Using Custom Risk Models for Enhanced Performance Attribution

Using Custom Risk Models for Enhanced Performance Attribution When Axioma introduced the Risk Model Machine about a year ago, clients quickly saw the obvious benefit, i.e., the ability to use their own factors to create custom risk models tailored to their own investment process. The resulting tight fit with the client's investment process enables custom risk models to deliver better forecasts of risk. And by enhancing the alignment between clients' risk and return factor models, undesired risk exposures can be avoided. But one of the largely unheralded features of the Risk Model Machine is its ability to drive enhanced performance attribution. Axioma Advisor spoke with Chris Canova, Vice President of Product Management and Strategy, to learn more... When you talk about the Risk Model Machine and the advantages of custom risk models, performance attribution is not a subject that typically springs to mind. Why is that? I think the main reason is that people tend to focus on other features the better risk estimates that CRMs deliver and how portfolio construction can improve. But performance attribution is definitely one of the unsung benefits of using custom risk models. Why does a custom risk model deliver better performance attribution than a conventional risk model? Well, say you're a portfolio manager and you are interested in decomposing the returns of your portfolio using a risk model. With a custom risk model, you can use the key factors that you yourself use to define any factor type. In fact, you may very well use factors in your analysis that Axioma doesn't use at all in its default model. Here we are talking about factors that typically fall into the category of valuation factors things like earnings, growth, value, and quality. By definition, you cannot use your own factors with an off-the-shelf risk model. And that's exactly why some portfolio managers look at performance attribution reports from a standard risk model and say, "That's a very interesting analysis, but it's not the way I think about value. So these results are not relevant to me. Can you produce this same analysis using my factors?"

The answer to this question is now, "yes." Performance Attribution Summary Comparing the Axioma US Fundamental Model with the Custom Model ASUS2-MH Custom Risk Model Return Risk IR Return Risk IR Factor Contribution 2.18% 3.32% 0.66 Factor Contribution 4.62% 3.38% 1.37 US2AxiomaMH.Style 2.66% 2.61% 1.02 MyRiskModel.Style(Custom) 5.37% 2.61% 2.06 MyRiskModel.Style(Axioma) -0.32% US2AxiomaMH.Industry -0.48% 1.53% (0.31) MyRiskModel.Industry -0.43% 1.36% (0.32) Axioma.Styles Contribution Avg Wtd Exp Medium-Term Momentum Risk MyRiskModel.Styles Contribution Avg Wtd Exp 2.00% 0.42 1.14% MyRiskModel.Quality 2.03% 0.56 1.16% Leverage 1.06% -0.20 0.82% MyRiskModel.Value 1.59% 0.78 1.48% Size 0.86% -0.40 1.12% MyRiskModel.Size 1.27% -0.40 1.13% Value 0.14% 0.02 0.15% MyRiskModel.Earnings 1.18% 0.56 0.74% Growth 0.73% 0.29 0.42% MyRiskModel.Growth 0.59% 0.29 0.44% Short-Term Momentum 0.35% 0.02 0.63% MyRiskModel.Momentum 0.57% 0.46 1.03% Liquidity 0.00% 0.07 0.17% MyRiskModel.Liquidity 0.02% 0.07 0.17% Market Sensitivity 0.01% 0.09 0.55% MyRiskModel.Market 0.02% 0.09 0.56% Sensitivity Exchange Rate -0.12% -0.06 0.52% MyRiskModel.Exchange Rate -0.01% -0.06 0.53% Sensitivity Sensitivity Volatility -2.37% 0.21 0.90% MyRiskModel.Volatility -2.20% 0.21 0.87% Risk So it's all about customization... Customization, yes, but also integration. The Risk Model Machine allows the portfolio manager to look at the risk of his portfolio, and to decompose the performance attribution of that portfolio, using his own custom factors. He's in effect plugging his own views into the Risk Model Machine. Based on that, the Risk Model Machine calculates the custom risk model, including all of the exposures, factor returns, and other information that feeds into performance attribution. Now here's where the integration comes in. Axioma's Performance Attribution module seamlessly picks up all of that data, just as if it were interacting with one of Axioma's own default risk models. So the reports received by the portfolio manager are customized to exactly the factor definitions that that manager has in place. It sounds like a very straightforward process... Axioma makes the process seamless. It's the platform it sits on that is sophisticated. And the integration between the Risk Model Machine and our Performance Attribution module is something that no one else in the marketplace has duplicated. The Risk Model Machine takes in the information and produces the custom risk model. That's one side of the equation. The Performance Attribution module has the ability to leverage that information and generate meaningful reports that clients can use to better understand their sources of returns and even to report the sources of their

outperformance or underperformance to clients, if they choose to do so. Asset managers are seeing a dramatic increase in questions from clients, and their ability to show sophistication and detail in their reporting to clients has become critical. The Risk Model Machine and Axioma Performance Attribution can help greatly in addressing those questions. No one else out there, I might add, can do this. Some can build the risk model, but they can't get it to translate well into reports in a timely or efficient manner. Others have very good reports, but they don't have a good way of generating custom risk models that meet the standards of quality Axioma can achieve. The Risk Model Machine builds custom risk models using our proven, high performance modelling platform, so clients know they are getting a quality risk model in their implementation. So what Axioma brings to the table is best-of-breed custom risk models that are seamlessly integrated into an effective performance attribution and reporting process a process that can be used to generate reports for the portfolio managers themselves, for the research teams or for clients. The new version of the Risk Model Machine v 7.3 is due out in August. What can we expect? We've got some pretty exciting new features to highlight. The enhancements in 7.3 focus mainly on reporting. We're improving the comprehensiveness of the performance attribution reports and adding a new integrated risk-analysis feature that lets users easily create automated Excel files and nicely tailored PDF reports, for both risk reporting and performance attribution. We're also making some very significant automation improvements, which will provide scale for many of our customers who are looking for the capability to roll out our reports for hundreds of portfolios. So, for clients that need a robust automated process, 7.3 will be able to do that. And in a matter of just a few months, we'll be able to talk about the exciting enhancements that await in 7.4! Using a Custom Risk Model for Performance Attribution in Axioma Portfolio Analytics When the factor risk model used in factor-based attribution does not include the portfolio manager s proprietary factor(s), the resulting reports often suggest that the source of active return is asset specific. This gives the manager very little insight into the actual sources of performance, other than identifying individual names that helped or hurt performance. With the Axioma Risk Model Machine, portfolio managers can create custom risk models using the style factors of their choice. If the manager s proprietary alpha factors

are included as style factors, the resulting model can be used to attribute both risk and performance in a more meaningful way. The use of custom risk models in performance attribution enables portfolio managers to offer their clients increased transparency by attributing a portfolio s risk and return to the proprietary investment factors used to construct and rebalance the portfolio. In Axioma Portfolio Analytics, custom risk models created with the Axioma Risk Model Machine can be used seamlessly in performance attribution analyses. The workflow is identical to using a standard Axioma-provided risk model. We illustrate with a simple example. Creating a Risk Model Using Proprietary Factors The Axioma Risk Model Machine makes it easy to add proprietary factors to Axioma fundamental factor models. In the Style Factor Settings section of the Risk Model Machine task set-up, you may choose any combination of standard Axioma-provided style factors and proprietary factors. The Style Factor Settings presents a list of all the risk model factors and user-imported factors present in the database. Simply highlight the factor(s) you wish to add to the risk model and click the right arrow to include them in the list of style factors as shown in Figure 1. Figure 1: Selecting style factors to add to the Custom Risk Model

If the proprietary factors are collinear with one or more of Axioma s style factors, it is advisable to remove those style factors from the risk model. Style factors are easily removed by highlighting them and clicking the left arrow. All risk models generated by the Axioma Risk Model Machine are updated daily, however, it is not unusual for proprietary factors to be updated less frequently. Factors that are provided less frequently can be accommodated by setting the lookback period for the factor. The lookback period specifies the number of days that the Risk Model Machine will lookback for user-provided factors. For example, for proprietary factors that are updated monthly, the lookback period could be set to 32 to ensure that the Risk Model Machine finds the factor for each day of the month. Figure 2: Setting Lookback days for a proprietary factor Once all the desired factors have been added to the model, set the desired date range for generating the models and click run to generate your history of custom risk models. This process may be time consuming depending on the number of factors in the model you are generating and the CPU and hardware characteristics of your computer. When your risk model generation task is complete, you will find a new risk model listed in the Content Explorer. Here the custom risk model is named CAMH with Alpha to indicate that it is a Canadian Medium Term model that includes an alpha factor.

Figure 3: The Custom Risk Model appears in the Content Explorer Running Performance Attribution with a Custom Model Once the custom risk model is present in the database, it can be used for subsequent performance attribution analyses just like an Axioma-provided risk model. In the factorbased PA task, custom risk models present in the database are listed in the Risk Model dropdown. Simply select the desired custom risk model to use it in the factor attribution computations.

Figure 4: Selecting the Custom Risk Model in Factor PA tab Once the Performance Attribution run is complete, the resulting summary report will explicitly list the contributions of each proprietary factor included.

Figure 5: Proprietary factor in the Attribution Summary A detailed analysis of each individual factor is available in the Factor Details Reports. Here we see detailed information about the exposure and return contribution of the proprietary factor.

Figure 6: Factor Detail for proprietary style factor Conclusion In Axioma Portfolio Analytics, custom risk models created with Axioma s Risk Model Machine can be used seamlessly in performance attribution analyses. Using custom risk models in concert with factor performance attribution provides better transparency to portfolio managers and their clients by attributing a portfolio s risk and return to the proprietary investment factors used to construct and rebalance the portfolio. Mark Your Calendar With Axioma's Upcoming Events Axioma is proud to partner with Attilio Meucci's SYMMYS to offer the... Advanced Risk and Portfolio Management Bootcamp August 13-18, 2012, Kimmel Center, New York University The ARPM Bootcamp provides in-depth understanding of buy-side modeling from the foundations to the most advanced statistical and optimization techniques, in nine gruesome, heavily quantitative hours each day, with theory, live simulations, review

sessions, and exercises. Topics include portfolio construction, factor modeling, copulas, liquidity, risk modeling, and much more. Also features world famous speakers, cocktail party, etc. For complete information and to register, visit the event website today. Clients and friends of Axioma are offered a discounted registration rate. When registering, select registration type "Partner," then select "Other" and input code "Axioma2012ARPM." To see photos and more from last year's ARPM Bootcamp, click here. Attendees will earn 40 CE units CFA Institute, 40 CPE units GARP. See you in August! Save the date! Axioma Breakfast Research Series returns in October 2012. London October 9 Stockholm October 10 San Francisco October 16 Chicago October 17 Boston October 23 New York October 24 More details to come soon.

The Transfer Coefficient, Custom Risk Models, and Portfolio Construction Robert A. Stubbs June 13, 2012 Introduction Portfolio managers who construct portfolios using an optimizer typically incorporate many constraints into the problem. These constraints are used to satisfy mandates, control trading, prevent undesirable bets, and protect against the unknown. While some constraints, such as risk targets, are necessary and others may be helpful (see Jagannathan and Ma, 2003), tight constraints can significantly detract from ex-post performance. The entire premise of optimized portfolio construction is that portfolios should have an optimal risk-return tradeoff. Expected returns and a risk model are the two elements that are used in the measurement of this tradeoff. Constraints can interfere with this optimal tradeoff by not allowing the risk model to play a significant role in determining the tradeoff. In extreme cases, the risk model may play no role at all. Recognizing the impact of constraints, Clarke et al. (2002) introduced the concept of the transfer coefficient to measure the extent of constraint interference. Specifically, the transfer coefficient (TC) is defined as the correlation between the risk-adjusted expected returns and risk-weighted portfolio. The transfer coefficient measures the degree to which expected return forecasts are transferred to a managed portfolio. In other words, it measures the efficiency of the portfolio construction process. In the absence of constraints, the transfer coefficient is one. More importantly, Clarke et al. (2006) expanded upon the Fundamental Law of Active Management introduced by Grinold (1989) by showing that IR = TC IC N, where IR is the information ratio, IC is the information coefficient, and N is the number of assets in the investment universe. This relationship shows that for a fixed IC and N, the risk-adjusted performance increases proportionally to an increase in the transfer coefficient. Quantitative managers will always search for greater IC, and they should. However, for the here and now, there is much to be gained by focusing on the TC. In many cases, increasing the transfer coefficient 20% or more is a real possibility and is far easier than increasing the IC by the same amount.

Increasing the Transfer Coefficient Mathematically, the TC is defined as (see Clarke et al., 2006): TC = α T (w b) αt Q 1 α (w b) T Q(w b), (1) where Q is the asset-asset covariance matrix, w is the vector of portfolio weights, and b is the vector of benchmark weights. 1 Improving the transfer coefficient can be accomplished in several ways. The most obvious is to loosen or eliminate constraints. While it is an effective method that should be considered, it is not always possible to do so. Yamamoto et al. (2011) increase the transfer coefficient by explicitly adding a constraint to the portfolio construction problem that forces the transfer coefficient to be sufficiently large. Using this constraint in historical backtests, they showed that the increased transfer coefficient led to improved ex-post performance as expected. Their methodology and empirical results used a special form of the transfer coefficient where the asset covariance matrix is the identity matrix. Such an approximation of the transfer coefficient does not truly measure the implementation efficiency of the risk-return tradeoff, however. Nevertheless, the idea is intriguing. Martin (2012) showed how the transfer coefficient can be constrained in its more general form. Martin (2012) showed that a lower bound on the transfer coefficient, θ, can be written as α T (w b) θ α T Q 1 α (w b) T Q(w b) 0, which fits the forms of a robust constraint (see Ceria and Stubbs, 2006) and can thus be formulated as a second-order cone constraint. This is a true constraint on the transfer coefficient and enforcing it does increase the transfer coefficient. But does it come at a cost? Suppose our portfolio construction problem is to maximize expected return subject to a tracking error constraint. We get a portfolio with a low transfer coefficient and would like to increase it, so we add a constraint. Now, look at the constraint closely. In order to increase the transfer coefficient, we either need more expected return or less tracking error. We maximized expected return in the original portfolio construction problem, so adding a constraint can certainly not increase this value any further. Therefore, the only way of increasing the transfer coefficient is to reduce tracking error. Enforcing the TC constraint will produce a solution with a lower predicted tracking error. This improves the transfer coefficient because as the predicted tracking error is reduced (essentially with a tighter tracking error constraint), fewer 1 Given a portfolio, w, and a risk model, Q, the implied alpha of w is defined to be the vector of expected returns that gives w as the optimal solution to an unconstrained mean-variance optimization problem. The implied alpha, γ, can be expressed as γ = Q(w b). Using implied alpha, we can rewrite the transfer coefficient as T C = α T Q 1 γ αt Q 1 α γ T Q 1 γ. Practitioners sometimes refer to the transfer coefficient as the correlation between alpha and implied alpha. However, this is true only if Q = I, i.e., only if the Euclidean distance is used as a distance measure.

of the other constraints are binding, thus allowing the maximal expected return to be reduced at a slower rate than the tracking error. So, increasing the TC via a constraint comes at the cost of reducing the target risk level. This is an option, but the tradeoff needs to be managed and considered along with the portfolio mandate. There is yet another method of increasing the transfer coefficient, one that is equally effective but more practical. We can increase the transfer coefficient by optimizing with a different risk model, Q. From (1), we can see that the risk model itself plays a critical role in the computation of the TC. Portfolio construction problems drive the optimal portfolio towards certain sources of systematic risk, such as value and momentum, while trying to eliminate other sources, such as size and liquidity. We claim that all sources of systematic risk that are under the control of the portfolio construction problem come into play in determining the optimal risk-return tradeoff and should thus be directly incorporated as factors in the risk model used in portfolio construction. To build intuition of this concept, we consider a simple two-dimensional example illustrated in Figure 1. The goal of the optimization problem is to construct a long-short portfolio that maximizes the expected return (α) subject to a risk constraint and simple asset bound constraints. In Panel 1a, the risk constraint is represented by the circle and the simple asset bounds by the square. The feasible region is represented by the intersection of the interiors (including boundaries) of the circle and square. The optimal solution is denoted by the point w. Note that a bound constraint became binding thus driving the optimal portfolio away from the direction of α creating a transfer coefficient less than one. Since the risk constraint is a circle in this particular example, the transfer coefficient is the cosine of the angle between the two rays leading to α and w from the origin. In Panel 1b, we consider what happens when we construct a custom risk model by adding α as a risk factor. We assume that α does contain some systematic risk that was otherwise not captured by the original risk model. Because the previously unknown source of risk is not included in the risk model, less exposure can be taken in α and still satisfy the risk constraint. In this particular example, the direction of α is restricted just enough such that the simple bound constraint is not binding. This leads to the optimal solution w and a transfer coefficient of one. While this example illustrated only the effect of including the expected returns in the risk model, it can be equally beneficial to incorporate systematic factors being constrained as well. Consider an expected return that naturally wants to take an exposure to market beta yet the investment strategy contains a constraint that forces the portfolio to be market-neutral (beta exposure of zero). The optimizer will then not be able to travel in the direction of α, but instead will travel in the direction of the projection of α onto β. When measuring the quality of a strategy (and risk model), it is necessary to look at expost performance. The reason is simple. A risk model that misses the sources of systematic risk at play in the portfolio construction problem will underestimate risk leading to potentially superior ex-ante results. This was even the case in our simple example illustrated in Figure 1. Even though the transfer coefficient was less than one when using the original risk model, the expected return of w is greater than that of w and both have the same predicted risk as determined by their respective risk models.

w 2 w 2 α w α w w w 1 w 1 (a) Initial Optimization (b) Optimization using CRM Figure 1: Graphical illustration of how a custom risk model can improve the transfer coefficient. Long-Short Example Consider the long-short portfolio construction problem: maximize α T w 1 2 wt Qw subject to Aw = 0, (2) where A is a m n matrix defining a set of linear constraints. Such a set of constraints is commonly found in market neutral portfolios since A can be specified to enforce neutrality across many dimensions such as dollar, beta, industry, and country. The optimal solution to (2), w, can be expressed as w = Q 1 α Q 1 A T [ AQ 1 A T ] 1 AQ 1 α. (3) In this particular case, the transfer coefficient can be expressed as TC = 1 αt Q 1 A T [AQ 1 A T ] 1 AQ 1 α. (4) α T Q 1 α It is easy to see that the transfer coefficient depends on the constraint matrix A as expected. It also depends on both the expected returns (α) and the risk model (Q). In fact, the transfer coefficient depends on the interaction of the expected returns, the constraints, and the risk model. To illustrate the effect of the risk model and constraints on the transfer coefficient, we consider two market-neutral strategies together with a few custom risk models. The first strategy maximizes utility expressed as the expected return less a multiple of total predicted variance. A normalized momentum factor is used as the expected returns. The portfolio is constrained to be dollar-, beta-, sector-, and size-neutral. That is, the net exposure to each of these factors must be zero. The second strategy adds additional constraints including asset bounds of ±2%, industry-neutrality, and a forced long and short value of 100%.

Risk Model Strategy 1 Strategy 2 Base 0.9187 0.6815 Base+Momentum 0.9710 0.8256 Base+Momentum+Beta 0.9759 0.8749 Base+Momentum+Beta+Size 0.9770 0.8765 Table 1: Transfer coefficients of two different strategies using four different custom risk models. The transfer coefficients were computed using the same custom risk model that was used in the strategy. Four different risk models were used in the objectives of the two strategies: the base model that contained only industry factors, base plus the same momentum factor used in the expected returns, base plus momentum and beta factors, and base plus momentum, beta, and size factors. These two strategies were optimized for each of the four risk models for one date to look at the effect on the transfer coefficient. The results are shown in Table 1. For both strategies, the transfer coefficient increases as the systematic risk factors that are at play in the strategy are added to the risk model. Incorporating systematic risks from constraint gradients and expected returns in the risk model improves the TC and becomes a more accurate reflection of the efficiency of the strategy implementation. But, it is not just the TC coefficient that changes. Incorporating the additional sources of systematic risks at play in the strategy into the risk model allows the risk model to play a greater role in the portfolio construction. While this allows for a greater transfer coefficient, that is not the main point. Rather, the transfer coefficient allows us to measure how much the risk model is coming into play and thus driving the optimal risk-return tradeoff. If the risk model is not a major player in the portfolio construction problem, we are violating the main premise of optimal portfolio construction and instead optimizing an expected return-constraint tradeoff. But not just any risk model will do. It is essential to accurately capture risk in exactly those dimensions that are driven by the portfolio construction problem. Those dimensions that are known should be incorporated into a custom risk model. Those that are unknown can be captured by the Alpha Alignment Factor (AAF). (See Saxena and Stubbs (2013, 2010, 2012) for details of how improved factor alignment and the AAF can lead to better risk estimation and ex-post performance.) Conclusions There are known knowns; there are things we know that we know. There are known unknowns; that is to say there are things that we now know we don t know. But there are also unknown unknowns - there are things we do not know we don t know. United States Secretary of Defense Donald Rumsfeld While this quote was made in reference to the Iraq war, it applies equally well to portfolio management. There are many unknown unknowns in the markets causing portfolio managers to heavily constrain their portfolios. However, with the exception of asset bound constraints, all of these constraints protect only against the known unknowns, such as industry risks. There were previously known unknowns, such as the understanding of the interaction of

constraints, expected returns, and risk model in portfolio construction. These unknowns have become known knowns and should be treated as such. By this, we mean that these known systematic risks should be incorporated into risk models so that a truly optimal riskreturn tradeoff can be made. Summer is almost here. Relax your constraints. Build a custom risk model. Increase your TC, and realize greater performance. References Sebastián Ceria and Robert A. Stubbs. Incorporating estimation errors into portfolio selection: Robust portfolio construction. Journal of Asset Management, 7:109 127, 2006. R. Clarke, H. de Silva, and S. Thorley. Portfolio constraints and the fundamental law of active portfolio management. Financial Analysts Journal, 58:48 66, 2002. R. Clarke, H. de Silva, and S. Thorley. The fundamental law of active portfolio management. Journal of Investment Management, 4(3):54 72, 2006. Richard C. Grinold. The fundamental law of active management. The Journal of Portfolio Management, 15(3):30 37, 1989. Ravi Jagannathan and Tongshu Ma. Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance, 58(4):1651 1684, 2003. Christopher Martin. How to constrain the transfer coefficient in Axioma Portfolio. Technical report, Axioma, Inc., March 2012. A. Saxena and R. A. Stubbs. Pushing frontiers (literally) using alpha alignment factor. Technical report, Axioma, Inc. Research Report #022, February 2010. A. Saxena and R. A. Stubbs. An empirical case study of factor alignment problems using the United States expected returns (USER) model. The Journal of Investing, 21(1):25 43, 2012. A. Saxena and R. A. Stubbs. Alpha alignment factor: A solution to the underestimation of risk for optimized active portfolios. Journal of Risk, 2013. Rei Yamamoto, Takuya Ishibashi, and Hiroshi Konno. Portfolio optimization under transfer coefficient constraint. Journal of Asset Management, 13(1):51 57, 2011.

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