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To what extent mathematical analysis of variation of stock prices in the time domain can be used to predict the ideal business transactions in stock market for Amazon Inc. using differentiation, first order and second derivative and calculus?

Applying what I learn has always been my passion. As an inquirer, I have always wondered about the application of the concepts or content that I study in the class to analyze facts and situations that I come across. Hailing from a business family background, the terms like stock prices, FPO and revenue has not been something unheard or unknown to me but what bothered me was always how this information can be compiled and organized to make successful predictions. Though am not sure but I have a desire to be an investment banker someday and to do that it is immensely important to understand the variation in stock prices and the ups and down of stock market with utmost clarity. Offlate the concepts of data science have become very popular and this subject finds an extremely useful application in the field of stock marketing. Once, while going through a Facebook video on how to analyze stocks and decide the ideal time to invest or sale a stock o encounter minimum risk factor, I realized that integration of data science or precisely mathematical tools is a key role in this area. Big business houses like – Amazon, Flipkart, Facebook depends a lot on secondary investors and thus the patterns in stock marketing have a great influence on their leverage as well as the profitability ratios. All of these are part of financial mathematics where various mathematical tools like – regression analysis, generating functions that relates revenue to quantity of product sold and cost of goods sold, integrating those functions to derive the marginal revenue and gross profit are incorporated. While studying the concept of function and maxima-minima in calculus, I was intrigued about the fact that analyzing the functions that relates stock prices with time over a period of years can be proved to be useful to decide and assess the exact and appropriate time to buy stocks or make investments. However, after going through some researches and case studies on financial mathematics, I realized that this is already a common practice and is a job done by business analysts or financial analysts as a part of the market research. Thus, I thought of exploring this idea and implementing that to the case study of a popular company-Amazon and this is how I arrived at my research question.

- Stocks, or portions of an organization, address proprietorship value in the firm, which give investors casting a ballot rights just as a leftover case on corporate profit as capital additions and profits.
- Stock markets are the place where individual and institutional financial backers meet up to purchase and sell partakes in a public setting. These days these trades exist as electronic commercial centers.
- Share costs are set by market interest in the market as purchasers and merchants place orders. Request stream and bid-ask spreads are frequently kept up with by subject matter experts or market producers to guarantee a precise and reasonable market.

A stock or offer (otherwise called an organization's "value") is a monetary instrument that addresses possession in an organization or company and addresses a proportionate case on its resources (what it claims) and income (what it produces in profits). While there are two primary sorts of stock normal and liked the expression "values" is inseparable from normal offers, as their consolidated market value and exchanging volumes are numerous sizes bigger than that of favored shares.

The primary differentiation between the two is that normal offers ordinarily convey casting a ballot rights that empower the normal investor to have something to do with corporate gatherings (like the yearly comprehensive gathering or AGM) where matters, for example, political decision to the governing body or arrangement of inspectors are casted a ballot upon while favored offers by and large don't have casting a ballot rights. Favored offers are so named in light of the fact that they have inclination over the normal offers in an organization to get profits just as resources in case of a liquidation.

Normal stock can be additionally ordered as far as their democratic rights. While the essential reason of normal offers is that they ought to have equivalent democratic rights one vote for every offer held a few organizations have double or various classes of stock with various democratic rights connected to each class. In a particularly double class structure, Class An offers, for instance, may have 10 votes for each offer, while the Class B"subordinate casting a ballot" offers may just have one vote for every offer. Double or various class share structures are intended to empower the organizers of an organization to control its fortunes, key heading and capacity to innovate.

The present corporate goliath probably had its beginning as a little private element dispatched by a visionary organizer years and years prior. Consider Jack Ma brooding Alibaba Group Holding Limited (BABA) from his condo in Hangzhou, China, in 1999, or Mark Zuckerberg establishing the most punctual rendition of Facebook, Inc. (FB) from his Harvard University apartment in 2004. Innovation monsters like these have become among the greatest organizations on the planet inside two or three decades.

In any case, developing at a particularly excited speed expects admittance to an enormous measure of capital. To make the progress from a thought developing in a business person's mind to a working organization, they need to rent an office or industrial facility, recruit representatives, purchase gear and crude materials, and set up a deals and dissemination organization, in addition to other things. These assets require huge measures of capital, contingent upon the scale and extent of the business startup.

Let *y* = *f *(*x*) be a single valued function of *x* defined in some interval. Let *x* be any value of *x* in the domain of definition of the function and the corresponding value of *y* is *y* = *f *(*x*). Suppose, for an increment ∆*x* of *x* the corresponding increment in *y* is ∆*y*.

Then \(\underset {\Delta x\rightarrow0}{lim}\frac{∆y}{∆x}\frac{dy}{dx}\) is called the derivative of the function *y* with respect to *x*, provided the limit exists. This is the first order derivative.

When the function *y* = *f*(*x*) is differentiated twice that is, \(\frac{d}{dx}(y)=\frac{d}{dx}\big(\frac{dy}{dx}\big)=\frac{d^2y}{dx^2}\) then, it is defined as the second order derivative of the function *y* = *f *(*x*).

Let *y* = *f*(*x*) be any function. Now, the minimum and maximum values of the function are called the Minima and Maxima of the function. This is an application of differential calculus. To determine minima and maxima of a given function, at first we differentiate the function and find the values where the the slope is equal to zero or \(\frac{dy}{dx}= 0. \) These points are the points where the function attains its maximum or minimum. To determine whether it is a maximum or a minimum, we different further and if the value turns out to be negative then then function attains its maximum at that point and vice versa.

**Sample calculation -**

Average Price of Stock in 1^{st} week of January 2020

\(=\frac{3272.00+3144.02}{2}= 3208.01\)

From section 4.0, it could be assumed that the price of stock of Amazon is a function of time only. Therefore, it could be written as;

*y* = *f *(*x*) = −0.114*x*^{3 }+ 4.719*x*^{2} − 45.62*x* + 3258

*y* = Price of stock in USD

*y* = Time (as number of week from January 2020)

According to the background information provided in section 3.3, to calculate the value of maxima and minima of the function *f*(*x*), the derivative of the function should be equated to zero.

\(\frac{dy}{dx}=\frac{d}{dx} (-0.114x3 + 4.719x2 - 45.62x + 3258)\)

\(=>\frac{dy}{dx}=-\frac{d(0.114x^3)}{dx}+\frac{d(4.719x^2)}{dx}-\frac{d(45.62x)}{dx}+\frac{d(3258)}{dx}\)

\(=>\frac{dy}{dx}= - 3 × 0.114x2 + 2 × 4.719x - 45.62 + 0\)

=> +0.342*x*^{2} - 9.438*x* + 45.62 = 0

\(=> x =\frac{9.432±\sqrt{(-9.438)^2+4×0.342×}45.62}{2×0.342}\)

\(=> x = \frac{9.432±\sqrt{89.075+62.408}}{0.864}\)

\(=> x =\frac{9.432±12.307}{0.864}\)

=> *x* = 25.16 Week or, *x* = - 3.32 week

Therefore, from the above calculation, it can be stated that, at *x = *25.16 week and at *x = - *3.327 week, the slope of the curve is zero. Thus, at these two instances of time, the price of stocks should be maximum or minimum. To determine the maxima and minima, the values of *x *is plugged into the equation of the variation of price that has been obtained from data processing as follows:

**Case 1 - x = 25.16 -**

*y* = *f*(*x*) = −0.114*x*^{3} + 4.719*x*^{2 }− 45.62*x* + 3258

=> *y* = *f *(25.16) = −0.114 × 25.16^{3} + 4.719 × 25.16^{2} − 45.62 × 25.16 + 3258

=> *y *= 3281.77* in USD*

**Case 2 - x = - 3.32 -**

*y* = *f *(*x*) = −0.114*x*^{3} + 4.719*x*^{2} − 45.62*x* + 3258

=> *y* = *f *(−3.32) = −0.114 × (−3.32)^{2} + 4.719 × (−3.32)^{2} − 45.62 × (−3.32) + 3258

∴ *y* = 3460.21 *in USD*

**Analysis -**

The maxima of the function is obtained to be at *x* = 25.16 week and the minima is obtained to be at *x* = -3.32 week (as the negative sign has no significance here, we can neglect it).

Hence, for maximum profit, an investor should sell the stocks at 5^{th} week from 1^{st} week of January 2020 as the price of the stock is maximum and sell the stocks at the 3^{rd} week as the price of the stock is the least.

To find the number of weeks at which the price starts to increase are found by the finding the double differentiation of the function *f*(*x*) and that should be equated to zero.

\(\frac{dy}{dx}= - 3 × 0.114x2 + 2 × 4.719x - 45.62\)

\(=>\frac{d^2y}{dx^2}=\frac{d}{dx}(− 3 × 0.114x2 + 2 × 4.719x − 45.62)\)

\(=>\frac{d^2y}{dx^2}=\frac{d}{dx} (-3 × 0.114x2 ) +\frac{d}{dx} (2 × 4.719x) - \frac{d}{dx} (45.62)\)

\(=>\frac{d^2y}{dx^2}= - 2 × 3 × 0.114x + 2 × 4.719 - 0\)

=> − 0.684*x* + 9.438 = 0

=> *x *= 1379 *week*

To verify the nature of the curve, the following table is constructed -

**Sample Calculations -**

Calculated value of price stock when *x* = 0

*y* = *f *(*x*) = −0.114*x*^{3} + 4.719*x*^{2} − 45.62*x *+ 3258

*y* = *f *(0) = - 0.114× 0^{3} + 4.719 × 0^{2} − 45.62 × 0 + 3258

*y* = 3258 *in USD*

**Analysis -**

From the above table, it can be observed that the price of stock at first decreases when the number of weeks increases from 0. The price of stock decreases from 3258 USD to 3281.77 USD (minima) which is the minimum value of stock obtained to be at week *x* = 3.32. With further increase in the number of weeks at a higher rate, the price of stock again starts to increase and reaches a point of *x *= 13.79 *week *after which the stock prices are increasing but the increase rate gradually decreases. This point is known as inflection point. The region of curve between *x* = 0 and *x = *13.79 is called concave up as the function initially decreases and then increases. After the inflection point the price of stock increases but at a slower rate and eventually at x = 25.16 weeks, the price of stock reaches its maximum value. From *x* = 13.79 to *x* = 25.16, the price of stock increases from 3281 USD to 3326 USD. After *x* = 25.16, the price of stock again decreases. This region after inflection point is called Concave down as the curve initially increases and once it reaches the maxima, the magnitude of the function starts to decrease.

**Real Life Significance -**

From the above analysis, we get to the point that any stock market broker should inform his clients at four different phases of time while purchasing a stock. Firstly, when the value of stock was decreasing as the nature of the curve was concave up, the broker should inform his clients regarding the market scenario as the stock prices are decreasing. This will help the investors to arrange and liquefy funds so that once the stock price approaches towards the minimum rate, they can immediately purchase the stocks.

Secondly, when the stock price reaches the minimum value, the broker should advice his clients to buy the stocks. Thirdly, as the concave down region approaches, i.e., at the inflection point, the broker should inform his clients regarding the market scenario as the stock prices are increasing. This will help the investors to sell the stocks once the price hits the maximum rate.

Fourthly, once the price starts to decrease after the maxima, the broker should inform the clients not to sell further stocks as it may hamper a significant amount of profit to a great extent.

To evaluate whether or not the stocks of Amazon Inc. is overvalued or undervalued, the market capitalization of Amazon Inc. with respect to its Revenue (per year) should be computed graphically. The relationship between Revenue (per year) and the market capitalization would conclude whether or not the stock of Amazon is overvalued or undervalued.

Year | Revenue (in billion USD) | Market Capitalization (in billion USD) |
---|---|---|

2012 | 61.09 | 113.89 |

2013 | 74.45 | 183.04 |

2014 | 88.99 | 144.31 |

2015 | 107.01 | 318.34 |

2016 | 135.99 | 356.31 |

2017 | 177.87 | 563.53 |

2018 | 232.89 | 737.46 |

2019 | 280.52 | 920.22 |

2020 | 386.06 | 1634 |

**Analysis -**

In the above graph, the variation of market capitalization (in billion USD) has been plotted with respect to the annual revenue (in billion USD) from 2012 to 2020. The annual revenue has been plotted along the X axis and the market capitalization has been plotted along Y axis. The nature of the graph is logarithmic that is, with increase of revenue from 61.09 billion USD to 386.06 billion USD; the market capital has increased from 113.89 billion USD to 1634 billion USD. The equation obtained, as shown in the graph is* **y* = 31.22ln(*x*) +94.66. So, if the annual revenue is 1 billion USD then, the market capitalization is 94.66 billion USD. Here, we can conclude that the rate of increase of revenue is not proportional to market capitalization.

To understand the slope of increase in market capital with respect to annual revenue, we differentiate the obtained equation with respect to the annual revenue.

*y* = 31.22 ln(*x*) + 94.66

\(=>\frac{dy}{dx}=\frac{d}{dx}(31.22 In (x) + 94.66\)

\(=>\frac{dy}{dx}=\frac{d}{dx}(31.22 In (x)) +\frac{d}{dx} (94.66)\)

\(=>\frac{dy}{dx}=\frac{31.22}{x}\) ...(1)

From the above equation, the slope or gradient of change of market capitalization with respect to annual revenue can be obtained. Using equation (1) and the values of market capitalization and annual revenue, we can determine the gradient of increase of market capital with respect to annual revenue.

Year | Annual revenue (in billion USD) (x) | Rate of increase of Market Capitalization \((\frac{31.22}{x})\) |
---|---|---|

2012 | 61.09 | 0.511 |

2013 | 74.45 | 0.419 |

2014 | 88.99 | 0.350 |

2015 | 107.01 | 0.291 |

2016 | 135.99 | 0.229 |

2017 | 177.87 | 0.175 |

2018 | 232.89 | 0.134 |

2019 | 280.52 | 0.111 |

2020 | 386.06 | 0.080 |

It is observable from figure 6 and Graph that the slope or the gradient of Market Capitalization decreases with respect to Annual Revenue. It decreases from 0.511 to 0.080 from the year 2012 to 2020. It can be concluded that with each passing year the revenue earned by Amazon is increasing but the market capital of Amazon is not increasing with respect to increase in revenue of the company. Revenue earned can be intepreted as the market demand or supply. So, purchasing an overvalued stock could lead to losses in the long run. As we have discussed earlier that stocks should be purchased when the stock price is low or at the lowest level. And, the stock price of Amazon is overvalued. Thus, it is adviced to not purchase the overvalued stocks of Amazon.

*To what extent mathematical analysis of variation of stock prices in the time domain can be used to predict the ideal business transactions in stock market for Amazon Inc. using differentiation, first order and second derivative and calculus?*

From the above study, we can conclude three main aspects related to buying and selling of stocks, market capitalization and annual revenue of a company. Firstly, investors should buy their stocks when the price of stocks reaches a minimum purchase rate in order to get significant profits. The investors should sell their stocks when the selling rate approaches the maximum.

Secondly, the brokers should advice their clients to buy stocks within appropriate time when the stock price decreases or when the nature of the curve is concave up. The broker should advice their clients to get prepared for selling the stocks once the curve reaches the point of inflection to get maximum profits.

Thirdly, with an increase in annual revenue, the market capital of the company increases. However, the slope or the gradient of market capitalization decreases with respect to annual revenue. This indicates that the growth of the company is not proportional to its annual revenue generated. This indicates that the stocks of the company are overvalued and hence, purchasing those stocks may incur significant losses in the future.

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“Stock Market Analysis: A Review and Taxonomy of Prediction Techniques.” International Journal of Financial Studies, vol. 7, no. 2, May 2019, p. 26. DOI.org (Crossref), https://www.mdpi.com/2227-7072/7/2/26