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Qiji Zhu

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Jim received his PH. D. degree in applied mathematics from Northeastern University in 1992 under the direction of Professor Jack Warga. He held visiting positions at University of Montreal, Center for Experiment and Constructive Mathematics at Simon Fraser University, University of Victoria, Guanghua School of Management at Peking University, Shanghai Logistic Institute at Fudan University, Risk Management Institute at Singapore National University, School of Mathematical and Physical Sciences at New Castle University and Courant Institute of Mathematical Sciences at New York University.Jim is interested in variational techniques and their applications in nonsmooth analysis, optimization, control theory and mathematical finance. He authored and coauthored one book and more than 60 papers in these areas. Jim is serving on the editorial boards of Abstract and Applied Analysis and International Journal of Mathematical Model and Methods in Applied Sciences.

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JULIO CESAR SOSA ALVA

Universidad de Buenos Aires

Thomas Cosimano

Henan University

Maurice Obstfeld

Tomasz Bielecki

Konstantin Avrachenkov

INRIA

Juvenal Bigirinama

Ravi Kashyap (King-Kong Shake-Sphere)

City University of Hong Kong

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Papers by Qiji Zhu

An Equilibrium Model for Term Structure of Interest Rates with Consumption Commitments

We study the term structure of interest rates in the presence of consumption commitments using an... more We study the term structure of interest rates in the presence of consumption commitments using an equilibrium model. Under reasonable assumptions we prove the existence and uniqueness of the equilibrium and develop computation methods. Examples are analyzed to illustrate the effect of consumption commitments on the term structure and its manifestations.

Convex Spectral Functions of Compact Operators

We consider functions on the space of compact self-adjoint Hilbert space operators. Specifically,... more We consider functions on the space of compact self-adjoint Hilbert space operators. Specifically, we study those extended-real functions which depend only on the operators' spectral sequences. Examples include the norms of the Schatten p-spaces, the Calder'on norms, the k'th largest eigenvalue, and some infinite-dimensional self-concordant barriers. We show how various convex and nonsmooth-analytic properties of such functions follow from the corresponding properties of the restrictions to the space of diagonal operators, and we derive subdifferential and conjugacy formulas. 2 1 Introduction Many optimization problems involve functions of bounded self-adjoint operators on a separable Hilbert space. Often these functions depend only on the spectrum of the operator involved. Many examples have their roots in physical phenomena such as vibrating systems. To optimize a process governed by such a system, we must study the spectra of such operators [14], often asking questi...

Clarke-Ledyaev Mean Value Inequalities in Smooth Banach Spaces

. We refine and extend the Clarke-Ledyaev mean value inequality to smoothBanach spaces.Keywords: ... more . We refine and extend the Clarke-Ledyaev mean value inequality to smoothBanach spaces.Keywords: Mean value inequalities, fuzzy sum rule and smooth spaces.Short Title: Mean Value Inequalities in Smooth Spaces.AMS (1991) subject classification: Primary 26B05.Acknowledgement: I thank A.&quot;Swiech and P. Wolenski for a stimulating conversation.1 IntroductionRecently, F. H. Clarke and Yu. Ledyaev proved a multidirectional mean value inequality [4,5]. It turns

Variational Analysis in Non-reflexive Spaces and Applications to Control Problems with L&sup1; Perturbations

ABSTRACT

Control problems with L/sup 1/ perturbations and variational analysis in non-reflexive spaces

Proceedings of 1994 33rd IEEE Conference on Decision and Control

ABSTRACT The value function plays an important role in optimization; it measures the sensitivity ... more ABSTRACT The value function plays an important role in optimization; it measures the sensitivity of the problem to perturbations of the objective function and the various constraints. Particularly interesting is the derivative of the value function, a measure of so called “differential stability”. When the value function is differentiable, it plays the role of a multiplier. In the context of dynamic optimization this observation establishes an heuristic relationship between the maximum principle and the dynamic programming approaches. Generally, however, the value function of a constrained optimization problem is far from being differentiable. To obtain a rigorous treatment of these heuristic relations one needs to apply the techniques of nonsmooth analysis

Lower semicontinuous Lyapunov functions and subdifferential calculus

Proceedings of the 41st IEEE Conference on Decision and Control, 2002.

We show that lower semicontinuous Lyapunov functions can be used to determine both stable and att... more We show that lower semicontinuous Lyapunov functions can be used to determine both stable and attractive sets of differential equations with a short proof similar to that of the original Lyapunov indirect method. Several examples illustrate the flexibility of using such lower semicontinuous Lyapunov functions.

Helly's Intersection Theorem on Manifolds of Nonpositive Curvature

Journal of Convex Analysis, 2006

The classical Helly’s theorem on intersection of convex sets in finite-dimensional Euclidean spac... more The classical Helly’s theorem on intersection of convex sets in finite-dimensional Euclidean space R , along with Caratheodory’s and Radon’s theorems, is one of the important results in combinatorial convex geometry. It claims (see [13, 9] for the precise statement) that, if for a family {Ca}a∈A of convex sets any N +1 sets have a common point, then all of the sets Ca have a common point. Note that in the case when the index set A is not finite we should assume that sets Ca are closed and at least one of them is bounded.

Mathematical analysis of investment systems

Investment systems are studied using a framework that emphasize their profiles (the cumulative pr... more Investment systems are studied using a framework that emphasize their profiles (the cumulative probability distribution on all the possible percentage gains of trades) and their log return functions (the expected average return per trade in logarithmic scale as a function of the investment size in terms of the percentage of the available capital). The efficiency index for an investment system, defined as the maximum of the log return function, is proposed as a measure to compare investment systems for their intrinsic merit. This efficiency index can be viewed as a generalization of Shannon’s information rate for a communication channel. Applications are illustrated. © 2006 Elsevier Inc. All rights reserved.

Investment system specific European option pricing intervals

Consider an investment system with a nonnegative expected return in a one period economy. We show... more Consider an investment system with a nonnegative expected return in a one period economy. We show that, for an option with a given strike price, there exists a pricing interval [pC , pW ] such that replacing the original investment with the option will benefit judging by the Kelly criterion only when the price of the option lies outside of the interval. More specifically, buying call options with a price less than pC or writing covered call options with a price greater than pW will improve the investment system. Bounds for these thresholds are established. This investment system specific option pricing interval is compatible with option valuation methods of using a replicating portfolio or a risk neutrality argument.

2 The Backtest Overfitting Demonstration Tool

In mathematical finance, backtest overfitting means the usage of historical market data (a backte... more In mathematical finance, backtest overfitting means the usage of historical market data (a backtest) to develop an investment strategy, where many variations of the strategy are tried on the same dataset. Backtest overfitting is now thought to be a primary reason why quantitative investment models and strategies that look good on paper (based on backtests) often disappoint in practice. Models suffering from this condition target the specific idiosyncrasies of a limited dataset, rather than any general behavior, and, as a result, often perform poorly when presented with new data. Backtest overfitting is an instance of the more general phenomenon of multiple testing in scientific research, where a large number of variations of a model are tested on the same data, without accounting for the increase in false positive rates. Standard overfitting techniques, such as the hold-out method, fail to identify this problem, because they are designed to evaluate the complexity of a model relativ...

Mathematik A General Framework for Portfolio Theory . Part III : Multi-Period Markets and Modular Approach by S

This is part III of a series of papers which focus on a general framework for portfolio theory. H... more This is part III of a series of papers which focus on a general framework for portfolio theory. Here we extend a general framework for portfolio theory in a one-period financial market as introduced in part I [12] to multi-period markets. This extension is reasonable for applications. More importantly, we take a new approach, the "modular portfolio theory", which is built from the interaction among four related modules: (a) multi period market model, (b) trading strategies, (c) risk and utility functions (performance criteria), (d) the optimization problem (efficient frontier and efficient portfolio). An important concept that allows to deal with the more general framework discussed here is a trading strategy generating function. This concept limits the discussion to a special class of manageable trading strategies, which is still wide enough to cover many frequently used trading strategies like for instance "constant weight" (fixed fraction). As application we discuss the utility function of compounded return and the risk measure of relative log drawdowns.

Mathematik A General Framework for Portfolio Theory . Part I : theory and various models by S

Utility and risk are two often competing measurements on the investment success. We show that eff... more Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of [Markowitz (1959)] and its natural generalization, the capital market pricing model [Sharpe (1964)], are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework we also recover and extend the results in [Rockafellar, Uryasev & Zabarankin (2006)] which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g. when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a...

A limiting example for the local “fuzzy” sum rule in nonsmooth analysis

Proceedings of the American Mathematical Society

We show that assuming all the summand functions to be lower semicontinuous is not sufficient to e... more We show that assuming all the summand functions to be lower semicontinuous is not sufficient to ensure a (strong) fuzzy sum rule for subdifferentials in any infinite dimensional Banach space. From this we deduce that additional assumptions are also needed on functions for chain rules, multiplier rules for constrained minimization problems and Clarke-Ledyaev type mean value inequalities in the infinite dimensional setting.

No-Arbitrage Principle in Conic Finance

In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbit... more In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid-ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “No-Arbitrage” principle in financial models with the bid-ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can buil...

A General Framework for Portfolio Theory—Part I: Theory and Various Models

Risks

Utility and risk are two often competing measurements on the investment success. We show that eff... more Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two-dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz (1959) and the capital market pricing model Sharpe (1964), are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework, we also recover and extend the results in Rockafellar et al. (2006), which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g., when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a general utility function allows for going beyond the "additive" performance measure to a "multiplicative" one of cumulative returns by using the log utility. As a result, the growth optimal portfolio theory Lintner (1965) and the leverage space portfolio theory Vince (2009) can also be understood and enhanced under our general framework. Thus, this general framework allows a unification of several important existing portfolio theories and goes far beyond. For simplicity of presentation, we phrase all for a finite underlying probability space and a one period market model, but generalizations to more complex structures are straightforward.

A General Framework for Portfolio Theory. Part II: Drawdown Risk Measures

Risks

The aim of this paper is to provide several examples of convex risk measures necessary for the ap... more The aim of this paper is to provide several examples of convex risk measures necessary for the application of the general framework for portfolio theory of Maier-Paape and Zhu (2018), presented in Part I of this series. As an alternative to classical portfolio risk measures such as the standard deviation, we, in particular, construct risk measures related to the “current” drawdown of the portfolio equity. In contrast to references Chekhlov, Uryasev, and Zabarankin (2003, 2005), Goldberg and Mahmoud (2017), and Zabarankin, Pavlikov, and Uryasev (2014), who used the absolute drawdown, our risk measure is based on the relative drawdown process. Combined with the results of Part I, Maier-Paape and Zhu (2018), this allows us to calculate efficient portfolios based on a drawdown risk measure constraint.

A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach

Risks

This is Part III of a series of papers which focus on a general framework for portfolio theory. H... more This is Part III of a series of papers which focus on a general framework for portfolio theory. Here, we extend a general framework for portfolio theory in a one-period financial market as introduced in Part I [Maier-Paape and Zhu, Risks 2018, 6(2), 53] to multi-period markets. This extension is reasonable for applications. More importantly, we take a new approach, the “modular portfolio theory”, which is built from the interaction among four related modules: (a) multi period market model; (b) trading strategies; (c) risk and utility functions (performance criteria); and (d) the optimization problem (efficient frontier and efficient portfolio). An important concept that allows dealing with the more general framework discussed here is a trading strategy generating function. This concept limits the discussion to a special class of manageable trading strategies, which is still wide enough to cover many frequently used trading strategies, for instance “constant weight” (fixed fraction)....

Nonconvex Separation Theorem for Multifunctions, Subdifferential Calculus and Applications

Set-Valued Analysis

We derive a nonconvex separation theorem for multifunctions that generalizes an early result of B... more We derive a nonconvex separation theorem for multifunctions that generalizes an early result of Borwein and Jofré and show that this result is equivalent to several other subdifferential calculus results in smooth Banach spaces. Then we apply this nonconvex separation theorem to improve a second welfare theorem in economics and a necessary optimality condition for a multiobjective optimization problem.

The Optimum Leverage Level of the Banking Sector

Risks

Banks make profits from the difference between short-term and long-term loan interest rates. To i... more Banks make profits from the difference between short-term and long-term loan interest rates. To issue loans, banks raise funds from capital markets. Since the long-term loan rate is relatively stable, but short-term interest is usually variable, there is an interest rate risk. Therefore, banks need information about the optimal leverage strategies based on the current economic situation. Recent studies on the economic crisis by many economists showed that the crisis was due to too much leveraging by “big banks”. This leveraging turns out to be close to Kelly’s optimal point. It is known that Kelly’s strategy does not address risk adequately. We used the return–drawdown ratio and inflection point of Kelly’s cumulative return curve in a finite investment horizon to derive more conservative leverage levels. Moreover, we carried out a sensitivity analysis to determine strategies during a period of interest rates increase, which is the most important and risky period to leverage. Thus, w...

Perturbed Differential Inclusion Problems with Nonadditive L 1-Perturbations and Applications

Journal of Optimization Theory and Applications

A modification to a theorem in Ref. 1 and its proof are given.

An Equilibrium Model for Term Structure of Interest Rates with Consumption Commitments

We study the term structure of interest rates in the presence of consumption commitments using an... more We study the term structure of interest rates in the presence of consumption commitments using an equilibrium model. Under reasonable assumptions we prove the existence and uniqueness of the equilibrium and develop computation methods. Examples are analyzed to illustrate the effect of consumption commitments on the term structure and its manifestations.

Convex Spectral Functions of Compact Operators

We consider functions on the space of compact self-adjoint Hilbert space operators. Specifically,... more We consider functions on the space of compact self-adjoint Hilbert space operators. Specifically, we study those extended-real functions which depend only on the operators' spectral sequences. Examples include the norms of the Schatten p-spaces, the Calder'on norms, the k'th largest eigenvalue, and some infinite-dimensional self-concordant barriers. We show how various convex and nonsmooth-analytic properties of such functions follow from the corresponding properties of the restrictions to the space of diagonal operators, and we derive subdifferential and conjugacy formulas. 2 1 Introduction Many optimization problems involve functions of bounded self-adjoint operators on a separable Hilbert space. Often these functions depend only on the spectrum of the operator involved. Many examples have their roots in physical phenomena such as vibrating systems. To optimize a process governed by such a system, we must study the spectra of such operators [14], often asking questi...

Clarke-Ledyaev Mean Value Inequalities in Smooth Banach Spaces

. We refine and extend the Clarke-Ledyaev mean value inequality to smoothBanach spaces.Keywords: ... more . We refine and extend the Clarke-Ledyaev mean value inequality to smoothBanach spaces.Keywords: Mean value inequalities, fuzzy sum rule and smooth spaces.Short Title: Mean Value Inequalities in Smooth Spaces.AMS (1991) subject classification: Primary 26B05.Acknowledgement: I thank A.&quot;Swiech and P. Wolenski for a stimulating conversation.1 IntroductionRecently, F. H. Clarke and Yu. Ledyaev proved a multidirectional mean value inequality [4,5]. It turns

Variational Analysis in Non-reflexive Spaces and Applications to Control Problems with L&sup1; Perturbations

ABSTRACT

Control problems with L/sup 1/ perturbations and variational analysis in non-reflexive spaces

Proceedings of 1994 33rd IEEE Conference on Decision and Control

ABSTRACT The value function plays an important role in optimization; it measures the sensitivity ... more ABSTRACT The value function plays an important role in optimization; it measures the sensitivity of the problem to perturbations of the objective function and the various constraints. Particularly interesting is the derivative of the value function, a measure of so called “differential stability”. When the value function is differentiable, it plays the role of a multiplier. In the context of dynamic optimization this observation establishes an heuristic relationship between the maximum principle and the dynamic programming approaches. Generally, however, the value function of a constrained optimization problem is far from being differentiable. To obtain a rigorous treatment of these heuristic relations one needs to apply the techniques of nonsmooth analysis

Lower semicontinuous Lyapunov functions and subdifferential calculus

Proceedings of the 41st IEEE Conference on Decision and Control, 2002.

We show that lower semicontinuous Lyapunov functions can be used to determine both stable and att... more We show that lower semicontinuous Lyapunov functions can be used to determine both stable and attractive sets of differential equations with a short proof similar to that of the original Lyapunov indirect method. Several examples illustrate the flexibility of using such lower semicontinuous Lyapunov functions.

Helly's Intersection Theorem on Manifolds of Nonpositive Curvature

Journal of Convex Analysis, 2006

The classical Helly’s theorem on intersection of convex sets in finite-dimensional Euclidean spac... more The classical Helly’s theorem on intersection of convex sets in finite-dimensional Euclidean space R , along with Caratheodory’s and Radon’s theorems, is one of the important results in combinatorial convex geometry. It claims (see [13, 9] for the precise statement) that, if for a family {Ca}a∈A of convex sets any N +1 sets have a common point, then all of the sets Ca have a common point. Note that in the case when the index set A is not finite we should assume that sets Ca are closed and at least one of them is bounded.

Mathematical analysis of investment systems

Investment systems are studied using a framework that emphasize their profiles (the cumulative pr... more Investment systems are studied using a framework that emphasize their profiles (the cumulative probability distribution on all the possible percentage gains of trades) and their log return functions (the expected average return per trade in logarithmic scale as a function of the investment size in terms of the percentage of the available capital). The efficiency index for an investment system, defined as the maximum of the log return function, is proposed as a measure to compare investment systems for their intrinsic merit. This efficiency index can be viewed as a generalization of Shannon’s information rate for a communication channel. Applications are illustrated. © 2006 Elsevier Inc. All rights reserved.

Investment system specific European option pricing intervals

Consider an investment system with a nonnegative expected return in a one period economy. We show... more Consider an investment system with a nonnegative expected return in a one period economy. We show that, for an option with a given strike price, there exists a pricing interval [pC , pW ] such that replacing the original investment with the option will benefit judging by the Kelly criterion only when the price of the option lies outside of the interval. More specifically, buying call options with a price less than pC or writing covered call options with a price greater than pW will improve the investment system. Bounds for these thresholds are established. This investment system specific option pricing interval is compatible with option valuation methods of using a replicating portfolio or a risk neutrality argument.

2 The Backtest Overfitting Demonstration Tool

In mathematical finance, backtest overfitting means the usage of historical market data (a backte... more In mathematical finance, backtest overfitting means the usage of historical market data (a backtest) to develop an investment strategy, where many variations of the strategy are tried on the same dataset. Backtest overfitting is now thought to be a primary reason why quantitative investment models and strategies that look good on paper (based on backtests) often disappoint in practice. Models suffering from this condition target the specific idiosyncrasies of a limited dataset, rather than any general behavior, and, as a result, often perform poorly when presented with new data. Backtest overfitting is an instance of the more general phenomenon of multiple testing in scientific research, where a large number of variations of a model are tested on the same data, without accounting for the increase in false positive rates. Standard overfitting techniques, such as the hold-out method, fail to identify this problem, because they are designed to evaluate the complexity of a model relativ...

Mathematik A General Framework for Portfolio Theory . Part III : Multi-Period Markets and Modular Approach by S

This is part III of a series of papers which focus on a general framework for portfolio theory. H... more This is part III of a series of papers which focus on a general framework for portfolio theory. Here we extend a general framework for portfolio theory in a one-period financial market as introduced in part I [12] to multi-period markets. This extension is reasonable for applications. More importantly, we take a new approach, the "modular portfolio theory", which is built from the interaction among four related modules: (a) multi period market model, (b) trading strategies, (c) risk and utility functions (performance criteria), (d) the optimization problem (efficient frontier and efficient portfolio). An important concept that allows to deal with the more general framework discussed here is a trading strategy generating function. This concept limits the discussion to a special class of manageable trading strategies, which is still wide enough to cover many frequently used trading strategies like for instance "constant weight" (fixed fraction). As application we discuss the utility function of compounded return and the risk measure of relative log drawdowns.

Mathematik A General Framework for Portfolio Theory . Part I : theory and various models by S

Utility and risk are two often competing measurements on the investment success. We show that eff... more Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of [Markowitz (1959)] and its natural generalization, the capital market pricing model [Sharpe (1964)], are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework we also recover and extend the results in [Rockafellar, Uryasev & Zabarankin (2006)] which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g. when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a...

A limiting example for the local “fuzzy” sum rule in nonsmooth analysis

Proceedings of the American Mathematical Society

We show that assuming all the summand functions to be lower semicontinuous is not sufficient to e... more We show that assuming all the summand functions to be lower semicontinuous is not sufficient to ensure a (strong) fuzzy sum rule for subdifferentials in any infinite dimensional Banach space. From this we deduce that additional assumptions are also needed on functions for chain rules, multiplier rules for constrained minimization problems and Clarke-Ledyaev type mean value inequalities in the infinite dimensional setting.

No-Arbitrage Principle in Conic Finance

In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbit... more In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid-ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “No-Arbitrage” principle in financial models with the bid-ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can buil...

A General Framework for Portfolio Theory—Part I: Theory and Various Models

Risks

Utility and risk are two often competing measurements on the investment success. We show that eff... more Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two-dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz (1959) and the capital market pricing model Sharpe (1964), are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework, we also recover and extend the results in Rockafellar et al. (2006), which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g., when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a general utility function allows for going beyond the "additive" performance measure to a "multiplicative" one of cumulative returns by using the log utility. As a result, the growth optimal portfolio theory Lintner (1965) and the leverage space portfolio theory Vince (2009) can also be understood and enhanced under our general framework. Thus, this general framework allows a unification of several important existing portfolio theories and goes far beyond. For simplicity of presentation, we phrase all for a finite underlying probability space and a one period market model, but generalizations to more complex structures are straightforward.

A General Framework for Portfolio Theory. Part II: Drawdown Risk Measures

Risks

The aim of this paper is to provide several examples of convex risk measures necessary for the ap... more The aim of this paper is to provide several examples of convex risk measures necessary for the application of the general framework for portfolio theory of Maier-Paape and Zhu (2018), presented in Part I of this series. As an alternative to classical portfolio risk measures such as the standard deviation, we, in particular, construct risk measures related to the “current” drawdown of the portfolio equity. In contrast to references Chekhlov, Uryasev, and Zabarankin (2003, 2005), Goldberg and Mahmoud (2017), and Zabarankin, Pavlikov, and Uryasev (2014), who used the absolute drawdown, our risk measure is based on the relative drawdown process. Combined with the results of Part I, Maier-Paape and Zhu (2018), this allows us to calculate efficient portfolios based on a drawdown risk measure constraint.

A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach

Risks

This is Part III of a series of papers which focus on a general framework for portfolio theory. H... more This is Part III of a series of papers which focus on a general framework for portfolio theory. Here, we extend a general framework for portfolio theory in a one-period financial market as introduced in Part I [Maier-Paape and Zhu, Risks 2018, 6(2), 53] to multi-period markets. This extension is reasonable for applications. More importantly, we take a new approach, the “modular portfolio theory”, which is built from the interaction among four related modules: (a) multi period market model; (b) trading strategies; (c) risk and utility functions (performance criteria); and (d) the optimization problem (efficient frontier and efficient portfolio). An important concept that allows dealing with the more general framework discussed here is a trading strategy generating function. This concept limits the discussion to a special class of manageable trading strategies, which is still wide enough to cover many frequently used trading strategies, for instance “constant weight” (fixed fraction)....

Nonconvex Separation Theorem for Multifunctions, Subdifferential Calculus and Applications

Set-Valued Analysis

We derive a nonconvex separation theorem for multifunctions that generalizes an early result of B... more We derive a nonconvex separation theorem for multifunctions that generalizes an early result of Borwein and Jofré and show that this result is equivalent to several other subdifferential calculus results in smooth Banach spaces. Then we apply this nonconvex separation theorem to improve a second welfare theorem in economics and a necessary optimality condition for a multiobjective optimization problem.

The Optimum Leverage Level of the Banking Sector

Risks

Banks make profits from the difference between short-term and long-term loan interest rates. To i... more Banks make profits from the difference between short-term and long-term loan interest rates. To issue loans, banks raise funds from capital markets. Since the long-term loan rate is relatively stable, but short-term interest is usually variable, there is an interest rate risk. Therefore, banks need information about the optimal leverage strategies based on the current economic situation. Recent studies on the economic crisis by many economists showed that the crisis was due to too much leveraging by “big banks”. This leveraging turns out to be close to Kelly’s optimal point. It is known that Kelly’s strategy does not address risk adequately. We used the return–drawdown ratio and inflection point of Kelly’s cumulative return curve in a finite investment horizon to derive more conservative leverage levels. Moreover, we carried out a sensitivity analysis to determine strategies during a period of interest rates increase, which is the most important and risky period to leverage. Thus, w...

Perturbed Differential Inclusion Problems with Nonadditive L 1-Perturbations and Applications

Journal of Optimization Theory and Applications

A modification to a theorem in Ref. 1 and its proof are given.