Negative Beta, Reference Markets, and Portfolio Diversification

# How can beta be negative?


# How can beta be negative?












I've been reading about the security market line and the definition of beta as $$\beta_i = \frac{Cov(R_i, R_m)}{Var(R_m)} $$ for any asset (doesn't have to be an efficient portfolio), and have read that beta can be positive or negative. My confusion is about how beta can be negative. Is it not the case that if $\beta_i$ is negative, then it could be possible that adding more of asset $i$ to the market portfolio would improve your Sharpe ratio (i.e. what you called the market portfolio isn't actually the market portfolio?)? Basically my question is how can the market portfolio have a negative covariance with anything? It seems to me that the only way this could be possible is if the asset in question has a high variance.

## Answer by Dimitri Vulis (score 2)

https://quant.stackexchange.com/a/58013

An index that represents all of the market is a CAPM assumption, but in reality $m$ is typically some stock index (like the S&P 500, which represent U.S. large cap stocks). It's not practical to build an index that would include all possible aset classes (stocks, bonds, FX, real estate...). Even a worldwide stock index isn't very practical.

The equivalent way to write the beta of a single stock to an index is $\beta_i = \rho(i, m)\frac{\sigma_{i}}{\sigma_m} $. The negative sign can come from $\rho$, the pearson correlation. The $\sigma$s don't reall affect the sign. What are some economic scenarios that can render $\rho$ negative?

- you can expect the stocks that do well when the economy is bad and vice versa to have negative correlation to the index sometimes. The textbook example is auto parts companies that are focused on used autos. When the economy is good and the stock index goes up, then consumers buy more new cars and junk old cars. When the economy is bad and the stock index goes up, then consumers delay buying new cars and fix old cars, which means buying old auto parts. Zoom Video Communications Inc (ZM) is a recent example of a single stock going up while the index went down due to the lockdown.

- "Flight to quality/safety" when the stock market crashes (like in March 2020), then you can expect some investors to panic-sell stocks and to buy instead the assets perceived as "safe" - gold and other precious metals (both the actual commodities and their miners), cryptoassets, government debt. Because of increased demand, these asset prices go up, and bond yields correspondingly go down.

- yield-like quoting conventions / construction. Some credit indices are quoted so an increase in the index (more or less - the extra yield on top of swap curve that investor demand for holding corporate debt) means the credit is more stressed. Of course this goes up if the stock market crashes. Some futures by construction go up when their underlying goes down.

Note that some stocks may or may not be in the S&P 500 index, but the other examples are not.

If you compute the $\rho$ using 5-year time series and monthly or daily returns, then the short stress periods when the assets moved in different directions are likely to be "drowned out" by perdiods of low volatility. If you want to focus on periods of high volatility, you can give more weight to observations with large moves, instead of vanilla $\rho$.

If $i$ is an asset that's part of the index $m$ (i.e. $\sigma_i$ contributes to $\sigma_m$) and historically has a negative $\rho$ to $m$ and is certain do so in the future - even under these conditions, a portfolio that's similar to $m$ but gives $i$ more weight, is likely to have lower $\sigma$ but will not necessarily have better Sharpe ratio than $m$ itself.

## Answer by demully (score 1)

https://quant.stackexchange.com/a/58001

A negative beta just means there is a negative covariance (and thus correlation) between your asset in question and your reference “market” portfolio.

Perhaps the most intuitive example of this is your “market” being stocks, and adding bonds or gold to the portfolio. These have positive (expected) rates of return, driven by macro effects that are maybe uncorrelated or maybe inverse to the drivers of stock returns. Which is precisely why people use them as portfolio diversifiers!

It is indeed hard to find stocks with a reliable negative market beta. But it’s perfectly normal looking across asset classes. Which is precisely the classic diversification argument, because stock portfolios are never effectively diversified.