Abridged version published in Risk, November 1999, available online.

Treynor-Black Revisited:
A New Application to Enterprise-Wide Portfolio Optimization

by

Ross M. Miller
Miller Risk Advisors
2255 Algonquin Road
Niskayuna, NY 12309 USA


February 1999

Just over twenty-five years ago Jack Treynor and his then-protégé, the late Fischer Black, published a paper in the Journal of Business deriving a model of portfolio selection—the Treynor-Black model—that elegantly captured the essence of how risk and return should be taken into account when constructing a portfolio of assets (Treynor & Black, 1973). Despite the increasingly sophisticated tools that have been developed to create and optimize portfolios since that paper was published, interest in this simple, elegant model increased dramatically over the past few years. Starting as a central example of active portfolio management in one of the major textbooks on investments (Bodie, Kane, & Marcus, 1989), it has increasingly been cited as a key quantitative tool within the arsenal of practitioners (Taggart, 1996). The two key "selling points" of the model relative to more complex portfolio optimization methods are that it requires relatively little information and that it expresses the solution for the optimal portfolio as a simple, algebraic formula. Even though the Treynor-Black model has always been associated with equity portfolio selection, if the proper care is taken it can applied at a more general level.

Some of the delay in the acceptance of the Treynor-Black model can be attributed to the fact that it arrived on the scene just as the efficient markets theory was gaining broad acceptance. (Burton Malkiel’s classic 1973 work on efficient markets, A Random Walk Down Wall Street was published the same year as the Treynor-Black model.) According to the reigning efficient markets model of the era, the Capital Asset Pricing Model (CAPM), the problem of portfolio selection as introduced by Harry Markowitz (1959) had become largely trivial because in most cases the optimal portfolio was always some combination of the "market" portfolio with a "risk-free" asset, e.g., U.S. T-bills. The assumption of market efficiency makes it suboptimal to hold any security or set of securities in isolation from the entire market portfolio because one will then bear any risk specifically associated with those securities (specific risk) without a receiving a compensating return.

The radical step taken by Treynor and Black in their model is to maintain the overall quantitative framework of the efficient markets approach to portfolio selection while simultaneously introducing a critical violation of the efficient markets theory, i.e., that individual portfolio managers could possess information about the future performance of certain securities that is not reflected in the current price or projected market return of the asset. The quantitative performance measure for a single asset used by Treynor-Black is alpha, the projected return of the security over-and-above its market risk-adjusted return. The optimal portfolio would then "tilt" towards securities with projected outperformance (alpha greater than zero) and away from securities with projected underperformance (alpha less than zero). Furthermore, Treynor and Black assume that alpha is determined in a subjective manner and they suggest a way for converting traditional Buy/Hold/Sell qualitative analyst recommendations into numerical alphas.

Under the assumptions made by Treynor-Black, which are described in more detail in the next section, the optimal share of each security is easily computed. This share is proportional to the security’s alpha and inversely proportional to the square of the security specific risk (measured as a standard deviation). This formulation neatly embodies the notion that (excess) returns matter, but (specific) risk matters even more.

Virtually all full-fledged portfolio optimization software in use today can be viewed as implementing some generalization of the Treynor-Black model that allows for more flexibility in the specification of returns and their correlation structure as well as the placing of constraints on the choice of assets. In order to reap the potential benefits of these more advanced models, one must sacrifice the intuitive simplicity of the closed-form solution provided by Treynor-Black, relying instead on complex optimization algorithms or simulations. One side effect of this added complexity is the likelihood that the optimal portfolio will exhibit instability, i.e., small changes in the parameters underlying the optimization can lead to large, discontinuous changes in the optimal portfolio. Such results are most common when securities with negative alphas are held in the portfolio simply to hedge risk introduced by securities with positive alphas. In addition, the perception that full optimization is too complex has led many financial institutions to limit themselves to risk management systems that focus only on risk—and not on return—and narrow their focus to the low-end tail of the probability distribution, often 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 an area where detailed information about risk and return is difficult to find in any reliable form, much less at the level of detail required for a full optimization. That area is the determination of a target product mix for a multi-product financial services enterprise. Building on a disciplined approach to product-line risk management previously developed by the author (Greene & Miller, 1996), Treynor-Black can be used as the basis for strategic planning, avoiding many of pitfalls of the "seat of the pants" approach to risk diversification. 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. Simply going through the modelling process causes one to focus on which risks are worth taking and what are appropriate levels of return for those risks.

This section presents an abridged version of the Treynor-Black model looks at the problem of actively allocating a portfolio among several assets. The Treynor-Black model also provides for the optimal division of assets between active and passive management (indexing) and the treatment of the model given here only addresses that problem in passing. The reader who desires more details can find them either in the textbook version of the model by Bodie, Kane, & Marcus (1989) or in the original Treynor-Black (1973) exposition.

The key to the Treynor-Black model is the recognition that the risk of holding an asset can be decomposed into two types: systematic (or market) and specific (or idiosyncratic or residual). Systematic risk is risk that cannot be eliminated by diversification because it is common to a large number of assets and so market participants must be paid to bear it. Specific risk, on the other hand, is specific to an asset and can essentially be eliminated solely 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 through diversification.

All systematic risk is assumed to be attributable to one or more market factors. An asset’s risk that cannot be associated with a market factor is taken to be risk specific to that asset. Hence, all correlation of risk between assets is induced by the factor structure, the specific risks are assumed, by definition, to be independent of one another. This factor model, developed by William Sharpe (1963) and also known as the index or diagonal model, was originally developed to deal with issues related to the limited computational power available back in the 1960s. For the applications of the Treynor-Black model that we examine, the real advantage of this approach is that it uses much less quantitative information than a fuller optimization method that requires the matrix of all pairwise asset correlations.

Finally, as with the original Markowitz portfolio selection model, all returns are assumed to follow a normal distribution. A method for softening the impact of this assumption is described in the final section of this article, which discusses extensions to the model.

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, e.g., appropriate returns for indices 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 that is not explained by the market factors serves as an estimate of the square of the specific risk. Finally, 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 premia for each of the factors can then be used to estimate a hurdle rate for the asset that takes systematic risk into account using CAPM or a more advanced factor model.

If all the statistical information described above is available, then the hurdle rate can be netted out of the projected total return for the asset to determine its excess return or alpha. Alternatively, alpha can be estimated using a subjective assessment of analyst input. 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 nonnegative 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 shortselling of assets with negative alphas, but in the contexts examined here taking a true short position may not be feasible.)

The Treynor-Black model works by finding the mix of assets whose associated alphas and specific risks generate the greatest possible benefit from active management. There is general agreement in the active portfolio management literature that the best measure of this benefit is the ratio of the portfolio alpha to the portfolio specific risk. (The portfolio alpha is the weighted average of the alpha for each asset, using the share in the portfolio as the weight, and the portfolio specific risk is the square root of the portfolio variance, where the portfolio variance is the weighted sum of the asset specific risks squared. We add specific risk together in this manner because it is, by definition, independent from product to product.) This ratio is known either as the appraisal ratio or information ratio. One important result of the full Treynor-Black model is that in order to maximize the performance—as measured by its Sharpe ratio—of a portfolio with both passive and active components, it is necessary to maximize the appraisal ratio of the actively managed component. Also, the Treynor-Black model assumes a fixed time horizon, which is generally taken to be one year; however, the model can be applied to longer or shorter time horizons as well.

Maximizing the appraisal ratio through the choice of assets turns out to be a straightforward optimization problem that generates a closed-form solution. This solution is to set the share the i^th asset in proportion to a _i /s ²(e_i), where a _i is the alpha of the i^th asset and s ²(e_i) is the square of its specific risk. (The term e_i is the random error term in the return of the i^th asset and s ² is the variance function.) Once a _i /s ²(e_i) has been determined for every asset, the exact share of the portfolio assigned to each asset is determined by dividing a _i /s ²(e_i) for that asset by its over all assets.

The actual workings of the Treynor-Black model are best demonstrated by an example. The numbers have been chosen not for their realism, but rather to illustrate some important features of the Treynor-Black model. Consider a portfolio that can be constructed from the following four assets:

The measure of total return for the i^th asset is its expected annual return, r_i, and the measure of total risk is the annual standard deviation of total return, s _i. In order to simplify the example, all systematic risk is assumed to be captured by a single systematic risk factor that is referred to as the market. The amount of systematic risk contained in the i^th asset is given by its beta, b _i. The only other information that is needed to apply the Treynor-Black model to these four assets are the risk-free rate of return (r_f), which is taken to be 5% per annum; the market rate of return (r_m), which is taken to be 10% per annum; and the market risk (s _m), which is taken to be 20% per annum.

Before proceeding with the calculation of optimal portfolio shares, we will notice how the four assets differ. The first asset has both high risk and high return, but is uncorrelated with the market (b ₁=0). Managers often view such an asset as an ideal addition to a portfolio because its lack of correlation with other assets helps to diversify risk. This observation is true, but as we shall see to a much more limited degree than one might suspect. In contrast, the second asset has a beta of two, characteristic of a highly leveraged asset. Its risk and return are even higher than the first asset, but in the same overall ratio. The final two assets have both lower risk and return than the first two assets, although the ratio of return to risk appears higher than that of the first two assets. Both have a modest beta of 0.5 and the latter of the two has both lower risk and return.

For the i^th asset, alpha (a _i) and the square of specific risk (a _i /s ²(e_i)) can be computed directly from the information above using the following standard formulas (and then substituting the appropriate market parameters):

a _i = r_i - (r_m-r_f)b _i -r_f
= r_i - .05b _i – .05

and

s ²(e_i) = s _i² - s _m²b _i²
= s _i² - .04b _i²

Performing these substitutions and then computing the Treynor-Black weights (and the share implied by them) generates the following table:

Notice that Asset 1, the great diversifier, receives the smallest weight in the portfolio. Although its zero beta keeps its alpha high, it does nothing to lessen its high risk, leaving it with a large amount of specific risk and a correspondingly small share of 7.78%. Asset 4 is the greatest beneficiary of the Treynor-Black model, even with the lowest alpha, its very low level of specific risk makes it the favored holding. Except for those rare instances where high return is accompanied by low risk (and such opportunities are frequently limited in quantity, an issue discussed below), the Treynor-Black model tends to favor assets with low risk and low return. Its aversion to high risk/high return assets tends to be greater than one would intuitively think it should be.

The basic properties of the portfolios selected by the Treynor-Black model can be seen from this example. First, the relative allocation among assets is independent of the amount of money to be allocated among them because allocations are expressed in terms of shares, not monetary amounts. In addition, adding or removing assets does not change the relative allocation among any of the existing or remaining assets. This property makes the Treynor-Black model suitable for use in decentralized applications. Assets can be partitioned into several groups, and the allocation decision within any individual group can be made without any knowledge of the assets in any other group.

Another important property is the stability of the model, its lack of sensitivity to small changes in the parameters of the model. In particular, all allocations are "well-behaved" functions of all the risk and return parameters, so that a small change in any parameter (or set of parameters) cannot lead to a fundamental change in the portfolio. As noted at the beginning of this paper, stability does not characterize more complex portfolio optimization models.

Because of its inherent modularity and modest data requirements, the Treynor-Black model can be extremely valuable as a strategic planning tool for a multiproduct or multidivisional financial enterprise. In this case an "asset" represents a product (or division) within the enterprise and the output of model is used to target the relative share of each product. This section will outline a process for applying Treynor-Black at the enterprise level and the following section will discuss some extensions to the model that can increase its value for this and other applications.

The application of the Treynor-Black model at the product level can be viewed as a three-step process that may need to pass through several iterations before it is complete. These steps are:

1. Risk-based product definition
2. Determination of product-level risk-adjusted excess returns (alphas)
3. Estimation of product-level specific risk

Risk-based product definition is a necessary first step in the process because the Treynor-Black model assumes that all correlation between products is captured by the systematic risk factors, leaving specific risk to be distributed independently from product to product. The potential for serious problems can be readily seen by considering the example above 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. Thus, 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 Treynor-Black model’s independence assumption has the effect of making the portfolio allocation depend on the way in which assets are defined—a difficulty that more complex models are designed to avoid. The diligence that is required to apply the model properly might be viewed as the price that is paid for its simplicity.

The independence requirement for specific risk means that most enterprises cannot use their existing definition of product lines or division 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. One way to approach the process is to create a product matrix where each dimension of the matrix represents a product attribute that affects specific risk. One can then create natural groupings of products starting by selecting products along the most important rows and columns of the matrix. These products groups will then define the "assets" for use in the Treynor-Black model. In general, some groups will be determined by geographic exposure (countries, regions, etc.), some by industry (financial, energy, etc.), and some by traditional product category (consumer, corporate, etc.). Also, the potential for overlap can be addressed by performing multiple runs of the model, one along each relevant dimension of risk so that separate target levels of exposure are set along each dimension.

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 so that fundamentally different risks appear separately, but not so fine that the duplication problem noted above starts to appear. As a rough guide, 20 to 100 groups tend to be a reasonable amount to work with; however, more or less than this may be appropriate in some circumstances.

The second step in the process is to determine the alpha for each of the products. The approach taken in the original Treynor-Black paper is 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 her or his research. This opinion is then converted into an estimate of the excess return. As noted above, Treynor and Black even suggest that existing recommendation scales (Buy/Hold/Sell) can be converted directly to alphas

The exercise of determining product-level alphas is essentially one of uncovering sources of competitive advantage and then estimating how they will contribute to returns in the future. In applying the Treynor-Black model the important thing is not what returns historically have been, but rather what they will be going forward. Existing mechanisms for computing risk-adjusted returns may have to be modified or supplemented in order to make them suitable for this application. Of particular concern are those cases where the excess return of a product is determined to be negative, i.e., its return does not exceed its hurdle rate. In the absence of overriding business reasons (relationship building, etc.) to continue the product, this is a clear signal to limit new business in the product and to consider mechanisms for laying off or hedging any existing risk from the product.

In those fortunate cases where products closely parallel positions in actively-traded assets, the hurdle rate for a product can be determined using regression analysis on the appropriate time series of historical returns. The statistical estimates of alpha that are generated this way should not be used as inputs to the Treynor-Black model because they are likely to be poor estimates of future returns. Instead, an independent assessment of future returns should be made, and the hurdle rate determined statistically should be netted out to compute an alpha.

The third and final step of determining the level of specific risk for each product can also be determined either subjectively or objectively. As noted above, when sufficient time series data relevant to a product 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 this easy. Building on the methodology for enterprise risk management developed in Greene and Miller (1996) one can construct a risk scorecard for each of the products that gauges the risk specific to the product on several dimensions. Even model risk can be included as a dimension in the analysis, where this risk will tend to increase as the experience one has with a product decreases. The individual scores on each dimension can then be aggregated using any of the standard methods ranging from the estimation of an aggregation function to the construction of an expert system. Regardless of the aggregation method used, the specific risk number produced must be calibrated so that it properly reflects the standard deviation of return that is not captured by systematic risk factors.

The risk scorecard for each product, along with the information used to estimate its alpha, can be summarized in an assessment of the product that can usually fit in a single-page format. The historical, actual, and target shares for the product can also be included. "One-pagers" constructed this way can provide a valuable tool for strategic management regardless of how seriously one takes the output of the Treynor-Black model.

As noted earlier, one may need 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. Also, as the product mix changes over time, the risk-based product definitions will need to be updated. There is no limit on how frequently or infrequently one may wish to update the estimates of alpha and specific risk. Major maintenance should probably be done annually, with minor changes incorporated on a quarterly or monthly basis.

Extending the Model

The Treynor-Black model, like any other optimization-based model, can be extended by the addition of constraints that can reflect real limitations on behavior. For example, we have already developed the model with a shortselling constraint by omitting products that do not cover their hurdle rates, giving them an automatic target share of zero. Likewise, it is usually the case that the amount of business that can be generated in any product line is constrained either by the internal capacity to originate that business or by the depth of the market for the product. It is quite easy to add capacity constraints for each product in the portfolio through an iterative maximization process.

In the example given above, suppose that 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 then 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, the underlying logic of allocating shares proportional to alpha and inversely proportional to the square of specific risk remains.

"Soft" capacity constraints lead to further complications. With a soft constraint, capacity can be increased, but only through a reduction in return, i.e., there is a measurable market impact to doing business. If the impact can be modeled as a few discrete "lumps," then an iterative process similar to that for the hard capacity constraint can be used. If the trade-off between share and pricing is more complex, e.g., it takes the form of an upward-sloping "supply" curve, the extension of Treynor-Black that results from the addition of this constraint can usually be solved analytically.

The constraint that is likely to be the most important one for the firm—and the one too often ignored in portfolio optimization—is that of economic viability, which in most cases is equivalent to the 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 is 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 point in time one or more of these constraints will be binding on the firm, so it is not possible to perform a true optimization without taking them into account. As noted in the introduction to this paper, the optimization performed by portfolio models such as Treynor-Black accounts for possibilities along the entire probability distribution of outcomes and gives credit for the return associated with risk. The issue of capital adequacy, on the other hand, 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 when the risk is skewed to the downside, as is the case for 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 with the capacity constraints considered above, capital constraints are trickier to incorporate into the analysis while maintaining its simplicity. At a technical level, each capital constraint has associated with it a Lagrange multiplier that measures how tightly the constraint bites and serves as a "shadow price" for capital associated with that constraint. Products that are significant consumers of one or more types of scarce capital have their Treynor-Black weights reduced to reflect their capital utilization.

In some cases it may be possible to retain the original simplicity of the Treynor-Black model simply by netting an implied charge for capital adequacy out of the excess return. Since the excess return already accounts for the risk-adjusted hurdle rate, this must be done carefully to avoid double-charging for capital. Also, although this paper has focused on the aspect of the Treynor-Black model that concerns active portfolio management, the overall optimal holdings for the firm will include passive holdings, most significantly, the capital held to deleverage or buffer the active portfolio The passive holdings of the firm are also influenced by the degree to which it decides to hedge the systematic risk that it has taken on in constructing the optimal active portfolio. In any event, the important point is that the Treynor-Black model can coexist with established risk management procedures and can assure that the firm not only meets the standards required for survival according to overall risk guidelines, but also attains its greatest potential for profitability on a quarter-to-quarter basis.

References