I have just read this paper http://www.cfapubs.org/loi/doi/abs/10.2469/faj.v66.n5.3

In the paper they define financial turbulence formula as:

enter image description here

Could anyone help me calculate/understand this formula, maybe with a simple numeric example?

Numeric example you could improve:

Let's say we want to find the financial turbulence of Dow Jones Industrial at time t, and I have DJI monthly returns of last 24 months (monthly returns created by using month end prices):

DJIR = [0.03, 0.01, -0.04, ..., 0.015]


DJIR[1] is 0.03 (it means 3% return)
DJIR[2] is 0.01
DJIR[3] is -0.04
DJIR[4] is -0.02
DJIR[5] is 0.05
DJIR[24] is 0.015

Now let's say I want to calculate the financial turbulence d at t = 5

$$d[5] = (DJIR[5] - \mu) * Covariance$$

So what's $\mu$ here at t = 5? How to calculate it? Is it the average of returns until this point, or is it a moving average of the last n returns?

And what's Covariante here at time t = 5, I know Covariance is calculated like this: $\frac{1}{N-1}\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})$

But who is $x_i$ and who is $y_i$ in the formula?

P.S. here is another article that explains the formula, but it's not clear to me.


1 Answer 1


You wrote: $$d[5] = (DJIR[5] - \mu) * Covariance$$ but you left out half of it (the inverse and the transposed vector on the right side). The correct formula is $$d[5] = (DJIR[5] - \mu)^2 / Var[DJIR]$$

The covariance "matrix" becomes the variance in a 1-dimensional case (in other words $x_i$ and $y_i$ are both equal to DJIR[i] in this case) and the "matrix inverse" of a number just becomes one over the number. [That is why you DIVIDE by variance, you don't multiply]. For mu you can use mean[DJIR] over a very long period of time (many years) but you could also set it to zero (which is commonly done and that is what I would recommend, it wont make much difference). Of course the variance is the variance up to this point.

But the formula is intended for more than 1 asset. You already have DJIA, add some other assets. Then you will have a true covariance matrix. The whole point is that there are a lot of markets and this formula lets you watch over all of them simultaneously in a clever way. With 1 market it is trivial.

  • $\begingroup$ Let's keep it in one dimension, so just one asset. If $\mu$ is commony set to zero, how do you calculate the $Var[DJIR]$ then? $\mu$ is needed to calculate also the $Var[DJIR] = \frac{1}{N-1}\sum_{i=1}^{n}(DJIR[i]-\mu)^2$. Moreover even by using the $Var$, shouldn't the function become: $$d[5] = (DJIR[5] - \mu) * Var[DJIR]$$ $\endgroup$ Commented Aug 21, 2015 at 12:05
  • $\begingroup$ (1) Make $mu$ equal to zero throughout. Another idea is use an intelligent value for $mu$ such as 0.0003 (3 basis point per day is about 8% a year which is reasonable for a stock market index). But use the same value everywhere. (2) As you will recall from linear algebra, the apostrophe means vector transpose and $v v'$ means the inner product of v with itself. If v is one dimensional that reduces to $v^2$, not $v$ as you have it. Also look at the charts you linked: the value of d is never negative, That is because they are taking the square! $\endgroup$
    – nbbo2
    Commented Aug 21, 2015 at 13:55
  • 1
    $\begingroup$ Look, it is very simple and can be described in words, with no formulas: Today's squared return is divided by the average squared return of the most recent T days. That's it. $\endgroup$
    – nbbo2
    Commented Aug 21, 2015 at 14:08

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