Probability Analysis of an Index Investing


Probability Analysis of an Index Investing


The most prominent theory to market assets modeling was advanced in 1952. It gave birth to many variations, but as far as asset classes indices are concern it remains more applicable and useful than any other. There are several reasons staying behind that.


A probability approach


The modern portfolio theory offers to deem assets' price returns random values and analyze them using the probability theory tools. The theory works well when we know probability distributions of price returns, but if we do not? In that case, many researchers begin to model such distributions using historical samples or other approaches. However, when you model a distribution for an individual stock, there is a room for a big mistake as any individual corporate event is able to reverse a returns' distribution. For instance, imagine we deal with a stock slowly sliding into bankruptcy state. That stock's price continuously decreases, and historical price returns' distribution will have a negative mean and some standard deviation. One day the company's peer decides to buy its stake and credit it, totally reversing its financial case. The stock price begins to rise, dramatically changing price returns' probability distribution.      

For any person far from mathematics the easy way to interpret a mean value is to view it as an average value. For price returns, it could be viewed as average of monthly, weekly or daily returns for some period. Standard deviation can be imagined as a variability of that average return - how close say daily returns lie around the average. Continuing with the terminology, a target return is a desired mean annual return investor would like to earn on her (his) portfolio in the long-term. 

The simplest probability distribution that is often used in a portfolio analysis is a Gaussian (or «Normal») distribution. It is the simplest as the only two parameters are fully characterize it: a mean value and a standard deviation. If you know that a financial asset's or a portfolio's returns are distributed normally, you can easily estimate the range of your investment outcomes for any time horizon. For instance, in a normal distribution 90% of annual returns lie in the range [mean value ± 1.65 standard deviation], 90% of five-year returns lie in the range [5 mean value ± 1.65√5 standard deviation]. This range is a kind of scenario analysis as it allows looking at risks before investing. The longer the period, the narrower range will be relatively its mean value. In other words, the longer investment horizon you aim the more concentrated possible portfolio returns will be around your target return.




An index investing makes your portfolio returns' distribution close to a «normal» one


Any asset class and representing it index is in essence a weighed sum of many assets, usually expressed by their prices. The Law of Large Numbers states that the probability distribution of a large sum of any random values will have a Gaussian (Normal) distribution. Therefore, index portfolios have a great advantage allowing to model them using just two probability parameters. Besides, the more asset classes you compose in your portfolio, the better «normality» its returns' probability distribution will have.


Why a regular model portfolio update is required even for index portfolios


The SparkIndex model estimates an each asset class' standard deviation using a mid-term sample of daily returns and a mean value using a long-term sample. Then the model uses these estimates as expected parameters of the future returns' distribution and works with them to find an optimal combination of several asset classes from the minimum risk/return ratio point of view. This approach is free from subjective views, and given the probability distributions normality we get a portfolio with reliable probability characteristics. 

However, despite the normality of index probability distributions, their parameters may fluctuate over time with changing macroeconomic and interest rates environment. These changes are continuously reflected in the SparkIndex model, as its historical sampling window is limited to several years and moves to the right with each new model portfolios update. For instance, an economic growth may transform an emerging market country into a developed one with lower level of interest rates and returns volatility. Although this process is slow the obvious question appears - how can we rely on a model portfolio which loses its optimization balance over time? The answer is a regular (once in several months) reiteration of the optimization process so that the portfolio would be relevant for three-four months. There is no need for a more frequent updates due to another interesting advantage of index investing. An expected return of any portfolio is the linear combination of expected returns of its constituents. With the passage of time fluctuations of expected returns of portfolio assets will partly compensate for each other, leaving a long-term expected return of the whole portfolio relatively stable.