Characteristics

Characteristics are properties that are capable of explaining the cross-section of returns. Alongside prices, they constitute the key input of our risk model. A key tenet of fundamental models is indeed that instruments sharing certain characteristics are exposed to the common market forces.

In the context of equity pricing models, a large body of literature is dedicated to what constitutes a valid characteristic (see e.g., ^[1] and references therein). For stocks, characteristics fall broadly into two categories: fundamental and price-based characteristics.

Fundamental characteristics correspond to financial ratios such as the price-to-book ratio and depend on the availability of accounting data and the frequency of earnings publications. Typically, fundamental ratios are published quarterly in the US and yearly anywhere else. Their comparability often requires re-adjustments depending on parameters such as the industrial sector of the company or the accounting standards used to format the company financial statement. Price-based characteristics, on the other hand, are simply statistics estimated from the stock's price. The momentum or the market beta are two well-known characteristics estimated from the stock's price.

The estimation of fundamental characteristics for portfolios of stocks is challenging for a number of reasons. First, characteristics should ideally be estimated from the fund's underlying components. However, many funds do not report their holdings. If they do, these reports are often outdated or incomplete, and do not take into account trades that may have significantly modified the fund's composition between reports. This makes the estimation of financial ratios from the fund's underlying instruments difficult and prone to potentially large errors, both in terms of content and timeliness. Second, the estimation of fundamental characteristics requires large and expensive databases, which can be operationally problematic. Next, financial statements of companies are updated at different frequencies depending on the firm and location of exchange. Firms traded in the US are required to report on a quarterly basis, while firms in the rest of the world tend to report on an annual basis. The market, however, reacts to new information at a much faster rate, and hence prices themselves may contain more up-to-date information than financial reports.

In order to circumvent the issues related to the estimation of fundamental characteristics, we estimate the characteristics of a portfolio as the exposure (commonly referred to as the beta) to a set of risk factors. These exposures are easily derived from time series data via a regression analysis. Measuring characteristics from risk factors differs from conventional practice. For instance, the "Profitability" of an instrument in equity models is usually measured through a combination of financial ratios, not from a regression to the "Profitability" risk factor. The approach chosen for the SP model can thus be said to "proxy" the characteristics of instruments through their exposures to a given risk factor.

Using price-based characteristics has many advantages. Unless large portions of historical prices are missing, characteristics obtained through regressions are generally available. Moreover, the large differences in amplitude between fundamental data points (e.g., company size in billions and price-to-book in fractions) that normally require normalization procedures to be used in the model's calibration procedure are not present when dealing with exposures. The timeliness of characteristics is also unproblematic. Contrary to financial ratios, characteristics reflect the latest information priced by the market. Finally, the estimation of exposures from regression yields a confidence interval around the value of the characteristics. As we will provide detail on later, confidence intervals are straightforward to obtain and provide information on the robustness of the estimate going forward, information that is often absent with financial ratios (although uncertainty around their values does also exist).

Risk Factors

The risk factors that we use to evaluate fund characteristics include the market itself, but also a number of fundamental and sector-based risk factors. The methodology used to evaluate characteristics will be detailed in the next section.

Generally speaking, risk factors correspond to the returns of a portfolio of stocks that are representative of the behavior of a certain group of stocks. A well-known set of risk factors is the Fama-French five factor model. There are many types of risk factors that are used to analyze the returns of portfolios. We consider so-called fundamental and sector-based factors.

Because we are dealing with portfolios of equities, the market factor is the de-facto prominent source of risk. As we will explain, all fundamental factors are designed to be orthogonal to the market factor. This means that the risk associated with fundamental factors does not contain any market risk. All of the market risk is indeed captured by the instrument beta to the market factor.

Fundamental risk factors

Fundamental risk factors (which are often referred to as rewarded risk factors or academic risk factors) are backed by an extensive body of academic research on asset pricing that recognizes them as persistent drivers of long-term performance. These risk factors represent sources of common risk that cannot be fully diversified away and that hurt a portfolio mostly in bad times, leading to a compensating risk premium. They have been subject to a high degree of academic scrutiny and challenge, providing investors with confidence in their use and ability to explain the long term risk and performance of equity portfolios. Their construction is based on straightforward stock selection criteria, making them actionable and intuitive, and their economic rationale (i.e., why they are deemed to represent a common source of systematic risk) is extensively documented ^[2]. The seven generally accepted fundamental risk factors are Market, Value, Size, Momentum, Low Risk, Investment and Profitability.

Figure 1 provides a summary of each fundamental risk factor (excluding the market risk factor):

While the market factor is cap-weighted, stocks in the long and short positions of each regressor are equally-weighted. Each regressor (with the exception of the market factor) is constructed by longing the 20% of stocks that are overperformers and shorting the 20% of stocks that are underperformers (overperformers and underperformers are shown in Figure 1). At the end of each quarter, the market exposure is computed for both the long and short positions using daily returns. The market beta is then applied retroactively to the relevant quarter to adjust the performance of both the long and short positions of each risk factor such that

Sector-based risk factors

Sector-based risk factors are often referred to as unrewarded risk factors since they are associated with sources of common risk that are not known to attract a risk premium. In other words, while sector-based risk factors contribute to the risk and short-term performance of a portfolio, they are not considered by the finance literature to contribute to its expected excess returns and long-term performance. Sector-based risk factors are included in the model due to their capacity to explain the risks to which an instrument is exposed.

We use sector-based risk factors derived from the S&P CIQ Industry Sector classification mapped onto the following aggregates: Basic Materials, Consumer Cyclicals, Consumer Non-Cyclicals, Energy, Financials, Health Care, Industrials, Technology, Telecoms and Utilities. These risk factors correspond to the track of a portfolio that is long the sector index and short the market index in such a way that the resulting time-series has no exposure to the market.

Estimation of characteristics

As we are working with equity portfolios, the market factor is de-facto the prominent source of risk. All other risk factors are therefore designed to be orthogonal to the market factor. In this way, the market risk of each instrument is captured exclusively by its market characteristic. The characteristics of instruments to non-market risk factors are thus considered active risk.

The characteristic

of the

instrument associated with the

factor is obtained at each time

by solving the following linear model:

where

denotes an instrument's active returns over a period of fixed size ending at time

and

represents the returns of the

risk factor over the same period. The active returns correspond to the residual returns of a univariate linear regression between an instrument's absolute returns

and the market factor, where the market factor is the long-only CW benchmark of the relevant investment universe. That is,

where

is the market factor.

The regression is done in two blocks, first estimating the exposures to the six fundamental risk factors through a multivariate regression, and then a separate multivariate linear regression against the ten sector-based risk factors (as will be discussed later, the cross-sectional model appropriately disentangles the informational overlaps between fundamental and sector-based betas and reduces the dimensionality of the model).

We estimate the parameters through a weighted least-squares regression (WLS) for which an analytical solution exists:

The weight of each day in the regression is stored in a diagonal matrix, denoted by

. To assign higher importance to more recent periods in the evaluation of characteristics, we use exponentially decaying weights. Specifically, we set

, where

. This weighting approach has two key benefits. First, it reduces the impact of the "window" effect, which can occur when a particularly volatile period is included in the evaluation window. Second, applying decaying weights gradually reduces the impact of past events over time, ensuring that the evaluation of characteristics remains relevant. On our platform, we use a half-life ττ of approximately 18 months.

Although risk factors are designed to be orthogonal, there may still be some residual correlation with the market factor. Estimating the market exposure first and using the residual returns that are orthogonal to the market factor to evaluate the other characteristics ensures that market risk is not transferred to other factors. This is particularly important since the market factor accounts for a relatively significant portion of the total risk.

Characteristics are calculated over the five years prior to time tt using daily returns. If the instrument has less than 5 years of returns, the characteristics are evaluated on the available data. Note that no fund with less than 3 years of data is analyzed on the platform.

Confidence intervals on characteristics

An important advantage of using exposures to risk factors as characteristics is that it is possible to compute a confidence interval that stems from the regression coefficient standard error, where

​​​As is shown later, the model provides an exact decomposition of risk and performance into contributions from each characteristic. For example, it is possible to show the level of risk that can be attributed to the exposure to the value factor. Having confidence intervals on the characteristics enables us to derive confidence intervals on the risk and performance decompositions, a feature that is not often available in analysis tools. These confidence intervals are also used in robustness tests and out-of-sample simulations.

Multi-Region portfolio characteristics

In the case of multi-region portfolios, the characteristics of instruments from different regions are determined using their respective local factors. Any exposure to risk factors from other regions is assumed to be zero. This approach results in a matrix of characteristics that resembles the following table:

In this fashion, the characteristics of all instruments are evaluated using their local risk factors. Regional models can thus be built using prices and matrices of characteristics such as the one above where both the instrument prices and local risk factors are translated to the investor's currency before evaluating the characteristics.

Research considerations on the estimation of characteristics

Static versus dynamic characteristics

In the model specification, instrument characteristics are dynamic, in the sense that they can have different values at each time

. In the context of equities, price-based characteristics change every period, whereas characteristics based on fundamental ratios remain the same between releases of company information.

When using regression analysis to obtain characteristics, the length of the historical track record may limit the availability of characteristics. In this case, it is possible to set the value of characteristics to a fixed value at each time

. This value may correspond to the regression coefficient over the whole period of interest, or the average of the available dynamical characteristics.

Note that models calibrated with dynamic characteristics cannot be used on instruments with only static characteristics. Separate risk models need to be calibrated either for static or dynamic characteristics. We have found that in practice, for periods of time of up to five years, static and dynamic models deliver equally good precision.

Multivariate versus univariate regressions

While we use a multivariate regression to compute characteristics, not all characteristics are estimated at once in a single multivariate regression. It is possible to obtain characteristics by performing the regression defined above for different sets of factors (e.g., style, industries) or even for each factor separately. This possibility is important because it enables the estimation of characteristics over small groups of risk factors.

Performing multivariate regressions instead of univariate regressions has a number of numerical advantages. First, it reduces the standard error of the estimates. Multivariate regressions involving either fundamental factors such as Value, Size and Momentum or industry factors tend to produce smaller residuals. Second, we found that in some cases, univariate regressions were highly correlated cross-sectionally. This is problematic as the cross-sectional covariances matrix between characteristics needs to be inverted in order to obtain the models' factors. This numerical operation is sensitive to the stability of the matrix, or in other words, its distance to singularity. In general, large cross-sectional correlations between characteristics tend to impair the stability of the matrix.

To illustrate the impact of the estimation method on the cross-sectional characteristics, we select 400 US funds and measure their exposures to a set of four fundamental risk factors between the end of March 2017 and the end of March 2022 using two methods. The first method estimates the model characteristics by regressing the active returns on each risk factor individually. The corresponding cross-sectional covariance matrix of the regression coefficients is displayed in the table below.

Cross-correlations between characteristics measured with univariate regression

The second method is a multivariate regression. The cross-sectional covariance matrix associated with the characteristics obtained using this method is displayed in the table below.

Cross-correlations between characteristics measured with multivariate regression