The correct probability distribution / way to identify large deviations in a set of daily changes to portfolio value

Question

I am working on a report which is being sent through to end users that should flag to them any "large changes" in the day-to-day values for the past 30 days. These values are day-to-day differences which we assume to have a mean of zero.

To shed a bit more light on the data, the values are proportional returns to a portfolio. They are used to measure the change in the riskiness of different aspects a portfolio which is constantly managed, so the riskiness should always stay at "round about the same level" based upon the investing strategy associated with that given portfolio.

That being said, the risk will fluctuate a bit from day-to-day, but it gets re-balanced and so the change in daily risk should fluctuate around zero - positively for roughly half the time & negatively for roughly half the time.

This analysis is to simply point out any large daily deviations so they can be examined.

So, assume we have the following data:

Day:    Change from previous day:
1       -40
2       30
3       15
4       12
5       -34
6       -2
...
30      12

And, as I stated earlier, they don't care about the direction of the change, just to flag any day where the change is of a "statistically significant" magnitude.

I'm curious as to how to calculate the statistics for this properly / what distribution would best be used to assign probability levels for this set of data.

What has been proposed is to, firstly, look at the square of the changes (rather than the original values since all we care about is magnitude):

1. To calculate the std dev as $\sqrt{ \dfrac{\sum \left( x_i - 0 \right)^2 }{29}}$ - In other words assume a zero expected value
2. To Calculate the std dev as $\sqrt{ \dfrac{\sum \left( x_i - \bar{x} \right)^2 }{29}}$ - In other words, use the mean we would calculate for this sample

And then to take each day's value and divide it by this std_dev to calculate it's magnitude.

I have quite a few problems with a few things here, but especially idea #2 since I believe the mean SHOULD BE ZERO.

Generally speaking, what is the correct way to analyze deltas with an expected value of zero and what distribution would you use for a 30-sample data-set to find statistically significant values? Note that historical data is available / can be easily gathered.

If we assumed the daily changes to be ROUGHLY normally distributed, would the square of them (only measuring deviation from zero, regardless of whether it's positive or negative) be distributed via Chi-Square? ... Or could a folded-Normal work if we took their absolute value?

I'm just looking for something to show how change magnitude might be distributed... Nobody would take any issue with assuming the changes to be normally distributed if necessary...

I hope this makes sense, but, if not, please let me know where I could clarify more.

THANKS!!!

(I originally asked this question here and was recommended to ask it here - Hopefully someone here can give me a good response)

Answer 1

A simple approach would be to apply a bootstrap technique to derive a distribution directly from your own real data. This has the advantage of being easy to visualize and explain to your constituents.

Such a process would look like this.

1. Rank order the magnitude of changes
2. Assign a percentile based on the rank and number of samples
3. Make a line plot of this data series with percentile on the y axis and magnitude of change on the x axis

Now (given a satisfactory number of data points) you can quickly see what percentile any given change falls in. The percentile threshold that you define as "statistically significant" is yours to make. If you set it to 0.95 you will see one change above your threshold every 20 days, on average - even if these changes are due to random noise.

Speaking of random noise, you'll want to try to suppress that. In business, one-day peaks are typically not as interesting as more sustained changes. I would suggest smoothing your data using a Gaussian (normal) distribution centered on the day of interest (NOT a straight rolling average). The point of this is to reduce "random" variations in daily changes and focus on someone longer-term shifts. This is called a low-pass filter in signal processing. (You can also apply a high-pass filter if you want to ignore very-long-term trends like BAU growth or decline and focus on medium-term changes limited to within your monthly scope)

Answer 2

Presently more of a comment but it's a bit long for a comment and likely to grow:

Usual models for returns suggest that the variability of returns (and hence the "outlyingness" a particular observation value might possess) varies over time (and in particular, is correlated over time).

With that in mind, you'd need to consider whether

(a) it's actually those changes in volatility you're trying to identify, or

(b) its outliers with respect to the changing volatility you're trying to identify, or

(c) you think that understanding of returns doesn't apply