Investment Time Horizon Research Articles

Multifractal returns and hierarchical portfolio theory

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. Inspired by an analogy between price dynamics and hydrodynamic turbulence, it models the time scale dependence of the probability distribution of returns in terms of a superposition of Gaussian laws, with a log-normal distribution of the Gaussian variances. This multifractal description of asset fluctuations is generalized into a multivariate framework to account simultaneously for correlations across time scales and between a basket of assets. The reported empirical results show that this extension is pertinent for financial modelling. Two sources of departure from normality are discussed: at large time scales, the distinction between discretely and continuously discounted returns leads to the usual log-normal deviation from normality; at small time scales, the multiplicative cascade process leads to multifractality and strong deviations from normality. By perturbation expansions of the cumulants of the distribution of returns, we are able to quantify precisely the interplay and crossover between these two mechanisms. The second part of the paper applies this theory to portfolio optimization. Our multiscale description allows us to characterize the portfolio return distribution at all time scales simultaneously. The portfolio composition is predicted to change with the investment time horizon (i.e. the time scale) in a way that can be fully determined once an adequate measure of risk is chosen. We discuss the use of the fourth-order cumulant and of utility functions. While the portfolio volatility can be optimized in some cases for all time horizons, the kurtosis and higher normalized cumulants cannot be simultaneously optimized. For a fixed investment horizon, we study in detail the influence of the number of rebalancing of the portfolio. For the large risks quantified by the cumulants of order larger than two, the number of periods has a non-trivial influence, in contrast with Tobin's result valid in the mean-variance framework. This theory provides a fundamental framework for the conflicting optimization involved in the different time horizons and quantifies systematically the trade-offs for an optimal inter-temporal portfolio optimization.

Time Diversification, Safety-First and Risk

The purpose of this article is to demonstrate the effect of investment time horizon on the choice of risky assets in a portfolio when the investor in question is optimizing a Safety-First (downside risk-aversion) utility function. It is shown, under standard assumptions, that although shortfall risk decreases exponentially with investment time horizon, the portfolio asset allocation proportions remain invariant. In fact, in some instances, the optimal allocation will not even depend on the drift of the underlying assets. Thus, we extend the classical results of Samuelson and Merton, derived under conventional utility assumptions, to an individual optimizing an A.D. Roy Safety-First objective; a discontinuous utility function that has been extolled as conforming to observed investor behaviour. A numerical example is provided.

NPV-COMPATIBILITY, PROJECT RANKING, AND RELATED ISSUES

ABSTRACT The presented paper addresses the question of the NPV-compatibility of the Overall Rates of Return (ORRs), and analyses two project ranking procedures that were advocated in the recent contribution “Overall Rates of Return: Investment Basis, Reinvestment Rates and Time Horizons” by D.M. Shull (The Engineering Economist, Vol.39, No.2, Winter 1994, pp. 139-163). The paper indicates that the current definitions of the ORRs are not fully NPV-compatible, and it provides a generalized, NPV-compatible ORR definition. The paper further shows, using numerical examples, that the two project ranking procedures adopted by Shull are conceptually inadequate. It presents the correct approach to project-ranking, and proves that this approach can be used in conjunction with any NPV-compatible profitability criterion. Also, the paper addresses and rectifies some inaccurate statements and formulas in Shull's contribution. Finally, it is noted that Shull's Scale-Adjusted Return Method was, in fact, introduced nineteen years ago by P.J. Athanasopoulos (“A Note on the Modified Internal Rate Of Return and Investment Criterion, ” The Engineering Economist, Vol.23, No.2, Winter 1978, pp. 131-133).

Emerging strategies in the venture capital industry

This article reports a study of the future direction of the venture capital industry by examining the basic strategies and strategic assumptions of a broad sample of venture capital firms. There are three main sets of results: First, the once homogeneous venture capital industry is rapidly dividing into several different “strategic groups.” Members of these “groups” are increasingly distinguishing themselves from other groups on four basic dimensions followed by member firms: 1. Financial Resources—Equity capital comes from a greater variety of sources (five major sources) resulting in fundamentally different demands on the mission of the receiving venture capital firm. 2. Staff Resources—The way venture capital firms use staff resources, particularly regarding investee management assistance, is becoming increasingly varied across different groups. Some firms provide fewer than 2-days per year, while others provide up to 450 man-days per year per client. 3. Venture Stages—While the overall industry retains a primary interest in stage 1,2, and 3 investment, specific firms vary considerably in the distribution of investment emphasis across these three stages. 4. Use of Financial Resources-Firms in the industry are becoming increasingly differentiated in the size of minimum investments they make ($100 M to $1000 M) and in their role as a direct investor versus a “broker” for institutional funds. Practicing venture capitalists should make use of this first set of findings in two ways. First, they may find it useful to compare their firm's orientation along these four strategic dimensions with those of the firm's that comprised this study. Second, they may seek to use these four strategic dimensions as a basis on which they might examine, clarify, and/or redefine the marketing strategy pursued by their firm. A second set of results identified three goals and priorities of venture capital firms that have neither changed over time nor across increasingly different strategic groups. Annualized, after-tax return on investments of between 25% and 40% remain the most common objective across all firms. A 5-to-6 year investment time horizon and a major emphasis on the quality of the management team in evaluating new deals were universal priorities across diverse venture capital firms. A third finding in this study was that venture capital firms profess greater “certainty” about the future direction of the venture capital industry than the direction of their firm. The most notable example of this is a strong sense that industry-wide rates of return are headed downward yet few senior partners expect their firm to experience this decline. Practicing venture capitalists may be interested to peruse these results to see what trends are predicted within the venture capital industry by this subsample of that industry. Second, they should consider the finding that industry-wide rates of return are headed downward in light of the first two sets of findings to develop their own opinion about the future performance of different strategic groups within the industry. It is important to note that the sample of venture capital firms on which this study was based did not include most of the larger, older funds. Some of these funds would be characterized as “industry leaders, pace-setters, and innovators.” The sample provides a solid representation of the “broad middle” of the venture capital industry and newer entrants into the industry. While larger, older funds are under represented, their impact on future trends and strategies in the industry is captured to some extent in the set of questions about “future direction of the venture capital industry.“ Finally, the emerging strategic groups in the venture capital industry that were identified by this study may be useful information for investors as well as users of venture capital. For investors, the opportunity to participate in venture capital activity should become more clearly understood and varied. Basically, this study should help investors differentiate the strategic posture of different venture capital firms and funds on four factors rather than simply industry/geographic considerations. For users of venture capital, the results of this study suggest a possibility for multiple options that are both more accessible and more catered to specific needs. Users of venture capital should find a clearer basis on which to differentiate venture capital firms in terms of venture stage priorities, staff utilization orientations, sources and uses of financial resources. This should make for more informed “shopping” among different venture capital sources and provide a basis on which to “shop” for the most compatible firm.

SYSTEMATIC RISK FOR HETEROGENEOUS TIME HORIZONS

THE OBJECTIVES Of this paper are two-fold. First, we will reiterate the existence of a significant problem in empirical analysis relating to the investment time horizon assumption of traditional return-systematic risk analysis models. Second, we will develop and empirically test a model for analyzing the return and systematic risk characteristics of investment performance data that does not require the assumption of a specific length investment holding period. The body of this paper is organized as follows. Section I provides a succinct review of existing methods of treatment of the time horizon problem. Section II provides an empirical examination of the existing return-systematic risk analysis models described in Section I. The results confirm the significance of the problems alluded to in Section I. Section III provides the development of the heterogeneous time horizon model. Empirical application of the heterogeneous time horizon model is described in Section IV, and the results are compared to the results from the homogeneous time horizon models. Finally, Section V provides the conclusions and implications of the study.