# Risk-neutral Probability, Risk-Adjusted Returns & Risk Aversion

When we employ the Fundamental Theorem of Asset Pricing and the existence of an equivalent probability measure, say $$Q$$ with respect to the historical probability $$P$$, we often say the expectation under this risk-neutral probability is "risk-adjusted" payoff or premium.

I often resort to an explanation that the "risk adjustment" occurs, because the actual world we live in, traders are not risk-neutral, so by assuming an arbitrage-free economy and the aforementioned fundamental theorem, we are essentially making use of the existence of the equivalent probability measure where the synthetic construction is uniquely possible (also assuming completeness).

1. Do you care to share a better and more intuitive way to explain the "risk-adjustment" in this context?

2. In finance, we often use "risk-adjusted" return on an investment, and some students without math finance background often get confused when these terms are equivocated. How would you relate two concepts on an intuitive level?

It would make total sense for you to quote the price of a financial instrument by discounting its future cash flows according to your own risk aversion ($$\Bbb{P}$$ measure, stochastic discount factor).

However, it would make it complicated for me and you to agree on that price hence developing a liquid market.

That being said, if we are both rational and agree on the existence of a so-called "risk-free asset" (+ market completeness) then we would both agree on what price to charge right (the price of the self-financing replicating strategy of the instrument). Basically what we've done here is eliminate our risk aversion by doing some "ketchup economics" (relative pricing) under the strong assumption that such an asset actually exists.

Of course we've now transposed the debate to whether this asset exist and do we agree on what it is? For collateralised transaction it is clear it's the PAI, but what about uncollateralised transactions? So I guess the question is whether we would like to price under $$\Bbb{P}$$ which is the true measure but "unknown" to, or under a mathematical construct $$\Bbb{Q}$$ which is known but only exists if we agree on some key assumptions, where our own risk-aversion disappear and we can "talk the same language".

• Excellent, thanks for the response. Only agents were risk-neutral.... :D – Frank Swanton May 12 at 23:50

My thoughts:

• Risk-neutral probability measure $${Q}$$ is a convenient mathematical tool that is used primarily for pricing derivatives

• The price of a derivative is essentially the price of the replicating portfolio. So to price a derivative, one can attempt to build a portfolio that replicates the derivative pay-off at maturity and then work backwards in time, to arrive at the price of the replicating portfolio at inception: this is then the price of the derivative (the usual delta-hedging method)

• Under the risk neutral measure $$Q$$ (if it exists & is unique), one can avoid the above cumbersome technique, and instead, one can simply take an expectation $$\mathbb{E}^{Q}$$ of the derivative pay-off at maturity and discount it, to obtain the same derivative price: the fact that this price is identical to the price arrived at using the replicating portfolio technique, shows that the risk-neutral measure is a mathematical tool that merely allows to simplify the computation of the derivative price by not having to compute the replicating portfolio directly at every time-step

• Under the risk-neutral measure, one doesn't care about the distribution of prices, one only cares about the expectation: that is because the discounted expectation under $$Q$$ gives prices, whilst the distribution doesn't really have a physical probabilistic meaning (i.e. let's consider B&S model vs. Bachelier model for a simple option: under $$Q$$, both give correct option price in expectation but the distributions are totally different and in a way, irrelevant).

Historical realized data can be regarded as a sample that can be used to estimate some real world probabilities, i.e. distributions under $$P$$. Some examples:

• Historical default frequencies of US corporate bonds are consistently lower than the credit spread of these bonds on US treasuries. In turn the credit spreads are driven by the prices at which these bonds trade: so under $$Q$$, the market over-estimates the credit risk on these bonds, but a better way of looking at it is that the market as a whole is able to estimate the default frequencies of these bonds under $$P$$ and it simply demands a premium to hold these bonds

• CDSs: again, CDS prices tend to over-estimate the default frequencies across all credit ratings. This shows that the CDS writer demands a risk-premium to write these CDSs

• Implied Vols: there is a proof that when an option seller delta-hedges an option, he will lose money if the realized vol is higher than the implied vol that he charged at option inception. This is an argument for why the implied vols contain a risk premium and why they should constantly over-shoot the realized vols of the underlying

So the above examples are meant to demonstrate that under $$Q$$, the market tends to charge some risk-premium, so that market participants are compensated for holding risky instruments.

This makes sense to me: a risk-neutral agent would not care between holding a portfolio of junk bonds or a AAA-rated bonds, as long as in expectation, these portfolios would provide the same return. But in the real world, the credit spread on junk bonds not only over-estimates the default frequency (same as the credit spread on AAA bonds), but the over-estimation is so pronounced that the realized return on the junk bonds is higher than on the AAA bonds: again, that is because in real world, people need an incentive to hold junk bonds, otherwise they'd just hold AAA bonds.

• Thanks for the response, Jan. Much appreciated! – Frank Swanton May 12 at 23:50
• In particular, your explanation on how taking the expectation under the risk-neutral probability to avoid the taxing process of computing the replicating portfolio is very intuitive. – Frank Swanton May 12 at 23:58