**Introduction**

In this article, I would like to quantitatively analyze Facebook (NASDAQ:FB) versus its benchmark index Nasdaq (NASDAQ:QQQ), to highlight interesting aspects of the co-relationship and propose some trading ideas as a result. The analysis will start with some graphs, and will then dive into a risk analysis, comparisons of different statistics, co-moments, risk and alpha premiums of Facebook, active returns, capture ratios, average length and size of Facebook's drawdowns, as well as probabilities for outperformance.

As a goal, Facebook investors may understand the co-relationship better and can identify statistic drawdown levels which can provide good trading setups. Using the information provided can also include hedging via the Nasdaq, amplifying mean active returns at favorable trading setups or even decreasing downside risk.

**Analysis**

As usual, I start off with some descriptive, introductory charts to highlight the relationship between the two securities. Below, you find the first one depicting the returns of both Nasdaq and Facebook. As everyone who somehow follows the company knows, Facebook had a rough start to trading as concerns about monetization of users resulted in lower trading prices.

Facebook, as a result, witnessed a roughly 50% drawdown from its opening prices but followed an almost linear upward trajectory since 2013. Below, we can see the cumulative returns, daily changes and drawdowns of both securities. One thing that certainly catches the eye is the relatively low drawdowns of Nasdaq when compared with Facebook, but then again that should be so by definition.

Looking at the 1-year price chart, however, we see that the drawdowns have become more similar and so have the returns. While Facebook still shows tremendous outperformance, at this point, we can assume the outperformance is getting less. Interesting to see is also that the up and down movement patterns are getting more similar, i.e. the correlation is getting higher.

To get a better understanding of the development of the return correlation between the two, please find the rolling correlation chart below:

**Plotting both in the risk-reward space**

It is needless to say that the Nasdaq per definition has a lower risk in almost any measurement than Facebook. However, to understand how the relationship plays out a quick look at the risk-return matrix should help:

While Facebook has posted an annualized gain of 32.13% since 2012, the Nasdaq has done half of it with 16.36%. The standard deviation of returns during that period was 41.15% and 15.61%, respectively. This results in a much better Sharpe Ratio for the Nasdaq of 1.05, while Facebook posts 0.78. Facebook with its beta of 1.08 thus has an active return of 15.77% over the Nasdaq.

For the YTD period, things have changed a little bit. While Facebook still posted phenomenal annualized returns of 27.81%, Nasdaq is now down to 4.84% as of last Monday's close. At the same time, the standard deviation of Nasdaq has gone up slightly to 17.46% while Facebook's has actually gone done to 30.34%. YTD, Facebook has the better Sharpe Ratio of 0.91 while the Nasdaq is down to 0.28. The beta remained at 1.08 and the active return increased to 22.97%

Over time, the risk-reward tradeoffs have developed for both as follows:

We can see that Facebook's standard deviation and returns have been steadily decreasing from 2013 until 2015, from when onwards we saw an increase again in both measurements up until last Monday. The Nasdaq-100, however, has little changed from 2013 until 2014, had a deteriorating Sharpe Ratio until 2015 and increased it slightly for the 12 months ending September 26th 2016. We can thus witness how Facebook, becoming a more and more mature and trusted company has mapped this development in the risk-return space.

On shorter time periods, namely for 12, 36 and 60 days, the comparison looks as follows:

**Return and risk comparison**

Facebook's average mean daily return is an astonishing 0.14% compared to 0.06% for the Nasdaq. The median return is 0.11% and 0.08%, respectively. The 1st quartile of returns (25% of returns are lower than that) are -1.08% and -0.41% for Facebook and Nasdaq, respectively. The 3rd quartile (75% of returns are lower than that) are 1.30% and 0.62%, respectively.

Year-to-Date, mean returns are 0.12% and 0.03%, respectively, with a median return of 0.12% and 0.07%. In the quartiles, we see a little shift in that the 1st quartile is now -0.60% and -0.46%, while the 3rd quartile is now 0.87% and 0.55%, essentially highlighting that both securities exhibit more centered returns for the year-to-date period.

For any given trading day, Facebook has a 47.2% likelihood of closing negatively, while the Nasdaq has a lower 45.68% likelihood. The histogram of daily returns corrected for outliers look for both as follows:

**Drawdown stats**

As stated in the introduction and quite frankly very intuitively, Facebook's drawdowns obviously have outsized the ones of the Nasdaq. But then again, if we look closer, we can also see that they have a tendency to getting more similar recently. For the 2016 period, Facebook's drawdowns seem to have actually been lower than Nasdaq's. This is certainly partly due to the fact that the macro concerns that dragged the benchmark indices lower do not concern Facebook's business model and investors that much. Facebook's five biggest drawdowns look as follows:

If we divide the drawdown size by the number of drawdowns, we learn that Facebook's average drawdown is 5.5%. The average length of a drawdown is 20.2 days, while the average recovery is 11.8 days. That means that it takes Facebook on average 8.4 days to reach the bottom of a drawdown.

Put differently, Facebook's stock spends 41.6% of the time on the way down and 58.4% of its drawdown time recovering from the bottom. Nasdaq, on the other hand, spends 43.6% of the time to get to the bottom and 56.4% to recover fully.

If we divide the mean return over the average drawdown, we get a further risk-return understanding of the two securities. Facebook has a mean return/drawdown ratio of 5.84, while Nasdaq has 7.49, thus rewarding investors more for its drawdowns.

If we weight the drawdown, however, by the number of days they took (and not simply taking an average of the number of drawdowns), Facebook has a mean drawdown of 11.6%. This is certainly very skewed as heavy weight is put on the first tremendous drawdown in 2012 and I would thus prefer the former number of 5.5% to count with.

Maximum drawdown ever, by the way, were 53.6% and 16.3%, respectively. Similarly, the expected tail loss (95%) are 5.36% and 2.30% for Facebook and Nasdaq.

**Upside versus downside ratios**

Let's analyze how much upside Facebook catches over the Nasdaq, or in other words: How much more can we earn on positive days when we go for Facebook instead of Nasdaq?

With Nasdaq as our benchmark and having a return correlation of 0.41, Facebook captures the factor of 1.24 to the upside, while only capturing a factor of 1.047 to the downside. Stated in a different way, on days when the Nasdaq traded positively, Facebook earned 1.24 times the return of the Nasdaq.

On the other hand, on days when the Nasdaq traded negatively, Facebook lost the factor of 1.047. While this is indicative of an elevated beta stock, the ratio of upside capture over downside capture (divide the two to get a value of 1.18) is highly favorable and enables the stock to be employed as a portfolio leverage.

When the Nasdaq had a positive day, in 70.6% of all cases, Facebook traded positively, too. On the other hand, when the Nasdaq was negative, Facebook was only negative in 68.4% of all cases.

Furthermore, on positive days in the Nasdaq, Facebook outperformed in 51.3% of all cases, while outperforming in 45.7% on days on which the Nasdaq was negative.

All of the above tells us great lengths about the relative relationship between the two. First of all, Facebook's returns are very much leveraging upside as compared to Nasdaq while leveraging the downside incomparably less. In 7 out of 10 cases, Facebook joined the Nasdaq in ending a trading day positively, and in these 7 cases, it did so remarkably well.

**Probability of outperformance over various time frames**

Using the previous return matrix of both securities, I analyzed 7 different time periods for calculating the likelihood for outperformance. The time periods analyzed contain the probabilities of outperformance on any single day, for a 3-day time frame, for 6 days, 9 days, 12 days, 18 days and 36 days.

The likelihood that Facebook outperforms the Nasdaq on any single day is 48.71%, which is intuitive as it relates to the stats above. However, for a 3-day time frame, the likelihood is 50.8%; for 6 days, it is slightly higher at 51.24%; for 9 days, 53.42%; for 12 days, it is 55.61%; for 18 days, it is 58.92%; and finally, for a 36-day time frame, the likelihood that Facebook is outperforming the Nasdaq is 61.00%.

So for an almost 2-month time frame, Facebook is more than 60% likely to achieve better returns than the Nasdaq. Combine that with the stats on upside capture then that's a bet I am willing to make, even if I am cautiously aware of the drawdown and company-specific risk topic.

Below, as usual, the graphical interpretation:

**Conclusions**

Facebook has shown immense strength supported by stellar top and bottom line growth and in my opinion, is likely to continue to do so. This is not a fundamental piece, but I strongly would assume that should Facebook continue its strength in the monetization domain that these favorable risk-return metrics will hold. From a technical perspective, an old saying is that strength begets further strength. Facebook is also a relatively high beta, high standard deviation stock.

However, we have seen lately that not only classic risk metrics such as standard deviation, but also drawdowns and recovery lengths have aligned more with the Nasdaq, all amid still spectacular recent gains. Thus, Facebook is an ideal candidate for generating active return or alpha and while being aware of company-specific market risk, Facebook can leverage a portfolio via an active trading strategy.

For instance, that could be one that shifts a greater proportion to Facebook's stock in the event of bottom forming during general market downturns and rebalancing to Nasdaq towards the end of a 36-day time frame.

Risks to the thesis are certainly that Facebook is a very crowded trade and many people have boarded the train. On the other hand, however, I believe there are also many people waiting to jump on as soon as drawdowns occur. In this article, I intended to provide some statistics and ratios for people who are already invested, but also statistics on drawdowns for those who are looking for reasonable risk-adjusted entry levels.

From a quantitative perspective, starting to add positions at 5.5% drawdown seems reasonable and is statistically supported. Thus, at current prices, entering the stock for those who have had an eye on it may be the support level of 122 USD.

**Disclosure:** I am/we are long FB.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.