Simulations

The objective of simulating the returns of equity portfolios is to eliminate the possibility of a sample-dependency, and move out of just the historical data easily available. Simulated returns offer an alternative view on the performance of a strategy by providing a set of plausible alternatives to its historical returns. Importantly, they introduce the notion that the characteristics of a strategy’s returns should be represented by a range instead of a fixed number, as should any process governed by a level of randomness.

Observable short-term historical return data and related factor returns offer limited insights into different market conditions. However, using the same data for simulations allows us to access out-of-sample information, which helps prevent the outcomes of a strategy from being confined to observed returns. Therefore, simulations play a crucial role in informing investment decisions.

Scientific Portfolio offers both conditional simulations which may be used to simulate realistic out-of-sample stress scenarios and unconditional simulations to simulate out-of-sample performances.

Conditional Simulations

The simulation of returns requires two key ingredients. First, a statistical distribution that reproduces the statistical properties of portfolios returns. Second, a sample of returns that will be used to calibrate the chosen distribution.

The choice of a statistical distributions fitting a particular returns sequence has been studied extensively. The so-called stylized facts of equity returns have been described in multiple places (^[1]; ^[2]; ^[3]), and good distributions are readily available to simulate them. We refer the reader to ^[4] and ^[5].

Choosing a sample to calibrate these distributions is far more problematic. In the long term, the statistical properties of financial instruments, such as their expected returns, are not consistent. Volatility too is not stable over time, and tends to cluster (^[6]; ^[7]) around its long term average. This variability is evidently problematic in the context of decision making. The use of regimes, with distinct, stable, and robust returns’ distributions, is therefore more practical. These regimes must also be economically meaningful, and their properties significantly different from that of the long-term returns.

Although the use of macroeconomic regimes is common among practitioners, there is no consensus on their actual definition. The academic literature is also mixed on the matter. It is widely documented that changes in macro variables influence equity returns (^[8]) but the identification of the relevant macroeconomic variables and the extent of their impact on equity returns is often challenged as varying periods, frequency and time horizons often lead to different results (^[9]; ^[10]).

In the context of equity portfolios, the works of (^[11]; ^[12]) identify a set of macroeconomic variables and market regimes relevant from the angle of equity portfolio returns. We examine the statistical properties of these regimes to shed light on their relevance for investments decisions.

Linear extrapolations and out-of-sample simulations

In the context of investment decisions, an understanding of the impact of regimes over several business cycles can be acquired by extrapolating the returns of the portfolio outside of its available historical sample. This is possible by using its exposures to different risk factors, which are available over longer periods of time.

We define extrapolated returns as follows:

where

corresponds to the exposures of the portfolio to the risk factors

measured using its historical track record, for each of the

factors considered.

Extending a portfolio’s return with this method is advisable under two conditions. First, the precision of the model, as measured by its R-squared for instance, must be high, i.e., it must provide a reliable description of the portfolio’s returns.

Secondly, the portfolio's strategy must be sufficiently stable and describable in terms of its exposure to risk factors. For example, a strategy focused on investing in undervalued stocks (a "Value" strategy) can typically be extrapolated beyond its historical data. Similarly, a strategy based purely on sectors can be extended using sector-specific risk factors.

It's crucial to treat risk factor exposures that are statistically insignificant with care. These often reflect uncontrolled or opportunistic aspects of the portfolio's strategy. Therefore, it's best to set them to zero, as their behavior cannot be reliably predicted using a basic factor model for future scenarios. Similarly, we set the residual returns

to zero (where

represents the portfolio's historical returns) because these residuals are typically unpredictable by nature. Thus, using historical data to simulate them is not dependable.

We tested the reliability of projecting a portfolio's long-term performance using funds with over 40 years of data and describing the portfolio's strategy with our fundamental risk factors. We split their history into a calibration set (6 months to 10 years of randomly selected daily returns) and a test set (remaining historical data). By estimating the portfolio's risk factor exposures in the calibration set and projecting returns onto the test set, we found that the average accuracy (R-squared) of projected returns was high, around 72%, close to the full-sample accuracy of 77%. Increasing the length of the calibration set improved accuracy, but gains were marginal beyond about five years. Overall, projecting returns through factor exposures, even with limited historical data, proved to be a reliable method for performance analysis.

Using a linear model approach helps us understand how different market conditions impact portfolio behavior and allows for reliable prediction of future returns. Risk factors, with their long-term historical data, are used to accurately simulate returns. This includes factors like correlations between investments and expected returns during specific market conditions.

In simulations, returns are calculated based on these factors and historical patterns, showing how portfolios might perform under various scenarios. This method gives us confidence in estimating portfolio behavior during different market environments, even when correlations between factors are relatively small.

Macroeconomic regimes and investment decisions

For regimes to provide useful and robust inputs into the decision-making process, they must satisfy the following conditions:

• Differentiation with full sample: The distribution of returns within regimes must be sufficiently different from long-term returns so that their occurrence impacts the portfolio’s returns and have thus reason to be considered an input in an investment process.

• In-sample stability: Regimes must be sufficiently stable so that any return sub-sample within each regime must share characteristics with the rest of the regime’s returns. Stability of parameters is also important to generate reliable simulations.

• Out-of-sample robustness: Investment decisions are forward-looking, hence it is important that within-regime return distributions remain stable in future periods. This crucial condition is often overlooked in the literature.

Definition of regimes

On our platform, we have selected macroeconomic variables following the guidelines of Esakia and Goltz (^[11]). This protocol assesses the relevance of these variables in the context of equity investments, ensuring they have a documented impact on equity returns with a solid theoretical basis.

The selected macroeconomic variables are shown in Table 1. The fundamental equity risk factors (ERF) are obtained from Scientific Beta and are designed such that there is no correlation between these ERF and the market factor ([*]).

Table 1: Macro-economic variables used to define regimes

Notes: Scientific Beta Equity Risk Factors are abbreviated by SB ERF. Tickers from the Federal Reserve Bank (FRED) are in parenthesis.

To define regimes, we use monthly times series going back to 1974. They are defined as the days belonging to months in the top quartile (‘high' regime) or bottom quartile (‘low’ regime). For instance, the high-volatility regime represents the months when market volatility is in its highest quartile at the end of the month. According to this definition, regimes are not only defined by the very days during which the value of the macro-indicator is at its highest (or lowest), but also the few weeks around that moment belonging to the same month. This is more in line with investment practice whereby investment views are typically implemented on a monthly or quarterly basis. Each regime contains about 3,000 daily returns, though the months belonging to the same regime are not necessarily consecutive.

Empirical results

To assess the in- and out-of-sample stability and robustness of regimes, we evaluated whether the within-regime expected returns and volatility were statistically different from long-term ones. Although the expected returns and volatilities within regimes are significantly different from their long-term returns, the difference in volatility is more systematically present. Hence, regimes impact more risk than returns. For instance, an increase of credit spreads, which indicates a higher risk of default, or a decrease in short term rates, tend to coincide with a significantly higher volatility in the market without a clear impact on direction. As expected, regimes related to the market itself and its volatility have the strongest impact on returns.

We then examined the stability of the returns’ distribution inside regimes by using the sample excess kurtosis. The kurtosis is a powerful indicator of stability, since it is directly proportional to the variation of the volatility within a sample, and indirectly to the variation of expected returns. Less excess kurtosis is therefore associated with more stable risk and performance within a sample. We find that the excess kurtosis present in almost all regimes is much lower than that measured in long-term market returns, thereby confirming that the properties of returns within each regime are often significantly more stable than generic long-term returns.

We also employ other empirical tests to confirm not only that equity returns within a regime have a distinct statistical signature, but also that these properties differ from those of long-term returns. For further details on the empirical tests conducted, consult the full paper by Vaucher et al. (^[13]).

Other considerations pertaining to the use of regimes

Two important factors affect the use of periods in an investment process: their typical length (or duration) and their historical representativeness. For example, a period of increased volatility might last only a few weeks or a couple of months at most. Most periods last about a quarter, with some exceptions like those related to inflation and interest rates, which can last more than a couple of years on average.

The historical representativeness of these periods also impacts their use for portfolio analysis. By historical representativeness, we mean that the phenomenon occurs repeatedly over time. This concept is illustrated in the figure below.