Interest is rising in the Treynor-Black model for portfolio selection, which can provide a new application to enterprise-wide portfolio optimisation, says Ross Miller

Just over 25 years ago, Jack Treynor and his then protégé the late Fischer Black developed a model of portfolio selection. The Treynor-Black model elegantly demonstrated how risk and return should be balanced when constructing a portfolio of assets (Treynor & Black, 1973). Despite recent advances in portfolio optimisation, interest in this model has increased dramatically in recent years.

Although the Treynor-Black model first reappeared as an example in an investments textbook (Bodie, Kane & Marcus, 1989), its use by practitioners is undergoing a resurgence (Taggart, 1996, and Kane, Marcus & Trippi, 1999). The two key advantages of the model, compared with more complex portfolio optimisation methods, are that it requires little information and that it can be expressed as a simple, algebraic formula. Even though the Treynor-Black model has always been associated with equity portfolio selection, it can also be applied at a more general level – for example, to manage credit risk in a fixed-income portfolio.

Model framework
The model maintains the quantitative framework of the efficient markets theory – or capital asset pricing model (CAPM) – with one critical violation: that portfolio managers may possess information about the future performance of securities that is not reflected in the price or projected return of the asset. The model uses alpha, the projected return of the security in excess of its market risk-adjusted return, as the measure of asset performance. Under the assumptions made by Treynor-Black, the optimal share of each security is found to be proportional to its alpha and inversely proportional to the square of its specific risk – the risk that cannot be attributed to a market or systematic component.

Advanced portfolio optimisation methods can be viewed as generalisations of the Treynor-Black model that allow for more flexibility in the specification of returns and their correlation structure. In order to reap the potential benefits of these models one must sacrifice the intuitive simplicity of a closed-form solution, relying instead on optimisation algorithms or simulations.

One side effect of this added complexity is the likelihood that the optimal portfolio will exhibit instability: small changes in the parameters underlying the optimisation can lead to large, discontinuous changes in the optimal portfolio.

In addition, the perception that optimisation is too complex has led to the use of risk management systems that seek only to limit extreme risk, ignoring the more probable events that determine the quarter-by-quarter earnings of the firm.

This article describes the application of the Treynor-Black model to a multi-product financial services enterprise, where detailed information about risk and return may be difficult to find. Building on a disciplined approach to product line risk management, previously developed by the author, that represents product lines using one-page "scorecards" (Greene & Miller, 1996), Treynor-Black can be used as the basis for strategic planning. Indeed, the disciplined approach to considering risk and return that is required by the Treynor-Black model may be as important to the successful management of portfolio risk as are the specific numerical results of the model.

Model structure
The abridged version of the Treynor-Black model presented here looks at the problem of actively allocating a portfolio among several assets. The model also provides for the optimal division of assets between active and passive management (see Bodie, Kane & Marcus, 1989, and Treynor & Black, 1973).

Treynor-Black divides the risk of an asset into two types: systematic (or market) and specific (idiosyncratic or residual). Systematic risk cannot be eliminated by diversification, so market participants must be paid a premium to bear it. Specific risk, on the other hand, is peculiar to an asset and can be eliminated by diversification.

As a result, with adequately functioning asset markets, which the Treynor-Black model assumes, any risk premium for bearing specific risk is competed away by those best able to mitigate it. All systematic risk is assumed to be attributable to one or more market factors. All correlation of risk between assets is induced by these factors (see Sharpe, 1963).

The advantage of this approach is that it uses much less quantitative information than an optimisation method, which requires the complete matrix of all pair-wise asset correlations. Finally, all returns are assumed to follow a normal distribution. A method for softening the impact of this assumption is described in the extensions to the model found in the final section of this article.

The model works by finding the mix of assets with the associated alphas and specific risks that gain the greatest benefit from active management

If time series with sufficient observations are available, the risk structure can be viewed statistically by considering a linear regression of asset returns over time against each of the market factors – for example, appropriate returns for indexes of the equity and fixed-income markets. The variance of the historical returns for an asset serves as an estimate of the square of its overall riskiness.

The variance not explained by the market factors serves as an estimate of the square of the specific risk. The regression coefficient for each market risk factor serves as an estimate of beta, the amount of the factor that the asset contains. These betas, along with estimates of the risk-free interest rate and the market risk premium for each of the factors, can then be used to estimate a "hurdle rate" for the asset. This rate takes systematic risk into account using CAPM or a more advanced factor model.

If all the statistical information detailed above is available, the hurdle rate can be netted out of the projected total return for the asset to determine its excess return or alpha. Alternatively, a subjective assessment of analyst input can give an estimate of alpha. Either way, systematic risk is not considered separately from return. Specific risk, on the other hand, is balanced against any alpha that remains. To simplify matters, we will only allow non-negative alphas to be considered. An asset with a negative alpha is automatically excluded from consideration, effectively giving it a zero share of the portfolio. The full Treynor-Black model allows for the short selling of assets with negative alphas.

The Treynor-Black model works by finding the mix of assets with the associated alphas and specific risks that gain the greatest possible benefit from active management. A standard measure of the benefit of active management is the ratio of the portfolio-level alpha to the portfolio-level specific risk. This ratio is known as the appraisal ratio or information ratio. An important result of the full Treynor-Black model is that, to maximise the performance – as measured by its Sharpe ratio – of a portfolio with both passive and active components, it is necessary to maximise the appraisal ratio of the actively managed component.

Maximising the appraisal ratio through the choice of assets turns out to be a straightforward optimisation problem that generates a closed-form solution. This solution is to set the share the ith asset in proportion to (ai/s2(ei)), where ai is the alpha of the ith asset and s2(ei) is the square of its specific risk. The term ei is the random error term in the return of the ith asset and s2 is the variance function. Once ai/s2(ei) has been determined for every asset, the exact share of the portfolio assigned to each asset is determined by dividing ai/s2(ei) for that asset by its sum over all assets.

Applying the model
Because of its inherent modularity and modest data requirements, the Treynor-Black model can be extremely valuable as a strategic planning tool for a multi-product financial enterprise. In this case, an asset represents a product or division within the enterprise. The output of the model is used to target the relative share of each product. Treynor-Black’s application at the product level can be viewed as a three-step process that may need to pass through several iterations before it is complete.

1. Risk-based product definition
Risk-based product definition is a necessary first step because the Treynor-Black model assumes that any correlation between products is captured by the systematic risk factors, leaving specific risk to be distributed independently. The potential for serious problems can be seen by considering our example and adding a fifth asset that is simply a "clone" of one of the other four assets. The insertion of this artificial asset has the effect of doubling the weight in the portfolio of the asset from which it was cloned. An enterprise with several products that contain virtually identical risks will tend to overload itself with that risk unless it appropriately consolidates the products when applying the Treynor-Black model. Despite its numerous virtues, the model’s independence assumption means that the portfolio allocation depends on how assets are defined – a difficulty that more complex models are designed to avoid. The diligence required to apply the model properly may be considered the price to pay for its simplicity.

The independence requirement for specific risk means that most enterprises cannot use their existing definition of product lines or divisions as the basis for input into the Treynor-Black model. Nonetheless, merely going through the process of examining a large enterprise’s risk from the perspective of the Treynor-Black model can be enlightening. It is common for a specific type of exposure to be found in a variety of forms throughout the enterprise. The number of risk-based products or groups of products generated by this process will depend on the size and diversity of the enterprise. The partition should be fine enough that fundamentally different risks appear separately, but not so fine that the duplication problem noted above starts to appear.

2. Determination of product-level, risk-adjusted, excess returns
Treynor & Black (1973) used securities analysis as the basis for determining alpha. Each asset is viewed as a security for which an analyst develops an opinion based on his research. This opinion is converted into an estimate of the excess return. Determining product-level alphas essentially entails uncovering sources of competitive advantage and estimating their link to future returns. Of particular concern are cases where the excess return of a product is forecast to be negative. Without overriding business reasons to continue the product, such as relationship building, this is a clear signal to limit new business and consider mechanisms for laying off or hedging any existing product risk.

3. Product-level specific risk
Determining the level of specific risk for each product can be done subjectively or objectively. When sufficient data are available, the specific risk can be obtained directly from a regression. It is the square root of the residual or unexplained variance of the regression. In many cases, however, the estimation of specific risk will not be so easy.

Building on the methodology for enterprise risk management developed in Greene & Miller (1996), one can construct a risk scorecard for each of the products to gauge the risk specific to the product on several dimensions. Even model risk can be included as a dimension in the analysis. It will tend to be inversely proportional to the experience that one has with a product. The individual scores on each dimension can then be aggregated, using methods ranging from the estimation of an aggregation function to the construction of an expert system.

In any event, the specific risk number produced must be calibrated so that it may be considered as a standard deviation. The risk scorecard for each product, along with the information used to estimate its alpha, can be summarised on a single page. The historical, actual and target shares for the product can also be included here. Such "one-pagers" can be a valuable strategic tool regardless of how seriously one takes the numerical output of the Treynor-Black model.

It may be necessary to repeat the three steps of the process until a final set of inputs for the Treynor-Black model is determined. The process of determining alphas and specific risk quantities for a given partition of the enterprise’s business into products can provide insights that lead to an even better partition. And as the product mix changes over time, the risk-based product definitions will need to be updated. The estimates of alpha and specific risk can be revised as often as necessary.

Extending the model
The Treynor-Black model may be extended by adding constraints that can reflect limitations on behaviour. For example, we have already developed the model with a short selling constraint, by omitting products with negative alphas. Likewise, the amount of business that can be generated in any product line is often constrained by the internal capacity to originate that business and by the depth of the market for the product. It is easy to add capacity constraints by using an iterative maximisation process.

In our example, suppose Asset 4 is limited to a 40% share of the portfolio, even though the model indicates that it should receive a 47.74% share. The excess 7.74% share is allocated to the other three assets according to their Treynor-Black weights. If multiple products are constrained, the reallocation process is performed iteratively until the entire portfolio is allocated. Although the Treynor-Black model no longer technically has a closed-form solution, it retains its other desirable properties.

The constraint that is likely to be most important for the firm, one too often ignored in portfolio optimisation, is that of economic viability. In most cases, this is equivalent to some notion of capital adequacy. In the original application of the Treynor-Black model to an investment portfolio, capital adequacy was not a problem because the capital structure of the portfolio allocator was assumed to consist of all equity and no debt.

When it comes to the constraint of maintaining adequate capital for its ongoing operations, the leveraged firm may find itself facing several constraints because of the demands of regulators and rating agencies, in addition to its own determination of economic capital.

It is likely that, at any given time, one or more of these constraints will be binding on the firm. This makes it impossible to perform a true optimisation without taking such constraints into account. Recall that the optimisation performed by portfolio models accounts for possibilities along the entire probability distribution of outcomes and gives credit for the returns associated with each risk. However, the issue of capital adequacy is entirely concerned with controlling the low-end tail of the distribution, without concern for the returns to be gained at the cost of expanding that tail.

The addition of a constraint related to the tail of the probability distribution is also very useful in cases where the risk is skewed to the downside, as with many debt or option-based assets. The addition of a downside penalty in the form of a capital adequacy constraint can compensate for serious departures from the normality assumptions of the Treynor-Black model.

In contrast to the capacity constraints considered above, capital constraints are more difficult to incorporate into the analysis. Each capital constraint has a Lagrange multiplier associated with it, measuring how tightly the constraint bites and serves as a "shadow price" for capital associated with that constraint. Products that use scarce capital must have their Treynor-Black weights reduced according to this estimate.

Note that by incorporating capital constraints the Treynor-Black model can co-exist with established risk management procedures. It does this by assuring that the firm not only maintains adequate capital but also achieves its greatest potential for profitability on a quarter-by-quarter basis.

Click here for a printer friendly version of this article