The signal of the day comes from finance. That is, we are going to look at an application
of signal processing to financial data. More specifically,
we are going to explore the time series of stock prices of
Lehman Brothers around 2008. Lehman Brothers was a global
financial services firm, and one of the largest
investment banks in the US. We have to use the past tense here,
because on September 15, 2008, after more than 150 years of
existence, the firm filed for bankruptcy. How could it be that such a large and
respected bank could suddenly go bust? The bankruptcy is just the final
act in a series of events that ultimately led to it. Lehman Brothers was highly
exposed to the US housing market, in particular to a type of loans
known as subprime mortgages. These loans were typically granted
to clients who were not able to afford standard loans due to their
limited financial capabilities. With the burst of the real
estate bubble in the US, these clients were the first ones
to default on their mortgages. That is,
they stopped paying back the banks. This led to huge financial losses for the
banks that were exposed to this market. In 2008, these losses had another even
larger impact on Lehman Brothers. Suddenly, investors and
other banks started wondering whether Lehman Brothers had the financial
means to face its own obligations. From there the situation
started spiraling, losses induced investors withdrawing their
money, which lead to even more losses. On September 15th 2008,
facing liquidity problems, Lehman Brothers had no other choice
than to file for bankruptcy. Now that we understand the origins
of this bankruptcy, let us have a look at the price of the Lehman Brothers
stock around this major event. In this plot, we are showing the Lehman Brothers stock
price between the years 2004 and 2014. This plot clearly shows a drastic drop in
the stock price at the time of bankruptcy. This happens because,
in the event of a bankruptcy, stockholders have the lowest priority in
claiming remaining assets of the company. Let us have another look at
the signal through a different lens, namely, we make a display
on the log scale. With this plot it is better to see how
the value evolves after the bankruptcy. The value does not go to zero. Why is that? Holding securities like this one is
associated with a high level of risk, due to the possibility that
the investor will never recover his investment after bankruptcy. Thus the securities trade
at the discounted value, typically a couple of cents. At just the same time, these securities
present high return potential. Indeed, just imagine you
bought the stock when it was worth only a thousandth of a dollar,
and selling back at one-tenth. In that case,
an investor would get a return of 1,000%. Certain investors specialize
in these type of investments, referred to as distressed investment. To cope with these situations, a tool able to detect abrupt changes in
stock prices would be highly desirable. Consider, for example, the simple spike detector
whose block diagram is depicted here. The first block is a moving average. This moving average, we have already
seen when we analyze temperature. It simply computes the average
of the signal over the last capital L observations. Its output is a smooth or so called low
pass version of the original signal. By subtracting the low pass version
from the original signal down here, we are extracting the high pass component. The later goes through a thresholding
operator that is triggered if a value is greater than
a certain threshold. Let us apply our spike detector on
the Lehman Brothers stock price. These plots show the low pass and high pass components for
a short moving average window. The window size is L is equal to five. With a short window,
our spike detector is prone to react. But we might as well be catching
false events, in other terms, noise. By increasing the length of
the moving average window, like here, L is equal to 50, the algorithm
becomes less sensitive to noise. This comes, however, at a certain price. The algorithm takes longer
to react to an event. Our spike detector is very rudimentary,
and we would not advise anyone to use it
to manage his or her investments. Besides the tradeoff between reactiveness
and sensitivity we have already mentioned, there is a major drawback
in this approach. Our algorithm is only backward-looking, so the detection of an event will always
take place after it has occurred, possibly a couple of hours or
even days after it has happened. At this point, market participants have
most likely already concluded that the major event must have happened. The other problem is that we might not
have all the good tools at hand to process such a signal. It is natural to introduce some randomness
in the modeling of stock prices. As evolution of the stock does
not exhibit clear patterns. But the underlying statistical
characteristic might be more stable over time. With random signals, certain tools that
we have developed are still valid, like filtering. But we also need to introduce new tools, which is the subject of another class
on statistical signal processing. Coming back to Lehman Brothers,
this signal is of particular importance, as it was the largest
bankruptcy filing in history, surpassing those of other
giants like WorldCom or Enron. The Lehman Brothers bankruptcy
is also thought to have considerably intensified
the 2008 financial crisis. It was really a crisis of trust between
the different market participants. Banks were reluctant to
lend money to each other, as another bank could
possibly also go bankrupt. This created a considerable drop in
the liquidities available to banks and other firms, and
contaminated the rest of the economy. What started out as a burst
of the real estate bubble in the US evolved into a worldwide
global financial and economic crisis.