by Peter J. de Marigny
“I cannot get there from here, baby
And I don’t care where I’m goin’
Change, Unchained… Nothing Stays the Same”
Lyrics from the Van Halen song, Unchained, may be applied to investors’ skeptical view of how asset returns are “chained” together for attributions. These “chained” returns are then used to create an attribution of returns to make statistical inferences. Monte Carlo simulations, scenario and sensitivity factor analyses, ANOVA using historical returns may be run on hypothetical investment portfolios to form “optimized” portfolios. A mean-variance efficient frontier then is represented as the curve upon which the highest return is attained for any level of risk (or the least risky portfolio for an expected return). Using mean-variance and correlation assumes a gaussian distribution versus a more realistic leptokurtic distribution and linearity.
Can we really “get there from here” as the song goes? The whole paradigm of financial consulting rests on the methodology of modern portfolio theory. Virtually every bank, broker, and wealth manager accepts the MPT model for construction of Investment Policy Statements. What about “Change, Unchained … Nothing Stays the Same”? The untold assumptions of MPT that may not appear in the course materials for advisers include that correlations are constant and that risk is variance. Are correlations truly constant or do they change? Should we be “chaining” returns? One improvement is factor based allocation that better defines the source of risk. This factor based approach leads to other developments in asset allocation specifically designed for liability driven investment policy that includes risk parity models. Developing factor based models usually includes regression models.
A main MPT chart used by advisers as the central theme to asset allocation to achieve objectives within risk tolerance and time horizon (among other) constraints is the efficient frontier often displayed with a tangential capital market line. The “CML” is used to illustrate the market portfolio mixed with a risk-free asset as the intercept and linear plot through the market portfolio at 100% and beyond which slope is the Sharpe Ratio. Conversely, factor-based investing can be used to actively position investment portfolios to achieve specific risk and return objectives. Both models can be judged against how well the respective models match target returns and risk outcomes using Chi-square (for risk) and t-tests.
The assumption of historical standard deviations on asset classes or on the market creates the phenomenon known as “Black Swan.” It is assumed that returns are normally distributed where one, two and three standard deviations from the mean are thought to be 68%, 95% and 99.7% probabilities, respectively. An observation outside three-sigma is said to be a Black Swan (rare) event. If we remove the paradigm that “chains” return series and did not assume a normal distribution of any series but allow that the observations determine the distributions, then we will have “killed the Black Swan.” If we consider that returns may not all be in the same series, then we will have changed the paradigm that uses chained historical returns for statistical inferences. Would we consider that the return of IBM for the quarter before the 2008 meltdown should be “chained” in a series with the two quarters ensuing the meltdown to make inferences about its future returns? Consider that statistical significance also depends on degrees of freedom (observations). A 25% correlation may be significant with high degrees of freedom whereas a small sample may increase the confidence interval so wide that even a 75% correlation may be insignificant. Therefore, correlation is a moving target and changes in certain market conditions and cannot be expected to be constant for truly optimizing portfolios.
The Black Swan only exists where events are completely chance-based such as in a dice game. We would not consider a long-shot horse winning a black swan event. This analogy is to make the point that like pari-mutuel wagering, probabilities are determined based on the wagering. This may or may not fairly represent the true underlying probabilities. By looking at return series and establishing probabilities of returns for investment policies we are using a second derivative of underlying risk. A stock price may be far removed from the underlying metrics of the company and may persist for an extended period. A stock with high financial and/or operating leverage may have a higher risk than the stock price reflects. In fact, often management will purposely manipulate financial leverage (using a high amount borrowing) to accentuate EPS. Would we say that a highly levered company experiences a black swan event when the company’s operating earnings declines the same as a non-levered company but realizes an outsized drawdown on its stock price and EPS? The point is that the black swan exists only in the minds of those who chain a series of stock returns (or other asset prices) establishing probabilities by pricing that may be completely incongruent to the true underlying risk of the asset. The “Death of the Black Swan” actually occurred many years ago, 1637 to be exact, when it was last seen in the collapse of Tulip Mania. *.*