RESEARCH
Leon Serfaty, CFA, Axioma Product Specialist
More than just a second risk number
STATISTICAL MODELS:
Fundamental models are the most widely used type of risk model in the investment industry. They have their philosophical basis in some of the foundational literature of investment theory, including the Capital Asset Pricing Model (CAPM)3 and the Arbitrage Pricing Theory (APT)4. While the CAPM hypothesized the relationship between an asset’s linear sensitivity to the market (systematic risk) and its expected return, APT posits that there could be many systematic factors determining asset pricing, without going so far as to specify what they might be. The basic idea informing all models of this class is that assets with similar characteristics will share similar return behavior. These common characteristics, once identified, are called pricing “factors”, and we can use the technique of multi-variate regression to determine the returns of these factors from the asset returns themselves. Fundamental multi-factor models typically include industry membership as a factor, as well as country of domicile or country of risk in regional or global models. They will also include thematic factors, typically called “Styles” or “Risk Indices” that capture return
This paper provides: .A discussion on why fundamental models remain the dominant class of risk models for equity investing .An overview of statistical models and the methodology behind them A case study to illustrate how fundamental and statistical models can be complementary and enhance a practitioner’s understanding of portfolio risk and exposure Recommendations on best practices for using fundamental and statistical models together Fundamental Factor Models
Multi-factor risk models have been widely used in the investment industry for over 40 years. There are three types of multi-factor risk models currently in use: macroeconomic, fundamental, and statistical1. Macroeconomic models may be the easiest to estimate, but their generally poor out-of-sample performance renders them less useful for risk forecasting and portfolio construction problems. Fundamental models are by far the dominant class of models because of the combination of reliability and interpretability that they offer. This leaves us with statistical models, which are often avoided or ignored by practitioners because of their apparent opacity. This whitepaper2 will seek to provide guidance on how a statistical risk model can be a valuable tool to enhance the risk analysis of any equity strategy when used as a complement to a fundamental risk model with the same forecast horizon.
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© SimCorp 2024
Risk decomposition Performance attribution
May miss short term trends, transitory factorsLacks intuition and difficult to interpret:
Weaknesses
Factors are not fixed Highly responsive Captures short term & transitory phenomena
Intuitive & widely used consistent framework for: Risk decomposition Performance attribution Portfolio Construction
Strengths
Factor exposures (loadings), factor returns, residual (specific) returns
Factor returns, residual (specific) returns
Estimated Outputs
Asset Returns
Asset returns, factor exposures (loadings)
Inputs
Statistical Risk Models
Fundamental Risk Models
differentiation on the basis of valuation and profitability metrics as well as more market-driven metrics such as price momentum, beta sensitivity, and liquidity. All models will leave some portion of the cross-sectional variation in returns in an unexplained residual. This residual, whether in the context of risk or return, is specific, or idiosyncratic, to the assets themselves and not related to systematic factors. Fundamental models’ utility and popularity stem from the manner in which the factors chosen by modelers tend to align quite closely with the ways in which investors evaluate their investment opportunity set: they tend to differentiate between more and less attractive investments on the basis of industry trends, valuation, profitability, liquidity and other fundamentals. Fundamental models achieve very strong results in explaining risk and return from their structure of well-understood and well-researched factors. However, this very same structure can be a drawback at times: what if there is a new, or temporary factor driving returns and risk in the market being modeled? The fundamental model won’t have this factor — and because it lacks this new, or temporary factor, the unexplained volatility in the assets being driven by it will appear as though it is idiosyncratic to the assets themselves and hence diversifiable. In the short run in particular, this can end up being a costly misspecification. Statistical Factor Models In contrast to fundamental models, statistical factor risk models do not impose or assume a fixed factor structure but instead use asset returns directly to mathematically construct an optimal set of factors explaining the current risk/ return environment, regardless of whether they represent short- or long-term phenomena or are associated with intuitive, well- known factors. The factors of a statistical risk model can evolve to fit the current market conditions every single day. This makes statistical models very good at forecasting risk in any market environment, but the lack of structure, consistency and interpretability in the factors can make them difficult to use in comparison to fundamental models.
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Where: R is a vector of asset returns, B is a matrix of factor exposures or factor loadings, f is a vector of factor returns, and u is a vector of asset-specific, idiosyncratic returns. While R is known, fundamental and statistical risk models approach the solution of the rest of the terms in this equation differently. With fundamental models, the factors and their exposures, B, are given, and the equation is solved for the factor return, f using weighted multi-variate regression. This approach permits risk modelers to select factors that are intuitive, well researched, and predictive. The factor returns as estimated are now directly related to the assets’ fundamental exposures, which are the “independent” variables in the regression. The factors used in a fundamental factor risk model are fixed, although the factor exposures can be updated daily. For statistical risk models, both the matrix of factor exposures, B, and the vector of factor returns, f, are solved for simultaneously so as to maximize the predictive power of the above equation. Statistical factors, factor exposures and returns are re-estimated independently for each daily risk model update. As a result, the factors and corresponding factor exposures may change substantially from one day to the next as they adapt to market conditions. When compared with fundamental factor risk models, statistical factor risk models have two key drawbacks. First, the factors have no obvious economic or investment meaning. They are simply numerical exposures that best explain the observed asset returns. Second, the factors change from one day to the next. This makes statistical factor exposures difficult to incorporate into a portfolio construction strategy or to use for the purpose of performance attribution. These drawbacks can also be turned to a practitioner’s advantage. During time periods when the factors in a fundamental risk model include all the key factors driving risks in the market, fundamental risk models work well. However,
R = Bf + u
Statistical Model Methodology A statistical factor model is a risk model whose factors are constructed by mathematically processing asset return time series, so that the set of factors chosen has the maximum possible explanatory power over that set of assets. Often the only major parameter chosen by the modeler is the number of factors to be fitted. The machine learning technique used is Principal Components Analysis (PCA), Asymptotic Principal Components Analysis (APCA), or a variant of these. Because these machine learning techniques maximize the commonality among the asset returns, the models are free to find factors not found in fundamental factor models. Statistical factors frequently capture short-term market trends that can be very significant even if they do not persist. Identifying and reacting to relevant market trends is, of course, an essential part of any investment process even if the trends do not last long enough to be included in a fundamental factor risk model. Mathematically, both fundamental and statistical risk models begin with the same linear representation of asset returns:
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Figure 1: Factor-based Performance Attribution5
0.13%-0.90%0.20%
Utilities
0.25%-0.40%0.29%
Real Estate
0.87%-0.81%0.21%
Materials
0.62%-0.35%0.39%
Information Technology
-1.03%-1.61%0.38%
Industrials
1.90%1.14%0.55%
Health Care
0.91%1.93%0.54%
Financials
2.22%0.41%0.40%
Energy
0.89%-0.49%0.35%
Consumer Staples
-0.05%-0.49%0.37%
Consumer Discretionary
-0.71%1.52%0.43%
Communication Services
5.98%-0.07%1.40%
Sectors
-0.02%-0.07%0.02%
Market Intercept
-0.15%-0.01910.12%
Value
-2.18%0.13341.01%
Size
1.09%0.00190.30%
Short-Term Momentum
-0.30%-0.00750.15%
Short Interest
-1.11%0.01570.24%
Residual Volatility
0.86%0.05140.21%
Profit Quality
-0.09%0.06150.11%
Profit Growth
0.13%-0.00190.15%
Opinion Divergence
2.73%-0.07450.33%
Nonlinear Residual
0.18%-0.02710.09%
MidCap
5.67%0.45422.26%
Medium-Term Momentum
-2.11%0.05820.56%
Market Sensitivity
-0.11%-0.02200.10%
Liquidity
-0.32%0.00060.12%
Leverage
-0.17%-0.02590.11%
Investment
0.70%0.03110.19%
Exchange Rate Sensitivity
0.38%0.01860.12%
Earnings Yield
-0.32%0.00670.26%
Downside Risk
0.21%-0.09320.21%
Dividend Yield
-0.46%-0.04370.11%
Crowding
2.67%
4.63%
Style
2.96%
10.60%
Factor Contribution
2.30%
2.25%
Specific Return
3.86%
12.85%
Active
17.35%
42.63%
Benchmark
18.04%
55.47%
Portfolio
Realized Risk
Avg Active Exposure
Source of ReturnContribution
Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH)
A Case Study We have simulated a US large cap Momentum tilt strategy rebalanced monthly with a constraint on active risk (tracking error) of 3% vs. the S&P 500 over the period from December 31, 2020 to February 29, 2024. The most recent Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH) was used to estimate the active risk in the optimized rebalancing each month. This strategy is expected to generate most of its active return relative to the benchmark via exposures to the factors we explicitly tilt on: Medium-Term Momentum and Size, and naturally we also expect that a significant fraction of the total active risk will also come from those sources. A look at the factor-based performance attribution using the US5.1-SH Model confirms this. In Figure 1, we see that over the 38-month period from January 2021 through February 2024, our “Big Momentum” Strategy (“Bigmo”) outperformed the S&P 500 by 1285 basis points, or 309 basis points annualized. We also see that the realized active risk, or tracking error was 386 basis points, substantially higher than the 3% limit we had set on forecast tracking error, also using the US5.1-SH Model. Highlighted in orange are our intentional factor tilts on Medium-Term Momentum and Size, which not only show the largest magnitude active exposures vs. the benchmark but also the largest realized risk in their contributions to active return4.
this is not always the case. Any risk practitioner knows that there are certain times when groups of related assets, or even the entire market to some extent, appear to be driven by a new and unexpected factor that is not included or well represented by the fixed set of fundamental factors. In this situation, the explanatory power of the fundamental risk model decreases. A statistical factor risk model, however, will adapt to changing market conditions, and the PCA process should capture any commonality in returns exhibited by cohorts of assets, regardless of the source of that commonality. In other words, the statistical model is free to pick up on any factors driving asset returns at the current moment. Groups of assets in a portfolio or market index will have high or low loadings (exposures) on the statistical factors, same as they would in the fundamental models. It is at this point that the practitioner may deploy some intuition regarding what these factors are: the assets that have exposure to them will be the guide.
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Non-linear Residual gets us part of the way to explaining risk and return that traditional linear factors miss, but not all the way. Even with this new, innovative factor, the Forecast Active Risk Spread between the statistical and fundamental variants of the US5.1-SH model (SH-S and SH, respectively) is quite wide at times, particularly from the last quarter of 2023 into 2024 (Figure 3).
Fundamental Active Risk ForecastStatistical Active Risk Forecast
Oct-23 Nov-23 Dec-23 Jan-24 Feb-24
Jul-23 Aug-23 Sep-23
May-23 Jun-23
Oct-22 Nov-22 Dec-22 Jan-23 Feb-23 Mar-23 Apr-23
Jul-22 Aug-22 Sep-22
May-22 Jun-22
Oct-21 Nov-21 Dec-21 Jan-22 Feb-22 Mar-22 Apr-22
Jul-21 Aug-21 Sep-21
May-21 Jun-21
Jan-21 Feb-21 Mar-21 Apr-21
5.00%
4.50%
4.00%
3.50%
3.00%
2.50%
Figure 3: Bigmo Strategy Active Short Horizon Risk Forecasts
Jan-24 Feb- 24
Nov-23 Dec- 23
Sep-23 Oct- 23
Jul-23 Aug- 23
May-23 Jun- 23
Mar-23 Apr- 23
Jan-23 Feb- 23
Nov-22 Dec- 22
Sep-22 Oct- 22
Jul-22 Aug- 22
May-22 Jun- 22
Mar-22 Apr- 22
Jan-22 Feb- 22
Nov-21 Dec- 21
Sep-21 Oct- 21
Jul-21 Aug- 21
May-21 Jun- 21
Mar-21 Apr- 21
Jan-21 Feb- 21
20.00%
15.00%
10.00%
0.00%
-5.00%
-10.00%
The tilt on Size was not a value add, detracting some 218 basis points despite the record high Size factor return in calendar year 2023. The factor highlighted in blue is the “Non-linear Residual” factor in the US5.1 Model. This factor explains some of the risk in the residual of the other factors by imposing a dynamic sub-model of higher order factor exposure effects (exposure squared, exposure cubed) along with cross-products of each factor pair (Size x Residual Volatility, Liquidity x Profit Growth, etc.) on the residual from the initial cross-sectional regression. Even though this exposure was not intentional, it appears to capture a significant fraction of the active return, even if the volatility of that contribution was quite low. Figure 2: Cumulative Active Return Contributions by Month
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Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH) and Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S) It’s clear from the chart that 20-day active return volatility is not a stable quantity, with several sharp peaks and valleys in the realized levels over this 38-month period. It is also clear that the statistical model appears to forecast the higher levels more accurately than the fundamental model, even if it doesn’t match the extremes of the peaks in the realized levels. Starting from the second quarter of 2023 and into 2024, it is instructive to see how both the proportion and level of factor risk changes in each model.
20-day forward Volatility of Active ReturnFundamental Active Risk ForecastStatistical Active Risk Forecast
Mar-23 Apr-23 May-23 Jun-23
Apr-22 May-22 Jun-22 Jul-22 Aug-22 Sep-22 Oct-22 Nov-22 Dec-22 Jan-23 Feb-23
Feb-22 Mar-22
Oct-21 Nov-21 Dec-21 Jan-22
Apr-21 May-21 Jun-21 Jul-21 Aug-21 Sep-21
Feb-21 Mar-21
Jan-21
6.00%
5.50%
2.00%
1.50%
1.00%
The active risk forecast from the fundamental model is clearly centered around 3%, as expected, since we rebalance the strategy monthly using that risk model and a 3% Active Risk constraint. Between rebalancings, the forecast can fluctuate substantially. We see no such anchoring tendency from the statistical model, although we do see periodic jumps in the forecast on the rebalancing dates. What we also see are prolonged periods of wide spreads between the two models. There are two periods where the models appear to be aligned: at the very beginning of the period in early 2021, and then from the third quarter of 2022 through October of 2023. We can look at how the models did relative to the 20-day forward volatility of active return – effectively showing us how good (or not good) the forecasts were on each date as in Figure 4 below. Figure 4: Bigmo Strategy: Short Horizon Active Risk Forecasts and 20-Day Forward Realized Active Return Volatility
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Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH) and Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S) The dates in these charts are at the beginning of each monthly period (risk forecast as of the previous trading day’s close). When the model active risk forecasts were largely in alignment in the April-October period, the fundamental model shows a higher proportion of factor risk, and higher levels of factor risk as well. The shift occurs in October as the levels of factor risk rise in the statistical model forecast and then in subsequent months both the levels and the proportion of factor risk increases and overtakes those of the fundamental model.
Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH) and Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S)
FundamentalStatistical
Mar-24
Jan-24 Feb-24
Nov-23 Dec-23
Oct-23
Sep-23
Jul-23 Aug-23
Jun-23
Apr-23 May-23
350
300
250
200
150
100
50
0
Dec-23
Oct-23 Nov-23
3.00
2.50
2.00
1.50
1.00
0.50
0.00
Figure 6: Factor Risk in Basis Points
Figure 5: Ratio of Factor to Specific Active Risk
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140.9
300.0
440.9
100.00%100.00%
Total
-87.1
73.4
-13.7
46.25%77.31%
All Others
228.0
226.6
454.7
53.75%22.69%31.06%
Subtotal Top 15
5.2
1.0
6.2
TDGTRANSDIGM GROUP INC1.20%0.14%1.06%
5.4
5.6
11.0
RCLROYAL CARIBBEAN GROUP1.07%0.08%0.99%
-0.2
5.3
2.38%1.04%1.34%
VVISA INC
5.7
3.0
8.7
2.50%1.75%0.74%
GOOGALPHABET INC
6.6
12.2
18.8
3.42%1.16%2.26%
LLYELI LILLY & CO
7.8
-0.5
7.3
2.37%0.50%1.87%
LINLINDE PLC
8.5
20.1
28.6
UBERUBER TECHNOLOGIES INC2.56%0.32%2.24%
8.9
17.6
2.10%0.06%2.04%
PHMPULTE GROUP INC
10.1
1.1
11.2
1.78%0.15%1.63%
FDXFEDEX CORP
10.9
7.6
18.4
2.96%0.35%2.61%
GEGENERAL ELECTRIC CO
11.7
10.7
22.4
3.04%1.72%1.32%
TSLATESLA INC
19.2
18.3
37.5
9.85%6.98%2.87%
MSFTMICROSOFT CORP
20.9
9.6
30.6
5.14%3.45%1.69%
AMZNAMAZON COM INC
36.2
56.4
92.6
6.40%3.06%3.34%
NVDANVIDIA CORPORATION
65.6
72.8
138.4
7.00%1.96%5.04%
METAMETA PLATFORMS INC
Spread Contribution
FUND Contribution
STAT Contribution
Weight Active Weight
Bmk
SymbolDescriptionNet Weight
The top 15 contributors to the active risk spread are quite closely related to the largest 15 active weights. In aggregate, the top 15 account for more than 100% of the total active risk forecast from the statistical model and 160% of the active risk spread. Whatever the previously hidden systematic risk in this active strategy is, it is coming from these positions. To some extent, this is a comforting realization. Because these positions are also the largest active weights, the hidden risk is aligned with the large-cap Momentum tilt we have intentionally made. If it were coming from somewhere else in the portfolio, say, from a group of smaller positions that are in the portfolio to meet our 3% tracking error target (in other words for diversification), it would be more troubling. By decomposing the asset level contributions further into factor contributions, we can see how this same group contributes to the active risk we intend to be taking in the Medium-Term Momentum and Size factors from the fundamental model.
What this tells us is that there was potentially “something else” driving active returns and risk in the fourth quarter of 2023 and into 2024. That “something else” was systematic in nature and appears to be captured by the statistical model in higher forecast factor risk. Decomposing the Factor Spread The risk forecast on the rebalancing date of December 29, 2023 shows the largest active risk and active factor spread in Figure 5 and Figure 6. The spread subsequently declined a bit, but remained steady, suggesting persistence in these hidden factors. To aid in understanding what might be driving the spread we start at the asset level, we compare each active position’s contribution to active risk from each model and the spread between them. Figure 7: Top 15 Active Risk Spread Contributors: Basis Points Annualized
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94.7
16.50.0
18.6
148.122.0
22.0
1.2-12.2
-2.2
53.710.7
72.7
15.312.2
20.8
94.411.3
Subtotal Top 1531.06%226.6
0.8
0.4
0.0
-1.1
1.3-0.4
TDGTRANSDIGM GROUP INC1.06%
1.5
0.3
3.6-0.5
RCLROYAL CARIBBEAN GROUP0.99%
0.6
0.5
-1.0
-0.8
1.34%
1.7
0.7
0.74%
8.2
0.9
-4.8
2.4
4.4
2.26%
-2.6
0.2
1.87%
3.2
9.9-0.1
UBERUBER TECHNOLOGIES INC2.24%
1.6
-0.3
6.7-1.4
2.04%
2.5
-1.5
-1.6
1.6-0.4
1.63%
3.3
-1.9
4.10.0
GEGENERAL ELECTRIC CO2.61%
4.0
1.32%
4.1
4.9
3.1
4.3
2.87%
2.6
1.4
1.69%
14.8
1.3
6.5
8.4
22.7
3.34%
21.4
2.0
11.8
4.7
3.4
29.6
5.04%
Fundamental Specific
Industry RiskMarket Risk
RiskSize RiskOther Style Risk
Momentum
Active Weight
SymbolDescription
When the factor risk contributions are shown this way, we can see that 35% of the total Momentum bet is coming from META and NVDA alone, while about 65% comes from the top 15. Nevertheless, the remainder of the Momentum risk is well spread out over the other 480+ active positions vs. the benchmark. With respect to the Size tilt, a little more than half is in these top 15 names, and the rest is diversified over the rest of the active weights. It’s also clear that the Momentum tilt is almost seven times the size of the Size tilt when it comes to the allocation of the 3% active risk budget imposed. All of this is reassuring until we recall that first of all, our realized risk over the past few months has been substantially higher than the 3% tracking error we have budgeted, and secondly, that the short horizon statistical model is forecasting active risk 141 basis points higher. Statistical Model Risk Decomposition Attributing active risk along the dimensions of a factor model is an essential step in the portfolio management feedback loop. When this is done with a fundamental model, the practitioner is generally looking to confirm that: .The risk being taken is consistent with the sources of return the manager has selected, and .There are no unintentional biases in the strategy that are adding unnecessary exposure to unwanted sources of volatility.
Where:
Asset Active Factor Risk Contribution is computed as:
Figure 8: Risk Spread Top 15 Contributions to Fundamental Risk
Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH), December 29, 2023
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The model is forecasting 441 basis points of active risk vs. the fundamental model which is forecasting 300. 314 of the 441 basis points are systematic. 81% of the 314, or 254 basis points are coming from six factors highlighted in blue. Notice the factor loadings appear vanishingly small compared to what we are used to when looking at a fundamental model. These loadings, like fundamental factor loadings, are linear sensitivities to the factor returns. Exposures this small imply enormous factor returns contribute this much in annualized volatility, but this is simply an artifact of the modeling process and doesn’t convey much meaning in the way we are used to thinking about factor returns or volatility. This is why simply looking at loadings, or exposures, is not sufficient to understand where the factor risk in the strategy is coming from. When we use the Asset Active Factor Risk Contribution method described above, we gain more insight into the positions that drive the factor risk.
This is not as straightforward with a statistical model since the factors defy interpretation and can change from period to period. However, the assets with the largest loadings (exposures) on the factors that have the biggest impact on the risk forecast can shed some light on what they might be capturing at any given time.
Figure 9: Bigmo Strategy Active Risk Decomposition (December 29, 2023)
Source: Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S)
28.91%
127.5
Specific Risk
0.35%
0.002
Factor 20
5.8
0.003
Factor 19
-0.07%
Factor 18
26.5
-0.010
Factor 17
0.49%
2.2
-0.003
Factor 16
0.16%
0.001
Factor 15
-0.09%
-0.4
Factor 14
-0.01%
-0.002
Factor 13
0.05%
-0.004
Factor 12
0.13%
-0.001
Factor 11
7.66%
33.8
Factor 10
5.80%
25.6
-0.009
Factor 9
1.27%
Factor 8
-0.04%
Factor 7
11.04%
48.7
0.012
Factor 6
5.83%
25.7
0.010
Factor 5
4.09%
18.0
-0.011
Factor 4
14.30%
63.1
-0.016
Factor 3
12.76%
56.3
-0.017
Factor 2
0.04%
Factor 1
71.09%
313.5
Statistical Factor Risk
Active Risk Contribution (bps)% of Total
Net Exposure
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440.956.363.125.725.633.826.582.6127.5
4.2-0.351.423.9
-13.7-41.8-7.1-52.88.6
31.06%454.798.170.178.516.929.526.831.2103.5
-0.7-0.20.6
.02.32.1-0.3
-0.80.5 RCLROYAL CARIBBEAN GROUP0.99%11.02.73.31.9-0.6-0.40.62.11.3 TDGTRANSDIGM GROUP INC1.06%
-0.6
5.31.7
0.80.70.8-1.2
1.23.01.0
8.72.4
5.13.3-0.87.3
2.26%18.83.1-3.74.00.6
7.32.12.53.30.5-0.3-0.2-1.91.3
1.41.02.07.8
UBERUBER TECHNOLOGIES INC2.24%28.68.74.94.7-2.0
-0.84.32.0
-5.82.2
2.04%17.67.4
0.81.20.64.5
1.9-0.1-0.6
1.63%11.22.9
-0.42.04.5
GEGENERAL ELECTRIC CO2.61%18.46.2
4.76.70.0-0.50.8-1.05.3
1.32%22.46.6
1.42.10.85.0
2.87%37.58.13.912.33.9
2.90.74.53.3
1.69%30.67.22.56.82.7
-1.75.53.921.2
NVDANVIDIA CORPORATION3.34%92.616.925.316.25.3
10.412.814.940.0
7.1
METAMETA PLATFORMS INC5.04%138.421.313.518.3
Stat Specific
Other Stat Factor
Statistical Factor 17
Statistical Factor 10
Statistical Factor 9
Statistical Factor 6
Statistical Factor 3
Statistical Factor 2
The top 15 contributors to the model risk spread naturally have some of the largest contributions to each of the significant factors driving the additional active risk. In fact, for Factors 2, 3, 6 and 17, the top 15 are more than 100% of the total. This tells us that the vast majority of this newly discovered systematic risk is concentrated in these same positions. This yields another clue in the process of determining what these factors are, as we can start to enumerate the fundamental, technical, and even “narrative” similarities of these names in the present moment. We won’t delve too far into any of those here, as those are not our areas of expertise. As a next step, one way of overlaying a quantitative “map” onto these statistical factors is to estimate the cross-sectional (point-in-time) correlations in asset level loadings between the factors in the fundamental model and statistical model as seen in Figure 11 on the next page. This map of “loadings on loadings” gives us more intuition about what these factors are, and also what they are not. For example, Factor 2, which accounts for almost 13% of the active risk forecast, is positively correlated with many of the Value-themed factors such as Dividend Yield, Earnings Yield, and Value. It is also negatively correlated with Momentum, and Market Sensitivity, which is a measure of risk associated with beta exposure to the market. The Bigmo strategy has a negative active loading on Factor 2. We can therefore assume that Factor 2 is low risk value that also appears to be related to companies that benefit from a weak dollar (Exchange Rate Sensitivity). Bigmo is effectively “short” this factor relative to the benchmark and thus it is a major source of active risk. Factor 6 is somewhat opposite to Factor 2 in that it has a positive correlation with Momentum and negative correlations with the Value themed factors. Bigmo has a positive active loading on Factor 6 which is responsible for 11% of the active risk.
Figure 10: Asset Active Factor Risk Contribution
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In order to complete the picture, we will look at some of the key industry factor overweights in relation to these statistical factors: Figure 12: US5.1-SH Industry loadings Correlation to US51-SH-S Statistical Factors
Figure 11: Style Loading Correlations
Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH) and Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S), December 29, 2023
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At first, it may seem counterintuitive that Factors 2 & 3 appear positively correlated with Banks and Electric Utilities – Bigmo is overweight these industries, and the active loading on Factors 2 & 3 are negative. But when we remember which positions in the portfolio are the source of this exposure and risk, it begins to make sense. More than 100% of the total Active Risk is coming from those top 15 names – and none of them are Banks or Utilities. But the second biggest source of risk is a semiconductor company and the fourth is a software company. It is also notable that the industries of three of the more significant active positions, namely META, GOOG, and AMZN (Interactive Media & Broadline Retail) do not have significant exposure overlap with any of these statistical factors. Reverting to the risk decomposition from the fundamental model shows about 9 basis points of active risk attributable to Interactive Media, but less than a basis point (not shown in table) from Broadline Retail. This could be telling us that industry membership is not a strong factor for these particular names and why their respective active risk contributions are so much higher in the statistical model. Figure 13: US 5.1-SH Industry Factor Risk Contributions
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“Popularity” vs. Crowding The new Hedge Fund Crowding factor in the US5.1 fundamental model uses SEC 13-F filings to determine the number of private funds holding a name, the percentage of the float held by these funds, and the percentage held relative to average daily trading volume. Some of the largest stocks, as popular as they may be, will not appear to be “crowded” trades by these metrics, due to their enormous floatation and trading volumes. The Bigmo strategy’s large-cap tilt results in miniscule active exposure to Crowding with commensurate minimal risk contribution. However, some of the smaller positions in the top 15 contributors do indeed have positive Crowding exposure, and their similarity in exposure to the key statistical factors suggest a relationship between them and the mega-caps that the fundamental model cannot capture: Figure 14: Top 15 Side-by-Side Factor Exposures
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The largest positions are all “not crowded” (with negative or neutral Crowding exposure) but many of the smaller ones have significant positive Crowding exposure. Note that their statistical factor exposures are much more similar, and typically in the same direction, particularly on Factors 2 & 3. While all these names either have positive Size or Momentum exposure (or both) – which tells us why the strategy has taken overweight positions – on the Crowding factor they can seem quite different. On the other hand, their similarities to one another in the statistical factors suggests a systematic relationship that could certainly be interpreted as “popularity” given how these stocks were trading at the time. For comparison, here are five smaller capitalization names that were part of the “AI trade” in late 2023 and early 2024:
Figure 15: Fundamental and Statistical Factor Exposure
Average-0.010.87-0.20-0.12-0.120.06-0.050.01-0.02
0.17-0.13-0.12-0.130.05-0.040.03-0.06
-0.10
Semiconductors & Semiconductor Equipment
KLACKLA CORP
-0.250.23-0.15-0.12-0.060.030.000.00-0.01
ARM HOLDINGS PLC ADR
ARM
0.100.56-0.14-0.12-0.080.10-0.030.02-0.01
Software
CADENCE DESIGN SYSTEM INC
CDNS
0.010.70-0.12-0.11-0.090.09-0.040.01-0.01
SNPSSYNOPSYS INCSoftware
0.182.70-0.48-0.12-0.230.03-0.150.01-0.03
Technology Hardware, Storage & Peripherals
SUPER MICRO COMPUTER INC
SMCI
MomentumSize
Medium-Term
TickerDescriptionIndustryCrowding
Statistical Factor Exposure
Fundamental Factor Exposure
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Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH) and Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S), February 29, 2024 An analysis of the asset level contributions to the spread reveals some persistence in the systematic risk we saw at the end of December.
On the date of the final strategy rebalance (February 29, 2024), we obtain a much smaller overall spread in the active risk forecast between the two models, just 44 basis points. The statistical model still shows much higher factor risk, and lower specific risk than the fundamental model even if the overall spread has narrowed. Figure 17: Bigmo Strategy Factor Active Risk Spread, in Basis Points Annualized
They show similar variation in Crowding exposure, strong Momentum loadings, and high loadings on the same statistical factors, particularly 2, 3, & 6. In fact, an equal-weighted portfolio of “AI” related companies shows nearly double the total risk in the statistical model relative to the fundamental model on December 29, 2023 (Figure 16). This portfolio, in Total Risk rather than Active Risk space, demonstrates that it is not a matter of the position sizing, rather the exposure to the key statistical factors that explains the large spread in risk forecasts between the two models – the statistical model is capturing systematic risk in these stocks that the fundamental model simply cannot. Figure 16: Equal Weight “AI Trade” Portfolio Risk Spread Factor Persistence
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44.5
344.5
Total100.00%100.00%
-86.2
120.8
34.6
All others54.60%76.26%
130.7
179.2
Subtotal Top 1545.40%23.74%21.67%309.9
3.7
0.84%
0.15%
TDGTRANSDIGM GROUP INC0.99%
3.9
11.6
15.5
1.06%
0.38%
UBERUBER TECHNOLOGIES INC1.44%
5.1
5.5
1.08%
0.18%
SHWSHERWIN WILLIAMS CO1.26%
6.7
1.92%
0.77%
COSTCOSTCO WHSL CORP NEW2.69%
3.5
0.58%
0.60%
1.18%
ADBEADOBE INC
8.1
1.03%
0.21%
1.24%
BXBLACKSTONE INC
9.4
0.32%
ISRGINTUITIVE SURGICAL INC1.40%
7.7
6.0
13.6
1.12%
2.75%
9.7
16.4
2.31%26.0
7.19%
9.50%
9.9
27.3
37.2
1.70%
2.54%
4.25%
11.1
21.9
2.04%32.9
1.33%
3.37%
AVGOBROADCOM LTD
9.5
21.1
1.20%
0.06%
BLDRBUILDERS FIRSTSOURCE INC1.26%
13.5
21.2
1.78%
1.83%
14.6
42.6
57.1
1.95%
4.57%
6.52%
17.8
36.1
1.99%
3.76%
5.74%
SymbolDescriptionNet Weight Bmk Weight
Source: Axioma US Fundamental Equity Factor Risk Model – Short Horizon (US5.1-SH) and Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S), February 29, 2024
The top 15 contributors to the active risk spread account for 131 basis points, which is 291% of the total spread. They are all overweights, but the correlation between active weight and risk spread contribution is not as strong as it was in December. According to the statistical model, nearly 90% of the active risk comes from these 15 positions, while in the fundamental model they account for about 60%. Even though the rest of the active portfolio has a risk spread contribution of -86 basis points, there are only five overweight positions with a negative risk spread contribution and in aggregate their contribution sums to just -2 bps. Of the remaining underweight positions, the largest negative risk spread contribution of -4.3 bps comes from Salesforce Inc., which was 0.7% of the benchmark on February 29, while Bigmo held none. This matters because it appears that the intentional, large cap Momentum tilt in Bigmo carries persistent systematic risk not captured by the fundamental model. It is largely the same names contributing to the spread, and the proportion of the risk spread attributable to them has grown, even as the overall spread has shrunk.
Figure 18: Risk Spread Decomposition
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28.96%
99.8
0.28%
-0.23%
1.82%
6.3
0.005
7.47%
-0.008
0.66%
2.3
-0.16%
0.34%
1.2
4.6
0.004
0.31%
0.85%
2.9
-0.005
5.05%
17.4
0.007
4.48%
15.4
-0.007
4.58%
15.8
0.39%
4.89%
16.8
0.008
8.52%
29.4
3.29%
11.3
22.07%
76.0
5.52%
19.0
-0.43%
71.04%
244.7
344.519.076.029.417.425.777.2
49.0
34.6-27.3-0.2-7.26.913.10.3
54.60%76.26%
All others
50.8
45.40%23.74%21.67%309.946.476.236.610.512.676.9
0.99%0.15%0.84%5.60.82.41.21.20.0-0.30.5
TRANSDIGM GROUP INC
TDG
1.44%0.38%1.06%15.53.33.70.90.91.31.93.4
UBER TECHNOLOGIES INC
UBER
1.26%0.18%1.08%5.51.50.31.60.7-0.21.10.6
SHERWIN WILLIAMS CO
SHW
2.69%0.77%1.92%11.82.11.52.20.10.92.62.4
COSTCO WHSL CORP NEW
COST
1.18%0.60%0.58%8.71.02.71.40.10.72.30.5
ADOBE INC
ADBE
1.24%0.21%1.03%8.11.81.00.1-0.30.63.81.1
BLACKSTONE INC
BX
1.40%0.32%1.08%9.43.01.01.20.9-0.93.01.3
INTUITIVE SURGICAL INC
ISRG
2.75%1.63%1.12%13.61.72.81.50.00.35.81.6
ALPHABET INC
GOOG
9.50%7.19%2.31%26.03.75.73.40.20.78.73.6
MICROSOFT CORP
MSFT
4.25%2.54%1.70%37.23.89.43.91.51.07.79.9
META PLATFORMS INC
META
3.37%1.33%2.04%32.95.311.54.2-2.61.57.65.4
BROADCOM LTD
AVGO
1.26%0.06%1.20%21.14.24.82.53.10.34.61.6
BUILDERS FIRSTSOURCE INC
BLDR
1.83%0.05%1.78%21.24.41.72.74.41.44.52.2
PULTE GROUP INC
PHM
6.52%4.57%1.95%57.15.320.77.0-1.42.912.010.6
NVIDIA CORPORATION
NVDA
5.74%3.76%1.99%36.14.47.22.81.52.111.76.3
Statistical Factor 5
Bmk Weight
Net Weight
Source: Axioma US Statistical Equity Factor Risk Model – Short Horizon (US5.1-SH-S), February 29, 2024
Recall that the statistical factors are not the same from period to period so whatever Factor 2 is on February 29 is not necessarily the same as it was on December 29. Nevertheless, Factors 2, 3, 5, 10 and 17 each still explain more than 5 percent of the active risk. Only Factor 9, which was above the 5% significance level on December 29 has dropped below that threshold. Figure 20: Risk Spread Top 15 Contributions to Statistical Risk
Figure 19: Stat Model Risk Decomposition (February 29, 2024)
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BEST PRACTICES
1. Monitor risk levels in all models The analysis shown in this paper demonstrates how spreads between fundamental and statistical models can arise from time to time and often reveal short-term systematic risk that the fundamental models are unable to see. Whether strategies are active or passive, or whether it is total risk or active risk that is more relevant, it is useful to have a “second set of eyes” on the risk levels. Whether or not the strategy requires monitoring risk at different forecast horizons, it may be useful to know when those forecasts diverge as well, even if the reasons are more obvious given recent market moves. The Axioma model packages give users access to four models in each geographic region modeled. The package consists of two types of models, and within each type two distinct forecast horizons are offered. The two types of models are fundamental multi-factor and statistical multi-factor, each variation offered in a short horizon version and a long horizon version.
The active portfolio loading on Factor 3 remains negative, so negative correlations to Market Sensitivity, Momentum, and Size make sense. Interestingly, Factor 3 appears to be positively correlated with the Non-Linear Residual factor, which Bigmo currently has a significant negative loading on. On the Industry factors, there is not a great deal of information from Factor 3 but it is negatively correlated with Semiconductors and Software where Bigmo has significant active allocations.
Particularly in Factors 2, 3 & 5, the top 15 account for more than 100% of the active risk contributions from those factors. Honing in on Factor 3, which accounts for 22% of the active risk, we can see how the asset loadings on that factor correlate to the fundamental factors: Figure 21: Factor Loading Correlations
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2. Decompose Risk Model Spread As we have shown here, once it is determined that there is a significant spread between fundamental and statistical models, use the differences in risk contribution at the asset level to determine what the largest contributors to the spread are. This can be done by simply using the % of (Active) Risk analytic from each model or by multiplying each position’s (active) weight and marginal contribution to (active) risk (MCTR, MCAR). This may immediately lead the observer to intuitive similarities between the assets that contribute most to the spread. Further analyzing the Asset Level Factor (active) Contribution will allow for the determination of which statistical factors are driving the spread, and then these can be mapped via cross-sectional correlation to the fundamental factor loadings on the same date. 3. Incorporate a second risk constraint While this paper did not explore portfolio optimization, in times when the statistical model shows significant spreads above the fundamental model used for portfolio construction, it may be worthwhile to constrain on (active) risk from the statistical model as well as the fundamental model in a rebalancing. Constraining statistical factor loadings may not be practical but constraining either total (active) or just factor (active) risk in the same-horizon statistical model may help mitigate undesirable and unplanned volatility in times of short-term disruption. While research and back-casting might be necessary, and if the strategy can absorb more trading, it also may make sense to use the statistical model forecast as the risk term in the objective or as the single risk constraint, while simultaneously constraining the fundamental factor exposures. The Axioma Portfolio Optimizer allows for several risk models to be loaded in a single optimization workspace. Get more information about Axioma factor risk models and Axioma Portfolio Optimizer for better risk forecasting and decomposition. Axioma tools are just some of the solutions available on SimCorp One, the platform that powers decisions at the speed of now.
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FOOTNOTES AND REFERENCES
Connor, G. (1995). The Three Types of Factor Models: A Comparison of Their Explanatory Power. Financial Analysts Journal (May-June), 42-46. This paper was originally published in July 2016 by Christopher Martin MFE, Anthony Renshaw, PhD and Chris Canova, CFA. Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. The Jour nal of Finance (September), 425-442. Ross, S. (1976). The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory vol 13, 341-60. The Realized Risk column in this table shows the independent annu alized standard deviation of each factor’s active return contribution. These cancel each other out to a great degree so that the total of the Realized Active Risk is 386 basis points, as shown.
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