ch3. measuring and monitoring volatility

[wooju7533]

Hi
I'm confused with fat tails in a return distribution & conditional and unconditional distribution
1.
p 9. "but fat tails could be explained by a conditional distribution"
→i thought conditional distribution is normal so it could be explained , is it right?

2.
p.9 "time variations in either mean or volatility could give rise to fat tails in the unconditional distribution, in spite of the fact that the conditional distribution is normal.
p.11 "it makes more sense to assume that asset returns are conditionally normally distributed. However, caution has to be applied, since in reality, asset returns are generally non-normal, whether unconditional or conditional
→ i thought conditional distribution is normal except asset return, is it right?

 

[David Harper CFA FRM]

Hi @wooju7533

1. "conditional" distribution does not necessarily mean conditional normal distribution: the conditional distribution could be any distribution. We could assume a heavy-tailed conditional distribution. But the normal GARCH(1,1) is our classic example of a model that happens to assume conditional returns are normal but implies unconditional returns are heavy-tailed.

2. a. Same concept: if the conditional returns are normal but either the mean and or constant are time-varying, then the unconditional (returns) distribution is probably heavy tailed

2b. This is just saying that when modelling we tend to assume conditional returns are normal, like we do with GARCH(1,1), however we do so knowing that empirically observed returns (conditional and unconditional) are not normal. This is just a statement that reality (empirical returns) does not match our clean parametric normal distribution.

So the key ideas here are:

• The difference between conditional versus unconditional (return) distributions, and the fact that our models tend to make convenience assumptions (e.g., normality) about conditional returns
• The difference between our parametric model assumptions (e.g., normal returns is a parametric or analytic distributional assumption) versus empirically observed (aka, non parametric) distributions.

please see https://forum.bionicturtle.com/threads/p1-t4-323-the-shape-of-asset-returns-allen.7149/post-81005 i.e.,

Hi @Soham Routh It's not my favorite question (323.3) that I ever wrote and there is a lot of theory underneath this (eg., GARCH) and I am currently lacking much time, so I need to be brief:

• First option is incorrect since conditional returns can be normal: Correct. Conditional returns can easily generate unconditional heavy tails
• (B) is plausible, it's just that L. Allen doesn't think it's the best explanation. If we have conditionally normal returns, but the mean varies, then the unconditional distribution can have heavy tails. So, a time-varying (conditionally) normal distribution can generate heavy tails.
• The problem with (C)--ie.., ("Conditional returns must be normal due to the central limit theorem (CLT)")--is that CLT doesn't apply: CLT doesn't make a statement about conditional returns. CLT is about the distribution of a sample average
• Re: "time-varying volatility imply heavy-tailed conditional returns?" I do not think so. Going the wrong direction. We typically specify the conditional distribution. Then its parameters either vary or are constant, over time. The unconditional distribution refers to the view over the whole multi-period. We can have a non-normal distribution (including light or normal tails) and it's variance/volatility changes over time such that both conditional and unconditional distributions are heavy tailed. It's hard to wordsmith time series. Time series is heavily mathematical; it is very difficult to discuss fully without math. I hope it's a start, i can't get too bogged down in this now. If you want to go deeper, I would recommend reading the source Linda Allen for an introduction. Then to really get technical, much of my understanding I owe to the excellent book Asset Price Dynamics, Volatility, and Prediction by Stephen Taylor. Thanks,

 

[Bluefox21]

Hi @Nicole Seaman - looks like a measuring volatility video has also been uploaded to the Operational Risk section of Valuation and Risk Models.

Kind regards,

Rob

 

[Nicole Seaman]

Hello @Bluefox21

Thank you for pointing this out. The video was mistakenly uploaded to the Advanced Part 1 study planner. I've removed it.

Thank you,

Nicole