{"id":802,"date":"2020-09-25T23:30:46","date_gmt":"2020-09-26T03:30:46","guid":{"rendered":"http:\/\/commoncents.blogwyrm.com\/?p=802"},"modified":"2022-03-27T20:23:08","modified_gmt":"2022-03-28T00:23:08","slug":"financial-arbitrage-redux","status":"publish","type":"post","link":"https:\/\/commoncents.blogwyrm.com\/?p=802","title":{"rendered":"Financial Arbitrage Redux"},"content":{"rendered":"

The previous blog introduced the notion of financial arbitrage and briefly explored the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) models for pricing an asset (e.g. stock).\u00a0 The CAPM correlates a particular asset with some macroeconomic factor (e.g. inflation or one of the indices) to determine the expected return on the arbitrage.\u00a0 The APT generalizes this 1-dimensional correlation to the case where multiple factors affect the asset price.\u00a0 The applicable formula that covers both cases is<\/p>\n

RA<\/sub>\u00a0= Rfree<\/sub>\u00a0+ \u03b21<\/sub>\u00a0( P1<\/sub>\u00a0- Rfree<\/sub>\u00a0) + \u03b22<\/sub>\u00a0( P2<\/sub>\u00a0- Rfree<\/sub>\u00a0) + ...\u00a0= Rfree<\/sub>\u00a0+ \u03b21<\/sub>\u00a0RP1<\/sub>\u00a0+ \u03b22<\/sub>\u00a0RP2<\/sub>\u00a0+ ...<\/p>\n

where:<\/p>\n

RA<\/sub>\u00a0is the expected rate of return of the asset in question,<\/p>\n

Rfree<\/sub>\u00a0is the rate of return if the asset had no dependence on the identified macroeconomic factors (free rate of return),<\/p>\n

\u03b2i<\/sub>\u00a0is the sensitivity of the asset with respect to the\u00a0i<\/em>th macroeconomic factor, and<\/p>\n

Pi<\/sub>\u00a0is the additional\u00a0risk premium<\/a>\u00a0associated with the\u00a0i<\/em>th macroeconomic factor with RPi<\/sub>\u00a0= Pi<\/sub>\u00a0- Rfree<\/sub>\u00a0being the actual risk premium.<\/p>\n

Obviously, setting all the \u03b2i<\/sub> beyond \u03b21<\/sub> to be zero in the APT recovers the CAPM.<\/p>\n

To use either of these models, the arbitrageur needs to set multiple free parameters (Rfree<\/sub>, RA<\/sub>, \u03b2i<\/sub>, Pi<\/sub>) using his judgement based on historical data and some of the aspects of this procedure will be the focus of this post.<\/p>\n

For simplicity, we\u2019ll limit the analysis to correlate one stock with one index and we\u2019ll follow the excellent article entitled CAPM Beta - Definition, Formula, Calculate CAPM Beta in Excel<\/a> by Dheeraj Vaidya for WallStreetMojo.\u00a0 I\u2019ll be adding only a few points here and there just to round out what Vaidya presented but it is worth emphasizing what a fine job he did in his presentation.<\/p>\n

The correlation we\u2019ll be exploring is between a company called MakeMyTrip (MMYT ticker symbol) and the NASDAQ Composite (^IXIC ticker symbol).\u00a0 To match, Vaidya\u2019s analysis, we confine our time frame from January 1st<\/sup>, 2012 to October 30th<\/sup>, 2014.\u00a0 Yahoo Finance served quotes under the historical data link that presents itself after entering a ticker symbol (see green ellipse in the figure below)<\/p>\n

\"\"<\/a><\/p>\n

Selecting the time span and downloading the data in CSV format are easy.\u00a0 I read the data for MMYT and ^IXIC in pandas data frames but since the average price of the NASDAQ Composite over that time span was $3563.91 compared to an average of $18.98 for MakeMyTrip, plotting each time series on a common plot won\u2019t work, even with a log scaling.\u00a0 Instead, taking a page from Z-scoring in statistics, I made a plot of the normalized stock price for each listing in which the instantaneous price was divided the average.<\/p>\n

\"\"<\/a><\/p>\n

There is no obvious correlation between the two time series. The NASDAQ Composite, more or less, rose steadily during this time span while MakeMyTrip shows a more of a parabolic behavior, with a downward trend during roughly the first third of the time span followed by minimum in the second third, and punctuated by rapid, and often volatile growth, in the third.<\/p>\n

These differences in the qualitative evolution of the two assets presents itself even more strongly in a scatter plot showing the adjusted closing price of each asset.<\/p>\n

\"\"<\/a><\/p>\n

Nonetheless, there is a reasonably good correlation between the two assets in terms of their fractional gain, defined as the difference between two successive days relative to the price of the earlier of the two days (i.e. (pi+1<\/sub> \u2013 pi<\/sub>)\/pi<\/sub> where pi<\/sub> is the price of the asset on the i<\/em>th<\/sup> day).<\/p>\n

\"\"<\/a><\/p>\n

There is a definite but weak positive correlation between the adjusted close of the NASDAQ Composite and MakeMyTrip.\u00a0 A linear regression, computed using numpy\u2019s polyfit routine (order 1), confirmed the same value of 0.9858 for the slope of a linear regression line that Vaidya reported.\u00a0 This value is then the \u03b2 between MakeMyTrip and the NASDAQ composite for this time span.<\/p>\n

But the fun doesn\u2019t stop there.\u00a0 We can use the power of the pandas package to extend Vaidya\u2019s presentation by randomly sampling the data to get an idea of the spread in the value of \u03b2 based on using different samples due to differences in time span or reporting interval.\u00a0 Running a Monte Carlo with 350 samples each (almost exactly half of the total number of available data points) for N = 10,000 trials gives the following statistics for \u03b2:<\/p>\n