Enhancing hedging performance with the spanning polynomial projection

Statistical time-series approaches to hedging are difficult to beat, especially out-of-sample, and are capable of out-performing many theory-based derivative pricing model approaches to hedging commodity price risks using futures contracts. However, the vast majority of time-series approaches to hedging discussed in the literature are essentially linear statistical projections, whether univariate or multivariate. Little is known about the potential hedging capabilities of nonlinear methods. This study describes how least-squares orthogonal polynomial approximation methods based on the spanning polynomial projection (SPP) can be used to enhance standard (linear) optimal hedging methods and improve hedging performance for a hedger with a mean–variance objective. Empirical analyses show that the SPP can be used effectively for hedging and gives better out-of-sample hedging performance than the benchmark VEC-GARCH hedging model. Results are robust to the inclusion of transaction costs and risk-aversion assumptions.     

Managing Beliefs about Monetary Policy under Discretion

In models of monetary policy, discretionary policymaking is typically constrained in its ability to manage public beliefs. However, when a policymaker possesses private information, policy actions serve as signals to the public about unobserved economic conditions and belief management becomes an integral part of optimal discretion policies. This article derives the optimal time‐consistent policy for a general linear‐quadratic setting. The optimal policy is illustrated in a simple New Keynesian model, where analytical solutions can be derived as well. In this model, imperfect information about the policymaker's output target leads to lower policy losses.     

Authors’ Reply: On the Time Value of Ruin - Discussion by F. Etienne De Vylder; Marc J. Goovaerts; David C.M. Dickson; Vladimir Kalashnikov; Gérard Pafumi

Abstract

The qualitative behavior of the optimal premium strategy is determined for an insurer in a finite and an infinite market using a deterministic general insurance model. The optimization problem leads to a system of forward-backward differential equations obtained from Pontryagin’s Maximum Principle. The focus of the modelling is on how this optimization problem can be simplified by the choice of demand function and the insurer’s objective. Phase diagrams are used to characterize the optimal control. When the demand is linear in the relative premium, the structure of the phase diagram can be determined analytically. Two types of premium strategy are identified for an insurer in an infinite market, and which is optimal depends on the existence of equilibrium points in the phase diagram. In a finite market there are four more types of premium strategy, and optimality depends on the initial exposure of the insurer and the position of a saddle point in the phase diagram. The effect of a nonlinear demand function is examined by perturbing the linear price function. An analytical optimal premium strategy is also found using inverse methods when the price function is nonlinear.     

Intertemporal asset allocation: A comparison of methods

This paper compares two recent Monte Carlo methods advocated for the computation of optimal portfolio rules. The candidate methods are the approach based on Monte Carlo with Malliavin Derivatives (MCMD) proposed by Detemple, Garcia and Rindisbacher [Detemple et al., 2003. A Monte-Carlo method for optimal portfolios. Journal of Finance 58, 401–406] and the approach based on Monte Carlo with regression (MCR) of Brandt, Goyal, Santa-Clara and Stroud [Brandt et al., 2003. A simulation approach to dynamic portfolio choice with an application to learning about return predictability. Working paper, Wharton School]. Our comparisons are carried out in the context of various intertemporal portfolio choice problems with two assets, a risky asset and a riskless asset, and different configurations of the state variables. The specifications studied include a linear model with a single state variable admitting an exact solution and a non-linear model with two state variables that requires a purely numerical resolution. The accuracies of the candidate methods are compared. We provide, in particular, efficiency plots displaying the speed–accuracy trade-off for various selections of the relevant simulation and discretization parameters. MCMD is shown to dominate in all the settings considered.     

Alternate programming structures for bank portfolios

Recently a number of mathematical programming models have been developed to assist banks in their portfolio (balance sheet) management decision making. Generally, the model structures used may be classified as either linear, linear goal, or two-stage linear programming. Of these, linear programming models are the most common. The purpose of this paper is to discuss the optimal bank portfolio management solutions produced by each of the above programming structures. In addition, a new model structure, two-stage linear goal programming, is developed and compared to the other structures. From a decision-making perspective, this new model structure is found to provide additional useful information.     

How to make regulators and shareholders happy under Basel III

In addition to the Basel II capital ratio, Basel III requires banks to respect additional ratios, such as leverage ratio, liquidity coverage ratio and net stable funding ratio. Banks are required to be compliant with all four constraints simultaneously. Our article provides a framework for banks to help their search for an optimal transition from Basel II to Basel III. Recognizing that banks’ return and the four constraints are of linear type, this search can be formulated as a linear program and solved by standard software. Incorporating uncertainty on future defaults, risk weights and withdrawals and formulating the problem as a Chance constrained model does not only yield optimal transition strategies but also determines the internal thresholds for the Basel III-ratios. Our approach needs two standard inputs from controlling: profit margins per product and non-financial adjustment costs to expand or cut back business. The adjustment cost can be used to calibrate the model to the current business mix. This calibration can be done by bank outsiders and allows the model to be used in impact studies to replace ad hoc strategies. To highlight its practicality, we apply our model to a typical German bank with a business mix that complies with Basel II, but not with the Basel III-, capital-, leverage- and net stable funding-ratio. Assuming that its business model is optimal under Basel II, we find that this bank would achieve compliance restructuring its funding side by replacing interbank funding by capital and retail deposits. Additional uncertainty would amplify the magnitude of the changes, but would still affect the same positions. These findings are robust against alternative margin definitions and adjustment cost levels.     

Olav Aabakken

Abstract

An empirical linear Bayes estimator is asymptotically optimal in the usual sense if its average risk converges to the risk of the corresponding linear Bayes estimator. The present paper demonstrates that the following result holds for the most commonly used models: If the unknown (structural) parameters are estimated in such a way that their mean square error converges at a certain rate, then the corresponding empirical linear Bayes estimator is asymptotically optimal with the same rate of risk convergence. In particular, this is the case for the random coefficient regression model, and for hierarchical models in the univariate case.     

Optimal taxation and public provision for poverty reduction

The existing literature on optimal taxation typically assumes there exists a capacity to implement complex tax schemes, which is not necessarily the case for many developing countries. We examine the determinants of optimal redistributive policies in the context of a developing country that can only implement linear tax policies due to administrative reasons. Further, the reduction of poverty is typically the expressed goal of such countries, and this feature is also taken into account in our model. We derive the optimality conditions for linear income taxation, commodity taxation, and public provision of private and public goods for the poverty minimization case and compare the results to those derived under a general welfarist objective function. We also study the implications of informality on optimal redistributive policies for such countries. The exercise reveals non-trivial differences in optimal tax rules under the different assumptions.     

Optimal Insurance Under the Insurer's VaR Constraint

In this paper, we impose the insurer's Value at Risk (VaR) constraint on Arrow's optimal insurance model. The insured aims to maximize his expected utility of terminal wealth, under the constraint that the insurer wishes to control the VaR of his terminal wealth to be maintained below a prespecified level. It is shown that when the insurer's VaR constraint is binding, the solution to the problem is not linear, but piecewise linear deductible, and the insured's optimal expected utility will increase as the insurer becomes more risk-tolerant. Basak and Shapiro (2001) showed that VaR risk managers often choose larger risk exposures to risky assets. We draw a similar conclusion in this paper. It is shown that when the insured has an exponential utility function, optimal insurance based on VaR constraint causes the insurer to suffer larger losses than optimal insurance without insurer's risk constraint.     

A comparison of MAD and CVaR models with real features

In this paper we consider two different mixed integer linear programming models for solving the single period portfolio selection problem when integer stock units, transaction costs and a cardinality constraint are taken into account. The first model has been formulated by using the maximization of the worst conditional expectation as objective function. The second model is based on the maximization of the safety measure corresponding to the mean absolute deviation. Extensive computational results are provided to compare the financial characteristics of the optimal portfolios selected by the two models on real data from European stock exchange markets. Some simple heuristics are also introduced that provide efficient and effective solutions when an optimal integer solution cannot be found in a reasonable amount of time.     

Dynamic Investment Strategy with Factor Models Under Regime Switches

A model for dynamic investment strategy is developed where assets’ returns are represented by multiple factors. In a mean–variance framework with factor models under regime switches, we derive a semi-analytic solution for the optimal portfolio with transaction costs. Due to the existence of transaction costs, the optimal portfolio is characterized as a linear combination of current and target portfolios, the latter of which maximizes the value function in the current regime. For some special cases of interest, we also derive simplified analytical solutions. To see the effect of regime switches, the proposed model is applied to US equity market in which small minus big and high minus low are employed as factors. Investment strategy based on our model demonstrates empirically that the regime switching models exhibit superior performance over the single regime model for such performance measures as realized utility and Sharpe ratio which are of particular interest in practice. Taking a close look at the time series of portfolio returns, the result shows the usefulness of the regime switching model as investors flexibly optimize asset allocations depending on the state of the market.     

Option pricing: A simplified approach

This paper presents a simple discrete-time model for valuing options. The fundamental economic principles of option pricing by arbitrage methods are particularly clear in this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.     

Least-squares Monte-Carlo methods for optimal stopping investment under CEV models

The optimal stopping investment is a kind of mixed expected utility maximization problems with optimal stopping time. The aim of this paper is to develop the least-squares Monte-Carlo methods to solve the optimal stopping investment under the constant elasticity of variance (CEV) model. Such a problem has no closed-form solutions for the value functions, optimal strategies and optimal exercise boundaries due to the early exercised feature. The dual optimal stopping problem is first derived and then the strong duality between the dual and prime problems is established. The least-squares Monte-Carlo methods based on the dual control theory are developed and numerical simulations are provided. Both the power and non-HARA utilities are studied.     

Personalized goal-based investing via multi-stage stochastic goal programming

In this paper, we propose a goal-based investment model that is suitable for personalized wealth management. The model only requires a few intuitive inputs such as size of wealth, investment amount, and consumption goals from individual investors. In particular, a priority level can be assigned to each consumption goal and the model provides a holistic solution based on a sequential approach starting with the highest priority. This allows strict prioritization by maximizing the probability of achieving higher priority goals that are not affected by goals with lower priorities. Furthermore, the proposed model is formulated as a linear program that efficiently finds the optimal financial plan. With its simplicity, flexibility, and computational efficiency, the proposed goal-based investment model provides a new framework for automated investment management services.     

Failure prediction: Sensitivity of classification accuracy to alternative statistical methods and variable sets

The purpose of this study was to assess the sensitivity of the prediction of business failure to the use of different variable sets and different statistical methods. This objective was accomplished by comparing the performance of failure prediction models using four alternative variable sets on firms which failed from 1966–1975. The four sets of variables were those used by Altman (1968), Deakin (1972), Blum (1974), and Ohlson (1980). A linear discriminant model, a quadratic discriminant model, and a logit model were developed for each of the four variable sets. For a given variable sets, the linear and logit models had comparable misclassification rates and performed at least as well as the quadratic models. Using linear discriminant analysis or logit analysis, all four variable sets performed comparably.     

Superreplication under model uncertainty in discrete time

We study the superreplication of contingent claims under model uncertainty in discrete time. We show that optimal superreplicating strategies exist in a general measure-theoretic setting; moreover, we characterize the minimal superreplication price as the supremum over all continuous linear pricing functionals on a suitable Banach space. The main ingredient is a closedness result for the set of claims which can be superreplicated from zero capital; its proof relies on medial limits.     

On a subjective approach to risk measurement

This study is based on the analogy between hedging a risky asset and keeping reserves to meet an unknown demand. The optimal hedging level, which depends on individual preferences, is regarded as a measure of risk. We determine the set of optimal levels and investigate the properties of the associated risk measures. This approach provides a new insight into Value at Risk (VaR). We consider it as a solution of a certain optimal inventory problem with linear cost and loss functions. We show that these functions determine the confidence level of VaR. In this way we obtain a simple model that helps us to choose a proper confidence level α and explains why supervisory institutions (such as the Basle Committee) choose a higher α than financial institutions themselves.     

Optimal execution with uncertain order fills in Almgren–Chriss framework

The classical price impact model of Almgren and Chriss is extended to incorporate the uncertainty of order fills. The extended model can be recast as alternatives to uncertain impact models and stochastic liquidity models. Optimal strategies are determined by maximizing the expected final profit and loss (P&L) and various P&L-risk tradeoffs including utility maximization. Closed form expressions for optimal strategies are obtained in linear cases. The results suggest a type of adaptive volume weighted average price, adaptive percentage of volume and adaptive Almgren–Chriss strategies. VWAP and classical Almgren–Chriss strategies are recovered as limiting cases with a different characteristic time scale of liquidation for the latter.     

A continuous-time model of self-protection

Finance and Stochastics - This paper deals with an optimal linear insurance demand model, where the protection buyer can also exert a time-dynamic costly prevention effort to reduce her risk...