# Risk Premiums vs Prediction Markets

aka **Why (some) market forecasts are biased and what can we do about them?**

tldr; When you care about an outcome, it will skew the odds you'll bet at. (To insure against bad outcomes). When everyone cares about an outcome, it skews the market odds.  
  
People (including [myself](https://www.lesswrong.com/posts/sAmkpdmfDMCohXiBX/scott-alexander-2021-predictions-market-prices)) often point at market forecasts as being the best predictors for some variable (inflation, the economy, whether England will win the Euros). 

However, market prices are more than just expectations. [In the words of an anonymous twitter account](https://twitter.com/macrocephalopod/status/1391867310592376834):

> Market price = Expectations + Risk Premium + Idiosyncratic Issues

This is a somewhat tautological framing however (since we don't know what risk premium or the idiosyncratic issues are other than the things which make this equality true). I'm going to try and explain what risk premium is, since people often focus too much on expectations and use idiosyncratic issues an excuse (fees are too high on PredictIt, you can't trade large size, etc..). Risk premium is important because unlike idiosyncratic issues which can be fixed with good market structure, risk premium will not go away. 

This matters, because **if we can't remove risk premiums and the idiosyncratic issues, then prediction markets are less accurate and less valuable for decision making**.

To this end, I'm going to start with some definitions:

**What are risk premiums?** Risk premium is *excess returns* which results from the *payout* being *correlated* with the rest of the world.  
  
**What are excess returns?** Excess returns are returns higher than 0 (or the risk-free rate depending on context) after accounting for the riskiness of the payout.  
  
Some examples:

*   a 1-1 bet on a 50/50 coin flip doesn't have excess returns
*   a 2-1 bet on a 50/50 coin flip has excess returns
*   a 10% 1y bond on a risky company may or may not have excess returns, depending on the default rate of the company and the recovery rate in event of default
*   the stock market has excess returns (probably)

**What is correlation with the rest of the world?**  
  
When an event happens, that says something about the world state we're in as well as the payout we get from the market. If the world state affects our utility in other ways than just the market, that is the correlation we're talking about.  
  
Some examples:

*   If I bet on a coin flip, aside from the money I have bet on the event, it has no effect on my well being.
*   If I bet on England to win the Euros, then I have some utility associated with the outcome *unrelated to my bet*. I will be happier if England win regardless of whether or not I bet.
*   If I insure my car against crashes then I have some utility associated with the outcome *unrelated to my bet*. I will be very sad if I crash my car.
*   If the economy starts doing badly, financial assets will start suffering, I will be suffering and so will everyone else.

### When does correlation lead to a risk premium?

(This is me writing out with examples what Cochrane explains more clearly [here](http://assets.press.princeton.edu/chapters/s7836.pdf) in Section 1.4. Particularly "Risk Corrections" and "Idiosyncratic Risk")

What matters is the size of the market, the ability to diversify and the participants involved.  
  
Let's start with the coin flip. If I am the only person in the world making a market in a certain coin flip, I might make the market .48 / .52 for $1 on heads. (That is if you want to buy heads, you need to pay 52c and I will buy heads from you at 48c). \[This is because 0.5 * U(wealth - .48) + 0.5 * U(wealth + .52) > U(wealth) (for my current wealth and utility)\] However, someone with much more money than me would look at my market and be willing to do the same trade for .49 / .51 because they are still going to earn some return. Someone with much more money than them will look at that market and offer .495/ .505 and eventually we end up with a very tight market around .50 the tightness effectively depending on how wealthy (and risk averse) the participants are.  
  
Let's now continue with the England example. Let's say we both agree the chances of England winning are 50%, my current worth is $10, I value England winning at $10 and England losing as -$8 and I have log utility.  
  
My current utility is $0.5 \cdot U($10+$10) + 0.5 \cdot U($10-$8) = 0.5 \cdot \log(20 * 2)$  
If we bet at 55/45 (me getting the raw end of the deal) then my expected utility is 0.5 * $0.5 \cdot U($10 + $10 - $.55) + 0.5 \cdot U($10 - $8 + $.45) = 0.5 \cdot \log(19.45 \cdot 2.45) ~ 0.5 \cdot \log(46)$. Therefore the market (between us) could end up clearing at the (incorrect) price of 55%. (Actually in this extreme example, I would be happy betting at up to 91%(!)). If everyone felt like me, the market might clear far from the probability everyone agrees on.  
  
However this market has many participants and they don't all have the same (extreme!) preferences. Therefore any difference between the clearing price here and the "fair" clearing price slowly gets arbitraged away until we have a good prediction market.  
  
What about the market for car insurance? Well, it's similar to the coin flip. I have some risk aversion associated with the cost of having to replace the cost of a car. On top of that I have the additional change in my utility which comes from having been in a car accident. So I am willing to pay more than the my estimated probability. However, there is someone else on the other side of this. When I go to renew my insurance there are many companies offering me car insurance. They are in competition, each with a large float and indifferent to whether or not I crash my car. Their only interest is their expected return. This competition drives the price closer to the true probability. (More on this later...)  
  
What about equities? Well it's similar to the coin flip. Uncertain money is worth less than certain money to me. It's also similar to the insurance example. More money in a good economy is less good than less money in a bad economy is bad for me. However, here we need to start thinking about the other side. Who is the other side of the equity market? Well, there are certainly people with much deeper pockets than me, so "uncertain money" doesn't drive them as much so the uncertainty value gets priced out. However, there are very few people out there who have pro-cyclical utility - (people who want more money in good times than they want less money in bad times). So the market clearing price (as a whole) will be less than the "fair" price. (This is one "solution" to the [Equity Premium Puzzle](https://en.wikipedia.org/wiki/Equity_premium_puzzle)).

### What does this mean for prediction markets?

Well for simple prediction markets: coin flip; sports events; weather(?); there are sufficiently many people who are either indifferent enough, or deep-pocketted enough that they will take on the risk for an ever decreasing edge. This means the market price should tend to the market participants' best estimate of the fair probability.  
  
For complicated markets: insurance, equities, inflation-protected bonds, it's not quite so simple to infer a probability/forecast/expectations from market prices.  
  
Let's start with **insurance**. We know that the probabilities which we infer from the insurance market are (broadly) an overestimate. (Insurance companies are profitable after all!) This would be fine if we knew how much of an overestimate they were. "Take the market prices... subtract off the profit margin from the insurers... done". Unfortunately, the insurance market is slightly more dynamic than that. (See the time varying loss ratios from [this fascinating set of slides](https://www.convexrisk.com/pirc/pirc-C.pdf)) 

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/02e38e5d3852af770314153418f2dedc22244a1adbd27f66.png)

Over time the amount of risk premium being demanded by the insurance companies waxes and wanes. Therefore we need to know the right adjustment to make for risk premium. (And arguably this is just as hard as predicting prices in the first place - no free lunch here).  
  
What about **equities**? If we know that equities broadly go up, can't we just subtract off that positive rate of return to use them as indicators for the real economy? Again, no. The problem is that the risk premium is time varying and knowing what the current risk premium is, is about as hard as valuing equities in the first place. No free lunch here.  
  
What about **TIPS**? TIPS are the classic example of using the financial markets as a forecasting tool. They are inflation-protected bonds issued by the US government. In theory, the difference between TIPS and nominal bonds will be the market's expectation for inflation. Certainly the difference* will realise to inflation. The question is, will the market clear at that price? The empirical evidence seems to be "not quite" **.

What about [**GDP-linked bonds**](https://brettongoods.substack.com/p/why-you-should-issue-gdp-linked-bonds)? Well, I can't think of anything more correlated to the real economy, so I would expect the risk premium of these (for global economies) to be very large (relative to their volatility). This would compare to nominal bonds which have a *negative* risk premium. This means for the country issuing them, they would be much more expensive than traditional debt issuance.

Their use as a forecasting tool would also be weakened, since they would systematically under-forecast GDP growth.

That said - I am still a fan of the idea. I would rather implement them as futures contracts (pretty much what Sumner advocates for [here](https://www.mercatus.org/system/files/Sumner_NGDPTargeting_v2.pdf)).   

### How bad can it get?

We can take some hope (or despair) from the [Hansen-Jagannathan bound](https://quant.stackexchange.com/questions/9340/intuitive-explanation-of-the-hansen-jagannathan-bound). This is usually used to say something about how volatile the state price density**** is. However, this volatility gives us an upper-bound for the Sharpe of any asset. (Including any asset we might use for prediction). If we are willing to assume*** that empirically the upper bound for the state price density is 0.5 then we know that $\frac{\mathbb{E}[\mathrm{estimate}]}{\sigma(\text{estimate})}  \leq \sim 0.5$. So there is at least hope that market prices will get us somewhere within an order of magnitude of the right answer.  
  
  
\* modulo some details regarding options  
  
\*\* there are a bunch of different factors driving this difference, some premium related some market inefficiency related, some to do with large flows in the market

\*\*\* this is a key assumption, but I'm quite willing to believe it, since no-one has found assets with much higher Sharpe than 0.5

\*\*\*\* For more information on the state price density and why this is relevant, the chapter of Cochrane [linked at the start](http://assets.press.princeton.edu/chapters/s7836.pdf) should be a good first port of call.