Mean reversion

[narayananvenkat]

David..

I request you to to eloborate the term means reversion. What it exactly means and how it impacts VaR.

Your editgrids are indeed elegant and educative. But unlike your webcasts exclusively I find it hard to follow (although I reapeatedly watch them) and learn. You indeed touch upon them in your tutorials but my diffculty is that while we concentrate on the concepts and AIMS content you push into our mind it is definitely and equally necessary to explain in detail these edit grids seperately like you did for distributions. Hope i am not taxing.

I still cherish your grid on Montecarlo in 2007 tutorials simply superb.

venkat

 

[David Harper CFA FRM]

Hi venkat,

On mean reversion, you might find this previous post helpful
(technically, it can have multiple meanings)

But I'd say that for our purposes, consistent with Linda Allen, we only mean (pun!) two things by "mean reversion:"

1. The mean reversion in GARCH(1,1); i.e., the volatility is reverting to a long-run average level. Say, 10%, and today the daily volatility is 16%, then GARCH(1,1) models a time series that "pulls down" the volatility toward the long-run volatility of 10%. Such that, in our use of GARCH to forecast the forward volatility, because it mean reverts, if we forecast forward far enough, we expect the forward vol to equal 10%

2. We also can mean that the *returns* are mean reverting (and recall the returns^2 pretty much constitute our volatility). This is a violation of i.i.d. (violates independence ). So, here mean reverting implies "returns are not independent from day to day, but rather we expect a high return to be followed by a low return." And we can also call this negative serial (auto) correlation or negative autocovariance between period returns.

So your impact on VaR can be seen by the very first 2009 XLS
see the first 2009 FRM learning XLS here.

if you d/load that into Excel, you will see cell C9 is an input for Autocorrelation. If you change that to a negative (e.g. -0.25), then you are simulating mean reversion and you will see below that the mean-reverting VaR is LESS THAN the VaR given by the square-root-rule scaling. So, as L. Allen says, this mean reversion implies the square-r-rule will overstate the actual VaR. (in extremis perfect mean reversion implies no risk, zero VaR).

Thanks for your kind feedback on the editgrids. I really will do my best to screencast the spreadsheets along with the regular tutorials. I made a note to add something to the site which allows this sort of request, so that I can try to screencast especially those that are wanted. Thanks again for your feedback and nice words!

David

 

[narayananvenkat]

Hi David...

Thanks for your crisp note on means reversion. I am clear now.
venkat

 

[kik92]

Hi David,

I have a problem with the following explanation of mean reversion (extracted from Part 2 Meissner, Chapter 2 : Empirical properties of correlation : how do correlations behave in the real world)

If S(t-1) increases by a very small amount, (St-St-1) will decrease by a certain amount, and vice versa. This is intuitive: If S(t-1) has decreased and is low at t− 1 (compared to the mean of S, ), then at the next point in time t, mean reversion will pull up S(t-1) to the long term mean of S, and and therefor increase S(t) - S(t-1).

I understand the fact that due to mean reversion, S(t-1) is pushed up, however how does this increase S(t) - S(t-1). Intuitively I would assume that if S(t-1) is now higher, it would reduce the difference.

Thanks in advance.

 

[David Harper CFA FRM]

Hi @kik92 My view is that it's just a case of unclear writing, because the math is easier to follow than his description, which is also hard for me to follow. Clearly he is not referring to the negative autocorrelation type of mean reversion (reversion in returns) but rather reversion to a long-run mean level (for example, as in the omega term of the GARCH model, that is reversion to a long-run variance, not a mean, but it's analogous). Mathematically, he's just showing an inverse relationship between the Δ[S(t) - S(t-1) and the level of S(t-1). I just entered the below quickly to model his math (I think), see same at see https://www.dropbox.com/s/ljm5ixrl6mihmm7/0802-meissner-mean-revert.xlsx?dl=0

Super quick and messy but here i am starting a zero for both series, Non mean reverting (Blue) and mean reverting (orange MR). They both start at initial value of zero. In first column are random standard normals. The Non series is white noisy, it just adds the scaled (i.e., multiplied by 10, which is just an input) standard normal. The MR series is given by S[t-1] + N(0,1)*10*(1 - 50%) + (LRM - S[t-1])*50%, where 50% is the rate of reversion ("gravity"). My "MR contr" column just pulls out a copy of the mean reverting component, (LRM - S[t-1])*50%, so we see it's non-random contribution. So hopefully you can see that i'm assigning 50% weight to the same random component as found in the Non series but the other 50% is assigned the distance from the S[t-1] and the LRM (which is arbitrarily zero here).

So that's a long way to interpret Meissner but when he writes:

• "If S t − 1 increases by a very small amount, S t − S t − 1 will decrease by a certain amount, and vice versa. " This is a confusing sentence in my opinion which I don't think is directly related to:
• "This is intuitive: If S t − 1 has decreased and is low at t − 1 (compared to the mean of S, μS ), then at the next point in time t, mean reversion will pull up S t − 1 to μS and therefore increase S t − S t − 1. " ... This is illustrated by (eg) my MR series from step 7 to step 8: at step 7, S(t-1) is very low at -11.35 such that Step 8 will see a high contribution due to mean reversion (the positive contribution is +5.68 as the series is getting pulled back strongly to zero)
• "If S t − 1 has increased and is high in t − 1 (compared to the mean of S, μS ), then at the next point in time t, mean reversion will pull down S t − 1 to μS and therefore decrease S t − S t − 1." ... This is illustrated (eg) my my MR series from steps 3 to 4, where the high 11.84 is getting pulled back to zero strongly in step 4 with a -5.92.

All in, I think this is nice example of where words can fail if they aren't very precise and also it shows why we need quantitative proficiency. It's almost impossible to write about this if we don't grapple with the calculations. For example, I'm not convinced his math supports the first statement exactly "If S t − 1 increases by a very small amount, S t − S t − 1 will decrease by a certain amount, and vice versa." He asserts MR as a negative first partial derivative, full stop and that's all. It's just a directional assertion, it actually doesn't speak to elasticity or the actual slope of the relationship. I hope that's helpful! Thanks for a thoughtful comment ...

 

[ad17171717]

Hello David,

I have a question in regards to Meissner's mean reversion calculation of Dow Jones returns.

On the X axis, is he plotting the correlation between the Dow's returns at (t) relative to (t-1)?

And I don't understand what he's plotting on the x-axis. Is this the difference between 2 types of correlations?

Thanks for the help!

 

[David Harper CFA FRM]

Hi @ad17171717 Great question! It was not obvious to me at first glance but it can be deduced (and confirmed) by the formulas and examples. His axis labels are accurate: he is regressing the (periodic) change in correlation against the previous level of correlation. First, the key variable here is (a time series of) average correlation. He says "the averaged correlation of the 30 × 30 Dow correlation matrices was 26.15%." Imagine the Dow pairwise correlation matrix, that's 30*29/2 = 435 pairwise correlations, then take the average; this is what I think he means by averaged correlation. (Although frankly I'm not hung up on this calculation per se, there is more than one way to aggregate a correlation, the point is that we have an average correlation and it's measured in a time series).

Where C_t is the today's correlation and C_t-1 is yesterday's correlation, the regression is C_t - C_t-1 = ΔC = α + β*C_t-1. His displayed sample regression coefficients are α = 0.2702 and β = -0.7751 such that the regression equation is given by ΔC = 0.2702 + (-0.7751)*C_t-1.

If we let opportunistically let regression alpha, α = -β*µ, where µ is the long-run average correlation, then:

• ΔC = α + β*C_t-1 = (-β*µ) + β*C_t-1, then ΔC = β*(C_t-1 - µ). But if we want to negate β, then ΔC = -β*(µ - C_t-1).
• Thusly, Meissner's Figure 2.4/Example 2.1 has regression β = -0.7751 but that's negated into a reversion parameter of 77.51% such that we don't actually need to be informed of the long-run average correlation because µ = -α/β = -0.2702/(-0.7751) = 34.86% (cool, right!).
• Then given C_t-1 = 26.15%, we can predict ΔC = β*(C_t-1 - µ) = -0.7751*(0.2615 - 0.3486) ≈ 6.75% such that predicted C_t = 26.15% + 6.75% = 32.90%; or negating β to use it as mean reversion parameter: ΔC = 0.7751*(0.3486 - 0.2615) ≈ 6.75%.
• I said daily, but I see now the chart says monthly, so we can see the detail behind understanding why this is a regression (on the Y-axis) of the monthly change in correlation against (on the X-axis) the previous month's correlation level.

To summarize, Meissner starts with a Vasicek model (his equation 2.2) for the evolution of correlation that, without the stochastic shock, reduces to C_t - C_t-1 = ΔC = R*(µ - C_t-1) where R is the mean reversion parameter. This is identical to ΔC = Rµ - R*C_t-1 such that in the regression of today's change in correlation against yesterday's correlation level, ΔC = α + β*C_t-1, β = -R → R = -β and since α = Rµ → µ = α/R; e.g., R = -(-0.7751) = 77.51% and µ = 0.2702/0.7751 = 34.85%. I hope that's helpful!