Question

What formula exists to determine whether the stock market will crash next week?

How can classical probability help us estimate the chance Amazon goes bankrupt in 2021?

What formula can help me determine whether my local newspaper sales ads are effective?\

Answer 1

What formula exists to determine whether the stock market will crash next week?

A stock market crash is a sharp and quick drop in total value of a market with prices typically declining more than 10% within a few days. Famous examples of major stock market crashes are the Black Monday in 1987 and the real estate bubble in 2008. A crash is usually attributable to the burst of a price bubble and is due to a massive sell-off that occurs when a majority of market participants try to sell their assets at the same time.

The occurrence of price bubbles implies that markets are not efficient. In inefficient markets prices do not always reflect fundamental asset values but are inflated or deflated based on traders’ expectations. These expectations are reinforced by traders’ subsequent actions which further inflate (or deflate) prices. This leads to positive (or negative) price bubbles which eventually burst. This phenomenon was described by George Soros as reflexivity and is the basic assumption for forecasting methods used in technical analysis.

Today, there is not much debate over the existence of bubbles in financial markets. However, understanding these inefficiencies and predicting when price bubbles will burst is a highly difficult task. Imagine you could identify a bubble that is about to build up and predict when the market will crash. You would not only be able to make a profit while prices are increasing but also to sell at the right moment to avoid losses.

Some mathematicians and physicists attempted to tackle this problem by investigating the mathematics behind price structures. One such physicist is Professor Didier Sornette who successfully predicted multiple financial crashes [1]. Sornette uses log-periodic power laws (LPPLs) to describe how price bubbles build up and burst. In essence, the LPPL fits the price movements leading up to a crash to a faster than exponentially increasing function with a log-periodic component (reflecting price volatility with increasing magnitude and frequency).

And this is where the idea for this project is coming from. If the recurring price structures found by researchers exist, should it not be possible for a machine learning algorithm to learn these patterns and predict crashes? Such an algorithm would not need to be aware of the underlying mathematical laws but would instead be trained on data with pre-identified crashes, and identify and learn these patterns on its own.

Data and Crashes

The first step was to collect financial data and identify crashes. I was looking for daily price information from low correlated major stock markets. Low cross-correlation is important for valid cross validation and testing of the model. The matrix below shows the cross-correlation of daily returns from 11 major stock markets.

Correlation matrix of daily price returns for 11 major stock market indices

To avoid having any two data sets with a cross-correlation greater than 0.5 in my collection, I proceeded with only data from the S&P 500 (USA), Nikkei, (Japan), HSI (Hong Kong), SSE (Shanghai), BSESN (India), SMI (Switzerland) and BVSP (Brazil).

To identify crashes in each data set, I first calculated price drawdowns. A drawdown is a persistent decrease in price over consecutive days from the last price maximum to the next price minimum. The example below shows three such drawdowns in the S&P 500 over the period from end of July to mid August 2018.

Example of three drawdowns. The first one shown lasted from July 25th to July 30th 2018 and has a total loss of approximately (2846–2803)/2846 = 1.5%

I considered two different methodologies to identify crashes. The first one follows a suggestion by Emilie Jacobsson [2] who defines crashes in each market as drawdowns in the 99.5% quantile . With this methodology I found drawdown thresholds that classify a crash ranging from around 10% for less volatile markets like the S&P 500 to more than 20% for volatile markets such as the Brazilian one. The second methodology follows the suggestion from Johansen and Sornette [3] who identify crashes as outliers, that is drawdowns that lie far from the fitted Weibul distribution when the logarithm of the rank of drawdowns in a data set is plotted vs the drawdown magnitude.

Distribution of drawdowns by rank as an example for the Shanghai index since 1996.

I tested my algorithms with both crash identification methodologies and concluded that the first methodology (Jacobsson) is advantageous for two reasons. First, Sornette does not clearly state how much deviation from the Weibul distribution classifies a drawdown as a crash, thus human judgement is necessary. Second, his methodology leads to the identification of fewer crashes which leads to heavily imbalanced data sets. This makes it harder to collect a sufficient amount of data for a machine learning algorithm to train on.

With the collection of the seven data sets mentioned above I accumulated a total of 59,738 rows of daily stock prices and identified a total of 76 crashes.

How can classical probability help us estimate the chance Amazon goes bankrupt in 2021?

Amazon Com Probability Of Bankruptcy is used to show its chance of financial distress over the next two years of operations under current economic and market conditions. Amazon Com Probability Of Bankruptcy is determined by interpolating and adjusting Amazon Altman Z Score to account for off-balance-sheet items and missing or unfiled public information. All items used in analyzing the odds of distress are taken from the Amazon balance sheet as well as cash flow and income statements available from the company's most recent filings. Please continue to Amazon Piotroski F Score and Amazon Altman Z Score analysis.2426.26 USD

For stocks, Probability Of Bankruptcy is the normalized value of Z-Score. For funds and ETFs it is derived from a multi-factor model developed by Macroaxis. The score is used to predict the probability of a firm or a fund experiencing financial distress within the next 24 months. Unlike Z-Score, Probability Of Bankruptcy is the value between 0 and 100 indicating the actual probability the firm will be distressed in the next 2 fiscal years.

What formula can help me determine whether my local newspaper sales ads are effective?

Local newspaper ads are great for small- to medium-sized businesses as they help increase brand awareness, product sales, new foot traffic, and repeat business. Although there are other advertising options such as radio, television, and the Internet, choosing print media like newspapers and magazines can make your investment more worthwhile. Here are some of the benefits of local newspaper advertising:

• More affordable than television, direct mail, or radio advertising.
• Flexible timelines and no limit to exposure with print ads as they can be viewed at leisure. Readers can take their time with the messages by browsing your ad in the local paper, and seeing it pop up every time they go back to finish reading a story.
• Proactive audience
• The average issue of a daily or weekly newspaper reaches more people than the average half-hour prime time TV show. Since you’re targeting a wide community of people who are likely to buy into your products and services, local print ads offer tremendous advantages.
• Establishes trust
• Room for last-minute changes
• There are a multitude of advertising options. You can experiment with your ad by adjusting the size, format, color, and style to achieve the desired impact.
• You will benefit from reaching an audience who already finds value in reading print media content as part of their daily routine.