Portfolio Optimisation and the Black Litterman Model

by Andrew D. Robertson
J.P. Morgan Investment Analytics & Consulting
andrew.d.robertson@jpmorgan.com

Portfolio optimisation has grown in popularity in investment management and now forms an important tool for portfolio managers to assist asset allocation. This article aims to introduce the concepts of portfolio optimisation and the Black Litterman model.

The development of Modern Portfolio Theory marked not only the dawn of financial economics, but also of quantitative finance as a field of study. In 1952, Harry Markowitz published the seminal paper “Portfolio Selection,” which paved the way to express the characteristics of portfolios mathematically. By showing that risk and return are equally important in portfolio construction and risk can be reduced through diversification (dependent on the correlation of assets via the variance of portfolio returns), Markowitz enabled the use of optimisation to take advantage of the risk-return relationship to create a set of optimal asset allocations within a portfolio.

Mean Variance Optimisation

Mean variance optimisation is a quantitative tool used to determine the ‘best’ allocation of assets within a portfolio.

The process begins by calculating the expected returns of the assets and creating a covariance matrix from the standard deviation of each asset and their correlation to each other. The optimiser then goal seeks to find a set of asset allocation weights that minimises the risk of the portfolio for a specific return.

The result is an optimal set of asset weights.

In practice, historical data is used to create the inputs for the mean variance model. While historical data is a good predictor of future covariances and standard deviations, past returns are not always good predictors of future returns. Michaud (1989) demonstrated this and argued that mean-variance optimisers can magnify input errors. This sensitivity in the model can potentially create a number of shortfalls, namely:

• Portfolios may not be practical to implement or recognisable to the investor,
• The asset allocation weights may not be well diversified, and
• Portfolios may suffer from large short positions.

As an example, the historical data of five MSCI world indices were sourced between 1988-2010, and returns, standard deviations and correlations were calculated. The portfolio weights were optimised using the traditional mean variance approach, without any constraints limiting the minimum/maximum allocation to individual asset classes. Exhibit 1 shows a chart of the resulting portfolio and the symptoms typical of using historical data to estimate expected returns. Note the allocation of extreme holdings, large short positions and impracticality of the portfolio.

Exhibit 1 — Mean Variance Example

Source: MSCI, J.P. Morgan Investment Analytics & Consulting estimates

Black Litterman Model

Fisher Black and Robert Litterman (1992) addressed the practical issues of the Markowitz model by creating a technique centred around a Bayesian approach to form a set of expected excess asset returns. Their aim was to make the mean variance theory more applicable to investment professionals by combining investors’ views with equilibrium returns, leading to potentially more reasonable, flexible and stable optimal portfolios.

Optimisation in the Black Litterman model begins with the equilibrium portfolio implied from the universe of assets in the market. Deviations from the equilibrium portfolio can then be applied to assets the user has views about. The model combines these to form a set of blended expected returns that are used in the mean variance optimisation to calculate a set of optimal portfolio weights. An outline of the steps involved can be seen in the table below.

Steps of Black Litterman Portfolio Optimisation

In the same way options traders imply volatility from market prices using the Black Scholes model, in the Black Litterman model, equilibrium returns are implied from market capitalisation weights using a process called reverse optimisation.

Reverse optimisation makes sense when using historical data to imply expected returns because rather than asking what allocation makes an efficient portfolio, reverse optimisation asks what returns make an efficient portfolio. The advantage is inputs may now be better defined, the asset allocation weights can potentially be measured with more certainty, and good estimates of future portfolio risk can be calculated from historical data.

Black and Litterman argue for the use of market capitalisation weights in the reverse optimisation process, which results in consensus excess returns consistent with the tangency portfolio of the Capital Asset Pricing Model. With the market forces of supply and demand in equilibrium, the weight allocation across the investment universe would be expected to be optimal and thus acts as the basis for the portfolio.

The equilibrium weights can be calculated using the dollar weight of the assets, as shown in Exhibit 2.

Exhibit 2 — Calculation of Market Capitalisation Weights 

The reverse optimisation process also requires a risk aversion parameter to indicate the level of risk against return of the market portfolio. This parameter can be derived from the Capital Asset Pricing Model and is the market price of risk. It can be estimated by calculating the excess return per variance using a proxy to the world market portfolio.

The ability of investors to explicitly express their views adds practical flexibility to the model. A view can be expressed as either absolute, relative or a combination of the two with an associated confidence level.

Exhibit 3 — Absolute and Relative Investor Views
 

Views are incorporated in the form of matrices as inputs into the model. (The reader should refer to Idzorek (2005) for a detailed explanation.)

The effect of a view on the asset allocation is to tilt the portfolio towards outperforming sectors and away from underperforming ones. Views are dampened by the blend with the equilibrium returns to limit the effect of extremes and to ensure greater consistency across the estimates. If the investor has no view, the model returns the optimal market capitalisation weights.

Once the equilibrium returns are calculated and the views expressed, the Black Litterman model forms a set of new combined expected returns. These returns are then used as inputs to the Markowitz mean-variance optimisation technique.

Black Litterman Process

Click here to view larger image.

Source: Idzorek (2005)

The objective of the model is to produce well-diversified, stable portfolios that make sense to the investor and incorporate his or her view of the world.

Exhibit 4 shows an example of Black Litterman optimisation. Using the equilibrium weights in Exhibit 2 and a covariance matrix calculated from historical data (sourced between 1988-2010) equilibriums returns were calculated. The relative view from Exhibit 3 was incorporated and new blended returns computed. The new returns together with the historical covariance matrix were inputted into the Markowitz mean variance optimiser, with the output being optimal allocation weights.

The results in Exhibit 4 show two scenarios, one where no views are expressed and the other with Exhibit 3’s relative view. Where no view was expressed, the model returns the market capitalisation weights as in Exhibit 2. With an incorporated view, the portfolio is tilted in favour of the MSCI Far East index and against the MSCI North America index, all other allocations remaining unchanged.

Exhibit 4 — Black Litterman Asset Allocation
 

Source: J.P. Morgan Investment Analytics & Consulting estimates

Conclusion

Mean variance optimisation provides a solution to the challenge of asset allocation, but can sometimes prove difficult to apply to certain real-world situations due to the nature of estimating expected returns from historical data. The Black Litterman model attempts to address this issue by using the world’s view as a basis of asset allocation and adds flexibility by allowing investors to express their own views.

This article is meant as an introduction to both approaches, which are highly technical fields of ongoing research. The techniques are appealing in a world where risk-controlled portfolios are more important than ever because they are based on tested and justifiable theories. However, they do come with their own set of risks, and successful implementation requires careful attention to the parameters influencing the expected returns and risk.

To view the next article, Multiple Asset Class Return Comparison, click here.

References
Black, F, and Litterman, R, (1992), “Global Portfolio Optimization,” Financial Analysts Journal, September-October, pp. 28-43.
Idzorek, T, (2005), “A step by step guide to the Black Litterman Model.”
Markowitz, H, (1952), “Portfolio Selection,” Journal of Finance, pp.77-91.
Merton, R, (1980), “On Estimating the Expected Return on the Market: An Exploratory Investigation,” Journal of Financial Economics, vol. 8, pp. 323-361.
Michaud, R (1989), “The Markowitz Optimisation Enigma: Is Optimised Optimal,” Financial Analysts Journal, January-February, pp. 31-42.