Time-consistent mean–variance hedging of longevity risk: Effect of cointegration

This paper investigates the time-consistent dynamic mean–variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality. Using an HJB framework, we solve the hedging problem in which insurance liabilities follow a doubly stochastic Poisson process with an intensity rate that is correlated and cointegrated to the index mortality rate. The derived closed-form optimal hedging policy articulates the important role of cointegration in longevity hedging. We show numerically that a time-consistent hedging policy is a smoother function in time when compared with its time-inconsistent counterpart.     

Theoretical and empirical estimates of mean–variance portfolio sensitivity

This paper studies properties of an estimator of mean–variance portfolio weights in a market model with multiple risky assets and a riskless asset. Theoretical formulas for the mean square error are derived in the case when asset excess returns are multivariate normally distributed and serially independent. The sensitivity of the portfolio estimator to errors arising from the estimation of the covariance matrix and the mean vector is quantified. It turns out that the relative contribution of the covariance matrix error depends mainly on the Sharpe ratio of the market portfolio and the sampling frequency of historical data. Theoretical studies are complemented by an investigation of the distribution of portfolio estimator for empirical datasets. An appropriately crafted bootstrapping method is employed to compute the empirical mean square error. Empirical and theoretical estimates are in good agreement, with the empirical values being, in general, higher.     

Global adapted solution of one-dimensional backward stochastic Riccati equations,with application to the mean–variance hedging

Backward stochastic Riccati equations are motivated by the solution of general linear quadratic optimal stochastic control problems with random coefficients, and the solution has been open in the general case. One distinguishing difficult feature is that the drift contains a quadratic term of the second unknown variable. In this paper, we obtain the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation. This solves the one-dimensional case of Bismut–Peng's problem which was initially proposed by Bismut (Lecture Notes in Math. 649 (1978) 180). We use an approximation technique by constructing a sequence of monotone drifts and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean–variance hedging problem with general random market conditions.     

Investment strategies and compensation of a mean–variance optimizing fund manager

This paper introduces a general continuous-time mathematical framework for solution of dynamic mean–variance control problems. We obtain theoretical results for two classes of functionals: the first one depends on the whole trajectory of the controlled process and the second one is based on its terminal-time value. These results enable the development of numerical methods for mean–variance problems for a pre-determined risk-aversion coefficient. We apply them to study optimal trading strategies pursued by fund managers in response to various types of compensation schemes. In particular, we examine the effects of continuous monitoring and scheme’s symmetry on trading behavior and fund performance.     

Optimal mean–variance selling strategies

Assuming that the stock price X follows a geometric Brownian motion with drift \(\mu \in \mathbb {R}\) and volatility \(\sigma >0\), and letting \(\mathsf {P}_{\!x}\) denote a probability measure under which X starts at \(x>0\), we study the dynamic version of the nonlinear mean–variance optimal stopping problem

$$\begin{aligned} \sup _\tau \Big [ \mathsf {E}\,\!_{X_t}(X_\tau ) - c\, \mathsf {V}ar\,\!_{\!X_t}(X_\tau ) \Big ] \end{aligned}$$

where t runs from 0 onwards, the supremum is taken over stopping times \(\tau \) of X, and \(c>0\) is a given and fixed constant. Using direct martingale arguments we first show that when \(\mu \le 0\) it is optimal to stop at once and when \(\mu \ge \sigma ^2\!/2\) it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for \(0 < \mu < \sigma ^2\!/2\) can be reduced to a family of linear problems. Solving the latter using a free-boundary approach we find that the optimal stopping time is given by

$$\begin{aligned} \tau _* = \inf \,\! \left\{ \, t \ge 0\; \vert \; X_t \ge \tfrac{\gamma }{c(1-\gamma )}\, \right\} \end{aligned}$$

where \(\gamma = \mu /(\sigma ^2\!/2)\). The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.

     

Near-optimal mean–variance controls under two-time-scale formulations and applications

Although the mean–variance control was initially formulated for financial portfolio management problems in which one wants to maximize the expected return and control the risk, our motivations stem from highway vehicle platoon controls that aim to maximize highway utility while ensuring zero accident. This paper develops near-optimal mean–variance controls of switching diffusion systems. To reduce the computational complexity, with motivations from earlier work on singularly perturbed Markovian systems [Sethi and Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhäuser, Boston, MA, 1994; Yin and Zhang, Continuous-Time Markov Chains and Applications: A Singular Pertubation Approach, Springer-Verlag, New York, 1998 and Yin et al., Ann. Appl. Probab. 10 (2000), pp. 549–572], we use a two-time-scale formulation to treat the underlying system, which is represented by the use of a small parameter. As the small parameter goes to 0, we obtain a limit problem. Using the limit problem as a guide, we construct controls for the original problem, and show that the control so constructed is nearly optimal.     

Optimality of excess-loss reinsurance under a mean–variance criterion

In this paper, we study an insurer’s reinsurance–investment problem under a mean–variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative Lévy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthermore, we obtain the explicit equilibrium reinsurance–investment strategy by solving the extended Hamilton–Jacobi–Bellman equation.     

The opportunity cost of mean–variance choice under estimation risk

Mean–variance portfolio choice is often criticized as sub-optimal in the more general expected utility framework. It is argued that the expected utility framework takes into consideration higher moments ignored by mean variance analysis. A body of research suggests that mean–variance choice, though arguably sub-optimal, provides very close-to-expected utility maximizing portfolios and their expected utilities, basing its evaluation on in-sample analysis where mean–variance choice is sub-optimal by definition. In order to clarify this existing research, this study provides a framework that allows comparing in-sample and out-of-sample performance of the mean variance portfolios against expected utility maximizing portfolios. Our in-sample results confirm the results of earlier studies. On the other hand, our out-of-sample results show that the expected utility model performs worse. The out-of-sample inferiority of the expected utility model is more pronounced for preferences and constraints under which in-sample mean variance approximations are weakest. We argue that, in addition to its elegance and simplicity, the mean–variance model extracts more information from sample data because it uses the covariance matrix of returns. The expected utility model may reach its optimal solution without using information from the covariance matrix.     

Algorithmic aspects of mean–variance optimization in Markov decision processes

We consider finite horizon Markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward. We show that either randomized or history-based policies can improve performance. We prove that the complexity of computing a policy that maximizes the mean reward under a variance constraint is NP-hard for some cases, and strongly NP-hard for others. We finally offer pseudopolynomial exact and approximation algorithms.     

Subjective mean–variance preferences without expected utility

Classical derivations of mean–variance preferences have all relied on the expected utility hypothesis. Numerous experimental studies have revealed that the expected utility model is systematically violated in practice. Such findings and the simplicity of the mean–variance framework have led researchers and practitioners to employ the mean–variance model without expected utility. However, the theoretical foundations of these models are scant.I provide behavioral foundations for a class of mean–variance preferences. My set of axioms characterizes an individual who assigns subjective probability to events and judges each portfolio solely on the basis of the mean and variance of its implied distribution over returns but does not necessarily rank the portfolios according to expected utility. I clarify the differences across specifications of my model. In addition, this model is robust to the consideration of a wide body of observed behaviors under uncertainty, which are inconsistent with the classical mean–variance model.     

Portfolio selection under possibilistic mean–variance utility and a SMO algorithm

In this paper, we propose a new portfolio selection model with the maximum utility based on the interval-valued possibilistic mean and possibilistic variance, which is a two-parameter quadratic programming problem. We also present a sequential minimal optimization (SMO) algorithm to obtain the optimal portfolio. The remarkable feature of the algorithm is that it is extremely easy to implement, and it can be extended to any size of portfolio selection problems for finding an exact optimal solution.     

Carter–Payne homomorphisms and Jantzen filtrations

We prove a q-analogue of the Carter–Payne theorem in the case where the differences between the parts of the partitions are sufficiently large. We identify a layer of the Jantzen filtration which contains the image of these Carter–Payne homomorphisms and we show how these homomorphisms compose.     

Stochastic goal programming: A mean–variance approach

We propose a stochastic goal programming (GP) model leading to a structure of mean–variance minimisation. The solution to the stochastic problem is obtained from a linkage between the standard expected utility theory and a strictly linear, weighted GP model under uncertainty. The approach essentially consists in specifying the expected utility equation corresponding to every goal. Arrow's absolute risk aversion coefficients play their role in the calculation process. Once the model is defined and justified, an illustrative example is developed.     

Optimal multi-period mean–variance policy under no-shorting constraint

We consider in this paper the mean–variance formulation in multi-period portfolio selection under no-shorting constraint. Recognizing the structure of a piecewise quadratic value function, we prove that the optimal portfolio policy is piecewise linear with respect to the current wealth level, and derive the semi-analytical expression of the piecewise quadratic value function. One prominent feature of our findings is the identification of a deterministic time-varying threshold for the wealth process and its implications for market settings. We also generalize our results in the mean–variance formulation to utility maximization with no-shorting constraint.     

Equilibrium in an ambiguity-averse mean–variance investors market

In a financial market composed of n risky assets and a riskless asset, where short sales are allowed and mean–variance investors can be ambiguity averse, i.e., diffident about mean return estimates where confidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a worst case max–min criterion. Then, in a market where all investors are ambiguity-averse mean–variance investors with access to given mean return and variance–covariance estimates, we investigate conditions regarding the existence of an equilibrium price system and give an explicit formula for the equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) mean–variance investors’ market has a deflationary effect on equilibrium prices with respect to a pure mean–variance investors’ market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambiguity-averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral.     

Optimal mean–variance asset-liability management with stochastic interest rates and inflation risks

This paper considers an optimal asset-liability management problem with stochastic interest rates and inflation risks under the mean–variance framework. It is assumed that there are \(n+1\) assets available in the financial market, including a risk-free asset, a default-free zero-coupon bond, an inflation-indexed bond and \(n-2\) risky assets (stocks). Moreover, the liability of the investor is assumed to follow a geometric Brownian motion process. By using the stochastic dynamic programming principle and Hamilton–Jacobi–Bellman equation approach, we derive the efficient investment strategy and efficient frontier explicitly. Finally, we provide numerical examples to illustrate the effects of model parameters on the efficient investment strategy and efficient frontier.     

Dynamic mean–variance portfolio selection with borrowing constraint

This paper derives explicit closed form solutions, for the efficient frontier and optimal investment strategy, for the dynamic mean–variance portfolio selection problem under the constraint of a higher borrowing rate. The method used is the Hamilton–Jacobi–Bellman (HJB) equation in a stochastic piecewise linear-quadratic (PLQ) control framework. The results are illustrated on an example.     

A new foundation for the mean–variance analysis

In this paper, I re-examine how the mean–variance analysis is consistent with its traditional theoretical foundations, namely, stochastic dominance and the expected utility theory. Then I propose a simplified version of the coarse utility theory as a new foundation. I prove that, by assuming risk aversion and the normality of asset variables, the simplified model is well behaved; indifference curves are convex and the opportunity set is concave. Therefore, there exist global optimal portfolios in the market. Finally, I prove that decision-making in accordance with the simplified model is consistent with the mean–variance analysis.     

Using sliced mean variance–covariance inverse regression for classification and dimension reduction

The sliced mean variance–covariance inverse regression (SMVCIR) algorithm takes grouped multivariate data as input and transforms it to a new coordinate system where the group mean, variance, and covariance differences are more apparent. Other popular algorithms used for performing graphical group discrimination are sliced average variance estimation (SAVE, targetting the same differences but using a different arrangement for variances) and sliced inverse regression (SIR, which targets mean differences). We provide an improved SMVCIR algorithm and create a dimensionality test for the SMVCIR coordinate system. Simulations corroborating our theoretical results and comparing SMVCIR with the other methods are presented. We also provide examples demonstrating the use of SMVCIR and the other methods, in visualization and group discrimination by k-nearest neighbors. The advantages and differences of SMVCIR from SAVE and SIR are shown clearly in these examples and simulation.