### Black-Litterman's Model: Portfolio's Friend or Foe?

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The traditional MVO (mean-variance optimization) of Harry Markowitz (1952) and the versions that followed constitute an analytical approach to identifying optimal asset allocations from the universe of investable securities. The data needed for the traditional MVO are the expected rates of return, the risks of individual securities, and the covariance among securities. The outcome of this approach is usually illustrated as a curve on which optimal portfolios lie. Each portfolio on this curve, known as the efficient frontier, has the smallest degree of risk for its level of expected return. Markowitz’s traditional MVO was extended and simplified by William Sharpe (1963), in Sharpe’s well-known CAPM (capital asset pricing model). For an intuitive explanation of CAPM and portfolio optimization, refer to Markowitz’s “Crisis Mode” in the Spring 2009 issue of the *Investment Professional*.

- Portfolio managers do not deal with the entire universe of securities. In most cases, a portfolio manager is only interested in a small sector of the market.
- The data for price movements among different securities and sectors (the variance–covariance matrix) are not easily obtainable. Different analysts may derive different variance–covariance structures for the same group of securities.
- The recommended weights obtained from MVO are highly sensitive to the expected rates of return. Just a small change in the expected rates of return would lead to different asset allocation recommendations.
- Estimation error in expected rates of return for MVO is 10 times more sensitive to the outcome of asset allocation than estimated risk and 20 times more sensitive than covariance (Ziemba 2003).

**MANAGING LIMITATIONS**

Since estimation error in expected rates of return is the most critical factor in an MVO, and those rates are the data that the analyst running the MVO will have the most difficulty obtaining, the traditional MVO is very sensitive to the quality and availability of market data. Three main methods have been proposed for optimizing the accuracy of an MVO: the VWARR (value-weighted average

rate of return); the HRR (historical rate of return); and MVO with constraints, which prohibits short selling. But, irrespective of the method used, the problems of extreme values and estimation error in the traditional MVO do not disappear in practice.

In the VWARR method, the recommended weights (how much to allocate among sectors, securities, or currencies) often become extreme. In one simulated case based on VWARR, for every $1,000 worth of assets in the portfolio, up to $400 were recommended to be in the utilities sector due to the low volatility of utilities as compared to other assets in the portfolio. The recommended weight for the financial sector turned out to be around only 5%. Such extreme relative ratios (8 to 1 in this case) are common when the VWARR method is used in traditional MVO.

Results derived from HRR may not be better than those from VWARR. HRR is not forward looking; it assumes that past return distributions will persist. It is not difficult to visualize how distorting the results would be if the momentum-trending market data of 2003 through 2007 were extrapolated toward 2008 through 2009. The recommended weights would be unacceptable to most analysts. In fact, Merton (1980), among others, has warned about the deficiency of historical data for the input requirements of the MVO approach. Specifically, when HRR is used, securities with higher rates of return, securities with lower or negative correlations, and securities with smaller volatilities become overweighted in the traditional MVO.

MVO with constraints is often used to alleviate the problem of extreme or illogical weights. However, the results of various simulations with no short selling demonstrate that this approach renders the efficient frontier less efficient, leading to corner portfolios. Graphically speaking, the locus of corner portfolios when short selling is not permitted is a frontier curve that is “kinked”—not continuous. As a result, the asset allocation weights, as suggested by the corner portfolios, are frequently unconvincing. One of the simulated cases for MVO with constraints resulted in the recommendation that an extreme weight go to a tiny sector of the market, a conclusion that it’s difficult to convince most clients to accept.

**BLACK-LITTERMAN AND THE ANALYSTS' VIEWS**

In an effort to mitigate these limitations, the Black–Litterman model (1992) and subsequent versions and modifications were developed. The Black–Litterman model does not simply use historical data to compute returns and the variance–covariance matrix. Instead, it starts with an equilibrium assumption that the global portfolio is well diversified and efficient. This is just the model’s starting point, a neutral initial stage. Through the use of a reverse optimization process, the model derives the returns of securities that constitute the global benchmark. More precisely, the model backs out the vector of implied excess equilibrium returns where the implied excess equilibrium returns are equal to the risk–return trade-off multiplied by the covariance matrix of excess returns and by market capitalization weights.

The risk–return trade-off is the coefficient for risk aversion, measuring the return that a client is willing to give up in order to carry less risk. It’s a scale factor in which increased aversion to risk results in a larger scale. However, in most cases, analysts may not agree with the derived returns of the securities of the global benchmark in the neutral stage. There are a number of reasons for that, including, in particular, the strong views an analyst may have about a small subset of the market.

Portfolio management is highly segmented in terms of industry, geography, and style. For example, an analyst may claim expertise in value investment for relatively neglected firms in certain regions. Or some managers may insist on incorporating their views about a given market, sector, or firm through micro attribution analysis, in which excess return is attributed to market allocation, security allocation, or the interaction of both. Unlike the traditional MVO model, Black–Litterman does permit analysts to convey agreement or disagreement with the derived returns of the securities in the benchmark. In short, Black–Litterman combines the traditional MVO of Markowitz and Sharpe with a flexible algorithm to quantify analysts’ views of markets, sectors, and securities.

Analysts and investors have the flexibility to express multiple views within the Black–Litterman model—by means of an absolute, relative, or combined deviation. For example, in an absolute deviation, the analyst may state that the financial sector, due to the latest deterioration of the assets, will have a negative excess return of 10%. A relative deviation, on the other hand, may represent the technology sector outperforming the S&P 500 by 10%. Or, in a combined view (which may take either an absolute or relative form), the analyst or committee may indicate that the financial and technology sectors combined would lose 5% or underperform a global benchmark by 3%

The incorporation of these views into the model is, in fact, a major limitation for those analysts that put Black–Litterman into practice. Expressing views casually is dangerous and may lead to model risk. Only the most naive analysts are confident in their views of markets, sectors, and firms. Once analysts have expressed their views, they must integrate those views into the model with the greatest of care. However, they can alleviate some of the pitfalls by using various approaches to estimating probability, including Bayesian methods.

**IMPLEMENTATION IN A NUTSHELL**

In Black-Litterman's most basic form, it requires the following vectors (columns of data):

- A vector expressing the analyst's views.
- A vector of random errors reflective of how confident or certain the views are. The random-errors vector is actually not incorporated in the model, but its variance affects computations and results.
- A vector of asset returns.

The main idea is to reconcile the matrix of asset returns with the refined views that have been adjusted for probability. Below is a sequential reverse optimization showing how the Black-Litterman model (or a modified version thereof) may be implemented in practice. Note that some of these steps are overlapped.

- Choose a global index or a well-diversified value-weighted index.
- Take the rate of the return of the chosen index as a given value.
- Assume the weights of the securities in the above index are the equilibrium market weights of assets in the portfolio.
- Back out the expected rates of return by back-solving for the implied returns of the securities in the above index. Caveat: This reverse approach is in contrast to the traditional MVO.
- Express views about returns.
- Refine the above expressed views to adjust for uncertainty. Use probability functions if you can meaningfully defend your views and justify your approach.
- Compute the view-adjusted market equilibrium returns.
- Run the mean-variance optimization.

**PLUS AND MINUS**

Carefully constructing the Black-Litterman model should endow the final portfolio with a significant edge over a traditional MVO:

- The recommended portfolio should be more efficient and less concentrated in securities or sectors than a portfolio optimized using MVO.
- The weights and types of securities may be more logical, at least with respect to the expressed views.
- Estimation error, a major limitation for MVOs using historical data, is usually reduced.
- Both the analyst and the client may relate to the results better, and recommended asset allocations may be more in line with the client's investment policy statement. For example, if market or other conditions place unique constraints on an institutional client in terms of partially considering (or not considering) an asset, sector, or currency, those constraints can be incorporated in the expressed views from the very beginning.

Equally important, and often ignored, is the question of whether Black-Litterman or its modifications would reduce or increase the tension of legal risk. Legal risk often arises from a mismatch between a client's situation on one hand, and capital market expectations and industry and company analyses on the other hand. In the absence of a reliable and formal algorithm to incorporate realistic views, clients may use the "prudent expert" requirement for portfolio management and institutions in court. Yet, if analysts' views are objectively defined, conceptually meaningful, and well documented, legal risk may decline.

A final drawback involves how best to implement Black-Litterman when the actual data for a security are insufficient. Inadequate data will always be a limitation, but one possible solution involves determining the security's covariance based on existing data. After constructing the covariance of other securities for which more data are available, the missing data could be "filled in" using a technique that would verify the correlation of that stock with the index. This proxy approach may help analysts who have to consider certain securities with limited available data.

**REFERENCES**

Black, Fischer, and Robert Litterman. September–October 1992. “Global Portfolio Optimization.” *Financial Analysts Journal*, vol. 48, no. 5. 28–43.

Markowitz, Harry M. March 1952. “Portfolio Selection.” *Journal of Finance*, vol. 7, no. 1. 77–91.

———. Spring 2009. “Modern Portfolio Theory Under Pressure.” *Investment Professional*, vol. 2, no. 2. 54–57.

Merton, Robert. 1980. “On Estimating the Expected Return on the Market: An Explanatory Investigation.” *Journal of Financial Economics*, no. 8. 323–361.

Sharpe, William F. January 1963. “A Simplified Model for Portfolio Analysis.” *Management Science*, vol. 9, no. 2. 277–293.

Ziemba, William T. 2003. *The Stochastic Programming Approach to Asset Liability and Wealth Management*. Charlottesville, VA. Research Foundation of AIMR.**–Ehsan Nikbakht, DBA, CFA, FRM, is a professor of finance at Hofstra University’s Zarb School of Business, vice chair of the New York Society of Security Analysts Derivatives Committee, and a member of the Investment Professional editorial board.**

*This article was originally published in the Spring 2009 issue of the *Investment Professional.

Some quants (especially those involved the hedge fund world) now use the Omega Ratio for asset allocation, i.e. they vary investment weights untill the Omega Ratio of a portfolio is maximized. This requires complex non-convex optimizers, but simple examples can be prototyped in Excel. See https://investexcel.net/929/omega-ratio-maximize/ for an example.

This minimize the risk of extreme losses but can result in a portfolio with higher variance than given by traditional MVO.

Posted by: Simon Connel | 11/09/2011 at 03:03 PM