**Introduction**

How to allocate capital across different asset classes is a key decision that all investors are required to make. It is widely accepted that holding one or few assets is not advisable, as the proverb *"Don't put all your eggs in one basket"* suggests. Hence, practitioners recommend their clients to build portfolios of assets in order to benefit from the effects of diversification.

An investor's portfolio is defined as his/her collection of investment assets. Generally, investors make two types of decisions in constructing portfolios. The first one is called asset allocation, namely the choice among different asset classes. The second one is defined security selection, namely the choice of which particular securities to hold within each asset class. Moreover, portfolio construction could follow two kinds of approaches, namely a top-down or a bottom-up approach. The former consists in facing the asset allocation and security selection choices exactly in this order. The latter inverts the flow of actions, starting from security selection.

No matter the kind of approach, investors do need a precise rule to follow when building a portfolio. In fact, the choice of asset classes and/or of securities has to be done rationally. The range of existing strategies is considerably wide. Indeed, one may allocate his/her own capital by splitting it equally among assets, optimizing several functions and/or applying some constraints.

Every day in the asset management industry, there are plenty of strategies that are proposed to investors all over the world. The aim of this article is to provide the reader with a comprehensive summary of those.

**Static and Dynamic Optimization Techniques**

To begin with, it is worth distinguishing the existing portfolio optimization techniques by the nature of their optimization process. In particular, static and dynamic processes are considered. In the former case, the structure of a portfolio is chosen once for all at the beginning of the period. In the latter case, the structure of the portfolio is continuously adjusted (for a detailed survey on this literature, see Mossin (1968), Samuelson (1969), Merton (1969, 1971), Campbell et al (2003), Campbell & Viceira (2002). Maillard (2011) reports that for highly risk-averse investors, the difference between the two is moderate, whereas it is larger for investors who are less risk averse.

**Markowitz Mean-Variance Optimization**

Within the static models, it is common knowledge that the mean-variance optimization suggested by Henry Markowitz represents a path-breaking work, the beginning of so-called Modern Portfolio Theory (MPT). In fact, Markowitz (1952, 1959) presents a revolutionary framework based on the mean and variance of a portfolio of "N" assets. In particular, he claims that if investors care only about mean and variance, they would hold the same portfolio of risky assets, combined with cash holdings, whose proportion depends on their risk aversion.

Despite of its wide success, this theory has been criticized by some researchers for issues linked to parameter uncertainty. In fact, the true model parameters are unknown and have to be estimated from the data, resulting in several estimation error problems.

The subsequent literature has focused on improving the mean-variance framework in several ways. However, two main approaches to the problem may be identified, namely a non-Bayesian and a Bayesian approach.

**Two Approaches**

As far as the former is concerned, it is worth reporting several studies. For instance, Goldfarb & Iyengar (2003) and Garlappi et al. (2007) provide robust formulations to contrast the sensitivity of the optimal portfolio to statistical and modelling errors in the estimates of the relevant parameters. In addition, Lee (1977) and Kraus & Litzenberger (1976) present alternative portfolio theories that include more moments such as skewness; Fama (1965) and Elton & Gruber (1974) are more accurate in describing the distribution of return, while Best & Grauer (1992), Chan et al. (1999) and Ledoit & Wolf (2004a, 2004b) focus on methods that aim to reduce the estimation error of the covariance matrix. Other approaches involve the application of some constraints. MacKinlay & Pastor (2000) impose constraints on moments of assets returns, Jagannathan & Ma (2003) adopt short-sale constraints, Chekhlov et al (2000) drawdown constraints, Jorion (2002) tracking-error constraints, while Chopra (1993) and Frost & Savarino (1988) propose constrained portfolio weights.

On the other hand, the Bayesian approach plays a prominent role in the literature. It is based on Stein (1955), who proved the inadmissibility of the sample mean as an estimator for multivariate portfolio problems. In fact, he advises to apply the Bayesian shrinkage estimator that minimizes the errors in the return expectations, rather than trying to minimize the errors in each asset class return expectation separately.

In following studies, this approach has been implemented in multiple ways. Barry (1974) and Bawa et al (1979) use either a non-informative diffuse prior or a predictive distribution obtained by integrating over the unknown parameter. Then, Jobson & Korkie (1980), Jorion (1985, 1986) and Frost & Savarino (1986) use empirical Bayes estimators, which shrink estimated returns closer to a common value and move the portfolio weights closer to the global minimum-variance portfolio. Finally, Pastor (2000), and Pastor & Stambaugh (2000) use the equilibrium implications of an asset-pricing model to establish a prior.

**Simpler Models**

To attempt portfolio construction throughout optimization is not the only alternative, though. In fact, alongside the wide range of portfolio optimization techniques, it is also worth considering other rules that require no estimation of parameters and no optimization at all. DeMiguel at al (2005) define them as "*simple asset-allocation rules*". For instance, one could just allocate all the wealth in a single asset, i.e., the market portfolio . Alternatively, investors may adopt the 1/N rule, dividing their wealth according to an equal-weighting scheme.

At this point, the reader may wonder why one should consider this kind of rules. In fact, techniques that require no optimization should not be optimal according to any measure. However, as far as the naïve 1/N is concerned, some researchers have reported some interesting results. For instance, Benartzi & Thaler (2001) and Liang & Weisbenner (2002) show that more than a third of direct contribution plan participants allocate their assets equally among investment options, obtaining good returns. Moreover, Huberman & Jiang (2006) find similar results. Similarly, DeMiguel et al (2009) evaluate 14 models across seven empirical datasets, finding that none is consistently better than the 1/N rule in terms of Sharpe ratio, certainty-equivalent return or turnover.

However, Tu & Zhou (2011) challenge DeMiguel et al. (2009) combining sophisticated optimization approaches with the naïve 1/N technique. Their findings confirm that the combined rules have a significant impact in improving the sophisticated strategies and in outperforming the simple 1/N rule.

Moreover, other naïve rules are reported by Chow et al. (2013), such as the 1/σ and the 1/β, included in the so-called low-volatility investing methods. In particular, they report that low-volatility investing provides higher returns at lower risk than traditional cap-weighted indexing, at the cost of underperformance in upward-trending environments.

**Smart Beta Strategies**

Finally, it is worth mentioning a special group of strategies that are extremely popular among asset management firms, known as Smart Beta strategies. Smart Beta strategies are virtually placed between pure alpha strategies and beta strategies, and emphasise capturing investment factors in a transparent way, such as value, size, quality and momentum. Examples of these strategies are risk parity, minimum volatility, maximum diversification and many others. Apart from the wide range of these kinds of techniques, it is critical to highlight why they are so diffuse among practitioners. Their enormous success is due to several interesting advantages, including the flexibility to access tailored market exposures, improved control of portfolio exposures and the potential to achieve improved return/risk trade-offs.

**Final Remarks**

This article aims to be a summary of the most notorious techniques considered in the existing literature, but the list is far from being complete. Moreover, the article does not analyze which strategy is the best, since I believe that the success of an investment technique depends on several factors, including the time frame considered, the kind of assets, the geography of the examined portfolio, the client's preferences, and it surely must rely on a quantitative application using real or simulated data.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.