# What is a good way to interpret covariance under risk neutral measure?

Shreve mentioned that the forward and futures spread depends on the covariance of the underlying and discount factor under the risk neutral measure. Can anyone explain how to interpret this covariance under risk neutral measure? What does that mean intuitively?

• Indeed, your question is an exercise !!
– user16651
Commented Sep 4, 2016 at 18:00
• Which one are you referring to? Commented Sep 4, 2016 at 18:24

Interpreting risk-neutral moments is tricky. Personally, I think the only good way to avoid heuristics and confusion in doing so is to always use the phrase "from the perspective of a risk-neutral agent". A risk-neutral agent is just a weirdo who thinks that the probability of a hurricane breaking out tomorrow is 10%. You would disagree with him on that, of course, but - interestingly - agree on the price of the hurricane insurance! For this weirdo, the covariance between the underlying of a futures and the discount factor is equal to the forward-futures spread; for you, it is not.

• Why would a risk-neutral agent think the probability of a hurricane breaking out tomorrow is 10%? Why would such an agent not be willing to pay an exorbitant amount for insurance? You seem to be confusing risk and danger. Commented Nov 2, 2017 at 22:46
• @Acccumulation This is just an example. 10% is to stress that the probability of a hurricane perceived by a risk-neutral guy who purchases the insurance for the same price as you do is much larger than the natural probability. Italics here is crucial: you should understand that the notion of risk-neutral in the asset pricing context is always a function of the observed prices: you and risk-neutral agents agree on the prices, but disagree on the probabilities. So, no exorbitant price is possible, just because you do not pay any.. Commented Nov 3, 2017 at 7:55

I don’t understand the complication of “risk neutrality” either temporally (risk, in the sense of uncertainty, is obviously dynamic) or in terms of agency (two agents may agree or disagree on a risk assessment; if they agree to disagree we call them counter parties). Covariance is most important as a step in building a diversified portfolio, you explore the covariance of potential assets to reduce overall portfolio risk while maintaining the expected value of overall portfolio return.