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arXiv2014

Mean-Variance Policy for Discrete-time Cone Constrained Markets: The Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure

The discrete-time mean-variance portfolio selection formulation, a representative of general dynamic mean-risk portfolio selection problems, does not satisfy time consistency in efficiency (TCIE) in general, i.e., a truncated pre-committed efficient policy may become inefficient when considering the corresponding truncated problem, thus stimulating investors' irrational investment behavior. We investigate analytically effects of portfolio constraints on time consistency of efficiency for convex cone constrained markets. More specifically, we derive the semi-analytical expressions for the pre-committed efficient mean-variance policy and the minimum-variance signed supermartingale measure (VSSM) and reveal their close relationship. Our analysis shows that the pre-committed discrete-time efficient mean-variance policy satisfies TCIE if and only if the conditional expectation of VSSM's density (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 findings indicate that the property of time consistency in efficiency only depends on the basic market setting, including portfolio constraints, and this fact motivates us to establish a general solution framework in constructing TCIE dynamic portfolio selection problem formulations by introducing suitable portfolio constraints.

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arXiv2014

Introduction to Risk Parity and Budgeting

Although portfolio management didn't change much during the 40 years after the seminal works of Markowitz and Sharpe, the development of risk budgeting techniques marked an important milestone in the deepening of the relationship between risk and asset management. Risk parity then became a popular financial model of investment after the global financial crisis in 2008. Today, pension funds and institutional investors are using this approach in the development of smart indexing and the redefinition of long-term investment policies. Introduction to Risk Parity and Budgeting provides an up-to-date treatment of this alternative method to Markowitz optimization. It builds financial exposure to equities and commodities, considers credit risk in the management of bond portfolios, and designs long-term investment policy. This book contains the solutions of tutorial exercices which are included in Introduction to Risk Parity and Budgeting.

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arXiv2014

Momentum Strategies with L1 Filter

In this article, we discuss various implementation of L1 filtering in order to detect some properties of noisy signals. This filter consists of using a L1 penalty condition in order to obtain the filtered signal composed by a set of straight trends or steps. This penalty condition, which determines the number of breaks, is implemented in a constrained least square problem and is represented by a regularization parameter ? which is estimated by a cross-validation procedure. Financial time series are usually characterized by a long-term trend (called the global trend) and some short-term trends (which are named local trends). A combination of these two time scales can form a simple model describing the process of a global trend process with some mean-reverting properties. Explicit applications to momentum strategies are also discussed in detail with appropriate uses of the trend configurations.

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arXiv2014

On asymptotic optimality of Merton's myopic portfolio strategies for discrete time market

This paper studies the properties of discrete time stochastic optimal control problems associated with portfolio selection. We investigate if optimal continuous time strategies can be used effectively for a discrete time market after a straightforward discretization. We found that Merton's strategy approximates the performance of the optimal strategy in a discrete time model with the sufficiently small time steps

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arXiv2014

Portfolio Optimization in Affine Models with Markov Switching

We consider a stochastic factor financial model where the asset price process and the process for the stochastic factor depend on an observable Markov chain and exhibit an affine structure. We are faced with a finite time investment horizon and derive optimal dynamic investment strategies that maximize the investor's expected utility from terminal wealth. To this aim we apply Merton's approach, as we are dealing with an incomplete market. Based on the semimartingale characterization of Markov chains we first derive the HJB equations, which in our case correspond to a system of coupled non-linear PDEs. Exploiting the affine structure of the model, we derive simple expressions for the solution in the case with no leverage, i.e. no correlation between the Brownian motions driving the asset price and the stochastic factor. In the presence of leverage we propose a separable ansatz, which leads to explicit solutions in this case as well. General verification results are also proved. The results are illustrated for the special case of a Markov modulated Heston model.

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arXiv2015

Reward-risk momentum strategies using classical tempered stable distribution

We implement momentum strategies using reward-risk measures as ranking criteria based on classical tempered stable distribution. Performances and risk characteristics for the alternative portfolios are obtained in various asset classes and markets. The reward-risk momentum strategies with lower volatility levels outperform the traditional momentum strategy regardless of asset class and market. Additionally, the alternative portfolios are not only less riskier in risk measures such as VaR, CVaR and maximum drawdown but also characterized by thinner downside tails. Similar patterns in performance and risk profile are also found at the level of each ranking basket in the reward-risk portfolios. Higher factor-neutral returns achieved by the reward-risk momentum strategies are statistically significant and large portions of the performances are not explained by the Carhart four-factor model.

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arXiv2014

Utility maximization in the large markets

In the large financial market, which is described by a model with countably many traded assets, we formulate the problem of the expected utility maximization. Assuming that the preferences of an economic agent are modeled with a stochastic utility and that the consumption occurs according to a stochastic clock, we obtain the "usual" conclusions of the utility maximization theory. We also give a characterization of the value function in the large market in terms of a sequence of the value functions in the finite-dimensional models.

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arXiv2014

Credit acceptance process strategy case studies - the power of Credit Scoring

The paper is aware of the importance of certain figures that are essential to an understanding of Credit Scoring models in credit acceptance process optimization, namely if the power of discrimination measured by Gini value is increased by 5% then the profit of the process can be increased monthly by about 1 500 kPLN (300 kGBP, 500 kUSD, 350 kEUR). Simple business models of credit loans are also presented: acquisition - installment loan (low price) and cross-sell - cash loans (high price). Scoring models are used to optimize process, to become profitable. Various acceptance strategies with different cutoffs are presented, some are profitable and some are not. Moreover, in a time of prosperity some are preferable whilst the inverse is true during a period of high risk or crisis. To optimize the process four models are employed: three risk models, to predict the probability of default and one typical propensity model to predict the probability of response. It is a simple but very important example of the Customer Lifetime Value (CLTV or CLV) model business, where risk and response models are working together to become a profitable process.

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arXiv2014

Pseudo Linear Pricing Rule for Utility Indifference Valuation

This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model, and provides two linear approximations for the utility indifference price. The key tool is a probabilistic representation for the utility indifference price by the solution of a functional differential equation, which is termed \emph{pseudo linear pricing rule}. We also provide an alternative derivation of the quadratic BSDE representation for the utility indifference price.

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arXiv2021

Maximum drawdown, recovery, and momentum

We empirically test predictability on asset price by using stock selection rules based on maximum drawdown and its consecutive recovery. In various equity markets, monthly momentum- and weekly contrarian-style portfolios constructed from these alternative selection criteria are superior not only in forecasting directions of asset prices but also in capturing cross-sectional return differentials. In monthly periods, the alternative portfolios ranked by maximum drawdown measures exhibit outperformance over other alternative momentum portfolios including traditional cumulative return-based momentum portfolios. In weekly time scales, recovery-related stock selection rules are the best ranking criteria for detecting mean-reversion. For the alternative portfolios and their ranking baskets, improved risk profiles in various reward-risk measures also imply more consistent prediction on the direction of assets in future. In the Carhart four-factor analysis, higher factor-neutral intercepts for the alternative strategies are another evidence for the robust prediction by the alternative stock selection rules.

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arXiv2018

Is It Possible to OD on Alpha?

It is well known that combining multiple hedge fund alpha streams yields diversification benefits to the resultant portfolio. Additionally, crossing trades between different alpha streams reduces transaction costs. As the number of alpha streams increases, the relative turnover of the portfolio decreases as more trades are crossed. However, we argue, under reasonable assumptions, that as the number of alphas increases, the turnover does not decrease indefinitely; instead, the turnover approaches a non-vanishing limit related to the correlation structure of the portfolio's alphas. We also point out that, more generally, computational simplifications can arise when the number of alphas is large.

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arXiv2014

Utility indifference pricing of derivatives written on industrial loss indexes

We consider the problem of pricing derivatives written on some industrial loss index via utility indifference pricing. The industrial loss index is modelled by a compound Poisson process and the insurer can adjust her portfolio by choosing the risk loading, which in turn determines the demand. We compute the price of a CAT(spread) option written on that index using utility indifference pricing.

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arXiv2014

Facelifting in Utility Maximization

We establish the existence and characterization of a primal and a dual facelift - discontinuity of the value function at the terminal time - for utility-maximization in incomplete semimartingale-driven financial markets. Unlike in the lower- and upper-hedging problems, and somewhat unexpectedly, a facelift turns out to exist in utility-maximization despite strict convexity in the objective function. In addition to discussing our results in their natural, Markovian environment, we also use them to show that the dual optimizer cannot be found in the set of countably-additive (martingale) measures in a wide variety of situations.

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arXiv2014

Two centuries of trend following

We establish the existence of anomalous excess returns based on trend following strategies across four asset classes (commodities, currencies, stock indices, bonds) and over very long time scales. We use for our studies both futures time series, that exist since 1960, and spot time series that allow us to go back to 1800 on commodities and indices. The overall t-stat of the excess returns is $\approx 5$ since 1960 and $\approx 10$ since 1800, after accounting for the overall upward drift of these markets. The effect is very stable, both across time and asset classes. It makes the existence of trends one of the most statistically significant anomalies in financial markets. When analyzing the trend following signal further, we find a clear saturation effect for large signals, suggesting that fundamentalist traders do not attempt to resist "weak trends", but step in when their own signal becomes strong enough. Finally, we study the performance of trend following in the recent period. We find no sign of a statistical degradation of long trends, whereas shorter trends have significantly withered.

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arXiv2014

$L_p$ regularized portfolio optimization

Investors who optimize their portfolios under any of the coherent risk measures are naturally led to regularized portfolio optimization when they take into account the impact their trades make on the market. We show here that the impact function determines which regularizer is used. We also show that any regularizer based on the norm $L_p$ with $p>1$ makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with $p<1$ do not. The $L_1$ norm represents a border case: its "soft" implementation does not remove the instability, but rather shifts its locus, whereas its "hard" implementation (equivalent to a ban on short selling) eliminates it. We demonstrate these effects on the important special case of Expected Shortfall (ES) that is on its way to becoming the next global regulatory market risk measure.

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arXiv2014

Signal-wise performance attribution for constrained portfolio optimisation

Performance analysis, from the external point of view of a client who would only have access to returns and holdings of a fund, evolved towards exact attribution made in the context of portfolio optimisation, which is the internal point of view of a manager controlling all the parameters of this optimisation. Attribution is exact, that-is-to-say no residual "interaction" term remains, and various contributions to the optimal portfolio can be identified: predictive signals, constraints, benchmark. However constraints are identified as a separate portfolio and attribution for each signal that are used to predict future returns thus corresponds to unconstrained signal portfolios. We propose a novel attribution method that put predictive signals at the core of attribution and allows to include the effect of constraints in portfolios attributed to every signal. We show how this can be applied to various trading models and portfolio optimisation frameworks and explain what kind of insights such an attribution provides.

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arXiv2014

Continuous time portfolio choice under monotone preferences with quadratic penalty - stochastic interest rate case

This is a follow up of our previous paper - Trybuła and Zawisza \cite{TryZaw}, where we considered a modification of a monotone mean-variance functional in continuous time in stochastic factor model. In this article we address the problem of optimizing the mentioned functional in a market with a stochastic interest rate. We formulate it as a stochastic differential game problem and use Hamilton-Jacobi-Bellman-Isaacs equations to derive the optimal investment strategy and the value function.

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arXiv2014

Stochastic Perron's Method for the Probability of lifetime ruin problem under transaction costs

We apply stochastic Perron's method to a singular control problem where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming principle (DPP), we characterize the value function as the unique viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.

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