Glossary

Absolute Emissions

Absolute emissions depend on a portfolio's weighted average carbon intensity normalized by enterprise value including cash (EVIC), but also take into consideration the value of assets under management of a portfolio. This indicator is expressed directly in tons of CO2 equivalent and can therefore be compared with carbon budgets. For a portfolio with $n$ constituents, absolute emissions is computed as $$\sum\limits_i^n\frac{\text{emissions}_i}{ \text{enterprise value}_i}\times \text{Assets Under Management}_i.$$

Brown Stocks

Brown stocks are those that have a carbon intensity above the universe median carbon intensity of the related Climate Policy Relevant Sectors (CPRS). It is important to note that each CPRS median has been calculated by investment zone.

Climate Active Ratio

The Climate Active Ratio measures the emissions reduction potential of a portfolio by normalizing its decarbonization by its active risk (relative to the regional capitalization-weighted benchmark). In this way, it quantifies the risk-adjusted efficiency of the portfolio’s decarbonization efforts.

The Climate Active Ratio is computed as: $$\text{Climate Active Ratio} = \frac{\frac{1}{1+\text{Emission Reduction}}}{\text{Active Risk}}$$ A low, inefficient Climate Active Ratio suggests greater potential for further decarbonization.

Climate Policy Relevant Sectors

The climate-policy relevant sectors (CPRS) classification identifies sectors whose primary economic activities "could be affected, either positively or negatively in a disorderly low-carbon transition [...] considering (i) the direct and indirect contribution to GHG emissions; (ii) their relevance for climate policy implementation [...] (iii) their role in the energy value chain" (Battiston et al., 2017) and has been used by several financial regulators to assess the exposure of financial institutions to transition risks.

We favor this classification for the design of our Climate Transition factor as it is both robust (the sectoral affiliation is easily accessible) and forward-looking (the classification is established based on disorderly transition scenarios).

CPRS sectors include:

• Energy Intensive
• Utility/Electricity
• Buildings
• Fossil Fuel
• Transportation

Climate Transition Risk Factor

The Climate Transition (CT) risk factor represents the returns of an equally weighted (EW) portfolio of stocks belonging to five Carbon Policy Relevant Sectors (fossil fuel, utility/electricity, energy-intensive, buildings, and transportation). The long (“brown”) leg is built as an EW portfolio of the 50% most Greenhouse Gas (GHG) emissions intensive stocks selected within each of the six main CPRS sectors. Symmetrically, the short (“green”) leg is built as an EW portfolio of the 50% least GHG emissions intensive stocks selected within each of the six main CPRS sectors. Both the long and the short positions are weighted so that the factor is market neutral.

To learn more about the Climate Transition risk factor, consult the Methodology Guide.

Completion approach

One method for adjusting a portfolio's capital allocation involves acquiring new instruments to add to a portfolio. This process is commonly known as completion. Portfolio completion therefore involves finding new instruments within a given investment universe that, when added to the portfolio, will result in an improvement of a specific investment objective, such as reducing risk or improving factor quality.

For portfolio completion, it is necessary to specify the amount of new capital dedicated to the acquisition of these new instruments.

An important point to note is that a completion task is always performed while keeping the weights of the current portfolio fixed.

Conditional Probability of Outperformance

Conditional probability of outperformance (CPO) assesses the likelihood of positive performance over a reference or risk-free rate in a given macroeconomic state. In the context of macroeconomic analysis, a time period, $t$, is defined as the day when an analyzed macroeconomic state, $S$, exhibits a surprise. CPO is therefore defined as $$\mathbb{P}[R_t - R_f > 0 | t \in S],$$ where $R_t$ represents the portfolio returns and $R_f$ the risk-free rate. In relative view, $R_f$ represents the reference return. CPO is computed for all macroeconomic states and for both upward and downward surprises.

To learn more about assessing the impact of macroeconomic shocks, consult the Methodology Guide.

Conditional Return

Conditional return is the conditional average return of a portfolio over upward or downward surprise dates. In the context of macroeconomic analysis, a time period, $t$, is defined as the day when an analyzed macroeconomic state, $S$, exhibits a surprise. Conditional return is defined as $$\mathbb{E}[R_t|t \in S],$$ where $R_t$ represents the portfolio returns. Conditional return is computed for all macroeconomic states and for both upward and downward surprises.

Conditional Risk

Conditional risk is the conditional average volatility of a portfolio over upward or downward surprise dates. In the context of macroeconomic analysis, a time period, $t$, is defined as the day when an analyzed macroeconomic state, $S$, exhibits a surprise. Conditional risk is defined as $$\sqrt{\text{Var}[R_t|t \in S]},$$ where $R_t$ represents the portfolio returns. Conditional risk is computed for all macroeconomic states and for both upward and downward surprises.

Conditional Sharpe Ratio

The conditional Sharpe Ratio is the Sharpe Ratio of a portfolio computed over upward or downward surprise dates. In the context of macroeconomic analysis, a time period, $t$, is defined as the day when an analyzed macroeconomic state, $S$, exhibits a surprise. The conditional Sharpe ratio is defined as $$\frac{\mathbb{E}[R_t-R_f|t \in S]}{\sqrt{\text{Var}[R_t|t \in S]}},$$ where $R_t$ represents the portfolio returns and $R_f$ is either a reference return or a risk-free rate. The conditional Sharpe ratio is computed for all macroeconomic states and for both upward and downward surprises.

Consensus Screen

The Consensus Screen is a set of exclusion criteria derived from an analysis of the exclusion policies of the 100 largest asset owners. The screen consists of the four most common exclusion criteria identified: involvement in tobacco, involvement in coal, involvement in controversial weapons, and controversies related to the United Nations Global Compact.

Constraints

Constraints are used in portfolio optimization to control the impact of the optimization process on a portfolio's weight changes and risk profile. Our platform supports two types of constraints:

1. Trading constraints control the maximum amount of capital that can be taken in and out of any particular instrument.
2. Risk constraints control the portfolio's characteristics in terms of relative risk or factor exposure.

The following table provides a summary of all available constraints. As detailed below, for a certain objective, a number of constraints may become unavailable.

Each optimization task performed on the platform is designed to guarantee success. This is achieved by proactively preventing constraints from making the optimization problem unfeasible. In such cases, the worst outcome for an optimization task is a return to the current portfolio, indicating that an improvement aligned with the investment objective was not identified. Consequently, not all constraints are universally available for every objective, and there may be instances where the scope or use of constraints is limited.

More details on constraints are provided in the Methodology Guide.

Country Allocation

Country allocation refers to the way that capital within a portfolio is distributed across different countries.

The weight assigned to each country is $$W^{\text{country}} = \sum\limits_{n=1}^N w_n^{\text{country}},$$ where, $w_n^{\text{country}}$ refers to the weight of stock $n$ in the specified country and $W^{\text{country}}$ refers to the sum of weights of all N stocks in the specified country.

The Scientific Portfolio platform shows the distribution of country weights in both absolute and relative terms (with respect to the loaded reference). The absolute view allows for the identification of concentrated positions or countries where there is little or no exposure while the relative view helps highlight the difference with respect to a reference.

Note that the results are dependent on the portfolio's stock weight profile that is available.

Currency allocation

Currency allocation refers to the way that capital within a portfolio is distributed across different currencies.

The weight assigned to each currency is $$W^{\text{currency}} = \sum\limits_{n=1}^N w_n^{\text{currency}},$$ where, $w_n^{\text{currency}}$ refers to the weight of stock $n$ in the specified currency and $W^{\text{currency}}$ refers to the sum of weights of all N stocks in the specified currency.

The Scientific Portfolio platform shows the distribution of currency weights in both absolute and relative terms (with respect to the loaded reference). Absolute analytics enable investors to assess their currency exposures based on the most recent rebalancing date, while relative analytics assist in identifying differences compared to the loaded reference.

Note that the results are dependent on the portfolio's stock weight profile that is available.

Diversification of fundamental risk exposures

Diversification of fundamental risk exposures assesses the level of diversification across fundamental factor exposures. It therefore measures whether exposures to fundamental risk factors are balanced across several fundamental factors.

When factor exposures are perfectly diversified (meaning that exposure to each risk factor is equally distributed), diversification of fundamental risk exposures is equal to six, the total number of fundamental factors. When factor exposures are concentrated into a small number of factors, diversification of fundamental risk exposures is lower.

First, the factor intensity is computed and is then used to compute the relative betas $$\beta_{relative,f}=\frac{\beta_f}{\text{Factor Intensity}},$$ where $f = \text{\{size, low volatility, profitability, momentum, value, investment}\}$.

The relative betas sum to 1 by definition. These relative betas are then used to compute the diversification of fundamental risk exposures, which is the inverse of the Herfindahl-Hirschman index. It is calculated as the inverse of the sum of squared relative betas, such that $$\text{Diversification of Fundamental Risk Exposures} = \frac{1}{\sum\limits_{f=1}^6(\beta_{relative,f}^2)}.$$ A balanced exposure to several fundamental risk factors enhances the potential for long term risk-adjusted returns because factors are often lowly correlated. While fundamental risk factors are expected to individually outperform on average, their outperformance does not generally occur at the same time.

Effective Number of Partitions

Effective number (EN) of stocks (as well as effective number of other partitions such as sectors, countries and currencies) measures the level of concentration of a specific partition within a portfolio. In other words, it explains how dollars are distributed across a partition. The EN is defined as the reciprocal of the Herfindahl-Hirschman index (HHI), where HHI is the sum of squared weights, $$\text{Effective Number of Partition} = \frac{1}{\sum \limits_{i=1}^n \text{w}_i^2}$$ where $\text{n}$ represents the number of constituents within the partition and $\text{w}_i$ represents the weight of constituent $i$.

A small EN indicates that some instruments have a disproportionate effect on the portfolio. On the other hand, a high EN figure is indicative of diversification across the partition.

EU Paris-Aligned Benchmark

The European Union (EU) Paris Aligned Benchmark (PAB) comprises a series of regulations designed to redirect capital toward climate-friendly investments, enhance transparency and prevent greenwashing. Portfolios must meet all the criteria outlined in the EU PAB regulations in order to receive the EU PAB label, indicating their compliance with the established standards.

The Paris Alignment Benchmarks Screen highlights companies that are not compatible with the Paris Agreement goals according to the EU climate transition and Paris-aligned benchmarks delegated regulation.

Extreme Losses and Conditional Value at Risk

On the SP platform, extreme risk is defined as the loss that can be expected in the worst week of the year. This definition of extreme risk is equal to the Conditional Value at Risk (CVaR) of the portfolio for a probability threshold of $\alpha=1/52\approx 2\%$. CVaR is a well know measure of extreme risk that corresponds to the average of all returns that fall in the worst $\alpha$-percentile of all returns, defined as $$CVaR_\alpha = E[\, r_t \, | \,r_t \leq r_\alpha],$$ where $r_t$ is the weekly return of the portfolio at time $t$ and $r_\alpha$ is the entry point of the worst $\alpha$-percentile of returns. A simple way to estimate the CVaR is to sort all returns and take the average of the worst $\alpha$-percentile, as shown in the figure below.

Like any measure of extreme risk, CVaR is difficult to estimate precisely because it is based on the average of a small fraction of the available returns, which makes it unstable and sensitive to the period chosen. Popular techniques that have been developed to create more data using historical distributions often rely on the assumption that portfolio returns are normally distributed, which results in underestimating the amplitude of extreme events.

On the Scientific Portfolio platform, extreme losses are evaluated in two steps. First, we use the portfolio's risk exposures and our risk factors to make realistic simulations of its behavior over the last 15 years. Second, we use this large collection of returns to implement recent techniques based on the Cornish Fisher expansion that make it possible not only to estimate extreme risk based on historical returns, but also to identify the risks that drive them.

Read more about the CVaR methodology in the Methodology Guide.

Factor Intensity

Factor intensity assesses the strength of a portfolio's exposures to fundamental risk factors. It is measured as the sum of beta coefficients for all fundamental factors (excluding the market) and can be expressed as:

$$\text{Factor Intensity} = \sum\limits_{f=1}^6\beta_f,$$

where $f = \text{\{size, low volatility, profitability, momentum, value, investment}\}$.

A portfolio with a high factor intensity has a high aggregate exposure to fundamental risk factors. A portfolio with a low factor intensity has a low aggregate exposure to fundamental risk factors.

For more information on how the fundamental factor exposures ($\beta$) are computed, please consult the Methodology Guide.

Fundamental Risk Factors

Exposures to fundamental, or rewarded, risk factors are expected to generate excess returns in the long term as a compensation for risk. They are backed by an extensive body of academic research on asset pricing that recognizes them as persistent drivers of long-term performance. They have been subject to a high degree of academic scrutiny and challenge, providing investors with confidence in their use. 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.

We include the following six fundamental risk factors (plus the Market) on our platform:

Fundamental risk factors are created by combining long and short positions. These positions are weighted so that each factor is market neutral.

Consult the Methodology Guide to learn more about fundamental risk factors.

Green Revenues - Enabling

In terms of green revenues, enabling activities are those "activities that enable other activities to make a substantial contribution to one or more of the six environmental objectives." The six objectives are: climate change mitigation, climate change adaptation, sustainable use and protection of water and marine resources, transition to a circular economy, pollution prevention and control, and protection and restoration of biodiversity and ecosystems. (EU taxonomy definition, source: Green revenue source EU taxonomie).

Green Revenues - Green

In terms of green revenues, green activities are those "activities that in and of themselves contribute substantially to one of the six environmental objectives." (EU taxonomy definition, source: Green revenue source EU taxonomie).

Green Revenues - Transition

In terms of green revenues, transition activities are "activities for which there are not technologically and economically feasible low-carbon alternatives, but that support the transition to a climate-neutral economy in a manner that is consistent with a pathway to limit the temperature increase to 1.5 degrees Celsius above pre-industrial levels, for example by phasing out greenhouse gas emissions.” (EU taxonomy definition, source: Green revenue source EU taxonomie).

Green Stocks

Green stocks are those that have a carbon intensity below the universe median carbon intensity of the related Climate Policy Relevant Sector (CPRS). It is important to note that each CPRS median has been calculated by investment zone (Developed Europe, Developed, United States).

Investment Risk Factor

The Investment risk factor measures the excess return of instruments with low levels of investment compared to instruments with high levels of investment. It therefore represents the returns of an equally weighted portfolio that is long the bottom 20% of stocks (low investment stocks) and short the top 20% of stocks (high investment stocks) sorted on total asset growth measured over a two year period in descending order.

The Investment risk factor is market beta neutralized ex-post on a quarterly basis. Every quarter, the market beta associated with the long and short leg of the risk factor is computed. The excess returns of the long ($R_{\text{long}}$) and short ($R_{\text{short}}$) legs are then adjusted by the respective market betas ($\beta_{\text{long, market}}$ and $\beta_{\text{short, market}})$ where $$\text{Investment Risk Factor} = \frac{R_{\text{long}}-R_{\text{risk free}}}{\beta_\text{long, market}} - \frac{R_{\text{short}}-R_{\text{risk free}}}{\beta_\text{short, market}}.$$ The Investment beta is the regression coefficient that is associated with the Investment risk factor.

Low Volatility Risk Factor

The Low Volatility risk factor measures the excess return of lower volatility instruments compared to instruments with higher volatility. It therefore represents the returns of an equally weighted portfolio that is long the bottom 20% of stocks (low volatility stocks) and short the top 20% of stocks (high volatility stocks) sorted on past volatility measured over a 2 year period in descending order. The Low Volatility risk factor is market beta neutralized ex-post on a quarterly basis. Every quarter, the market beta associated with the long and short leg of the risk factor is computed. The excess returns of the long ($R_{\text{long}}$) and short ($R_{\text{short}}$) legs are then adjusted by the respective market betas ($\beta_{\text{long, market}}$ and $\beta_{\text{short, market}})$ where

$$\text{Low Volatility Risk Factor} = \frac{R_{\text{long}}-R_{\text{risk free}}}{\beta_\text{long, market}} - \frac{R_{\text{short}}-R_{\text{risk free}}}{\beta_\text{short, market}}.$$ The Low Volatility beta is the regression coefficient that is associated with the Low Volatility risk factor.

Macroeconomic State Variables

Macroeconomic State Variables are derived from tradeable instruments that are linked to characteristics of the economy. These variables are forward-looking, reflect investor expectations, come at an arbitrarily low frequency and are well documented in modern quantitative finance. The following macroeconomic state variables are included on the SP platform:

To read more about the methodology behind macroeconomic state variables, consult the Methodology Guide.

Market beta

The market beta (commonly called risk exposure) measures the sensitivity of a portfolio to the market risk factor. A beta value of 1 signifies that the portfolio is projected to move in line with the market. A beta value below 1 implies that the portfolio is expected to be less volatile than the market. A beta value exceeding 1 indicates that the portfolio is expected to be more volatile than the market. The market beta is computed as $$R_{i,t} = (c_{t})_{i,m}\bold{F}_{m,t}+r_{i,t},$$ where $R_{i,t}$ represents an instrument's absolute returns, $(c_{t})_{i,m}$ is the beta of the $i^{th}$ instrument associated with the market risk factor, $r_{i,t}$ denotes an instrument's active returns over a period of fixed size ending at time $t$ and $\bold{F}_{m, t}$ is the market risk factor.

Market Risk Factor

The Market risk factor measures an instrument's sensitivity to the market, where the market factor is the long-only cap-weighted benchmark of the relevant investment universe.

With equity portfolios, the market factor is de-facto the prominent source of risk. All fundamental 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 beta. The characteristics of instruments to non-market risk factors are thus considered active risk.

Maximum Drawdown

Maximum drawdown is a risk measure that calculates the maximum observed loss experienced by a portfolio. It measures the largest drop from peak to trough before a new peak is achieved.
$$\text{Maximum Drawdown} = \frac{\text{Trough Value - Peak Value}}{\text{Peak Value}}$$

When instruments contain different periods of data, the construction of the maximum drawdown process can become problematic. To mitigate this issue, we use the long-term factors of the SP risk model to simulate missing returns based on an instrument's exposure to these factors, to ensure that we have 15 years of data. We then combine the simulated returns obtained by the risk model with observed returns to compute the portfolio's maximum drawdown.

Note that as portfolio losses become larger, it takes a much larger gain than the loss to reach recovery from a trough. For example, a maximum drawdown of 50% means that it is necessary to double the value of the investment (therefore, a 100% return from the trough) to fully recover the losses.

Read more about the maximum drawdown methodology in the Methodology Guide.

Momentum Risk Factor

The Momentum risk factor measures the excess return of instruments whose prices have recently increased compared to instruments whose prices have recently dropped. It therefore represents the returns of an equally weighted portfolio that is long the top 20% of stocks and short the bottom 20% of stocks sorted on the past 52 weeks compounded returns excluding the most recent month in descending order.

The Momentum risk factor is market beta neutralized ex-post on a quarterly basis. Every quarter, the market beta associated with the long and short leg of the risk factor is computed. The excess returns of the long ($R_{\text{long}}$) and short ($R_{\text{short}}$) legs are then adjusted by the respective market betas ($\beta_{\text{long, market}}$ and $\beta_{\text{short, market}})$ where $$\text{Momentum Risk Factor} = \frac{R_{\text{long}}-R_{\text{risk free}}}{\beta_\text{long, market}} - \frac{R_{\text{short}}-R_{\text{risk free}}}{\beta_\text{short, market}}.$$ The Momentum beta is the regression coefficient that is associated with the Momentum risk factor.

Objective

In portfolio allocation, an objective refers to a specific financial goal or target that an investor aims to achieve. The Scientific Portfolio platform offers three investment objectives:

1. Reduce a portfolio's volatility
2. Reduce a portfolio's tracking error
3. Improve a portfolio's factor quality

The impact of an investment objective on a portfolio can be controlled by a set of constraints.

Peer Group

A peer group is a group of instruments which are comparable to the portfolio under analysis.

A default peer group is generated for the main portfolio. We make use of the Scientific Portfolio risk model to compute the distance between a portfolio's risk factor exposures and those of funds within the portfolio's investment zone. The five instruments with the smallest distance (and therefore, most similar behavior) are assigned to the peer group. Note that it is also possible to define the peer group manually.

Peer group figures shown within the application correspond to the average of the members.

Portfolio

A portfolio represents a weighted list of instruments. A portfolio can contain funds, indices, ETFs or strategies entered manually. The weight of instruments within the portfolio must sum to ~100%.

A portfolio can represent a current position or a simulation.

Position Turnover

In portfolio allocation, position turnover is considered a trading limit constraint. It can be used to limit the amount of capital traded in or out of any position to a given percentage of the portfolio value so that changes are not concentrated in any single position.

Profitability Risk Factor

The Profitability risk factor measures the excess return of higher profitability instruments compared to instruments with lower profitability. It therefore represents the returns of an equally weighted portfolio that is long the top 20% of stocks (high profitability stocks) and short the bottom 20% of stocks (low profitability stocks) sorted on past 52 weeks gross profit / total assets in descending order.

The Profitability risk factor is market beta neutralized ex-post on a quarterly basis. Every quarter, the market beta associated with the long and short leg of the risk factor is computed. The excess returns of the long ($R_{\text{long}}$) and short ($R_{\text{short}}$) legs are then adjusted by the respective market betas ($\beta_{\text{long, market}}$ and $\beta_{\text{short, market}})$ where $$\text{Profitability Risk Factor} = \frac{R_{\text{long}}-R_{\text{risk free}}}{\beta_\text{long, market}} - \frac{R_{\text{short}}-R_{\text{risk free}}}{\beta_\text{short, market}}.$$ The Profitability beta is the regression coefficient that is associated with the Profitability risk factor.

Ranking

On the SP platform, a vignette with a given metric is provided on each page. Within the vignette, a bar is displayed that shows quintiles based on the relative position of the portfolio, reference and peer group in comparison to all other funds from within the portfolio's investment zone. In order to compute the quintile, funds are ranked from worst to best based on the given metric. Once the data is in order, it is divided into five equally sized groups, with each group comprising 20% of the data. Note that only funds from within the portfolio's investment zone are included in the ranking.

Reference

The reference provides a single point of comparison for the portfolio under analysis.

The reference can either be assigned:

• automatically. In this case, the reference is set to Scientific Portfolio's default Cap-Weighted benchmark of the portfolio's investment zone; or
• manually. In this case, it could range from a previous allocation of the same portfolio, a different portfolio, funds, ETFs, indices or any uploaded time series.

The only constraint for the reference is that it shares the same investment zone as the portfolio.

Relative Conditional Value at Risk

Relative conditional value at risk represents a portfolio's conditional value at risk in comparison to a reference.

The characteristics associated with relative returns correspond to the difference between the characteristics of the portfolio and those of the reference, that is $$c_t^{\text{relative}} = c_t^{\text{portfolio}} - c_t^{\text{reference}}$$ Once the relative characteristics are calculated, they are used as input into the risk model, alongside relative returns, to compute the relative conditional value at risk.

Relative Maximum Drawdown

Relative maximum drawdown represents the maximum loss experienced by the portfolio with respect to the reference. The daily price series for the portfolio and for the reference is used to create a new price series. This new series is constructed by calculating the daily ratio between the price of the portfolio $P_{P_t}$ and the price of the reference $P_{R_t}$: $$\text{relative prices(t)} = \frac{P_{P_t}}{P_{R_t}}$$ The relative prices are then used to compute the relative maximum drawdown.

Relative Return

Relative return represents a portfolio's return in comparison to a reference for a given period.

In order to compute the relative return, we calculate the difference between the total returns of the portfolio annualized and the total returns of the reference annualized: $$\text{Relative Return} = \bigg(\prod_{t=1}^T (1+R_{P,t})\bigg)^{52/T}-\bigg(\prod_{t=1}^T (1+R_{R,t})\bigg)^{52/T}$$ where $R_{P,t}$ represents the returns of the portfolio, $R_{R_t}$ is the returns of the reference and $t$ is the number of weeks.

The relative performance contribution is computed as the difference between the contributions of the portfolio and the reference.

Relative Time Under Water

Relative time under water represents the length of time of the relative maximum drawdown. It measures the amount of time it takes to recover the relative loss with respect to the reference.

Reshuffle

One method for adjusting a portfolio's capital allocation involves modifying the weights assigned to existing instruments within a portfolio. This process is commonly known as reshuffling. It is important to note that depending on the user-defined constraints, certain weights could potentially be reduced to zero, effectively resulting in a divestment.

Return

Return measures the gain or loss of a portfolio. The return provided on the platform is calculated by compounding weekly portfolio returns over a set time period and annualizing them with the 52 weeks per year convention.

The returns are presented in annualized form so that comparisons can be made across different portfolios and different investment horizons. $$\text{Annualized Return} = \bigg(\prod_{n=1}^n (1+R_{P,n})\bigg)^{52/n}-1$$ where $R_{P,n}$ represents the weekly returns of the portfolio and $n$ is the investment horizon.

Note regarding missing data: when time series data is absent, prices are forward-filled for a period of up to 15 days. In the event that more than 15 days of data is missing, the instrument is not considered valid and returns are not computed.

Risk Analysis

The Scientific Portfolio platform offers two distinct types of risk analysis which we call “historical” and “point-in-time”.

The historical approach analyzes a strategy based on a daily time series upload, allowing for an ex-post analysis that effectively encapsulates the effects of all historical portfolio compositions, including rebalancing. This option proves valuable when explaining risk over a historical period, such as the last three years.

The point-in-time approach analyzes a portfolio based on a holdings-based composition upload, which provides an instantaneous, ex-ante snapshot of the portfolio at the current time. In essence, it offers insights into the risks the portfolio faces at present and is useful when interested in analyzing short-term risk, assuming that the portfolio will not be rebalanced.

It is important to note that these two options will produce different risk results and therefore, it is not advisable to directly compare a point-in-time snapshot with a historical analysis in a side-by-side manner. Since historical analysis relies on a richer set of information (a strategy reflects all historical portfolio compositions), all the pre-loaded financial instruments (e.g., indices, ETFs, mutual funds) in the Scientific Portfolio platform’s investment universe are analyzed as strategies when it comes to risk; note that some non-risk related analytics (e.g., ESG or climate metrics) specifically require holdings data and are computed based on the latest portfolio composition of the instrument.

Risk Decomposition

Risk Decomposition breaks down systematic risk into contributions that can be grouped by time, instrument or characteristic to obtain different granular views of the drivers of risk. Summing all risk contributions over every dimension gives the exact systematic risk of a portfolio.

Systematic risk, $\sigma$, is given by: $$\sigma_w = \sum_t \sum_i^N\sum_\ell^L (RC_t)_{\ell,i} +\sum_t \sum_i^N (RC_t)_{\varepsilon,i}$$ where $(RC_t)_{\ell,i}$ is the risk contribution at time $t$ for the $\ell^{th}$ characteristic of the $i^{th}$ instrument. As an example, the contribution to risk associated with exposure to the value factor over time is given by $\sum\limits_{t,i}(RC_t)_{\text{value},i}$. Note that risk contributions are annualized.

Further information on Scientific Portfolio's risk decomposition methodology is provided in the Methodology Guide.

Robust Active Risk Diversification

Active Risk Diversification (ARD) robustly assesses the historical spread of active risk contributions. Therefore, it represents the effective number of active risk contributions. A low ARD figure indicates that active risk is concentrated into a small number of factors while a high ARD figure indicates that active risk is relatively well-spread across risk factors.

Since analytics are provided on long-only equity instruments, the Market factor is expected to consistently be a very large risk contributor, giving an impression of concentration for every instrument. It is therefore necessary to correct for this natural bias by using the active (i.e., relative to the regional capitalization-weighted benchmark) performance and risk of an instrument.

The Scientific Portfolio risk model decomposes systematic active risk - that is, the portion of active risk explained by the systematic risk factors- into individual factor-based active risk contributions. The active risk is then computed as the sum of each risk factor's contribution to active risk $$\sigma_\alpha = \sum \limits_{i}^L RC_i,$$ where $RC_i$ is the risk contribution of the $i^{th}$ characteristic to risk over the last five years and $L$ is the number of risk factors. We then measure the effective number of contributors to active risk: $$\text{Active Risk Diversification} = \frac{\sigma_\alpha^2}{\sum \limits_{i=1}^L (RC_i)^2}.$$ If active risk were equally distributed across all risk factors (this theoretical allocation is often called a “risk parity” approach), the value of ARD would be seventeen.

ARD is based on characteristics which are subject to uncertainty. In order to understand how the uncertainty of the characteristics impacts ARD, we use a technique called the delta method to approximate a confidence interval around the metric. The robust ARD is then calculated by adjusting the metric to the lower confidence limit. This makes the figure more robust to different realizations of returns. This method can be explained visually by:

When the confidence interval includes zero, the robust ARD is set around zero, keeping the lower confidence limit ranking order, as shown below:

Given that a confidence level of one sigma is used, it can be assumed that the robust ARD will be higher than the figure stated on the platform approximately 2 out of 3 times should different realizations of returns occur. Read more about the robust ARD in the Methodology Guide.

Robust Factor Quality

Factor quality assesses the extent to which a portfolio is exposed to fundamental risk factors and whether such exposures are well diversified. It is calculated as the product of two components: the factor intensity measure and Diversification of fundamental risk exposures measure. $$\text{Factor Quality = Factor Intensity} \times \text{Diversification of Fundamental Risk Exposures}$$ A low Factor quality value indicates that the portfolio is either not materially exposed to fundamental factors, or that it is heavily concentrated into a small number of fundamental factors, or both. A portfolio with both a high factor intensity and diversification of fundamental risk exposures will have a high factor quality, therefore enhancing the long-term performance potential of the portfolio.

Factor quality is based on characteristics which are subject to uncertainty. In order to understand how the uncertainty of the characteristics impacts the factor quality, we use a technique called the delta method to approximate a confidence interval (CI) around the metric. The robust factor quality is then calculated by adjusting the metric downward to the lower confidence limit. This adjustment ensures that the most conservative figure is used and makes it more robust to different realizations of returns. More specifically,

• when factor quality is $> 0$, the confidence interval is subtracted to compute the robust factor quality.

• when factor quality is $<0$, the confidence interval is subtracted to compute the robust factor quality.

This method can be explained visually by:

Given that a confidence level of one sigma is used, it can be assumed that the factor quality will be higher than the figure stated on the platform approximately 2 out of 3 times should different realizations of returns occur.

Read more about robust factor quality in the Methodology Guide.

Robust Sharpe Ratio

The Sharpe ratio is computed from expected returns and volatilities, unknown quantities which need to be statistically estimated. Consequently, the sample Sharpe ratios are subject to some estimation error.

We use a Heteroskedasticity and Autocorrelation Consistent (HAC) estimator to calculate the standard error. In turn, this allows us to construct confidence intervals for the Sharpe ratio estimate and provide a robust Sharpe ratio that is set to the lower end of the confidence limit (with a confidence level of 1 sigma). This can be described visually as:

When the confidence interval includes zero, the robust Sharpe ratio is set to zero, as shown below:

More information on the robust Sharpe ratio and the technique used can be found in the Methodology Guide.

Science-Based Targets: Committed

Commitments demonstrate an organization’s intention to develop targets and submit these for validation within 24 months. They are indicated by the word ‘committed’ in the platform.

For more information about the commitment compliance policy, see: https://sciencebasedtargets.org/resources/files/Commitment-Compliance-Policy.pdf

Science-Based Targets: Long-Term Targets

“Long-term targets indicate the degree of emission reductions organizations need to achieve net-zero according to the SBTi’s Corporate Net-Zero Standard criteria. These targets must be achieved no later than 2050 (or 2040 for the power sector).” (SBTI definition, source: https://sciencebasedtargets.org/companies-taking-action)

Long-term targets are developed by companies wishing to set net-zero targets under the “Corporate Net-Zero Standard” (SBTI). For more information about Corporate Net-Zero Standard, see: https://sciencebasedtargets.org/resources/files/Net-Zero-Standard.pdf.

Science-Based Targets: Near-Term Targets

“Near-term Targets outline how organizations will reduce their emissions, usually over the next 5-10 years. These targets galvanize the action required for significant emissions reductions to be achieved by 2030. Near-term targets are also a requirement for companies wishing to set net-zero targets.” (SBTI definition, source: https://sciencebasedtargets.org/companies-taking-action)

The Target Validation Protocol for Near-term Targets describes the fundamental principles, procedures, and criteria utilized for evaluating targets and establishing compliance with the Science-Based Targets Initiative (SBTi) Criteria for Near-term Targets.

Some sectoral distinctions are made. For more information, see the target validation protocol: https://sciencebasedtargets.org/resources/files/Target-Validation-Protocol.pdf.

Science-Based Targets: Neutral before 2050

Set net-zero targets involves (a) minimizing scope 1, 2, and 3 emissions to zero or a residual level in line with achieving net-zero emissions globally or within the sector in eligible 1.5°C scenarios or sector pathways, and (b) offsetting any remaining emissions at the net-zero target date, along with any greenhouse gas emissions released subsequently into the atmosphere. Companies seeking to define net-zero goals within the framework of the Corporate Net-Zero Standard have their both near- and long-term targets validated by the Science-Based Targets Initiative (SBTi).

For more information about Corporate Net-Zero Standard, see: https://sciencebasedtargets.org/resources/files/Net-Zero-Standard.pdf.

Science-Based Targets: Targets Set

When a company submits a target, the Science-Based Targets Initiative (SBTi ) thoroughly assesses the target to ensure it is compliant with SBTi Criteria and aligned with climate science. If the target is compliant, they are validated and marked on the Climate alignment page as “Targets set”. A science-based target must encompass a minimum of 95 percent of the company's overall scope 1 and 2 emissions. If a company’s relevant scope 3 emissions are 40% or more of total scope 1, 2, and 3 emissions, they must be included in near-term science-based targets.

Most companies follow this four-step process to set science-based targets:

1. Commit: Submit a letter establishing your intent to set a science-based target
2. Develop: Work on an emissions reduction target in line with the SBTi’s criteria
3. Submit: Present your target to the SBTi for a complete validation
4. Communicate: Announce your target and inform your stakeholders

The SBTi requires that 100% of scope 1 and 2 emissions be covered under a target.

For more information, see: https://sciencebasedtargets.org/step-by-step-process.

Scope 1 Emissions

Scope 1 emissions correspond to direct emissions from sources owned or controlled by the company in the manufacture of its products and/or the production of its services.

Scope 2 Emissions

Scope 2 emissions correspond to indirect emissions linked to the consumption of purchased energy.

Scope 3 Emissions

Scope 3 emissions correspond to other indirect emissions in the corporate value chain (upstream and downstream).

Screened Equities

Screened equities are those companies within a portfolio that contribute negatively to a target associated with a Sustainable Development Goal.

Sector Allocation

Sector allocation refers to the way that capital within a portfolio is distributed across different sectors. The weight assigned to each sector is $$W^{\text{sector}} = \sum\limits_{n=1}^N w_n^{\text{sector}},$$ where, $w_n^{\text{sector}}$ refers to the weight of stock $n$ in the specified sector and $W^{\text{sector}}$ refers to the sum of weights of all N stocks in the specified sector.

The Scientific Portfolio platform shows the distribution of sector weights in both absolute and relative terms (with respect to the loaded reference). The absolute view allows for the identification of concentrated positions or sectors where there is little or no exposure while the relative view helps highlight the difference with respect to a reference.

Scientific Portfolio uses The Refinitiv Business Classification (TRBC) from 2012 for sector classification. Sectors include:

• Utilities
• Financials
• Industrials
• Telecoms
• Non-Cyclical Consumer
• Basic Materials
• Energy
• Healthcare
• Cyclical Consumer
• Technology

Note that the results are dependent on the portfolio's stock weight profile that is available.

Sector-Based Risk Factors

Sector-based risk factors are often called unrewarded risk factors as they contribute to the risk and short-term performance of a portfolio but are not considered by the finance literature to contribute to the expected excess returns and long-term performance. They are sometimes referred to as tactical, meaning that although these risk factors are not expected to deliver long-term premiums, they are often used tactically to gain short-term excess returns. However, these risks can have a strong influence on the volatility and maximum drawdown (absolute figures), as well as the relative maximum drawdown and tracking error (relative figures).

Scientific Portfolio incorporates ten sector-based risk factors derived from The Refinitiv Business Classification (TRBC) Level 1 sectors. These include: Basic Materials, Consumer Cyclicals, Consumer Non-Cyclicals, Energy, Financials, Health Care, Industrials, Technology, Telecoms and Utilities.

Sector-based risk factors represent the returns of a portfolio that is long the sector index and short the market index.

Read more about sector-based risk factors in the Methodology Guide.

Sharpe Ratio

The Sharpe Ratio evaluates an instrument's risk-adjusted performance. It is calculated by dividing the excess returns over the risk-free rate by the standard deviation of excess returns, offering a measure of return per unit of risk.

$$\text{Sharpe Ratio} = \frac{R_P-r_f}{\sigma_P},$$ where $R_p$ is the return, $r_f$ is the risk-free rate and $\sigma_P$ is the standard deviation, representing the volatility. The higher the Sharpe Ratio, the better the instrument's risk-adjusted performance.

Scientific Portfolio computes annualized Sharpe Ratios by using the difference between the arithmetic annualized return of the strategy $R_P$ and the arithmetic annualized risk-free return $r_f$. The excess returns' standard deviation $\sigma_P$ is also annualized from daily data.

Size Risk Factor

The Size risk factor measures the excess return of smaller market-cap instruments compared to larger market-cap instruments. It therefore represents the returns of an equally weighted portfolio that is long the bottom 20% of stocks (small market-cap stocks) and short the top 20% of stocks (large market-cap stocks) sorted by free-float adjusted market capitalization in descending order.

The Size risk factor is market beta neutralized ex-post on a quarterly basis. Every quarter, the market beta associated with the long and short leg of the risk factor is computed. The excess returns of the long ($R_{\text{long}}$) and short ($R_{\text{short}}$) legs are then adjusted by the respective market betas ($\beta_{\text{long, market}}$ and $\beta_{\text{short, market}})$ where $$\text{Size Risk Factor} = \frac{R_{\text{long}}-R_{\text{risk free}}}{\beta_\text{long, market}} - \frac{R_{\text{short}}-R_{\text{risk free}}}{\beta_\text{short, market}}.$$ The Size beta is the regression coefficient that is associated with the Size risk factor.

Surprises

In the context of macroeconomic analysis, a surprise represents an unexpected change in a macroeconomic state. It is measured by the difference between the actual value of the factor and its predicted value.

For every macroeconomic state variable, surprises can be defined as upward or downward.

• An upward surprise represents an event where the macro series increases. For example, an upward surprise for long-term interest rates corresponds to a date when the recorded change in the long-term rate is higher than predicted by the model.

• A downward surprise represents an event where the macro series decrease. For example, a downward surprise in volatility corresponds to an unexpected decrease in the broader market volatility as measured by VIX.

Particularly, decreases in yields, inflation and volatility constitute downward surprises. Conversely, unexpected changes in the opposite direction (i.e., increase in yields, inflation and volatility) comprise upward surprises.

To learn more about surprises and how they are identified, consult the Methodology Guide.

Sustainable Development Goals

The United Nations Sustainable Development Goals came into force in 2016 and should be completed by 2030. They build on the eight Millennium Development Goals set in 2000 but differ in two ways: 1) whereas the latter focused on social issues, the Sustainable Development Goals cover social, environmental and economic issues, and 2) they are the result of negotiations involving all stakeholders, including companies. In addition to the 17 goals, the framework defines 169 targets and 232 indicators to monitor progress. The Sustainable Development Goals framework is increasingly used by institutional investors to manage, monitor and report on their extra-financial impact.

The Sustainable Development Goals Screen highlights any company whose behaviors and activities undermine the achievement of one or more of the SDGs.

Time Under Water

Time under water is related to maximum drawdown and refers to the amount of time it takes for a portfolio to achieve a new peak that surpasses the previous peak. A short period of time under water indicates a short recovery period while a long period of time under water indicates that it takes longer to recover losses.

Total Turnover

In portfolio allocation, total turnover is considered a trading limit constraint. It can be used to optimize a portfolio by trading no more than a given percentage of the total portfolio value to limit trading costs.

Tracking Error

Tracking error represents the standard deviation of the difference in returns between a portfolio and reference and therefore measures how closely a portfolio tracks a reference. $$\text{Annualized Tracking Error} = \sqrt{52 \times \frac{\sum\limits_{t=1}^T [(R_{P,t}-R_{R,t})^2}{T-1}}$$ where $R_{P,t}$ and $R_{R,t}$ represent the weekly returns of the portfolio and benchmark, respectively and $T$ is the investment horizon. Annualized tracking error is used to measure the consistency of a portfolio's tracking performance over time. A low tracking error indicates that a portfolio's performance is closely aligned with the reference while a high tracking error indicates that a portfolio's performance is not closely aligned with the reference.

In order to compute relative risk contributions, we estimate the relative returns between the portfolio and reference. The characteristics of the relative strategy correspond to the difference between the characteristics of the portfolio and those of the reference: $$c_t^{\text{relative}} = c_t^{\text{portfolio}} - c_t^{\text{reference}}$$ Once the relative characteristics are calculated, they are used as input into the risk model to compute the relative risk contributions.

Unconditional Simulations

One of the challenges of portfolio risk analysis is the high dependence on historical data. To counteract this issue, we use the past data available to calculate exposures to risk factors and use them to extrapolate a portfolio's returns back in time using the historical factor returns observed as far back as the 1970s.

We define extrapolated returns as follows:

$$\hat{r}_t = \sum\limits_{l=1}^{L}\beta_lf_{lt},$$ where $\beta_l$ corresponds to the exposures of the portfolio to the risk factors $f_l$ measured using its historical track record, for each of the $l =1,2,…,L$ factors considered.

Next, we simulate a range of other possible outcomes that could have occurred and compare these to the realized performance. These simulations are obtained through a Gaussian Model based on the exposure of the portfolios' long-term fundamental (rewarded) risk factors. At each point in time, 1,000 simulated returns are aggregated to build the 1 sigma (envelopes 2/3 of all simulated returns) and 2 sigma (envelopes 95% of all simulated returns) confidence intervals, which allow us to identify periods of abnormal positive or negative returns.

Value Risk Factor

The Value risk factor measures the excess return of instruments with high intangible-adjusted book-to-market values compared to instruments with low intangible-adjusted book-to-market values. It therefore represents the returns of an equally weighted portfolio that is long the top 20% of stocks (value stocks) and short the bottom 20% of stocks (growth stocks) sorted on the intangible-adjusted book-to-market value in descending order.

The Value risk factor is market beta neutralized ex-post on a quarterly basis. Every quarter, the market beta associated with the long and short leg of the risk factor is computed. The excess returns of the long ($R_{\text{long}}$) and short ($R_{\text{short}}$) legs are then adjusted by the respective market betas ($\beta_{\text{long, market}}$ and $\beta_{\text{short, market}})$ where $$\text{Value Risk Factor} = \frac{R_{\text{long}}-R_{\text{risk free}}}{\beta_\text{long, market}} - \frac{R_{\text{short}}-R_{\text{risk free}}}{\beta_\text{short, market}}.$$ The Value beta is the regression coefficient that is associated with the Value risk factor.

Volatility

Volatility measures the degree of variation of a portfolio's returns from its average value. The statistical measurement used to measure volatility on the Scientific Portfolio platform is standard deviation.

Volatility is presented in annualized terms (using the 52 weeks per year convention). $$\text{Annualized Volatility} = \sqrt{\frac{52}{T} \times \bold w^T\bold \Omega \bold w}$$ where $T$ covers the investment horizon, $\bold w$ is the vector of weights of each instrument within the portfolio and $\bold \Omega = \bold \Omega_{sys}+\bold\Omega_{\varepsilon}$. The matrix comes from the risk model which is described in the Methodology Guide.

Weighted Average Carbon Intensity

The weighted average carbon intensity (WACI) is a portfolio metric that measures the level of exposure to carbon-intensive companies. It depends on the weight of each company in a portfolio and on the emissions. First, in order to take into account the size of the companies, emissions are normalized by company revenues or enterprise value including cash (EVIC). Then, the average of the intensities of the companies in the portfolio is computed, weighting them by the (financial) weight they represent. Therefore, for a portfolio with $n$ constituents, $$\text{WACI} = \sum\limits_i^n\frac{\text{emissions}_i}{ \text{normalization metric}_i}\times \text{w}_i$$ where $\text{w}_i$ is the weight of constituent $i$ in the portfolio.