Beta: An Unseen Force Shaping Portfolios

Have you ever pondered why some portfolios seem to move against the market trend? The answer could lie in beta, a seemingly innocuous yet powerful number behind your investments. However, what happens when we take beta out of its theoretical realm and into the real world? Let's explore the fascinating journey of realized beta and beta equal to 1.

Understanding Beta

Beta, a key component of modern portfolio theory, measures a stock's volatility relative to the market. A beta of 1 indicates that the stock moves in line with the market; anything above 1 means it's more volatile, below 1 suggests it's less so. But when we aim to create portfolios with a beta of 1, what does beta look like once these portfolios are implemented?

Beta Behavior in the Real World

In statistical analysis, there's an 'estimation period' and then there's reality, or 'out-of-sample period'. It was wondered if beta might behave differently when we move from one to the other. Thus, betas for S&P 500 constituents were estimated in two periods: January to June 2010 (estimation) and January to June 2011 (out-of-sample).

Figure 1: Estimates of beta for S&P 500 constituents for 2011 H1 vs. 2010.

Source: Portfolio Probe

As shown, betas don't remain static; they shift around somewhat. Some stocks become more volatile relative to the market (their beta increases), while others become less so (beta decreases). But when portfolios with beta equal to 1 are created using these shifting betas, what happens next?

Realized Beta: The Unpredictable Twin

Calculating realized beta for individual stocks is straightforward – it's just another regression. However, for portfolios? It's a bit trickier. Realized beta depends on the weights of assets in the portfolio, which change as prices fluctuate throughout the period. So, we need to decide: do we use weights at the start, end, or midpoint of the period?

Figures 2 & 3: Realized beta for portfolios using weights at start vs. end (20 assets and 200 assets).

Source: Portfolio Probe

Surprisingly, portfolio betas remain remarkably stable, hovering around 1 – even when weights from different points in time are used. This resilience is notable given how variable stock betas can be (Figure 6 of "4 and a half myths about beta in finance").

Beta Equal to 1: A Closer Look

Now, let's consider portfolios constructed with the express purpose of having a beta equal to 1. What do these look like, and how do they perform?

Figures 4 & 5: Portfolio returns vs. realized beta (20 assets and 200 assets).

Source: Portfolio Probe

As expected, using midpoint weights tends to strike a balance between start and end weights. But what's striking is how stable portfolio betas remain, even as individual stock betas fluctuate.