Beta and its features

2 questions from Beta: 1. Can beta be negative? I guess, yes. If yes, then what’s the implication of a negative beta? It actually means that a particular stock is shown to have an opposite co-movement with the market in general, right? How do we interpret this negative beta on the ground that Beta is a measure of systematic risk? (How can risk be negative? Actually this is my primary question here) 2. If Beta is higher than 1 for a particular stock, we immediately conclude that this stock is risky from a systematic framework. But, the formula for beta calculation says that it is equal to [Covariance between a stock’s returns and the market returns/ Variance of market returns]. Now, say, if my calculation shows that the stock is highly correlated with the market, correlation coefficient will be high and this is why covariance term will also be high. Hence, an obvious paradox is a higher-than-average Beta says that the stock is risky; however, the same higher-than-average beta if originated from a higher covariance and/or correlation with the market, says that there will be no benefit to diversify our investment from the stock to the market in general, right? So, what will we do in this case?

It’s unlikely that a stock would have a negative beta but short-bias hedge funds would have negative beta. Short-bias hedge funds reduce systematic risk of an equity portfolio (thus, risk contribution is negative). Beta is important because it gives a sense of what risk premium you should expect to collect (given zero alpha which is CAPM). If a stock has high beta, the expected risk premium would be high due to higher systematic risk. Short-bias hedge funds have to have high positive alpha to provide positive returns - really challenging task.

If a stock has really high correlation with the market that means that its idiosyncratic risk is low. Diversification can substantially reduce idiosyncratyc risk but has no impact on systematic risk. Does that help?

Thanks for your explanation to the first part. But I didn’t understand your second para. Why should a stock, highly correlated with the market, have a low non-systematic or idiosyncratic risk? Can you clarify?

Negative beta means that if the market is doing good, it’s doing bad and vice versa (it was one of the technical questions for one of my interviews). Beta just means how correlated a stock is with the market, so if there is a negative correlation then the beta is negative. 0 Beta means it doesn’t have anything to do with the market (649, horse racing). Academic examples for negative beta are gold and defense stocks (Walmart).

Diversifciation is all about non-systematic risk. If there isn’t non-systematic risk then there is no need for diversifying (assuming 0 alpha and all the other stuff). You can adjust your beta using leverage.

What maratikus is saying is that high correlation = low non-systematic risk, since 1 – correlation = non-sytematic risk.

Say, a stock has a perfect positive correlation with the market. Correlation is 1, which should be equal to Cov(Ra,Rm)/(STDEVa*STDEVm). So, according to this formula it could easily be shown that correlation can perfectly be 1, even when STDEVa is higher than STDEVm, right? Higher correlation doesn’t mean low non-sytematic risk in that STDEVa can be higher than STDEVm, meaning that total risk of stock (a) is higher than total risk of the market (m). Hence, the positive risk differential for the stock (a) originates from the non-systematic portion. What do you say?

I also didn’t get your equation: “1 – correlation = non-sytematic risk”. Can you plz clarify what’s the underlying economic principle to hold this equation?

For example, if stock (a) has the following returns: 8%, 4%, 6%, 10% and the market has 4%, 2%, 3% and 5% returns over the same period, the correlation statistic is +1. But it does not mean that the stock has no non-systematic risk portion because it has a higher standard deviation than the market. But your equation shows that non-systematic risk eauals 0. Can you explain?

Variance = eplained variance (systematic risk squared) + unexplained variance (idiosyncratic risk squared).

Stdev(a)^2 = beta^2*StDev(m)^2+epsilon^2, epsilon - is the idiosyncratic risk squared.

Correlation = beta*StDev(m)/StDev(a)

Therefore, epsilon^2/Stdev(a)^2=1-Correlation^2=1-R^2. If correlation is high (R-square is high), epsilon is low relative to the stock volatility. Does that help?

Thank you so much. Yes, that helped a lot. :slight_smile: