# Portfolio Dynamics Are Difficult to Understand

“*The risk of a portfolio is not a linear function of the vector of its components. Rather, the variance of a portfolio is a quadratic function of its composition. This thwarts the intuition of most Analysts and Investors. Indeed, the nature of risk may be the single most important argument for the use of quantitative analysis in investment management. Neither Investors nor Analysts can be blamed for this fact. Nor can Harry Markowitz. Nature made risk a quadratic function. Markowitz only discovered it*.” *William Sharpe*

Studies conducted by psychologists have consistently illustrated how human beings have difficulty with correctly integrating multiple sources of information. Even when factors can be combined in a linear function, human experts across disparate fields such as law and medicine fail to perform as well as a simple regression model. This problem becomes especially prevalent when dealing with convex or non-linear problems. William Sharpe won the Nobel Prize for simplifying an otherwise highly counterintuitive quadratic solution (brought forth by Harry Markowitz) into a linear framework. This model is known to many as the CAPM, and unfortunately the model required too many simplifying assumptions to make it work. Subsequent research showed that several of these assumptions did not appear to hold true in capital markets. The greatest violation was the assumption of a positive expected linear relationship between beta and future returns (systematic risk). In practice, research showed that beta was either negatively related or had no clear relationship to future returns and displayed more of a non-linear profile.

The original Markowitz (MPT) framework is very difficult to conceptualize, and leads to portfolios that would otherwise not appear to be logical. Several counter-intuitive properties exist, here are two interesting ones: 1) when the sharpe of most portfolios are high, adding assets that reduce portfolio volatility will increase the sharpe regardless of their returns 2) when the sharpe of most portfolios are close to zero, adding assets with a positive return will increase the sharpe regardless of their volatility. If a human being were to form a portfolio using judgment it is highly unlikely that they would account for these properties. This is where discretionary managers get into trouble– they may have a good understanding of how to forecast returns or generate possible scenarios, but integrating this information is not as straightforward as it appears. This is where portfolio math tends to have most of its value.

Despite the strengths of MPT , it is also important to understand where the math can go wrong. Primarily these flaws stem from the difficulty in estimating noisy time series data and the exacerbation of error that occurs as a function of generating “optimal” solutions. MPT optimization can become highly concentrated and inadvertently maximize allocation mistakes as a function of the degree of instability with inputs such as returns, correlations, and volatilities. However, the math remains functionally correct and is closed related to the math underlying multiple regression models. In the next post, I will show how the different optimization algorithms are related. Most share a common theme that is not clearly discussed in the research.

I’m not sure why you think the two properties you list are counter-intuitive. Suppose I’m creating a portfolio of things to eat and I care about tastiness and nutrition. If what I have on my plate is nutritious but not very tasty, then I’m going to want to add something very tasty without regard to its nutrition. And vice versa if my plate is full of tasty but non-nutritious things.

Pat, I don’t neccesarily disagree but for the most part a lot of the “correct” methods for handling different situations given a set of expectations for return/vol/correlations is definitely counterintuitive for numerous highly intelligent but discretionary traders that I have had conversations with. For quants, most of these things are actually very straightforward and seem second nature since most of what a quant has to know in finance is related to these basic concepts in one way or another.

best

david