Best Practices in Factor-Based Analytics

Transcription

1 Best Practices in Factor-Based Analytics Phil Martinelle Axioma, Inc. November 7, 2016 Introduction As a portfolio manager, have you ever been surprised by a bad return period? Or wondered if there is a better way to identify the risks in your portfolio? Have you wanted to look for sources of return beyond sector breakdowns? If so, this paper will provide an overview into how you can address these questions and more. Most managers decompose their portfolio return at the sector and asset levels. While this provides important insight, it does not capture the complete returns decomposition, leaving a lot of valuable information out of their analysis. And many managers monitor their portfolio s historical tracking error they may even be required to periodically report on it. Unfortunately, knowing historical tracking error does not always correlate with what the tracking error will be in the next period. And many managers struggle to identify where their tracking error came from historically, before even considering where it is likely to come from in the future. This paper will help guide you through decomposing returns and tracking error, otherwise known as Active Risk, and will provide a concise overview of Axioma s tools to better understand return and predict and manage risk. We will do this with a case study analysis of a real-life portfolio.

2 Overview of risk Let s begin with an overview of risk in general and some ways it is measured. Most people think of risk as an exposure to danger or, more specifically, the probability of a negative occurrence. For instance, someone might consider the risk, or probability, of not surviving a skydiving trip. In this case, calculating this risk is fairly straightforward, assuming the data is available, by dividing the number of unsuccessful skydiving trips by the total number of skydiving trips. When measuring investment risk, the calculation is somewhat more complicated. Instead of a simple division, we look at standard deviation. Standard deviation also employs a probability, yet it is a probability band around a mean. To give an example, let s say we have two sets of portfolio returns. Figure 1 shows two different portfolios returns versus the same benchmark, monthly, for a year. They each have an average monthly excess return of 0.19%. By looking a bit deeper, we can see that Portfolio A is clearly tracking the benchmark much closer with a monthly high excess return of 0.80% and low of -0.40%, while Portfolio B has a monthly high excess return of 2.30% and low of -2.10%. Obviously if the investor is trying to track somewhat closely to the benchmark, then Portfolio A is significantly less risky than Portfolio B. As expected, this manifests itself in the observed annualized standard deviation (multiply the monthly standard deviation by the square root of 12 to annualize). Portfolio A shows 1.34% observed annualized standard deviation while Portfolio B shows a much larger 4.86%. In other words, for Portfolio A we would expect that with 68% confidence the annual excess return will be within -1.15% and 1.53%. Portfolio B s annual excess return risk has a much wider band, from -4.67% to 5.05%. Again the variability of returns shows that Portfolio A is less risky than Portfolio B, since we can be confident that Portfolio A s returns will display less variance then Portfolio B; and thus that the probability of an extreme loss is less with Portfolio A than with Portfolio B. We call the standard deviation of excess return Active Risk. Figure 1 Portfolio A Benchmark Excess Return Average 0.19 High 0.80 Low St Deviation 0.39 Annualized St Dev 1.34 Portfolio B Benchmark Excess Return Average 0.19 High 2.30 Low St Deviation 1.40 Annualized St Dev 4.86 Best Practices in Factor-Based Analytics page 2

3 Factors Thus far we have looked at observed data and calculated some metrics that help us better understand the data and its riskiness. This is very important and instructive analysis, yet it does not tell us where risks lie. For instance, in Figure 1 we can clearly see Portfolio B is riskier than Portfolio A. What we don t know is why Portfolio B is riskier. In order to determine that, we need to look at the portfolio s risk factors, or what are simply called factors. Factors are common and observable characteristics of every security. For example, every security will have a price-to-earnings ratio and will belong to a certain industry. We will look into this concept a bit more below. Let s take health insurance, a non-financial example, to help better understand risk factors. We will look at two potential risk factors: smoking and peanut butter brands. We know that smoking can lead to lung cancer, which is expensive to treat. Conversely, we would imagine that the brand of peanut butter someone eats does not affect their likelihood of contracting an illness that requires expensive treatment. Therefore, when deciding how much to charge for a policy, we would expect an insurance provider to inquire if an individual smokes tobacco products but not ask if the individual eats Jif or Skippy peanut butter. What we would say then is that the explanatory power of the factor smoker is much higher than that of peanut butter brand when it comes to predicting if an individual will require expensive treatment, with the peanut butter brand factor probably having no explanatory power at all. Factor-Based Regression Analysis Using a statistical method called regression, we are able to look at a list of potential factors to determine which ones are best able to explain observed return and then create an equation to model the observed return. To do this, we list in matrix form the exposure each security has to each factor and regress that matrix of securitylevel exposures against the vector of each security s return. The complete list of securities that are used in the regression is called the estimation universe. In this case, we use an estimation universe of roughly 3,000 U.S. securities. Exposures are the observable characteristics, such as the book-to-price ratio, of each security. Once we normalize that data (by subtracting the universe mean and dividing by the universe standard deviation), we call that factor Value. Once the regression has been run for day X, it will produce a set of slopes (or factor returns) for each factor, along with a metric called a T-stat, that tells how useful that factor is to explaining the overall observed return. Based on the T-stats, among other metrics, we can determine which factors to keep because they have more explanatory power (just like the smoking factor in the health insurance example) and which ones to remove (the peanut butter brand factors). Ultimately what this means in practice is that the final regression model structure will provide a slope for each factor, for each day. By multiplying a security s exposure to each factor by the factor return, we are able to determine the factor s contribution to the security s return. We can sum all these contributions to produce the model-expected security s return for day X. And taken at the portfolio factor level, Best Practices in Factor-Based Analytics page 3

4 we can sum the factor contributions in a portfolio to produce the model-expected portfolio return for day X. The difference between the model s expected return and the true holdings-based return is called selection or residual. Next we take this new information and create the process to look at Predicted Active Risk. We should take a moment to distinguish between two different definitions of Active Risk (and standard deviation in general). Our portfolio example in Figure 1 looks at Observed Active Risk. This means that we take the observed excess returns for the previous X dates and then calculate a standard deviation of the excess returns. Managers will commonly call this the portfolio s historical tracking error. This is an important calculation when looking at portfolio risk, although it does not tell us why the risk is what it is. There is another concept that is called Predicted Active Risk. This method is somewhat different in that it takes the historical regression-produced factor returns and creates a covariance matrix. One can think of this as an n x n matrix where the n s are each factor. The diagonal of this matrix is called the variance, and this is the volatility of the individual factor returns over the last Y days. The off-diagonal elements are the covariance terms, and this shows how all the different factor return pairs vary together. The important thing to remember is that the covariance matrix constructed using historical factor returns is used to produce a forward looking or Predicted Active Risk (as opposed to a simple Historical or Observed Active Risk), and it has the benefit of being able to show how each factor contributed to risk. This Predicted Active Risk will be the basis of our overall discussion. Aggregating the sources of Active Risk is slightly more complicated than aggregating returns. Based on the matrix algebra employed above, each security will have a Marginal Contribution to Active Risk (MCAR). By multiplying the net weight of each security (the security s weight in the portfolio minus the same security s weight in the benchmark) by each security s MCAR, we will receive each security s contribution to Predicted Active Risk. If we sum all of the securities contributions, we will arrive at the portfolio s total Predicted Active Risk. Factor Discovery Through significant research, we have uncovered a list of factors that we use in our risk model. (A detailed overview of the risk model is presented in the Appendix in the attached US4 Factsheet.) These factors help us better understand why one portfolio is riskier than another or, as is more often the case, where the sources of return and risk lie in a single portfolio, versus a benchmark. Dividend Yield Earnings Yield Exchange Rate Sensitivity Growth Industry classification Leverage Liquidity Market Sensitivity Medium Term Momentum Midcap Profitability Short Term Momentum Size Value Volatility Best Practices in Factor-Based Analytics page 4

5 Use Case Now that we have identified the factors that will be used to better understand the sources of risk and return, we can use a sample portfolio to illustrate the concepts. Our sample portfolio holds roughly 35 assets, benchmarked against the Russell 1000 Growth Index for a one-year period from June 2015 to June Portfolio Performance Below in Figure 2, we see a high-level snapshot of the full-period return decomposition. The summary table shows the portfolio returned -8.00% while the benchmark returned 2.97%, leading to a negative excess return of %. The Risk column shows the observed annual standard deviation of returns, as opposed to the model predicted risk that we will look at later. The portfolio clearly carries more observed risk at 23.39% versus the benchmark of 17.34%, with an observed Active Risk of 10.11%. Clearly, the portfolio did not track the benchmark very well over this period. We will look at why. The Return Decomposition table shows the Active Return of % broken into Specific (-3.30%) and Factor (-7.67%) return contributions. The Specific is the residual we noted earlier, and this is the segment of return that the model cannot explain, which is more specifically the manager s stock-picking ability. The Factor contribution is the amount that the model can explain, and this is decomposed into the Style (-6.31%) and Industry (-1.37%) contributions. The portfolio s negative excess return was greatly influenced by the Style factor bets (which may or may not have been intentional) and the manager s skill at stock-picking, while the Industry bets provided a somewhat smaller negative contribution. Figure 2 With further digging, we uncover more information on why the portfolio performed so poorly. Figure 3, the Contributors to Active Return by Style table, shows the Style factors sorted by their contribution to return. We can see that summing all the contributions arrives at the -6.31% Style contribution we noted above. The Average Weighted Exposure column shows the portfolio had a number of active bets, and some paid off while some did not. The first one, Leverage, shows the portfolio was underexposed compared with the benchmark by -0.55, which led to a positive contribution of 1.52%. Put another way, the portfolio benefitted Best Practices in Factor-Based Analytics page 5

6 because it owned less-leveraged securities than the benchmark, and during this period less-leveraged securities performed better than more highly leveraged securities. On the other side of the spectrum, the portfolio was exposed to higher volatility securities than the benchmark by and during this period lower volatility securities outperformed, which led to a large -3.62% contribution to return. The Profitability, Growth, and Dividend Yield bets all negatively contributed to the portfolio s excess return by large margins. The management of this portfolio does not appear to incorporate Style factors, which significantly detracted from the portfolio s excess return. Figure 3 We already looked at the Average Weighted Exposure at the factor level. Of course, averages and even totals can be misleading. Since the attribution was run over a year (about 252 days), there might be some significant movements over that period. There are a few ways we will dig into this. Figure 4, the Common Factor Contributions table, shows the portfolio s day-by-day Active Return in blue and the portion of that which could be decomposed into Style, Industry, and Specific contributions. Looking at Figure 4, we can see that the portfolio did a fair job of tracking the benchmark over the first six months, with a midpoint Active Return around -2.5%. The second half of the year was marked by a quick two-month drop to almost -16% and then a final four-month claw-back to get the portfolio to the % it ended with. Clearly, something went quite badly over those two months. Best Practices in Factor-Based Analytics page 6

7 Figure 4 Figure 5, the Style Contributions table, breaks the Style component into the 14 Style factors. The portfolio s active exposure to Volatility consistently provided a strong negative return, while Leverage consistently benefitted active return. Interestingly, in black we can see that Market Sensitivity (Beta) started the period off with minimal impact, yet contributed significantly to the mid-period negative return and subsequent positive return claw-back. Profitability and Growth cumulative returns also began the period somewhat flat, yet became more negative as time went on. Before we look at Industries, let s take a peek at what some of these Style factors did day-by-day. Figure 5 Best Practices in Factor-Based Analytics page 7

8 Figures 5-10 show the day-by-day portfolio level active exposures by Style factor (gray bars) as well as the day-by-day Style factor contributions to return (blue bars). These combine to show the day-by-day cumulative contribution to excess return from each Style factor (black line). Figure 6 Figure 7 Best Practices in Factor-Based Analytics page 8

9 Figure 8 Figure 9 Best Practices in Factor-Based Analytics page 9

10 Figure 10 Beginning with Leverage, the figure clearly shows consistent underexposure to this factor over the year. This, coupled with fairly consistently negative factor returns, provided a constant and beneficial excess return contribution. Volatility was a bit more nuanced, with the active exposure moving around a fair bit, yet staying constantly overexposed compared to the benchmark. The early period factor returns were quite negative, which led to large early losses, yet flattened out after the midpoint. Profitability was somewhat reversed in that the portfolio was constantly under-exposed to more profitable securities, with under-exposure increasing as time went on. Early on, this did not have a big impact on the portfolio, but the contribution to return became much more negative after the mid-point of the analysis. Growth showed a relatively constant overexposure to higher growth securities. The return contribution looked initially similar to Profitability by starting off muted, but then became more negative, before flattening out in the later months. Market Sensitivity, Figure 9, made an interesting return contribution. The portfolio was constantly in higher beta securities and these initially didn t contribute much to excess return. Starting around the end of 2015, higherbeta securities did quite poorly, which led to a large negative return. This reversed around first-quarter 2016 and high-beta securities performed well, with some large swings at the end. Based on the above Figures, the portfolio appeared to have fairly consistent active exposures to the Style factors, likely due to limited trading, where the day-to-day movement of the Style factors themselves generally incurred significant losses on the portfolio. In this case, the manager might have considered doing more to limit unintended Style factor bets, while focusing on intended bets, in order to better track the benchmark. Best Practices in Factor-Based Analytics page 10

11 Figure 11 Figure 12 Best Practices in Factor-Based Analytics page 11

12 Asset Company Industry Total Return Average Portfolio Weight Average Benchmark Weight Average Active Weight "Portfolio Contribution Benchmark Contribution Active Contribution ALXN ALEXION PHARMACEUTICALS INC Biotechnology % 4.34% 0.34% 4.00% -1.56% -0.14% -1.53% BIIB BIOGEN IDEC INC Biotechnology % 3.46% 0.63% 2.83% -1.58% -0.36% -1.30% VRTX VERTEX PHARMACEUTICALS INC Biotechnology % 3.87% 0.25% 3.62% -1.26% -0.09% -1.28% ICPT INTERCEPT PHARMACEUTICALS IN Figure 13 Biotechnology % 2.44% 0.03% 2.42% -0.91% -0.02% -0.97% Total Return Average Portfolio Weight Average Benchmark Weight Average Active Weight "Portfolio Contribution Benchmark Contribution Active Contribution Asset Company Industry Electronic FIT FITBIT INC Equipment, Instruments & % 0.95% 0.00% 0.95% -1.89% -0.01% -1.94% Components Life Sciences ILMN ILLUMINA INC Tools & % 3.78% 0.24% 3.55% -1.66% -0.11% -1.65% Services ALXN ALEXION PHARMACEUTICALS INC Biotechnology % 4.34% 0.34% 4.00% -1.56% -0.14% -1.53% BIIB BIOGEN IDEC INC Biotechnology % 3.46% 0.63% 2.83% -1.58% -0.36% -1.30% VRTX VERTEX PHARMACEUTICALS INC Biotechnology % 3.87% 0.25% 3.62% -1.26% -0.09% -1.28% BIDU ICPT BAIDU INC ADR INTERCEPT PHARMACEUTICALS INC Internet Software & Services Figure % 0.84% 0.00% 0.84% -1.19% 0.00% -1.23% Biotechnology % 2.44% 0.03% 2.42% -0.91% -0.02% -0.97% Next, we will look at the securities industries to see how they contributed to active return. Figure 11 shows limited contribution to return from most Industries, with Biotechnology as the exception. This is collaborated in Figure 12, which showed Biotechnology was by far the largest detractor from return, at -2.00%. The average active exposure shows a large overweight to Biotech at an average of 11.15%. Presumably the manager intended to greatly overweight Biotech with unfortunate results. This can be seen in more detail when looking at the asset-level analysis in Figure 13. Here we see that the top four overweight Biotech securities combined to produce a -5.08% active return, with Figure 14 showed these are four of the top seven negative contributors to active return. Thus far, we have taken a detailed look into how the portfolio performed from an Active Return standpoint. It will be further insightful to see how much risk the portfolio took on to achieve its Active Return. Best Practices in Factor-Based Analytics page 12

13 Figure 15 Active Risk Figure 15 provides the decomposition of the predicted annual Active Risk of the portfolio as of June 30, 2015, the first date of analysis. A few things stand out. The Predicted Active Risk of 4.93% is roughly half the observed Active Risk of 10.11% we saw earlier. Further, we see that Style s contribution to Predicted Active Risk is actually smaller than the Industry s contribution to Predicted Active Risk. This emphasizes the importance of monitoring risk on a regular basis and highlights the fact that market movements can have a large impact on predictions, which makes a yearly prediction especially susceptible to changes. Figure 16 Figure 16, Factor Group Contribution to Active Risk, shows the contribution to Predicted Active Risk from Style and Industry over the year. Noticeably, Style started out as a small contributor, as we saw in Figure 15, yet as the year progressed Style jumped significantly, while Industry began to taper off. Figure 17 provides the day-by-day Active Predicted Risk for the year. As expected, the Active Predicted Risk of the portfolio increased significantly as the year progressed and as movements in the market influenced forward predictions. Again, this demonstrates the need to constantly monitor a portfolio s risk, as market movements and transactions can have a significant impact on the portfolio s positioning. Best Practices in Factor-Based Analytics page 13

14 Figure 17 Adding to this point, Figures provide a graphical view of the daily active exposure and daily percent of Predicted Active Risk for some of the more volatile Style factors. Similar to the exposure Figures we reviewed for attribution, these figures provide an informative look at how the exposures changed over the month and how that impacted the percent of Active Risk. Special attention should be paid to the magnitude of the axis. For example, Volatility reached a maximum 7% of Active Risk, where Market Sensitivity was about 20%. Looking at Figure 18, we can see the level of risk from Short-Term Momentum dramatically increased early in 2016, while the active exposure became quite negative, which indicates that the securities short-term returns of the portfolio fell quite a bit compared with the benchmark. Market Sensitivity also showed a dramatic and sudden jump in percent of Active Risk around the end of the first quarter in 2016, even with a fairly gradual increase in exposure. Again, the Figures confirm that risk and exposure can change quickly and in unexpected ways, which again shows how important it is to frequently monitor the portfolio s risk profile. Thus far, we have looked at Active Risk from a Factor level. Now let s take a look deeper at the security level. Figure 18 Best Practices in Factor-Based Analytics page 14

15 Figure 19 Figure 20 Figure 21 Figure 21 shows the top five securities ranked by percent of Active Risk as of June 30, 2015, the first date of the analysis. Additionally, the figure shows the Active Weight (the weight of the security in the portfolio minus that same security s weight in the benchmark) and MCAR. The MCAR provides helpful information as noted earlier, the total Active Risk is the sum of all of the individual securities active weights multiplied by their MCAR. Best Practices in Factor-Based Analytics page 15

16 Thus, a large MCAR and-or higher active weight will lead to a higher contribution to Active Risk and thus a higher percent of Active Risk for any given security. In Figure 12, we observed that three of the worst performing securities for the year in Figure 14 show up as having the top five highest levels of predicted percent of Active Risk at the start date of analysis. This clearly shows our model s ability to indicate how a potential concentration of risk could lead to unexpected or undesirable returns. A manager should look at these riskier securities in conjunction with a factor-level risk analysis and consider if these active bets are appropriate or if they should be reduced to avoid excess risk exposure. To put it another way, factor- and security-level risk analysis will provide a means to identify sources and concentrations of risk. The manager should then use that information to determine if the weightings of securities and by extension the portfolio s exposure to risk factors is too high and should be reduced. The manager should extend this same analysis to the Style factors as well. Figure 22 shows the Style factor active exposures and the percent of Active Risk at the beginning of the analysis. Unsurprisingly, Volatility contributes the largest percent of Active Risk among Style factors, despite having a fairly moderate active exposure. By looking at Figure 23, we can further drill to determine where the sources of Volatility came from. Figure 22 Figure 23 Best Practices in Factor-Based Analytics page 16

17 Figure 23 shows the top security-level contributors to Factor Risk for the Volatility factor. Once again, we see three of the same Biotechnology securities listed earlier. Additional concepts So far, we have decomposed the portfolio s Active Return and Active Risk and uncovered some important information. We have seen that managing risk from the top level down to the security level is an essential part of portfolio construction and management. And we witnessed the connection between sources of predicted risk and subsequent sources of return. There are a few additional tools, such as statistical models and stress testing, that a manager can use when monitoring his or her portfolio s risk. In addition to analyzing a portfolio using a factor risk model, there is benefit to using multiple models, including statistical models. Statistical models are different from factor models because in a statistical model the factors are not chosen in advance and are indeed unknown. Stat models, by nature of not being fixed to pre-specified factors, have the flexibility to observe sources of risk that the factor models may not pick up. The model process itself will isolate independent factors that have explanatory power. While the limitations of these models are clear (they cannot be used to determine overweight or underweight to observable factors and thus cannot explain what factors are adding or subtracting risk), they are very useful at the topline and for securitylevel analysis. Figure 24 In Figure 24, we can see the portfolio s predicted annualized Active Risk over time, using a factor model in blue and a statistical model in red. One can see that over different periods the different models had varying views of Predicted Active Risk. Initially, the statistical models showed the portfolio having a higher Predicted Active Risk than the factor models, sometimes by quite a bit. For a brief period in the second half of the analysis period, the factor model shows higher Predicted Active Risk. And at the end of the analysis period, the statistical model show quite a bit higher Predicted Active Risk. This leads to the obvious question of what model is right. The Best Practices in Factor-Based Analytics page 17

18 simple answer is that they each provide insight into the portfolio s risk profile. At times when the statistical model s Predicted Active Risk is higher, it makes sense to look at the security level percent of Predicted Active Risk to see if certain securities have a higher risk profile than the factor models show. This could mean that there is something going on in the market that the factors models are not picking up, but the statistical model is. In general, it benefits the portfolio analysis process to look at all available models to get a better picture of the sources of risk. By comparing Figure 25, which used the statistical model, with Figure 21, which used the factor model, we see that each model produced a somewhat different prediction of percent of Active Risk for each security. And in fact, each also produced a slightly different list of the top highest-ranked risk contributors. Splunk Inc., for instance, had a much higher predicted percent of Active Risk in the statistical model than it did in the factor model. This is a potential source of hidden risk that the factor model may not have picked up, and this is something the manager might want to consider when making weighting decisions. For further details, please see Axioma s paper More than Just a Second Risk Number: Understanding and Using Statistical Models. Figure 25 Stress Testing In addition to decomposing risk, managers often have an interest in seeing how their portfolios would perform in certain situations. This is commonly referred to as scenario analysis or stress testing. Below we can see two different stress-test methods. The first, as seen in Figure 26, takes the current portfolio factor exposures, as of June 30, 2015, and applies the factor returns as they existed during certain market conditions. For instance, had the portfolio been similarly constructed during the Black Monday Stock Market Crash in 1987, it would have dropped by around -20%, while the Russell 1000 Growth index would have fallen by around -18%, thus underperforming by around 2%. Conversely, had the portfolio been similarly constructed during the Subprime Crisis in 2007, the portfolio would had dropped by around -8% versus the benchmark decline of -9%, and thus would have outperformed by around 1%. Best Practices in Factor-Based Analytics page 18

19 Figure 26 Figure 27 shows a somewhat different method of computing a stress test. In this case, we are shocking specific factors today using the current risk model. We can see that shocking inflation by 5% leads to a positive portfolio return of around 1%, versus a roughly flat benchmark return under that same shock. These two types of stress tests are helpful by using actual occurrences, as well as potential changes to factors, to see how the portfolio would perform. Figure 27 Optimization One final concept worth discussing is optimization. While the background on optimization is outside the scope of this paper (see appendix for further details on optimization), it does have some relevance as optimization, in this case, is trying to solve for minimizing Predicted Active Risk versus the Russell 1000 Growth, while Best Practices in Factor-Based Analytics page 19

20 maintaining the same securities as the initial portfolio. The idea is to maintain the portfolio s securities (with the assumption that the manager is trying to capitalize on a high conviction list), while letting the optimizer decide on the Style and Industry factor exposures that best minimize Active Risk. In Figure 28, we can see the results of this optimization rebalance run quarterly over the same period of analysis. By comparing Figure 28 with Figure 2, we see the optimized portfolio returned 2.67%, again by keeping the securities held the same, compared with the original portfolio that returned -8.00%. In other words, the one-year return improved by more than 10% by focusing on minimizing the risk from Style and Industry factors while maintaining the original list of securities. Figure 28 Conclusions Analysts, managers, and risk officers that can understand risk and how to decompose it have a powerful tool to better understand their investments. Risk models and risk analysis can prevent a portfolio from taking on undesired and/or unintended bets and can prevent intended bets from imposing a greater-than-desired amount of risk on a portfolio. Axioma s solutions isolate, describe and mitigate the risks in a portfolio from the top, factor, and individual security levels. This level of insight is immensely valuable to financial professionals in their efforts to limit and control their investment risks. Best Practices in Factor-Based Analytics page 20

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