Probabilistic Features of Index Investing

The most important step towards the digitization of financial markets was taken in 1952 with the advent of Modern Portfolio Theory. Its main achievement was the ability to "classify" investments and compose them in a portfolio that most closely match a client's risk profile. Modern Portfolio Theory gave birth to a multitude of other theoretical works, but few of them gave something useful to an investor to improve they portfolios return/risk ratio. Modern Portfolio Theory has its assumptions and idealizations as well, but they can be approximated by portfolios consisting of asset class indices. The advent of ETFs has made Modern Portfolio Theory 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 is symmetrical one — the probabilities of the same price deviations above and below their mean value are equal.

In practice, this is not the case, especially when dealing with individual assets (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 basis for modeling the dynamics of single assets.

However, the situation changes when dealing with asset class indices. As an index is a weighted sum of a large number of assets, distributed in any manner, the aggregate distribution of these values will, according to the Law of Large Numbers, tends toward a Gaussian distribution.

The main advantage of a Gaussian distribution is that it is fully described by two parameters: the expected value (mean) and the variance. In financial terms, these are the asset's average return and its average risk (standard deviation) over a specific time period. In 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 probabilistic 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 of indices and their correlations; 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 asset classes is a "floating" variable that changes depending on current market conditions — such as interest rates and the prevailing level of risk (the current state of the economy). This is precisely why the optimal portfolio composition changes from month to month, a factor that must be accounted for in real-world portfolios.