Optimal Investment and Consumption with Default Risk: HARA Utility

In this paper, we consider a portfolio optimization problem in a defaultable market. The representative investor dynamically allocates his or her wealth among the following securities: a perpetual defaultable bond, a money market account and a default-free risky asset. The optimal investment and consumption policies that maximize the infinite horizon expected discounted HARA utility of the consumption are explicitly derived. Moreover, numerical illustrations are also presented.     

Optimal Portfolios with Credit Default Swaps

Using a continuous-time, stochastic, and dynamic framework, this study derives a closed-form solution for the optimal investment problem for an agent with hyperbolic absolute risk aversion preferences for maximising the expected utility of his or her final wealth. The agent invests in a frictionless, complete market in which a riskless asset, a (defaultable) bond, and a credit default swap written on the bond are listed. The model is calibrated to market data of six European countries and assesses the behaviour of an investor exposed to different levels of sovereign risk. A numerical analysis shows that it is optimal to issue credit default swaps in a larger quantity than that of bonds, which are optimally purchased. This speculative strategy is more aggressive in countries characterised by higher sovereign risk. This result is confirmed when the investor is endowed with a different level of risk aversion. Finally, we solve a static version of the optimisation problem and show that the speculative/hedging strategy is definitely different with respect to the dynamic one.     

Alternate approach to unify CAPM and APT

We utilize the joint elliptical distribution to model a multi-factor return generating process and derive an equilibrium multi-beta capital asset pricing model (CAPM) in which the market portfolio and a set of nonelliptical factors are sufficient to price all financial assets. Most important, it is shown that the market portfolio, while generally nonelliptical, can proxy all elliptical factors and hence: including elliptical factors in addition to the market portfolio in the pricing equation contribute nothing to asset pricing. While the representative investor prices the exposure of aggregate wealth to various nonelliptical systematic risk factors, individual securities are priced in accordance to their contributions to different aspects of the risk of aggregate wealth. The present model collapses to the Sharpe-Lintner CAPM when either the market investor is neutral to nonelliptical risk factors or when all risk factors follow a joint spherical distribution. When residuals cancel out of the market portfolio, the present model collapses to Conner (1984) pricing model.     

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

Robust Portfolio Rules and Asset Pricing

I present a new approach to the dynamic portfolio and consumptionproblem of an investor who worries about model uncertainty (inaddition to market risk) and seeks robust decisions along thelines of Anderson, Hansen, and Sargent (2002). In accordancewith max-min expected utility, a robust investor insures againstsome endogenous worst case. I first show that robustness dramaticallydecreases the demand for equities and is observationally equivalentto recursive preferences when removing wealth effects. Unlikestandard recursive preferences, however, robustness leads toenvironment-specific "effective" risk aversion. As an extension,I present a closed-form solution for the portfolio problem ofa robust Duffie-Epstein-Zin investor. Finally, robustness increasesthe equilibrium equity premium and lowers the risk-free rate.Reasonable parameters generate a 4% to 6% equity premium.     

HARA utility maximization in a Markov-switching bond–stock market

We present a flexible multidimensional bond–stock model incorporating regime switching, a stochastic short rate and further stochastic factors, such as stochastic asset covariance. In this framework we consider an investor whose risk preferences are characterized by the hyperbolic absolute risk-aversion utility function and solve the problem of optimizing the expected utility from her terminal wealth. For the optimal portfolio we obtain a constant-proportion portfolio insurance-type strategy with a Markov-switching stochastic multiplier and prove that it assures a lower bound on the terminal wealth. Explicit and easy-to-use verification theorems are proven. Furthermore, we apply the results to a specific model. We estimate the model parameters and test the performance of the derived optimal strategy using real data. The influence of the investor’s risk preferences and the model parameters on the portfolio is studied in detail. A comparison to the results with the power utility function is also provided.     

Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility

Given an investor maximizing utility from terminal wealth with respect to a power utility function, we present a verification result for portfolio problems with stochastic volatility. Applying this result, we solve the portfolio problem for Heston's stochastic volatility model. We find that only under a specific condition on the model parameters does the problem possess a unique solution leading to a partial equilibrium. Finally, it is demonstrated that the results critically hinge upon the specification of the market price of risk. We conclude that, in applications, one has to be very careful when exogenously specifying the form of the market price of risk.     

Asset allocation and liquidity breakdowns: what if your broker does not answer the phone?

This paper analyzes the portfolio decision of an investor facing the threat of illiquidity. In a continuous-time setting, the efficiency loss due to illiquidity is addressed and quantified. For a logarithmic investor, we solve the portfolio problem explicitly. We show that the efficiency loss for a logarithmic investor with 30 years until the investment horizon is a significant 22.7% of current wealth if the illiquidity part of the model is calibrated to the Japanese data of the aftermath of WWII. For general utility functions, an explicit solution does not seem to be available. However, under a mild growth condition on the utility function, we show that the value function of a model in which only finitely many liquidity breakdowns can occur converges uniformly to the value function of a model with infinitely many breakdowns if the number of possible breakdowns goes to infinity. Furthermore, we show how the optimal security demands of the model with finitely many breakdowns can be used to approximate the solution of the model with infinitely many breakdowns. These results are illustrated for an investor with a power utility function.     

Limits of Arbitrage: Theory and Evidence from the Mortgage-Backed Securities Market

“Limits of Arbitrage” theories hypothesize that the marginal investor in a particular asset market is a specialized arbitrageur rather than a diversified representative investor. We examine the mortgage‐backed securities (MBS) market in this light. We show that the risk of homeowner prepayment, which is a wash in the aggregate, is priced in the MBS market. The covariance of prepayment risk with aggregate wealth implies the wrong sign to match the observed prices of prepayment risk. The price of risk is better explained by a kernel based on MBS market‐wide specific risk, consistent with the specialized arbitrageur hypothesis.     

Optimal portfolios when volatility can jump

We consider an asset allocation problem in a continuous-time model with stochastic volatility and jumps in both the asset price and its volatility. First, we derive the optimal portfolio for an investor with constant relative risk aversion. The demand for jump risk includes a hedging component, which is not present in models without volatility jumps. We further show that the introduction of derivative contracts can have substantial economic value. We also analyze the distribution of terminal wealth for an investor who uses the wrong model, either by ignoring volatility jumps or by falsely including such jumps, or who is subject to estimation risk. Whenever a model different from the true one is used, the terminal wealth distribution exhibits fatter tails and (in some cases) significant default risk.     

Überlegungen zu einem fairen Risikomanagement-Mix in Versicherungsunternehmen

This article deals with the question of how a ?fair risk management mix“ that does not lead to a wealth transfer between shareholders and policyholders can be achieved in a joint-stock insurance company. In our financial model of an insurer, the ?fair“ situation, it is assumed that there is no wealth transfer between shareholders and policyholders when both parties receive a net present value of zero on their investments. Taking the default risk of the insurance company into account, we first model a ?fair“ situation for the insurer’s existing portfolio. Surprisingly, closing a new insurance contract that has been priced on a fair basis and then included in the insurer’s existing portfolio leads to a disequilibrium situation because the net present value for the shareholders is no longer zero. This new net present value can be viewed as the fair price of any risk management measure the insurer must take so as to reestablish an equilibrium for both parties, the shareholders and the policyholders.     

Portfolio optimization under convex incentive schemes

We consider the terminal wealth utility maximization problem from the point of view of a portfolio manager who is paid by an incentive scheme, which is given as a convex function g of the terminal wealth. The manager’s own utility function U is assumed to be smooth and strictly concave; however, the resulting utility function U°g fails to be concave. As a consequence, the problem considered here does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal portfolio in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has a continuous law. In many cases, this existence and uniqueness result is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As examples, we discuss (complete) one-dimensional models as well as (incomplete) lognormal mixture and popular stochastic volatility models. We also provide a detailed analysis of the case where the unique optimizer of the dual problem does not have a continuous law, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor.     

Valuation Model of Defaultable Bond Values in Emerging Markets

This paper develops a model to value defaultable bonds in emerging markets. Default occurs when some signaling process hits a pre-defined default barrier. The signaling variable is considered to be some macro-economic variables such as foreign exchange rates. The dynamics of the default barrier depend on the volatility and the drift of the signaling variable. We derive a closed-form solution of the defaultable bond price from the model as a function of a signaling variable and a short-term interest rate. The numerical results show that the model values generated by using foreign exchange rates as the signaling variables can broadly track the market credit spreads of defaultable bonds in South Korea and Brazil. Given an expected level of the foreign exchange rate, defaultable bond values under a stressed market situation can be obtained.     

Risk aversion and skewness preference

Empirically, co-skewness of asset returns seems to explain a substantial part of the cross-sectional variation of mean return not explained by beta. This finding is typically interpreted in terms of a risk averse representative investor with a cubic utility function. This paper questions this interpretation. We show that the empirical tests fail to impose risk aversion and the implied utility function takes an inverse S-shape. Unfortunately, the first-order conditions are not sufficient to guarantee that the market portfolio is the global maximum for this utility function, and our results suggest that the market portfolio is more likely to represent the global minimum. In addition, if we do impose risk aversion, then co-skewness has minimal explanatory power.     

Alternative Defaultable Term Structure Models

The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives. Nicola Bruti-Liberati: In memory of our beloved friend and colleague.     

Optimal portfolio allocation with higher moments

We model the risky asset as driven by a pure jump process, with non-trivial and tractable higher moments. We compute the optimal portfolio strategy of an investor with CRRA utility and study the sensitivity of the investment in the risky asset to the higher moments, as well as the resulting wealth loss from ignoring higher moments. We find that ignoring higher moments can lead to significant overinvestment in risky securities, especially when volatility is high.      

Mean-Variance Asset Liability Management with State-Dependent Risk Aversion

This article investigates the asset liability management problem with state-dependent risk aversion under the mean-variance criterion. The investor allocates the wealth among multiple assets including a risk-free asset and multiple risky assets governed by a system of geometric Brownian motion stochastic differential equations, and the investor faces the risk of paying uncontrollable random liabilities. The state-dependent risk aversion is taken into account in our model, linking the risk aversion to the current wealth held by the investor. An extended Hamilton-Jacobi-Bellman system is established for the optimization of asset liability management, and by solving the extended Hamilton-Jacobi-Bellman system, the analytical closed-form expressions for the time-inconsistent optimal investment strategies and the optimal value function are derived. Finally, numerical examples are presented to illustrate our results.     

DURATION,DEFAULT RISK,AND THE TERM STRUCTURE OF INTEREST RATES

We examine the interactive effect of default and interest rate risk on duration of defaultable bonds. We show that duration for defaultable bonds can be longer or shorter than default‐free bonds depending on the relation between default intensity and interest rates. Empirical evidence indicates that in most cases duration for defaultable bonds is much shorter than for their default‐free counterparts because of the negative relation between default risk and interest rates. Results suggest that the duration measure must be adjusted for the effects of default risk and stochastic interest rates to achieve an effective bond portfolio immunization.     

A Benchmark Approach to Portfolio Optimization under Partial Information

This paper proposes a filtering methodology for portfolio optimization when some factors of the underlying model are only partially observed. The level of information is given by the observed quantities that are here supposed to be the primary securities and empirical log-price covariations. For a given level of information we determine the growth optimal portfolio, identify locally optimal portfolios that are located on a corresponding Markowitz efficient frontier and present an approach for expected utility maximization. We also present an expected utility indifference pricing approach under partial information for the pricing of nonreplicable contracts. This results in a real world pricing formula under partial information that turns out to be independent of the subjective utility of the investor and for which an equivalent risk neutral probability measure need not exist.