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The performance of mutual fund or pension fund managers is often evaluated by comparing the returns of managed portfolios with those of a benchmark. As most portfolio managers use dynamic rules for rebalancing their portfolios, we use a dynamic framework to study the optimization of the tracking error–return trade-off. Following these observations, we assume that the manager minimizes the tracking error under an expected return goal (or, equivalently, maximizes the information ratio). Moreover, we assume that he/she complies with a stochastic hedging constraint whereby the terminal value of the portfolio is (almost surely) higher than a given stochastic payoff. This general setting includes the case of a minimum wealth level at the horizon date and the case of a performance constraint on terminal wealth as measured by the benchmark (i.e. terminal portfolio wealth should be at least equal to a given proportion of the index). When the manager cares about absolute returns and relative returns as well, the ri...

With a large number of securities (N) and fewer observations (T), deriving the global minimum variance portfolio requires the inversion of the singular sample covariance matrix of security returns. We introduce the Break-Down Free Generalized Minimum RESidual (BFGMRES), a Krylov subspaces method, as a fully automated approach for deriving the minimum variance portfolio. BFGMRES is a numerical algorithm that provides solutions to singular linear systems without requiring ex-ante assumptions on the covariance structure. Moreover, it is robust to illiquidity and potentially faulty data. US and international stock data are used to demonstrate the relative robustness of BFGMRES to illiquidity when compared to the “shrinkage to market” methodology developed by Ledoit and Wolf (2003) . The two methods have similar performance as assessed by the Sharpe ratios and standard deviations for filtered data. In a simulation study, we show that BFGMRES is more robust than shrinkage to market in the presence of data irregularities. Indeed, when there is an illiquid stock shrinkage to market allocates almost 100% of the portfolio weights to this stock, whereas BFGMRES does not. In further simulations, we also show that when there is no illiquidity, BFGMRES exhibits superior performance than shrinkage to market when the number of stocks is high and the sample covariance matrix is highly singular.

The performance of active portfolio managers who must comply with a weights constraint is often assessed against a benchmark. The weights constraint is common as the funds are committed by their own prospectus to a minimum (or maximum) portfolio concentration. We characterize the optimal asset allocation and analyze the implications of the weights constraint on the manager's performance and on the relevance of the information ratio. We obtain that because of the weights constraint, at the optimum, the information ratio often decreases when the manager is free to deviate more from the benchmark.

Two pervasive features of industries experiencing rapid technological progress are uncertainty (with regard to the technological feasibility and marketabilility of an innovation) and networks (the dense web of research alliances and joint ventures linking firms to each other). This paper connects the two disparate phenomena using the notion of real options . It visualizes firms as nodes and the links connecting them as call options that give each pair of interlinked firms the right, but not the obligation, to sink additional resources into a project at some future date conditional on favorable technical/market information. The formation of networks is endogenous as firms establish links with others by appraising their value using option pricing methods. Our model explains the following: why networks are particularly ubiquitous in industries that are subject to high uncertainty; why networks often display an interconnected “hubs and spokes” architecture; why small (or peripheral spoke) firms often sink resources into relatively higher risk higher return investment projects (and those too with only large, or hub firms); and why so many research alliances are continuously formed and dissolved. Our paper also outlines the conditions under which ex-ante symmetric firms end up ex-post forming complex asymmetric networks.

Endowment fund managers face an asset allocation problem with several particularities: they are more interested in spending for current and future beneficiaries than growing value, although the trade-off between these two alternatives needs to be understood; they have to consider longest-term investment, typically an infinite horizon. We do address these allocation constraints in a dynamic framework where minimum subsistence levels (introducing the idea that a minimum spending amount needs to be made at every time period) are introduced in the objective function. We derive explicit formulas for the optimal spending stream, endowment value, spending rate and portfolio strategy in a simple Black/Scholes type economy. We analyze the effects of parameter changes on asset allocation decisions and provide simulations on bearish, median and bullish paths.

Closed‐form solutions for HARA optimal portfolios are obtained in a dynamic portfolio optimization model in three assets (stocks, bonds, and cash) in a Vasicek‐type model of stochastic interest rates with correlated stock prices. The HARA is a buy‐and‐hold combination of a zero‐coupon bond with maturity matching the investor’s horizon and a “CRRA mutual fund.” This simple characterization facilitates insights about investor behavior over time and provides explanations on the rational use of convex versus concave investment strategies. The model illuminates clearly the role of the different market parameters and relative risk aversion in portfolio strategies.

Should the proportion of risky assets in the risky part of an investor’s portfolio depend on the investor’s risk aversion? According to basic financial theory, in particular the mutual-fund separation theorem with a riskless asset, the answer is no. The theorem states that rational investors should divide their assets between a riskless asset and a risky mutual fund, the composition of which is the same for all investors. Risk aversion affects only the allocation between the riskless asset and the fund. However, Niko Canner et al. (1997), CMW hereafter, observed that popular investment advice does not conform to this theory. They reported the stocks, bonds, and cash allocations recommended by four advisors for conservative, moderate, and aggressive investors. As shown in Table 1, which is reproduced from CMW, the advisors recommend a bond/stock ratio that varies directly with risk aversion. For example, Fidelity recommends a bond/stock ratio of 1.50 for a “conservative” (more riskaverse) investor, a ratio of 1.00 for a “moderate” (less risk-averse) investor, and a ratio of 0.46 for an “aggressive” (still less risk-averse) investor. The inconsistency between such advice and the separation theorem is called an asset allocation puzzle by CMW. They attempted to solve the puzzle by relaxing key assumptions in the theory, but finally reached a negative conclusion: “Although we cannot rule out the possibility that popular advice is consistent with some model of rational behavior, we have so far been unable to find such a model” (p. 181). However, they suggested that consideration of intertemporal trading might help resolve the puzzle. In the present paper, we provide theoretical support for the popular advice. The two key insights are that the investor’s horizon may exceed the maturity of the cash asset and that the investor rebalances the portfolio as time passes. If the investor’s horizon exceeds the maturity of cash, which might be a money-market security with maturity of one to six months, then cash is not the riskless asset as is commonly assumed in the basic theory. In a theory allowing portfolio rebalancing, as opposed to a buy-and-hold framework, it is not unreasonable to assume that the investor can synthesize a riskless asset (a zero-coupon bond maturing at the horizon) using a bond fund and cash. Then bonds will be both in the (synthetic) riskless asset and in the risky mutual fund and we show that in this case the theoretical bond/stock ratio varies directly with risk aversion for any hyperbolic absolute risk aversion (HARA) investor. As an example of the type of results that a specific model can produce, we provide a continuous-time model with closed-form solutions, which produces theoretical bond/stock ratios similar to the popular advice. The present paper is organized as follows: in the next section, we analyze the popular advice in terms of the theory of mutual-fund separation of David Cass and Joseph E. Stiglitz (1970). We show that this theory is relevant both in static and dynamic frameworks and use it to analyze the popular advice in complete and incomplete markets. In Section II, we analyze the popular advice in the context of Robert C. Merton’s (1971) continuous-time statement of mutualfund separation and present an illustrative model in which a CRRA investor makes continuous-time portfolio decisions under interest rate and stock price uncertainty. In Section III, numerical results are compared with the popular advice. Section IV is a conclusion. * Bajeux-Besnainou: Department of Finance, School of Business and Public Management, George Washington University, 2023 G Street NW, Washington, DC 20052; Jordan: National Economic Research Associates, 1255 23rd Street NW, Washington, DC 20037; Portait: CNAM and ESSEC, Finance Chair CNAM, 2 Rue Conte, Paris, France. This research was supported by a grant from the Institute for Quantitative Investment Research. We thank two anonymous referees for their comments. 1 HARA functions include quadratic utility, which is one way of justifying mean-variance preferences, and constant relative risk aversion (CRRA) utility. Both quadratic and CRRA utility were considered in the CMW analysis.

The martingale approach to pricing contingent claims can be applied in a multiple state variable model. The idea is used to derive the prices of derivative securities (futures on stock and bond futures, options on stocks, bonds and futures) given a continuous time Gaussian multi-factor model of the returns of stocks and bonds. The bond market is similar to Langetieg's multi-factor model, which has closed-form solutions. This model is a generalization of Vasicek's model, where the term structure depends on state variables following correlated mean reverting processes. The stock market is affected by systematic and unsystematic risk.

The numeraire portfolio, also called the optimal growth portfolio, allows simple derivations of the main results of financial theory. The prices of self financing portfolios, when the optimal growth portfolio is the numeraire, are martingales in the ‘true’ (historical) probability. Given the dynamics of the traded securities, the composition of the numeraire portfolio as well as its value are easily computable. Among its numerous properties, the numeraire portfolio is instantaneously mean variance efficient. This key feature allows a simple derivation of standard continuous time CAPM, CCAPM, APT and contingent claim pricing. Moreover, since the Radon-Nikodym derivatives of the usual martingale measures are very simple functions of the numeraire portfolio, the latter provides a convenient link between the standard Capital Market Theory a la Merton and the probabilistic approach a la Harrison-Kreps-Pliska.

Ross (1976) has shown, in a static framework, how options can complete financial markets. This paper examines the possible extensions of Ross's idea in a dynamic setup. Surprisingly enough, we find that the answer is very sensitive to the choice of the stochastic model for the underlying security returns. More specifically we obtain the following results: In a discrete-time model, classical European options typically become redundant with some probability (Proposition 2.1). Obnly path dependent (“exotic”) options may generate dynamic spanning (Proposition 4.1). In a continuous-time model with stochastic volatility of the underlying security, and under reasonable assumptions, a European option is always a good instrument for completing markets (Proposition 5.2).

Active portfolio manager performances are commonly assessed against a benchmark. In this case, his/her performance is often measured by the Information Ratio, the maximization of which is equivalent to the maximization of an expected return under a tracking error constraint. In addition, asset managers often deal with weights constraints (for instance, no more than 10% in equity). These constraints are regulatory or inherent to the fund's policy. We consider a fund manager complying simultaneously with a tracking error (computed for instance, vis-a-vis a bond index) and a weights constraints. These two constraints are not necessarily redundant even when the benchmark complies with the weights constraint. We show, theoretically and through numerical examples that the weights and the tracking error constraints can be simultaneously binding, we consider both equality and inequality weights constraints, derive the analytical and geometrical solutions in both cases and provide financial interpretations based on funds separation. We compute the loss in the Information Ratio due to a weights constraint and analyze the implications on asset allocation and performance measures. In particular, due to the weights constraint, the asset manager may operate under a smaller Information Ratio when free to deviate more from the benchmark (higher Tracking Error). This result undermines the coherence of the Information Ratio as a measure of the ability of asset managers.

The aim of this article is to analyze the portfolio strategies that are mean-variance efficient when continuous rebalancing is allowed between the current date (0) and the horizon (T). Under very general assumptions, when a zero-coupon bond of maturity T exists, the dynamic efficient frontier is a straight line, the slope of which is explicitly characterized. Every dynamic mean-variance efficient strategy can be viewed as buy and hold combinations of two funds: the zero-coupon bond of maturity T and a continuously rebalanced portfolio. An appropriate dynamic strategy defining the latter is explicitly derived for two particular price processes and comparisons of the Efficient Frontiers (Static versus Dynamic) are provided in these cases.