty, which is ignored under classical risk minimization.     

MULTIVARIATE RISK MEASURES: A CONSTRUCTIVE APPROACH BASED ON SELECTIONS

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set‐valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set‐valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set‐valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set‐valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant, and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set‐valued risk measures considered by Kulikov (2008, Theory Probab. Appl. 52, 614–635), Hamel and Heyde (2010, SIAM J. Financ. Math. 1, 66–95), and Hamel, Heyde, and Rudloff (2013, Math. Financ. Econ. 5, 1–28).     

LARGE DEVIATIONS IN MULTIFACTOR PORTFOLIO CREDIT RISK

The measurement of portfolio credit risk focuses on rare but significant large-loss events. This paper investigates rare event asymptotics for the loss distribution in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes. The first limit examines the tail of the loss distribution at increasingly high loss thresholds; the second limiting regime is based on letting the individual loss probabilities decrease toward zero. Both limits are also based on letting the size of the portfolio increase. Our analysis reveals a qualitative distinction between the two cases: in the rare-default regime, the tail of the loss distribution decreases exponentially, but in the large-threshold regime the decay is consistent with a power law. This indicates that the dependence between defaults imposed by the Gaussian copula is qualitatively different for portfolios of high-quality and lower-quality credits.     

Who should sell stocks?

Never selling stocks is optimal for investors with a long horizon and a realistic range of preference and market parameters, if relative risk aversion, investment opportunities, proportional transaction costs, and dividend yields are constant. Such investors should buy stocks when their portfolio weight is too low and otherwise hold them, letting dividends rebalance to cash over time rather than selling. With capital gains taxes, this policy outperforms both static buy‐and‐hold and dynamic rebalancing strategies that account for transaction costs. Selling stocks becomes optimal if either their target weight is low or intermediate consumption is substantial.     

DISCRETE TIME HEDGING OF THE AMERICAN OPTION

In a complete financial market we consider the discrete time hedging of the American option with a convex payoff. It is well known that for the perfect hedging the writer of the option must trade continuously in time, which is impossible in practice. In reality, the writer hedges only at some discrete time instants. The perfect hedging requires the knowledge of the partial derivative of the value function of the American option in the underlying asset, the explicit form of which is unknown in most cases of practical importance. Several approximation methods have been developed for the calculation of the value function of the American option. We claim in this paper that having at hand any uniform approximation of the American option value function at equidistant discrete rebalancing times it is possible to construct a discrete time hedging portfolio, the value process of which uniformly approximates the value process of the continuous time perfect delta‐hedging portfolio. We are able to estimate the corresponding discrete time hedging error that leads to a complete justification of our hedging method for nonincreasing convex payoff functions including the important case of the American put. This method is essentially based on a new type square integral estimate for the derivative of an arbitrary convex function recently found by Shashiashvili.     

Does the More Risk‐Averse Investor hold a Less Risky Portfolio?*

We study the suitability of using absolute risk aversion as a measure of willingness to take risk in the Arrow–Debreu portfolio framework. We define a global measure of risk for the Arrow–Debreu portfolio, which is measured by the sensitivity of an individual's Arrow–Debreu portfolio payoff to the change in the market return. We call this measure ‘conservatism’ and show that the concept of ‘more conservative’ is stronger than that of ‘more risk‐averse.’ A higher absolute risk aversion is only necessary but not sufficient to induce a less risky Arrow–Debreu portfolio. Our results not only challenge the well‐accepted notion that a more risk‐averse investor holds a less risky portfolio, but also suggest a stronger measure – conservatism – for evaluating the riskiness of portfolio.     

MAXIMIZING THE GROWTH RATE UNDER RISK CONSTRAINTS

We investigate the ergodic problem of growth-rate maximization under a class of risk constraints in the context of incomplete, Itô-process models of financial markets with random ergodic coefficients. Including value-at-risk , tail-value-at-risk , and limited expected loss , these constraints can be both wealth-dependent (relative) and wealth-independent (absolute). The optimal policy is shown to exist in an appropriate admissibility class, and can be obtained explicitly by uniform, state-dependent scaling down of the unconstrained (Merton) optimal portfolio. This implies that the risk-constrained wealth-growth optimizer locally behaves like a constant relative risk aversion (CRRA) investor, with the relative risk-aversion coefficient depending on the current values of the market coefficients.     

A Combination Rule for Portfolio Selection with Transaction Costs

We propose a new portfolio rule for portfolio selection problems in the presence of transaction costs. The new portfolio rule is formed by combining an extant portfolio rule with the no‐rebalancing portfolio rule, which specifies the current portfolio weights before rebalancing as the desired portfolio weights. The new portfolio rule can be applied into most extant portfolio rules. Simulation and out‐of‐sample evidence show that the new portfolio rule can greatly improve portfolio performance, in comparison with the extant portfolio rules to be combined.     

TRANSFORM ANALYSIS FOR POINT PROCESSES AND APPLICATIONS IN CREDIT RISK

This paper develops a formula for a transform of a vector point process with totally inaccessible arrivals. The transform is expressed in terms of a Laplace transform under an equivalent probability measure of the point process compensator. The Laplace transform of the compensator can be calculated explicitly for a wide range of model specifications, because it is analogous to the value of a simple security. The transform formula extends the computational tractability offered by extant security pricing models to a point process and its applications, which include valuation and risk management problems arising in single‐name and portfolio credit risk.     

FIRE SALES FORENSICS: MEASURING ENDOGENOUS RISK

We propose a tractable framework for quantifying the impact of loss‐triggered fire sales on portfolio risk, in a multi‐asset setting. We derive analytical expressions for the impact of fire sales on the realized volatility and correlations of asset returns in a fire sales scenario and show that our results provide a quantitative explanation for the spikes in volatility and correlations observed during such deleveraging episodes. These results are then used to develop an econometric framework for the forensic analysis of fire sales episodes, using observations of market prices. We give conditions for the identifiability of model parameters from time series of asset prices, propose a statistical test for the presence of fire sales, and an estimator for the magnitude of fire sales in each asset class. Pathwise consistency and large sample properties of the estimator are studied in the high‐frequency asymptotic regime. We illustrate our methodology by applying it to the forensic analysis of two recent deleveraging episodes: the Quant Crash of August 2007 and the Great Deleveraging following the default of Lehman Brothers in Fall 2008.     

Asset Allocation of Two‐Person Households under Different Longevity Expectations

This study examines asset allocations of near‐elderly couples when spouses have different longevity expectations. Since the risk‐adjusted return on equities increases with investment horizon, a spouse who expects longer retirement period has an incentive to hold riskier portfolio. Using data from the Health and Retirement Study, we show that portfolio riskiness increases with the subjective survival probability of the decision‐making spouse. As predicted by the bargaining model, portfolio outcomes are uncorrelated with the horizon of the spouse who has less bargaining power. Results also show that the extent expected horizon is incorporated into asset allocation depends on the decider's gender. The share of equities depends on the husband's expected horizon when he leads decision making but not on the wife's horizon when she has more power. These findings contradict the prediction that wife‐led households may hold more equities than do husband‐led households because of their longer lifespan.     

MARTINGALE MEASURES FOR DISCRETE-TIME PROCESSES WITH INFINITE HORIZON

Let ( S_t ) _tεI be an R^d-valued adapted stochastic process on (Ω, , ( _t ) _tεI , P ). A basic problem occurring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on  equivalent to P such that ( S_t ) _tεI is a martingale with respect to Q. It is known (see the fundamental papers of Harrison and Kreps 1979; Harrison and Pliska 1981; and Kreps 1981) that there is an intimate relation of this problem with the notions of "no arbitrage" and "no free lunch" in financial economics. We introduce the intermediate concept of "no free lunch with bounded risk." This is a somewhat more precise version of the notion of "no free lunch." It requires an absolute bound of the maximal loss occurring in the trading strategies considered in the definition of "no free lunch." We give an argument as to why the condition of "no free lunch with bounded risk" should be satisfied by a reasonable model of the price process ( S_t ) _tεI of a securities market. We can establish the equivalence of the condition of "no free lunch with bounded risk" with the existence of an equivalent martingale measure in the case when the index set I is discrete but (possibly) infinite. A similar theorem was recently obtained by Delbaen (1992) for continuous-time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous-time case when the process ( S_t ) _{t εR+} is bounded and, roughly speaking, the jumps occur at predictable times. In the infinite horizon setting, the price process has to be "almost a martingale" in order to allow an equivalent martingale measure.     

RISK‐MINIMIZATION FOR LIFE INSURANCE LIABILITIES WITH DEPENDENT MORTALITY RISK

In this paper, we study the pricing and hedging of typical life insurance liabilities for an insurance portfolio with dependent mortality risk by means of the well‐known risk‐minimization approach. As the insurance portfolio consists of individuals of different age cohorts in order to capture the cross‐generational dependency structure of the portfolio, we introduce affine models for the mortality intensities based on Gaussian random fields that deliver analytically tractable results. We also provide specific examples consistent with historical mortality data and correlation structures. Main novelties of this work are the explicit computations of risk‐minimizing strategies for life insurance liabilities written on an insurance portfolio composed of primary financial assets (a risky asset and a money market account) and a family of longevity bonds, and the simultaneous consideration of different age cohorts.     

AN ONLINE PORTFOLIO SELECTION ALGORITHM WITH REGRET LOGARITHMIC IN PRICE VARIATION

We present a novel efficient algorithm for portfolio selection which theoretically attains two desirable properties:

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