Probabilistic Features of Index Investing

The most important step toward the mathematization of financial markets was taken in 1952 with the advent of Modern Portfolio Theory (MPT). Its main achievement was the ability to quantify investments and assemble them into a portfolio that closely matches a client's risk profile. While MPT gave birth to a multitude of subsequent theoretical works, few of them provided investors with practical tools to improve their portfolio's return-to-risk ratio. Modern Portfolio Theory relies on certain assumptions and idealizations, but these can be closely approximated by portfolios composed of asset-class indices. Ultimately, the advent of ETFs has made MPT far more practical.

A Probabilistic Approach to Portfolio Construction

Modern Portfolio Theory proposes treating financial asset price changes as random variables and analyzing them from a probability theory standpoint. The theory works most elegantly with Gaussian (normal) distributions, which are symmetrical - meaning the probabilities of equal price deviations above and below the mean value are identical.

In practice, this is not the case, especially when dealing with individual assets such as specific stocks and bonds. Defaults, bankruptcies, and sector-specific shocks make the probability of sharply negative returns much higher than that of sharply positive ones. This is precisely why the historical return distributions of individual assets cannot be used as a reliable basis for single-asset dynamic modeling.

However, the situation changes when dealing with asset class indices. Because an index is a weighted sum of a large number of assets distributed in any manner, the aggregate distribution of these values tends toward a Gaussian distribution, according to the Central Limit Theorem.

The main advantage of a Gaussian distribution is that it is fully described by just two parameters: the expected value (mean) and the variance. In financial terms, these represent an asset's average return and its risk (standard deviation) over a specific time period. Within a Gaussian distribution, 90% of all possible returns fall within a range of ±1.65 standard deviations from the mean, and over time, this range narrows in relative terms compared to the mean value.

Consequently, portfolios built on asset class indices exhibit greater stability in their statistical behavior.

Why Regular Optimization and Rebalancing of Index Portfolios is Important

Numerous studies show that historical data can be used to estimate the standard deviations and correlations of indices; the main challenge lies in estimating expected returns. IskraIndex employs its own proprietary approach to solving this problem, based on the long-term stability of asset class characteristics. However, the expected return of an asset class is a floating parameter that changes depending on current market conditions - such as interest rates and the prevailing level of risk in the economy. This is precisely why the optimal portfolio composition shifts from month to month, which must be accounted for in real-world portfolios.