# 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:

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Maybe I'm missing some point here, but I'm not sure how you're able to "squeeze out" the risk premium by using the proposed construct. Could you elaborate a bit? – Karol Piczak Jun 5 '11 at 21:13
@Karol: see my editing the post. – vonjd Jun 6 '11 at 11:05

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@vonjd Just to clarify one point, the reason for the asymmetry between long and short is that the equity market in aggregate must be held long by all participants. This is not the case, for example, with commodity futures, which is why we do not find a consistent risk premium to holding commodities. – Tal Fishman Jul 9 '12 at 15:50

## 2 Answers

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 $F_0 = S_0 \exp(r T)$.

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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.

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