Why Risk-Free Carry and Risk Premiums Are Not Contradictory
Summary
The document examines whether a systematic risk premium could be eliminated by buying an underlying asset and selling a futures contract, and whether this would imply that expected returns equal the risk-free rate. The accepted response separates this proposed hedge from bearing uncertain market risk: a fully hedged cash-and-carry position has no uncertainty and, under no-arbitrage pricing, earns the risk-free rate when the futures price reflects financing over the holding period. That relation does not establish that risky assets themselves have no premium.
Other responses question how broadly the argument applies. One notes that an argument for expected return equaling the risk-free rate relies on a limiting setup with many uncorrelated, diversifiable stocks; another points out that a hedge requires a suitable relationship between the positions. The thread is conceptual and provides no empirical test of the equity premium. Its discussion also does not settle how risk premiums behave across assets or markets, so its conclusions should be read as an explanation of the hedged futures example rather than a general pricing theory.
Key ideas
- A fully hedged underlying-and-futures position can eliminate market uncertainty and earn a financing return under no-arbitrage pricing.
- The cash-and-carry relation does not show that an unhedged risky asset must have an expected return equal to the risk-free rate.
- Hedging depends on the relationship between the positions and cannot remove every kind of risk by assumption.
- The cited argument for eliminating risk premiums applies only under restrictive assumptions about diversifiable assets.
- The discussion is conceptual and does not provide empirical evidence resolving the risk-premium debate.
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# Debunking risk premium via "hedging" argument? (or why even in the real world $\mu$ should equal $r$)
# Debunking risk premium via "hedging" argument? (or why even in the real world $\mu$ should equal $r$)
Since I began thinking about portfolio optimization and option pricing, I've struggled to get an intuition for the risk premium, i.e. that investors are only willing to buy risky instruments when they are compensated by an add-on above $r$ (the risk free interest rate).
On the one hand this is understandable and backed by most empirical data.
On the other hand there is this divide between the risk-neutral world of derivatives pricing and the real world with real world probabilities.
The bridge between both worlds goes via a hedging argument.
I wonder (just to give one possibility of debunking the classic risk premium argument) if there really was something like a systematic risk premium $\mu-r$ nothing would be easier as to squeeze it out by simply buying the underlying and selling the futures contract against it. The result should be a reduction of the risk premium until it becomes $0$ and the trading strategy wouldn't work anymore.
The result should be that even in a risk-averse world probabilities become risk-neutral (i.e. the growth rate $\mu$ equals $r$) like in option pricing.
Is there something wrong with this reasoning?
EDIT OK, my original argument doesn't work. Anyway: I'm still feeling uncomfortable with the idea. My reasoning is that the whole idea of an inherent mechanism for compensating people for taking risk within a random process nevertheless (!) seems to be build on shaky ground. If it was true it would also be true for shorting the instrument since you are also taking risk there. But in this more or less symmetrical situation one side seems to be privileged. And the more privileged one side is the more disadvantaged the other side must be. It all doesn't make sense...
I think the only compensation that makes sense in the long run is $r$.
Please feel free to comment and/or give some references on similar ideas. Thank you again.
EDIT2 After a longer quiet period on that issue see this follow-up question and answers: Why should there be an equity risk premium?
## Answer by Richard Herron (score 5, accepted)
https://quant.stackexchange.com/a/1288
If you're long the underlying and short the futures contract, then you have no risk and earn the risk-free rate. You get into the position at $S_0$ and will be able to get out of the position at $F_0$ at time $T$. By a no arbitrage argument it must be that $F_0 = S_0 \exp(r T)$. I imagine Hull has a pretty good exposition on this.
The risk premium is because of uncertainty, and in this case there's no uncertainty. But imagine that everyone knows that 1/2 the time I will not deliver the underlying at $T$. Then you'll likely only agree to $F_0 < \frac{1}{2} S_0 \exp(r T)$ and if everyone knows that I 1/2 the time I won't deliver, then arbitrage won't restore $F_0 = S_0 \exp(r T)$.
## Answer by vonjd (score 3)
https://quant.stackexchange.com/a/25724
The most rigorous approach I have seen so far eliminating the risk premium is this one:
Emanuel Derman: The Perception of Time, Risk and Return During Periods of Speculation (2002)
Equation 2.23 on page 11 derives $\mu$ ~ $r$ but it only holds in the limit when you hypothesize countless uncorrelated stocks in a diversifiable market.
Still an interesting approach.
## Answer by mccarren (score 2)
https://quant.stackexchange.com/a/1282
I also find the "risk premium" idea unsatisfying, but I don't think your hedging argument works. Because hedging implies correlation, the risk free asset needs to be well correlated with the risky one in order for you to make any spread between the two. Are you saying that someone should be able to earn the risk premium (without taking any risk!) by being long equities and short treasuries, for example? I don't think that's possible.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.