# Why do stocks with a negative beta return less than the risk free rate?

Let's say we have two stocks, Stock A and Stock B.

Both of them have the same standard deviation $\sigma$, and therefore have the same risk.

The only difference is that Stock A has a perfect positive correlation $\rho=1$ to the market ($\beta>0$), while Stock B has a perfect negative correlation $\rho=-1$ to the market of ($\beta<0$).

According to CAPM, Stock B should pay me less than the market risk-free rate while Stock A should pay me more. If both have the same amount of risk, i.e. standard deviation, then why does Stock B pay me less than Stock A?

I can only think of two reasons:

1. There is less market supply of negative $\beta$ stocks than positive $\beta$ stocks, and therefore a higher price for negative $\beta$ stocks and lower returns.
2. Since the market generally has positive returns (and a positive E[r]), a stock with a market correlation ($\rho$) of -1 has generally negative returns (and a negative E[r]).

Anyone care to give their opinion on this?

Focusing on intuition rather than theory, $$\beta$$ can also be thought of as the "risk premium" of that specific asset relative to the market.

In general, market risk premium links two very important aspects of the world: Consumption & Return. So if we look at the world in two states, an "Up State" & "Down State", here is what we would see:

## States:

• Up State: Consumption High Probability: 50%
• Down State Consumption Low Probability: 50%

## Risk Free Asset:

Price: 10

• Up State: 10.1
• Down State: 10.1

## Stock A

Price : 20

• Up State: 40
• Down State: 10

## Stock B

Price: 20

• Up State: 10
• Down State: 40

We can calculate the $$\mathbb{E}[r]$$ for each asset by using the formula $$\mathbb{E}[r] \triangleq \frac{EV}{BV}-1$$. If we multiply the final value by the probabilities, we can see that each asset has a return of $$\frac{40\cdot .5 + 10\cdot .5}{20} - 1 = 0.25$$.

So what's the difference? The difference between the two assets are what they pay during different levels of consumption.

In the "Up State," Consumption is high. In historical terms, think of early 2007 or 2000 -- nobody wants a stock that is paying a 1% return (recall how frothy the stock market was and how compressed yield spreads were). In the down state, Consumption is low, think of 2002 or 2008, people weren't spending -- they were paying down their debts and increasing savings. So an investment with a "positive risk premium" (Asset A) or "positive $$\beta$$" pays you a high return when consumption is high and a low return (i.e. loss) when consumption is low.

That's why Stock A pays you more, you must be compensated for taking the added risk of really bad returns when consumption is low, that's what "compensatory" or market risk is based on -- illustrated by an upward sloping, convex utility function of consumption.

• Stock B does have the same risk of bad returns... but they occur in states of converse consumption levels. Think of an insurance policy, I pay 10 per mo and if I die my spouse gets 100,000. In the event I die, my wife will have made the return of $\frac{100,000}{10 x num premiums I paid} - 1$. Otherwise, her rate of return is a loss of every penny we've invested in the policy. You refer to a $\beta$ "to the market" which has a very specific meaning: "correlated to the aggregate market, i.e. systematic risk." Remember, Stock B only pays you less in the event the market appreciates. Nov 19, 2013 at 19:55
• If they have the same risk then why does on pay more than the other? Nov 19, 2013 at 20:01
• They don't have the same "risk"... risk (from a perspective of $\beta$) is having a low payoff when consumption is low, and a high payoff when consumption is "high." Nov 19, 2013 at 21:22
• I guess when I talk about risk I think about the chance you will not get your expected return a.k.a the standard deviation. Maybe the market doesn't look at it this way. Nov 19, 2013 at 22:16

The standard deviation of an asset is less relevant if you hold a portfolio and not a single asset.

In the CAPM, every investor holds the same portfolio of risky assets - the market portfolio.

The CAPM implies further that the market portfolio has the highest expected portfolio return per unit of portfolio standard deviation of all possible portfolios.

This property leads mathematically to a positive linear relationship between expected return and beta, with expected returns for stocks with negative beta being negative as well.

Beta is negative if the correlation of the stock with the market portfolio is negative. So with your suggestion 2. you are probably not too far off.