Portfolio selection with a drawdown constraint

When identifying optimal portfolios, practitioners often impose a drawdown constraint. This constraint is even explicit in some money management contracts such as the one recently involving Merrill Lynch’ management of Unilever’s pension fund. In this setting, we provide a characterization of optimal portfolios using mean–variance analysis. In the absence of a benchmark, we find that while the constraint typically decreases the optimal portfolio’s standard deviation, the constrained optimal portfolio can be notably mean–variance inefficient. In the presence of a benchmark such as in the Merrill Lynch–Unilever contract, we find that the constraint increases the optimal portfolio’s standard deviation and tracking error volatility. Thus, the constraint negatively affects a portfolio manager’s ability to track a benchmark.     

Optimal portfolio allocation under the probabilistic VaR constraint and incentives for financial innovation

We characterize the investor’s optimal portfolio allocation subject to a budget constraint and a probabilistic VaR constraint in complete markets environments with a finite number of states. The set of feasible portfolios might no longer be connected or convex, while the number of local optima increases exponentially with the number of states, implying computational complexity. The optimal constrained portfolio allocation may therefore not be monotonic in the state–price density. We propose a type of financial innovation, which splits states of nature, that is shown to weakly enhance welfare, restore monotonicity of the optimal portfolio allocation in the state-price density, and reduce computational complexity. We are grateful to Ken Kavajecz and seminar participants at Harvard Business School, London School of Economics, Maastrict University, ZEI Bonn, and Danske Bank Symposium on Asset allocation and Value-at-Risk: Where Theory Meets Practice for comments on an earlier version of this paper. We also benefitted from the suggestions of two anonymous referees. Our papers can be downloaded from www.RiskResearch.org.     

Model uncertainty and VaR aggregation

Despite well-known shortcomings as a risk measure, Value-at-Risk (VaR) is still the industry and regulatory standard for the calculation of risk capital in banking and insurance. This paper is concerned with the numerical estimation of the VaR for a portfolio position as a function of different dependence scenarios on the factors of the portfolio. Besides summarizing the most relevant analytical bounds, including a discussion of their sharpness, we introduce a numerical algorithm which allows for the computation of reliable (sharp) bounds for the VaR of high-dimensional portfolios with dimensions d possibly in the several hundreds. We show that additional positive dependence information will typically not improve the upper bound substantially. In contrast higher order marginal information on the model, when available, may lead to strongly improved bounds. Several examples of practical relevance show how explicit VaR bounds can be obtained. These bounds can be interpreted as a measure of model uncertainty induced by possible dependence scenarios.     

Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochastic investment model is reformulated as a deterministic programme which allows the calculation of the optimal constrained growth wagers at discrete points in time.     

Conditional Value-at-Risk,spectral risk measures and (non-)diversification in portfolio selection problems – A comparison with mean–variance analysis

We study portfolio selection under Conditional Value-at-Risk and, as its natural extension, spectral risk measures, and compare it with traditional mean–variance analysis. Unlike the previous literature that considers an investor’s mean-spectral risk preferences for the choice of optimal portfolios only implicitly, we explicitly model these preferences in the form of a so-called spectral utility function. Within this more general framework, spectral risk measures tend towards corner solutions. If a risk free asset exists, diversification is never optimal. Similarly, without a risk free asset, only limited diversification is obtained. The reason is that spectral risk measures are based on a regulatory concept of diversification that differs fundamentally from the reward-risk tradeoff underlying the mean–variance framework.     

Portfolio Optimization under Solvency Constraints: A Dynamical Approach

We develop portfolio optimization problems for a nonlife insurance company seeking to find the minimum capital required that simultaneously satisfies solvency and portfolio performance constraints. Motivated by standard insurance regulations, we consider solvency capital requirements based on three criteria: ruin probability, conditional Value-at-Risk, and expected policyholder deficit ratio. We propose a novel semiparametric formulation for each problem and explore the advantages of implementing this methodology over other potential approaches. When liabilities follow a Lognormal distribution, we provide sufficient conditions for convexity for each problem. Using different expected return on capital target levels, we construct efficient frontiers when portfolio assets are modeled with a special class of multivariate GARCH models. We find that the correlation between asset returns plays an important role in the behavior of the optimal capital required and the portfolio structure. The stability and out-of-sample performance of our optimal solutions are empirically tested with respect to both the solvency requirement and portfolio performance, through a double rolling window estimation exercise.     

The performance of model based option trading strategies

This paper analyzes returns to trading strategies in options markets that exploit information given by a theoretical asset pricing model. We examine trading strategies in which a positive portfolio weight is assigned to assets which market prices exceed the price of a theoretical asset pricing model. We investigate portfolio rules which mimic standard mean-variance analysis is used to construct optimal model based portfolio weights. In essence, these portfolio rules allow estimation risk, as well as price risk to be approximately hedged. An empirical exercise shows that the portfolio rules give out-of-sample Sharpe ratios exceeding unity for S&P 500 options. Portfolio returns have no discernible correlation with systematic risk factors, which is troubling for traditional risk based asset pricing explanations.     

Tractable hedging with additional hedge instruments

In an uncertain volatility model where only the stock and the money market account are traded, the upper price bound of a European claim is given by the solution of a Black-Scholes-Barenblatt equation. If an additional hedge instrument is available, the price bound can be tightened. This is also true if the set of admissible strategies is restricted to tractable strategies, which are defined as sums of Black-Scholes strategies. We study the structure of both strategies, the general strategies and the tractable strategies, when an additional convex instrument is available. For a call and a bullish vertical spread, we give closed-form solutions for the optimal tractable hedge when the additional instrument is a call option. We show that the position in the additional convex claim as well as the reduction in the price bounds allow to capture the amount of convexity risk a claim is exposed to.     

Optimal investment for insurers

We consider a stochastic model for the wealth of an insurance company which has the possibility to invest into a risky and a riskless asset under a constant mix strategy. The total insurance claim amount is modeled by a compound Poisson process and the price of the risky asset follows a geometric Brownian motion. We investigate the resulting integrated risk process and the corresponding discounted net loss process. This opens up a way to measure the risk of a negative outcome of the integrated risk process in a stationary way. We provide an approximation of the optimal investment strategy, which maximizes the expected wealth under a risk constraint on the Value-at-Risk.     

Nonlinear portfolio selection using approximate parametric Value-at-Risk

As the skewed return distribution is a prominent feature in nonlinear portfolio selection problems which involve derivative assets with nonlinear payoff structures, Value-at-Risk (VaR) is particularly suitable to serve as a risk measure in nonlinear portfolio selection. Unfortunately, the nonlinear portfolio selection formulation using VaR risk measure is in general a computationally intractable optimization problem. We investigate in this paper nonlinear portfolio selection models using approximate parametric Value-at-Risk. More specifically, we use first-order and second-order approximations of VaR for constructing portfolio selection models, and show that the portfolio selection models based on Delta-only, Delta–Gamma-normal and worst-case Delta–Gamma VaR approximations can be reformulated as second-order cone programs, which are polynomially solvable using interior-point methods. Our simulation and empirical results suggest that the model using Delta–Gamma-normal VaR approximation performs the best in terms of a balance between approximation accuracy and computational efficiency.     

Equilibrium relations in a capital asset market: A mean absolute deviation approach

We consider the equilibrium in a capital asset market where the risk is measured by the absolute deviation, instead of the standard deviation of the rate of return of the portfolio. It is shown that the equilibrium relations proved by Mossin for the mean variance (MV) model can also be proved for the mean absolute deviation (MAD) model under similar assumptions on the capital market. In particular, a sufficient condition is derived for the existence of a unique nonnegative equilibrium price vector and derive its explicit formula in terms of exogeneously determined variables. Also, we prove relations between the expected rate of return of individual assets and the market portfolio.     

Dynamic optimal portfolio choice in a jump-diffusion model with investment constraints

We consider the dynamic portfolio choice problem in a jump-diffusion model, where an investor may face constraints on her portfolio weights: for instance, no-short-selling constraints. It is a daunting task to use standard numerical methods to solve a constrained portfolio choice problem, especially when there is a large number of state variables. By suitably embedding the constrained problem in an appropriate family of unconstrained ones, we provide some equivalent optimality conditions for the indirect value function and optimal portfolio weights. These results simplify and help to solve the constrained optimal portfolio choice problem in jump-diffusion models. Finally, we apply our theoretical results to several examples, to examine the impact of no-short-selling and/or no-borrowing constraints on the performance of optimal portfolio strategies.     

Beliefs regarding fundamental value and optimal investing

Standard optimal portfolio selection models take no account of the special information that active investors believe they possess. For example, active investors who believe they can place bounds on the price of a security will want to use that information when assessing risk and expected return in order to construct an optimal portfolio. In this paper, we use two continuous-time models to analyze how placing boundaries on the price of a stock affects assessed risk, expected returns, and the optimal holdings of an active investor, and how those vary as a function of the relation between the stock price and the boundaries. In particular, the optimal strategy takes significant long/short positions as the price nears its lower/upper boundary.     

Do hedge funds have enough capital? A value-at-risk approach

We examine the risk characteristics and capital adequacy of hedge funds through the Value-at-Risk approach. Using extensive data on nearly 1,500 hedge funds, we find only 3.7% live and 10.9% dead funds are undercapitalized as of March 2003. Moreover, the undercapitalized funds are relatively small and constitute a tiny fraction of total fund assets in our sample. Cross-sectionally, the variability in fund capitalization is related to size, investment style, age, and management fee. Hedge fund risk and capitalization also display significant time variation. Traditional risk measures like standard deviation or leverage ratios fail to detect these trends.     

The 1/N investment strategy is optimal under high model ambiguity

The 1/N investment strategy, i.e. the strategy to split one’s wealth uniformly between the available investment possibilities, recently received plenty of attention in the literature. In this paper, we demonstrate that the uniform investment strategy is rational in situations where an agent is faced with a sufficiently high degree of model uncertainty in the form of ambiguous loss distributions. More specifically, we use a classical risk minimization framework to show that, for a broad class of risk measures, as the uncertainty concerning the probabilistic model increases, the optimal decisions tend to the uniform investment strategy.To illustrate the theoretical results of the paper, we investigate the Markowitz portfolio selection model as well as Conditional Value-at-Risk minimization with ambiguous loss distributions. Subsequently, we set up a numerical study using real market data to demonstrate the convergence of optimal portfolio decisions to the uniform investment strategy.     

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.     

Bayesian Value-at-Risk with product partition models

In this paper we propose a novel Bayesian methodology for Value-at-Risk computation based on parametric Product Partition Models. Value-at-Risk is a standard tool for measuring and controlling the market risk of an asset or portfolio, and is also required for regulatory purposes. Its popularity is partly due to the fact that it is an easily understood measure of risk. The use of Product Partition Models allows us to remain in a Normal setting even in the presence of outlying points, and to obtain a closed-form expression for Value-at-Risk computation. We present and compare two different scenarios: a product partition structure on the vector of means and a product partition structure on the vector of variances. We apply our methodology to an Italian stock market data set from Mib30. The numerical results clearly show that Product Partition Models can be successfully exploited in order to quantify market risk exposure. The obtained Value-at-Risk estimates are in full agreement with Maximum Likelihood approaches, but our methodology provides richer information about the clustering structure of the data and the presence of outlying points.     

Portfolio credit-risk optimization

This paper evaluates several alternative formulations for minimizing the credit risk of a portfolio of financial contracts with different counterparties. Credit risk optimization is challenging because the portfolio loss distribution is typically unavailable in closed form. This makes it difficult to accurately compute Value-at-Risk (VaR) and expected shortfall (ES) at the extreme quantiles that are of practical interest to financial institutions. Our formulations all exploit the conditional independence of counterparties under a structural credit risk model. We consider various approximations to the conditional portfolio loss distribution and formulate VaR and ES minimization problems for each case. We use two realistic credit portfolios to assess the in- and out-of-sample performance for the resulting VaR- and ES-optimized portfolios, as well as for those which we obtain by minimizing the variance or the second moment of the portfolio losses. We find that a Normal approximation to the conditional loss distribution performs best from a practical standpoint.     

Short-horizon regulation for long-term investors

We study the effects of imposing repeated short-horizon regulatory constraints on long-term investors. We show that Value-at-Risk and Expected Shortfall constraints, when imposed dynamically, lead to similar optimal portfolios and wealth distributions. We also show that, in utility terms, the costs of imposing these constraints can be sizeable. For a 96% funded pension plan, both an annual Value-at-Risk constraint and an annual Expected Shortfall constraint can lead to an economic cost of about 2.5–3.8% of initial wealth over a 15-year horizon.     

Ethical Investing Has No Portfolio Performance Cost

We guide investors in three ethical investment applications by comparing ethically constrained versus unconstrained optimal portfolio methods that force well-behaved weights. With optimal readjustment upon constrained investing, in Sharpe ratio analysis, we find no evidence of a performance cost for sin-free, carbon-free, or Shariah portfolios even though, in the most exacting case, Shariah investing excludes roughly 65% of common shares from security selection. In each case, we identify the specific portfolio adjustments needed to prevent an ethical portfolio performance cost.