## Why Investors Need Multiple Betas

Segmented up side and down side betas can be used to for better risk management and to perform non-linear stress testing

Beta analysis has become a staple of the investment industry because it provides a simple way of encapsulating expectations about both relative return and relative risk.

But virtually all measures of beta assume that the fund and its benchmark have the same relationship when making money as when losing money. Possibly even more egregious is the built-in assumption that the relationship is linear across all returns.

Betas should be measured for different zones of returns to capture differences not only in up markets and down markets but also in extreme markets.

Measuring upside/downside statistics is well established in financial services: downside volatility has been a standard measure for decades, and some firms extend the idea to upside and downside correlation.

But few firms consider upside/downside beta, perhaps because they limit themselves to a fundamental factor framework in which market side plays no role. However, in a statistical or regression approach, computing such betas is rather straightforward, especially when dealing with single factor regressions.

In the case of only one index or benchmark, we could divide the dataset in two parts: one subset covering only those days on which the index suffered a loss for which we compute β-, and another subset covering only those days on which the index returned a gain, for which we might compute β+. This would allow for a comparison of how differently, if at all, the fund is sensitive to the index in up markets and in down markets.

Just as it is desirable to have a relatively large beta to upward markets, it’s also desirable to have a small beta to downward markets. Funds that show larger values of β- than β+, on the other hand, would lose more in downward markets than they make in upward markets.

Taking this concept one step further, we propose computing not two betas but four: β – -, β-, β+, and β++, each of which covers a specific zone of index return.

For normal markets, defined as those within one standard deviation of the index’s average return, we calculate β-, and β+ as described above. But we further segment the index’s returns into extreme markets – those outside the one standard deviation band. For days when the index is up more than one standard deviation, we compute β++ and for those days when the index suffers a loss greater than one standard deviation, we compute β – -. The light grey lines shows the beta if only one regression were performed, with a value of 0.89. The red lines show the betas of the four separate zones which are significantly different from the overall beta.

Figure 1 shows an example using recent returns of the S&P 500 with returns of a hypothetical fund, demonstrating the zones and values of the four betas. For comparison with our technique, we show the all-inclusive beta of 0.89 in the light grey line going through all the data. Using our segmented beta approach, the ‘normal betas’ have values β- = 0.8 and β+ = 1.05, showing that the fund is slightly less sensitive to benchmark movements on the downside than on the upside. To compute those values, we considered only the days on which the fund was up/down and having returns within one standard deviation.

We further divided the data into extreme zones, defined as returns larger than one standard deviation beyond the mean. While each of those zones has only 11 data points, they visibly demonstrate a different relationship with the fund than the data within the one standard deviation zones. The positive extreme beta, β++, has a value of 1.3 while the negative extreme beta, β – -, has a value of only 0.5. As shown through in multiple beta analysis, this hypothetical fund is more sensitive to index/benchmark movements on the upside than the downside and exhibits non-linear behavior.

While segmenting the data set by standard deviation naturally limits the number of data points in the extreme subsets to only 11% of the total, we believe it is superior to using other methods, for example an equal segmentation of data (e.g. 25% for each zone), because of the canonical nature of standard deviation. Rather than using only 100 days’ history as we did in this example, we suggest using 200 days’ history in practice, giving 22 data points each for the computation of β – – and β++.

Beta analysis is often used in simulating market stresses since, for a given shift in the index’s value, beta can be used to estimate the fund’s likely response. For the example shown above, had only one beta been computed, the estimated result for any shift in the S&P 500 would be 0.89 times the S&P move. For example, for a 1% move up in the S&P, we would estimate a 0.89% rise in the fund. For a -1% move in the S&P, we would estimate a -0.89% move in the fund. Instead if we had used β+ and β-, we would arrive at slightly different answers: -0.8 for the downside and +1.05 for the upside. Similarly, using the four betas would result in still further differences. The table below summarizes the results of using just one beta, two betas and four betas for both small and large movements in the index.

Certain investment vehicles, such as hedge funds, are supposed to provide non-linear returns that might be picked up by such a multi-beta analysis. Measuring funds’ responses to the markets with multiple betas has the potential to add a layer of useful analysis both for return generation and risk management.