Macroeconomic Surprises, Zero-Mean Assumptions, and Fat-Tailed Risk

# Why do surprises in macroeconomic variables average out to zero?


# Why do surprises in macroeconomic variables average out to zero?












In the book Investments (Bodie, Kane, Marcus), in chapter 8, the authors discuss index models (page 247) and, in its context, systematic risk. The authors state, without explanation, that the market factor m of the unanticipated part of the realized return will have a mean of zero because, over time, "surprises will average out to zero".

I am unable to understand how the surprises would average out to zero. I believe my confusion can be understood through these questions:

- The statement implies that, when looked at over a long time horizon, firms come close to accurate predictions of expected return. However, it has been widely claimed that this is not the case. So how can we reconcile these two facts?

- Considering the above question, how can any macroeconomic variable reasonably proxy for unexpected developments in the economy?

- Could you please also show how the variance of the market factor can be estimated?

To give you an understanding of my background, I am an undergraduate enrolled in an introductory investments course. As such, my understanding is limited, and I would appreciate it very much if you could point me to other resources that might help me tackle my skepticism for security analysis and understand this concept more thoroughly.

Thanks for your time! I look forward to reading your answer.

## Answer by Martin Vesely (score 2, accepted)

https://quant.stackexchange.com/a/53615

Your confusion is probably caused by these two facts:

- In theory, a surprise is descirbed by a random variable with so-called standard normal distribution having standard variation equal to zero. This comes from observation that during good times this is the case and any market fluctuation can be described by such distribution.

However,

- In practise surprises (or rather shock), have a huge influence on market (think about Great depression in 1930's, oil shocks in 1970's , crisis after Lehman's fail, Covid-19 pandemic etc.). Such shocks cannot be described by normal distribution as large changes are extremelly impropable under normal distribution assumption. Moreover, the real distribution is skewed, i.e. negative shock are much bigger than positive - have a look at any share index development - huge drop during crisis (in days and weeks) and only gradual recovery after (in years).

So, the problem is the difference between theory and reality. To have a better model of reality, you should switch from normal distribution assumption to so-called fat tail distributions.

But there are some reasons why normal distribution is still widely used (personally, I do not agree with them and I am an advocate of fat-tail distribution models):

- when there is no crisis, models with normal distribution works well

- normal distribution is mathematically pretty, i.e. models with it are simple and easy to calculate

- in contrast to fat-tailed distribution, the normal distribution is easy to understand even for non-mathematican - you have probably already heard about Gauss or bell-shaped curve (i.e. a graphical ilustration of normal distribution)

I hope this shed some light on your issue.