Math Exploration Kushal Trademark Stock Analysis Contents Introduction ................................ ................................ ................................ ........... 2 Aim & Rationale: ................................ ................................ ................................ 3 Data Presentation: ................................ ................................ .............................. 4 Modeling: ................................ ................................ ................................ .............. 5 Kushal Tradelink Accumulation: ................................ Error! Bookmark not defined. Pred icting Future Potential ................................ ................................ .................... 8 Conclusion: ................................ ................................ ................................ .......... 13 Bibliography ................................ ................................ ................................ ......... 13 Introduction A stock market is a market within a stock exchange which consists of an area where securities are bought and sold. Countries all around the world contain a stock exchange where the market consists of for the common market which pertains to them. The point of a stock market is , essentially, for any individual regardless of class, gender, and ethnicity to be able to buy a proportion of a commodity, business, and or product to be able to sell in order to make a profit. The basic principle of the stock market is for an individual to buy the stock at the minimum value and to sell the stock at the maximum in order to make the largest profit possible. But, Is this always the case? Many people around the world are paid millions of dollars to predict the future stock price of a commodit y due to the potential value being greater in the long run. The trading typhoon: Warren Buffet states the basic principle of trading is, ‘buy low and sell high.’ However, based on this principle what should we regard as low? What if the stock is said to be high now but it will continue to rise in the future? There are many questions about the stock market that still yet to be answered. One of the major assumptions, however, is that the market will continue to increase regardless of current issues such as th e COVID - 19 pandemic or the 2008 Financial Crisis. An example of one of many stock market locations is the Indian stock exchange. The Indian stock exchange is located in Dalal Street in Mumbai, India and has a market cap of 2.27 trillion rupees as of April 2018. Over the years the Indian market has grown exponentially due to the rising demand for new commodities as well as the exponential growth in population size. Due to billions of dollars being circulated day to day suggests that the Indian stock market i s generally quite volatile and has the potential to grow exponentially. The Indian stock market also lays parallel to the cultural events that occur within the nation. The stock market is quite unique due to the large number of days they are closed for as within the Indian culture there are almost 20+ governmental holidays within the year – which don’t include culture specific events – which suggests that the amount of time the market is open is very little in comparison to other prominent exchanges such as the New York stock exchange. Key Terms in the Stock Market Minimum value The lowest value of the stock (usually the best time to buy as it takes the lowest proportion of an individual’s disposable income) Maximum value The highest value of the stock (u sually the best time to sell the stock as its value is at its highest point) Accumulation value Accumulation is the increase in transactional assets gained over a time period. This gives you the potential benefit over the period analyzed. Future Potential The potential future price of a stock market. This is usually inferred though mathematics. Close Price The price at which the stock is at when the market closes in India at 4pm. Market Cap The market value of a publicly traded company’s outsta nding shares. Maximum Profit The maximum value – minimum value (total monetary value gained from transaction) Aim & Rationale: The Indian stock market is a world of infinite money supply where everyday billions of rupees are circulated. Ever since the age of 10, I have dreamed about standing in Dalal street – the location of the Indian stock exchange - and watching how stocks rose to unprecedented values. Coming from India and having a father in the financial sector, valuations and stock returns were am ongst many conversations I would have at the dinner table. For my stock analysis I wanted to mix my future goals with my true heritage by analyzing an Indian stock, specifically, Kushal Ltd. Kushal Tradelink Limited is a commodity manufacture, trade, and supply company based in India. Over the years the company has changed their production methods and have become one of the leading importers of wastepaper which they use for recycling purposes. However, this shift did increase costs, which could suggest the overall decrease in the value of the stock over the year and a half analyzed. But my rational for analyzing such a stock is due to my interest and large belief in sustainable business which I hope to study in the future. Growing up in an emirate driven to be the sustainable hub for the future, environmental preservability has proliferated around me from a very young age. Furthermore, due to commodities being more of a volatile sector of the stock exchange makes this stock quite interesting to analyze as we ll as the fluctuations in price make the future potential promising, especially for this company, as manufacturing and trade are still on the rise due to globalization continuing to occur in the modern age. (Kushal Tradelink Limited , 2020) The aim of this report is to create a model to predict the future stock value of Kushal Limited using past trends . For this report, the maximum amount of data to create a fair analysis of the stock market will be used. From August 1 st , 2018 to December 31 st , 2019 a period of 343 trading days. A scatter graph will be used to display the data. Data Presentation: 2018 2018 - 2019 August 1st - October 31st November 1st - January 31st Days Close Date Close 60 27.3 123 43 59 27.85 122 43.85 58 28.35 121 44.7 57 28.9 120 45.6 56 30.4 119 46.5 55 32 118 47.4 54 30.6 117 48.35 53 32.2 116 49.3 52 33.85 115 50.3 51 35.6 114 49.65 50 37.45 113 50.65 49 39.4 112 51.65 48 41.45 111 52.7 47 43.6 110 53.75 46 45.85 109 54.8 45 48.25 108 53.75 44 50.75 107 52.7 43 53.4 106 52.9 42 53.4 105 53.95 41 53.35 104 55.05 40 53.55 103 56.15 This is a screenshot of my data; the all data fields are displayed in the appendix of the report. From directly looking at the graph it is clear the biggest change in closing price value is in between day 48 and day 101, thus, the exploration will look at this specific period to see wh ere if I did invest I would make the greatest return. However, the question does arise why did this sudden change occur? The information displayed below illustrates the causes of the sudden change as well as my reasoning for future potential. (Yahoo Finance, 2020) 0 10 20 30 40 50 60 70 0 50 100 150 200 250 300 350 400 Close Price of Kushal Tradelink Trading Days Graph 1.1: Stock Price Kushal Tradelink 0 ≤ x ≤ 343 On Day 42, Sandeep Agrawal , CEO of Kushal Ltd. state d that the manufacturing industry had a huge rise in their material costs, raising their cost of production. This led to a fall in demand of the stock and thus the change occurred. On this day , the minimum point, there was a change or turning point in the stock value. This was because the Indian rupee had increase in value against the dollar by almost 1.5 rupees. This could be due to increased in trade within the country. Kushal Tradelink reaches its maximum price on this day, due to the rise in value of rupee as well as decrease in cost of production due to the company increasing rate of production, which decreased cost s. There is a gen eral decrease in the value of the stock over time, however, as previously said there is no best time to buy the stock as the stock market in theory should keep rising. If the stock value did potentially increase again, would this be the best time to buy as technically this is the minimum price. The general decrease over the years in the stock value I believe was due to the switching over to a more sustainable business plan. But as the world changes their mindset towards sustainability it can be inferred the stock value will potentially rise. Modeling: As I was analyzing the stock, I was curious about how I could make my own model between the days 40 and 111 as it showed the biggest change in the stock value. I recognized a clear quadratic equation bet ween days 43 and 94 and then a negative quadratic equation between 94 and 104. This led me to research into piecewise functions: function comprised of multiple sub functions. I decided to model my own piecewise functions as it would give me the best overview of this time period. Also, by finding multiple subfunctions I would be able to differentiate them individually to find the maximum and minimum values. Using Desmos, I modeled the graph creating a piecewise function between day 40 and 111. The scat terplot illustrates the relationship between trading days and close price of Kushal Tradelink. Graph 1.2: Sto ck Price Kushal Tradelink 40 ≤ 𝒙 ≤ 110 𝑓 ( 𝑥 ) { ( 1 24 ( 𝑥 − 70 ) 2 + 22 5 ) , 43 < 𝑥 ≤ 94 ( − 1 3 5 ( 𝑥 − 100 5 ) 2 + 58 5 ) , 94 ≤ 𝑥 ≤ 104 ( 1 3 ( 𝑥 − 106 5 ) 2 + 53 ) , 104 ≤ 𝑥 < 109 Reflection 1 By creating my own piecewise function, I can carefully analyze the stock value over the 75 days analyzed and this will also allow me to differen tiate the quadratic equation to see the optimum price I should buy and sell at. Moreover, the fact that the piecewise function is so accurate in terms of the correlation to the scatterplot means that the values obtained are quite accurate. Trading Days Close Price Kushal Tradelink As soon as I saw the quadratic above, I could tell that it was based off of the 𝑦 = 𝑥 2 Following a series of transformations from y = 𝑥 2 , I created the function below. It was stretched by a scale factor of 1 24 so the quadratic matches the scatterplot. Al so, it translated 70 units to the right and 22.5 units up as the minimum point of the graph is 22.5. I could also tell that the second quadratic equation (the maximum) was based off of the 𝑦 = − 𝑥 2 function. This led me to create a series of transformations, stretched by 1 − 3 5 and then the graph is translated to the right of 100.5 units and a movement upward by 58.5 as this is the highest of maximum value of the stock. The equation below illustrates the inverse quadratic equation: To find the percentage change in my investment I will use the formula below. 𝐶𝑙𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒 𝑁𝑒𝑤 − 𝐶𝑙𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐶𝑙𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 x 100 Reflection 3 If I had made the investment at 22.5 rupees and then sold at 58.5 rupees, I would have made a profit of 35 rupees suggesting a 160% rise in my original investment. However, this would only occur if I bought at the perfect time and sold at the perfect time. If I did, however, miss the maximum point of sale, then my returns would not be as large. Moreover, the stock does not return to a value higher than the purchase price, thus, suggesting I would most likely lose money if I missed the maximum value on graph 1.2. Reflection 2 Due to the function moving from the graph of 𝑦 = 𝑥 2 and 𝑦 = − 𝑥 2 the functions found are in vertex form suggesting that the minimum and maximum point are already in the function and, thus, I don’t need to differentiate to find the minimum and maximum point. Predicting Future Potential As you probably know, the past has happened and there is nothing I can really do about it. However, predicting the stock market is one of the many question’s economists have been trying to figure out for generations. I was very curious about if I could pre dict the stock value based of past data as my father always said the best time to invest is today . The question arose that if the theory that ‘the market always went back up’ was correct then on day 343 would be the lowest (minimum) price to buy. To create a model for this I could create a piecewise function from day 16 1 to 343 then reflect it onto the 𝑦 axis at 𝑥 = 343 Graph 1.3: Stock Price Kushal Tradelink 161 ≤ 𝒙 ≤ 𝟑𝟒𝟑 𝑓 ( 𝑥 ) { 1 2 | 𝑥 − 172 | + 19 85 161 2 ≤ 𝑥 ≤ 176 35 1 1 5 | 𝑥 − 194 | + 13 3 176 35 ≤ 𝑥 ≤ 205 34 − 1 2 5 𝑥 + 103 205 34 ≤ 𝑥 ≤ 223 𝑓 ( 𝑥 ) { 1 2000 ( 𝑥 − 343 ) 2 + 6 6 223 ≤ 𝑥 ≤ 343 Trading Days Close Price Kushal Tr adelink Reflection 5 The modulus notation was used in this piecewise function due to the nature of the stock value being quite unique between day 161 and 176 and 176 and 205. There is a sharp change in the value of the stock between these two points which the modulus notation provides. Now that I had modeled my own piecewise function between days 161 and 343, if I used passed trends to predict the future growth and I assume that the ‘stock market always goes back up’ then I can reflect the piecewise function, specifically the quadratic function 1 2000 ( 𝑥 − 343 ) 2 + 6 6 in the 𝑦 axis at 𝑥 = 343 . This will give me an exponential functio n up to day 570. Graph 1.3: Stock Price Kushal Tradelink 161 ≤ 𝒙 ≤ 𝟓𝟕𝟎 This graph does look very good, however, is it realistic? Well due to recent COVID - 19 events this graph sadly does not support the actual trend of what will most likely happen. Due to India’s population size of approximately 1 billion people the effect of COVID - 19 will cause a decrease in consumer spending as a lot of businesses will make redundancies as their costs in crease significantly – including Kushal Limited as the manufacturing sector was hit very hard. But if this was the case and the COVID - 19 virus did not occur, this graph could very much be a possibility due to India recently increasing trade with China and USA (both countries imports impacted largely due to COVID - 19). Moreover, the 160% profit made during the fluctuation between day 70 and 101 was stuck in my head. I decided to find the value at which I could make that 160% profit again, based on if I purcha sed on day 343 at 6.6 rupees. To find the day I would make the 160% profit based on the graph above I found 160% of 6.6 = 17.16 rupees. If I equated that to the quadratic function 1 2000 ( 𝑥 − 343 ) 2 + 6 6 I should be able to find the day to sell to hit that perfec t 160% Trading Days Close Price Kushal Tradelink 𝑓 ( 𝑥 ) = 1 2000 ( 𝑥 − 343 ) 2 + 6 6 𝑓 ( 𝑥 ) = 17 16 Therefore, 1 2000 ( 𝑥 − 343 ) 2 + 6 6 = 17 16 ( 𝑥 − 343 ) 2 2000 + 33 5 = 429 25 𝑥 2 − 686 𝑥 + 343 2 + 13 , 200 = 34320 𝑥 2 − 686 𝑥 + 343 2 − 21 , 120 = 0 − 𝑏 ± √ 𝑏 2 − 4 𝑎𝑐 2 𝑎 𝑥 = 198 𝑥 = 488 According to the data above, it is clear that day 198 has passed, so this tells me that day 488 would be my optimum time to sell based on the largest change in past trading events between day 70 and 101 On the other hand, I’m assuming that the stock would gradually rise over time. However, if trade with China an d USA continued and COVID - 19 did not occur, I would think it would be more of a logarithmic function, and after day 343 there would be more of a logarithmic increase. There are implication of this which will be discussed. Gr aph 1.3: Stock Price Kushal Tradelink 161 ≤ 𝒙 ≤ 𝟓𝟎𝟎 Trading Days Close Price Kushal Tradelink Instead of using a quadratic function from day 223 and 343 I used the logarithmic function below: − 3 5 𝑙𝑛 ( 𝑥 − 200 ) + 23 9 Following this I created an inverse log function and reflected the function above in the 𝑦 axis at 𝑥 = 343 − 3 5 𝑙𝑛 ( − 𝑥 + 500 ) + 24 24 Th e re is a general increase in the rate of change between day 343 to day 500. However, implications of doing this is that there will be a vertical increase at day 500 . This is unrealistic (the vertical increase) but the exponential growth till day 500 could very be p ossible. Due to this graph being more exponential in its growth, theoretically, I should get to that 160% increase from 6.6 rupees to 17.16 rupees. If I equate the logarithmic function to 17.16, I can find the day I should sell the function. 𝑓 ( 𝑥 ) = − 3 5 ln ( − 𝑥 + 500 ) + 24 24 𝑓 ( 𝑥 ) = 17 16 Therefore, − 3 5 ln ( − 𝑥 + 500 ) + 24 24 = 17 15 ln ( − 𝑥 + 500 ) = 354 175 − 𝑥 + 500 = 𝑒 354 175 − 𝑥 + 500 = 𝑒 2 √ 𝑒 4 175 𝑥 = − 𝑒 2 √ 𝑒 4 175 + 500 𝑥 = 492 From this, it becomes clear that the quadratic function would be more beneficial as I would receive my 160% increase earlier if I invested on day 343. However, following this 160% increase it would definitely be better for the s tock market to potentially have a logarithmic increase as after day 499 there will be a complete vertical increase, but this isn’t completely realistic with modern day stock markets. Graph 1.3 is a more realistic graph to illustrate the effect of the graph . But, if COVID - 19 does affect India to a large extent there could be a linear or exponential decrease in the value of Kushal Limited Stock. Reflection 6 In order to equate the logarithmic function to 17.16 I had to create a radical expression in which I converted an 𝑙𝑛 function, once moved to the right side of the = sign, to an 𝑒 function. Moreover , what I found very interesting is that the 𝑥 value is actually greater than the 𝑥 value found in the quadratic equation in graph 1.3. This could make sense as If I compare the graph side by side, both functions gradually increase, but when I look closely as graph 1.3 the rate of change is larger initially in comparison to graph 1.4 where there is a slower increase initially but does exponentially increase when 𝑥 = 400 An anomaly is clearly day 299, where the stock price fell from 9.66 to 6.54 rupees due to the shortage of the manufacturing industry. This was due to implications occurring with the company – insider trading – which led to the CEO making a statement which stated the individual who participated in such an act to be fired. This did cause a slight ris e in stock, however, overall, there was a general decrease due to the loss of demand in the manufacturing industry. Even if I purchased the stock at the minimum value on day 299 my profit would not be high due to the stock only rising by approximately 3 ru pees the next day. Also, the general decrease in the stock value over the forceable future – after day 299 - as illustrated on the graph would mean I would end up with a diminishing valuation 0 10 20 30 40 50 60 70 0 50 100 150 200 250 300 350 400 Close Price of Kushal Tradelink Trading Days Graph 1.5: Stock Price Kushal Tradelink 0 ≤ x ≤ 343 Conclusion: Overall, it is clear that the investment in the Stock Kushal Tradelink will not be a beneficial investment if I hadn’t bought in the optimum area of purchase where the accumulation value is the highest. Due to the general decrease in the stock value, the investment in such a commodity would not have a p ositive outcome as it is generally decreasing throughout this period of time. However, as previously sad there is not point in thinking in the past as the best time to invest is now. So, based on the logarithmic and quadratic graph illustrated in Graph 1. 3 and 1.4 the optimum time to buy would be on Day 43 as that is a minimum point. But this is based on the assumption that there will be a general increase in the value of Kushal Tradelink and it is not based on physical data. If I took the real value of Kushal Tradelink after Day 343 there is an actual decrease all the way down to 3.5 rupees, which does suggest that the recent COVID - 19 impact did have a large influence on the data as the stock plummeted exponentially due to the COVID - 19 impacting global t rade as many countries borders had closed down Also, with India being a huge trade - hub and having a population at approximately 1 billion people the impact can clearly be inferred based of these facts. There were positive outcomes of the report , by using piecewise functions, I was able to find the exact values for the maximum and minimum , by finding the 160% increase between day 43 and 101 it allowed me to implement these results and use past trends to predict future growth which is then illustrated in Gr aph 1.3 and 1.4. T o improve the on the other hand I believe I could have implemented the Gaussian Theorem to see if I could find the daily stock return as it would allow me to see the change between the closing prices of consecutive days. Furthermore, seei ng how much I gained or lost of the stock, each day once I made an investment into the commodity, would give me the ability to analyze the market in more depth. A major assumption of the study is that the stock will continue to fall, thus, suggesting that the optimum time to buy the stock was the time period stated throughout the report. However, if there is a sudden increase in demand for supply and manufacturing of infrastructure products then it can be inferred that there will be a rise in the stock val ue and, thus, the optimum time to buy may be imminent instead of assuming that the stock will just continue to fall. Also, based on the basic stock market theory that ‘the market will always increase’ the optimum level to invest would be at the lowest poin t on day 343, however, I am again making the assumption that the market will continue to rise. In the end the report does successfully models the stock value of the commodity company over the year and a half correctly. Even though many aspects could have contributed to the fall in the value of the stock, the careful analysis and research into the stock manipulation within the Mumbai based company suggests the inferences made throughout the report are correct. Bibliography Kushal Tradelink L imited. (2020, 9 28). Home Page . Retrieved from Kushal Limited: https://www.kushallimited.com/whatwedo.php MoneyLife Digital Team. (2019, November 26). Stock Manipulation: Kushal Tradelink . Retrieved from MoneyLife: https://www.moneylife.in/article/stock - m anipulation - kushal - tradelink/48611.html Yahoo Finance. (2020, September 25). Kushal Limited . Retrieved from Yahoo Finance.