Mathematics AA SL

Sample Internal Assessment

Table of content

Rationale

Aim

Background information

Data collection and processing

Process of calculation

Reflection

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Mathematics has always inspired me to go beyond the scope of the curriculum because of its diversity in application across various majors of science, technology, commerce, humanities and many more. As a matter of fact, IB has a great role to play behind this inspiration. As the course structure has always pushed me towards procuring the education in real life scenarios, it has developed a curiosity in my every observation of any event. This curious behavior has landed me to conclude mathematics as one of the most versatile subjects amongst those I have chosen in my curriculum.

Very recently, I have seen a web series in an OTD platform, namely, SonyLIV which was based on stock market and the scams involving it. Being one of the most viewed and top-rated web-series, like others, I really liked the series. Being an inquirer, there were few questions which struck my mind after watching the web-series. How does the stock market broker advice their clients the exact time of buying or selling stocks? Is it an assumption based on news reporting of several events that affect the market? Is there any definite pattern in increase and decrease in stock with respect to time which the brokers observe?

To get rid of the above and many more other questions, I started my research on prices of stock. After a few sleep-less nights of study, I found that mathematics has a great role to play in stock market. This was the reason why I have mentioned it as the most versatile subject. I have studied a few more research journals and articles where I understood that the mathematics that involves prediction of exact time of buying and selling stocks involve a concept of mathematics that is being taught in IB. However, there are few sections of mathematics that involves the process, though, those are mere extension of the curriculum of IB. This interest in deciphering the procedure and calculations involve prediction of proper time of buying and selling stock has landed me to the aim of this exploration.

The main objective of this exploration is to determination the exact time of buying or selling of stocks by a consumer to ensure maximum profit.

Stock Marketing is a standard term used to represent buying and selling of stocks of any business, corporate, educational institution, health care, government sectors and many more. It is a platform where any individual can purchase a share of any company or the above-mentioned entities and earn the percentage of profit or loss acquired by the company over time. Companies, start-ups, government sectors, and many other institutions approach brokers who runs an agency, often termed as investor center to sell their stocks to the consumers. The brokers efficiently advice the consumers who are interested in any investment in stocks, in choosing the appropriate stock based on the budget of the consumer and return on investment over time of the stock. If a consumer buys a stock at a lower price and sells the stock at a higher price, then the consumer earns profit. In a contrary, if a consumer buys a stock at a higher price and sells the stock at a lower price, then the consumer incurs a loss.

One of the most important services that the consumers or the clients expect from their brokers is to provide a notification right before buying or selling stock to ensure maximum profit. To achieve accuracy in estimating the correct time of stock operations, the brokers analyze the data of variation in stock price with respect to time.

The maximum value of any function *y *= *f* (*x*) is called the maxima of the function, and the minimum value of the function is called the minima of the function. This is an application of differential calculus. Any function *y *= *f *(*x*) could be graphically represented in cartesian coordinate system. The slope at any point of the function could be determined by finding the derivative of the function at the particular point. Slope of any curve equal to zero signifies that the tangent at that point of the curve is parallel to X – axis. As a result, if the equation of slope of the curve is equated to zero, and the root of the equation is determined, then it would be the value of *x** *at which the function is either maximum or minimum. In order to determine the maxima or minima, the equation of slope is further differentiated and if the value of equation is obtained to be negative, then the point is maxima, and vice versa.

This exploration involves prediction of time instances (in terms of days) on which buying or selling of stock would be beneficial for the stock investors. This exploration involves two steps of operation. Firstly, a function varying with respect to time has been determined which would indicate the variation of stock with respect to time by graphical exploration of variation in price of stock over time. Secondly, by using differential calculus and concepts of maxima and minima, the dates at which the price of stock was maximum and minimum would be calculated and followed by that, the time instant (date) at which the nature of the curve changes would be calculated. These points are known as Inflection points. These points are essential to state the instant after which the price would be increasing or decreasing.

In this exploration, stock of Facebook Inc. has been considered. This is because Facebook is one of the companies which has sustained several ups and downs in the past year 2020. Facebook Inc. has taken over few other companies as well in the pandemics.

This exploration involves case study based on past data. In order to predict the variation of stock prices in future, more mathematical tools, concepts of economics and rigorous statistical concepts, analysis and interpretations are required which are beyond the scope of this exploration.

Lastly, the stock price of Facebook Inc. was evaluated on the basis of overvaluation and undervaluation by comparing the market capital with respect to annual revenue earned.

Sample Calculation:

Average Price of Stock in 1st week of January 2020

\(= {219.88 + 206.52\over 2} \)

= 213.20 in USD

Standard Deviation of Price of Stock in 1st week of January 2020

\(= {\sqrt{(219.88 - 213.20)^2\ + (206.52-213.20)^2\over 2}} \)

= 9.45 in USD

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

*y* = *f *(*x*) = - 0.0029*x*^{3} + 0.2589*x*^{2} - 4.1663*x* + 211.72

where,

y = price of stock in USD

x = time (as number of week from January 2020)

According 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 0.

\({dy\over dx} = {d\over dx}\bigl(-0.0029x^3\ + \ 0.2589x^2\ - \ 4.1663x\ + \ 211.72 )\)

\({dy\over dx} = - {d(0.0029x^3)\over dx}\ + \ {d(0.2589x^2)\over dx} \ - \ {d(4.1663x)\over dx}\ + {d(211.72)\over dx}\)

\({dy\over dx}=- 3×0.0029x^2 \ + \ 2×0.2589x - 4.1663\ + \ 0\)

\({dy\over dx} = - 0.0089x^2\ + \ 0.5178x\ - \ 4.1663\)

- 0.0089*x*^{2} + 0.5178*x* - 4.1663 = 0

0.0089*x*^{2} - 0.5178*x* + 4.1663 = 0

\(x = {0.5178 ± \sqrt{(-0.5178)^2+ 4×0.0089×4.1663} \over 2×0.0089}\)

\({ x = \frac{0.5178±\sqrt{0.268-0.148}}{0.0178}}\)

\({ x = \frac{0.5178±\sqrt{0.120}}{0.0178}}\)

\(x = {0.5178±0.346\over 0.0178}\)

*x* = 29.09 ± 19.46

*x*_{1} = 48.55 week

*x*_{2} = 9.36 week

Therefore, from the above calculation, it can be stated that, at *x* = 9.63 week and *x *= 48.55 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:

*y* = *f *(*x*) = - 0.0029*x*^{3} + 0.2589*x*^{2} - 4.1663*x* + 211.72

*y* = *f *(48.55) = - 0.0029 (48.55)^{3} + 0.2589 (48.55)^{2} - 4.1663 (48.55) + 211.72

= - 331.87 + 610.25 - 202.27 + 211.72

= 287.83 in USD

Case 2: *x* = 9.63:

*y* = *f* (*x*) = - 0.0029*x*^{3} + 0.2589*x*^{2} - 4.1663*x* + 211.72

*y* = *f* (9.36) = - 0.0029(9.36)^{3} + 0.2589 (9.36)^{2} - 4.1663(9.36) + 211.72

= - 2.59 + 24.01 - 40.12 + 211.72

= 193.02* in USD*

Analysis:

The maxima of the function is obtained to be at *x* = 48.55 week* *and the minima of the function is obtained to be at *x* = 9.63 week.

Hence, to ensure maximum profit, an investor should buy the stocks at 9^{th} week from the 1^{st} week of January 2020 as the price of the stock is the least and sell the stocks at 49^{th} week from the 1^{st} week of January 2020 as the price of the stock is maximum.

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

\({dy\over dx}= - 0.0089x^2 + 0.5178x - 4.1663\)

\({d^2y\over dx^2}= {d\over dx}\biggl(-0.0089x^2\ + 0.5178x - 4.1663\biggl) \)

\({d^2y\over dx^2} = - {d(0.0089x^2)\over dx} + {d(0.5178x)\over dx} - {d(4.1663)\over dx}\)

\({d^2y\over dx^2}= -2×0.0089x \ + \ 0.5178 - 0\)

\({d^2y\over dx^2}= - 0.0178x + 0.5178\)

- 0.0178*x* + 0.5178 = 0

- 0.0178x = - 0.5178

\(x = {-0.5178\over -0.0178}\)

*x* = 29.09 week

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

Sample Calculation

Calculated value of price of stock when *x* = 0

*y* = *f *(*x*) = - 0.0029*x*^{3}+ 0.2589*x*^{2 }- 4.1663*x *+ 211.72

*y* = *f *(0) = - 0.0029 (0)^{3} + 0.2589(0)^{2} - 4.1663 (0) + 211.72

*y *= 211.72 in USD

Analysis

From the above table, it can be observed that the price of stock initially decreases when the number of weeks increases from 0. The price of stock decreases from 211.72 USD to 193.02 USD (minima) which is the minimum value of stock obtained to be at week *x* = 9.63. 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* = 29.09 week after which the price of stock increases but the rate gradually decreases. This point is known as inflection point. The region of curve between *x* = 0 and *x *= 29.09 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* = 48.55 weeks, the price of stock reaches its maximum value. From *x* = 29.09 to *x* = 48.55, the price of stock increases from 234.70 USD to 287.83 USD. After *x* = 48.55, the price of stock again decreases. This the 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, any stock market broker should inform his clients at four different instances of time. Firstly, when the value of stock was decreasing as the nature of the curve was concave up, the broker should notify his clients regarding the market scenario as the price of stocks are decreasing. This will help the investors to arrange and liquify funds so that once the stock price hits the bare minimum rate, they can immediately purchase the stocks.

Secondly, when the price of stock reaches the minimum value, the broker should advice his clients to buy the stocks.

Thirdly, at the onset of concave down region, i.e., at the inflection point, the broker should inform his investors regarding the market scenario as the price of stocks are increasing. This will help the investors to have a mindset of selling 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 incur significant less profit.

To evaluate whether or not the stocks of Facebook Inc. is overvalued or undervalued, the market capitalization of Facebook 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 Facebook is overvalued or undervalued.

Year

Revenue (in billion USD)

Market Capitalization (in billion USD)

2012

5.08

65

2013

7.87

139

2014

12.46

218

2015

17.92

296

2016

27.63

332

2017

40.65

513

2018

55.83

562

2019

70.69

585

2020

85.96

778

Analysis

In the above-mentioned graph, the variation of market capitalization (in billion USD) (as per fourth quartile balance sheet) 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 of the graph and the market capitalization has been plotted along Y – Axis. It has been observed that with an increase in revenue from 5.08 billion USD to 85.96 billion USD, the market capital has increased from 65 billion USD to 778 billion USD. Despite an increasing trend has been obtained, however, the nature of the trend is logarithmic. The equation of obtained trend is shown: *y* = 230.05 ln *x* -350.02, where y denotes market capitalization (in billion USD) and *x* denotes annual revenue (in billion USD). Hence, it can be concluded that the rate of increase in revenue is not proportional to market capital.

To understand the gradient of increase in market capital with respect to revenue could be determined by differentiating the equation of obtained trend with respect to revenue.

\({dy\over dx} = {d\over dx}(230.05 \ In \ in \ x \ - \ 350.02)\)

\({dy\over dx}\ = {d\over dx}( 230.05\ In\ in \ x) - {d\over dx}(350.02) \)

\({dy\over dx}= {230.05\over x}...(1)\)

From the above equation, the gradient of change of market capitalization with respect to revenue could be determined. The values of the revenue have already been computed in Table No. 3. Using the equation (1) and plugging in the values of Revenue (as in Table No 3), in place of *x* (as in equation 1), the gradient of increase of market capital could be determined:

Year

Annual Revenue (in billion USD) (*x*)

Rate of increase of Market capitalization \(230.05\over x\)

2012

5.08

45.29

2013

7.87

29.23

2014

12.46

18.46

2015

17.92

12.84

2016

27.63

8.33

2017

40.65

5.66

2018

55.83

4.12

2019

70.69

3.25

2020

85.96

2.68

It is clearly observed from Table 4 and Graph 3, the gradient of market capitalization decreases from 45.29 to 2.68 with respect to Annual Revenue from 2012 to 2020. Hence, it can be concluded that, with passing years, the revenue earned by the company is increasing; however, the market capital of Facebook Inc., i.e., the potential of the Facebook is not increasing with respect to the increase in revenue. This signifies that the stocks of the company are overvalued. This is because, increase in revenue signifies increase in selling of stocks. However, the market capital of Facebook Inc. is not increasing to maintain the revenue earned. Revenue earned could be interpreted as the market demand or supply. As a result, purchasing an overvalued stock might result in incurring loss in a long run. As per the discussion in section 5.2, it was evaluated that stocks should be purchased when the stock price is low. On the other hand, the stock price of Facebook is overvalued, i.e., more than it should be. Thus, brokers should recommend their clients, not to purchase stocks of Facebook Inc. as it is overvalued.

To determine the exact time of buying or selling of stocks by solving the first order derivative, second order derivative and analysis of asymptotes of price of stock function with respect to time.

Based on the collected data regarding the price of stocks of Facebook Inc. from the 1^{st} week of January 2020 till 2^{nd} week of March 2021, the following assumptions as claimed:

- The investors should have bought the stocks of Facebook Inc. on 10
^{th}week from the 1^{st}week of January 2020 to avail minimum purchase rate of stocks and consequently, getting a chance to enjoy maximum profit. - The investors should have sold the stocks of Facebook Inc. on 49
^{th}week from the 1^{st}week of January 2020 to avail maximum selling rate of stocks and consequently, getting a chance to enjoy maximum profit. - For a broker, when the value of stock was decreasing as the nature of the curve was concave up, he should advice his clients to arrange funds to buy stocks in proper time.
- For a broker, when the price of stock reaches the minimum value, he should advice his clients to buy the stocks.
- The broker should inform his investors to prepare to sell the stocks once the curve reaches the inflection point to avail maximum selling rates.
- With an increase in revenue from 5.08 billion USD to 85.96 billion USD, the market capital has increased from 65 billion USD to 778 billion USD.
- The gradient of market capitalization decreases from 45.29 to 2.68 with respect to Annual Revenue from 2012 to 2020. This indicates that the growth of the company is not significant with respect to its revenue earned. Moreover, as the growth of company is not proportional to the revenue (revenue increases invariable), it signifies that the stocks of the Facebook Inc. is overvalued.
- The stock price of Facebook Inc. is overvalued and hence, purchasing of stocks of Facebook may incur loss in future. Because price of overvalued stocks decrease and price of undervalued stocks increase because of the potential of the company.

- The range of data collected for this observation is significantly wide. It has included the variation of stock prices over 60 weeks. As a result, all the three phases of stock marketing have been observed in the graph – increasing rate, decreasing rate, and stagnant rate. It has made the exploration more coherent.
- The average of the highest and the lowest value of stock for each week has been considered in this exploration to represent an overall price of the stock of the week to make the exploration more coherent as the highest and the lowest rates of each week exists for a fraction of second.
- The source of data is one of the best platforms that continuously assess the price of stocks. Thus, the website could be considered to be error free and authentic.

- The prices of stock with respect to time were collected from a commercial website. Despite the renounced reputation of the website, there is no scope of verifying the data available in the open source. This might lead to the presence of discrepancy in the collected data which would make the exploration less coherent.
- It was assumed that the variation of stock prices in future would follow the same pattern which was shown in the data set that was considered in this exploration. However, this might not be the condition in real-life.
- Method of triangulation was not performed in this exploration due to lack of authenticated data sources. This might lead to the presence of discrepancy in the collected data which would make the exploration less coherent.

Stock market trading is a genre of commerce which involves extensive use of mathematics. Similarly, there are other trading options which involves similar strategy of earning money. One of such type of investment is investment on gold. Price of gold increases as well as decreases with time. However, stock market trading has significantly more risk factor than that of investing in gold. As an extension of this exploration, another exploration could be framed to determine the proper time of buying and selling gold to earn the optimum profit. This exploration will involve two steps of operation. Firstly, a function varying with respect to time will be determined which would indicate the variation of gold with respect to time by graphical exploration of variation in price of gold over time. Secondly, by using differential calculus and concepts of maxima and minima, the dates at which the price of gold will be maximum and minimum will be calculated and followed by that, the time instant (date) at which the nature of the curve changes would be calculated. These points will be essential to state the instant after which the price would be increasing or decreasing.

The aim of the exploration could be framed as follows:* “To determine the exact time of buying or selling of gold by solving the first order derivative, second order derivative and analysis of asymptotes of price of gold function with respect to time.”*

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