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I have advanced degrees, Master’s (S. M.) and Doctoral (Sc. D.), in Materials Engineering from the Massachusetts Institute of Technology, Cambridge, MA, USA and Bachelor's and Master's degree in Mechanical Engineering, from the University of Poona (now Pune) and the Indian Institute of Science,... More
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  • Tesla Motors: Linear And Nonlinear Analysis For Profitability 5 comments
    Mar 2, 2013 5:09 AM | about stocks: TSLA


    Couldn't help it folks, had to post this!

    Like many others, after the recent negative press received by Tesla Motors Inc. (as a result of the negative review of the Model S performance by a NYT reporter), I took interest in the company for the first time and started analyzing its financial data using a new methodology that I have described in several articles uploaded at scribd.com. At least on my computer, I can adjust the three parameters, "m", "n", and "a", in the mathematical equation, y = m*(x^n)*exp (-ax), which describes the costs-revenues data for Tesla Motors and extrapolate to produce any desired level of profits! Here x is revenues and y is costs and x^n means x raised to the exponent n. If n = 1/2, we have the square root of x, for n = 2, we have square of x, if n = 3, we have the cube and so on for any general value of n.

    If we consider the cumulative revenues and costs for 2012 (these are the 3 month, 6 month, 9 month, and full year values), we find that costs have decreased dramatically in the fourth quarter of 2012, see Figure 1, although the company has not yet reported a profit since it became a public company in 2012. This nonlinear decrease in the costs, with increasing revenues, can be modeled using the mathematical equation give above, where the numerical values of the constants m, n, and a can be deduced, as I have shown, from the financial data. This is discussed in detail in two articles that interested readers can find, see links given below (or click here and here).

    1. http://www.scribd.com/doc/127767159/Tesla-Motors-Nonlinear-Model-for-Profitability-Analysis Linear analysis and introduction to the model for breakeven analysis.
    2. http://www.scribd.com/doc/127767159/Tesla-Motors-Nonlinear-Model-for-Profitability-Analysis Nonlinear model and breakeven revenue calculations.

    I have posted several short posts on my Facebook page describing these findings. The simpler power-law model, y = m(x^n) is seen to offer an adequate description for the costs-revenues data. Although costs exceed revenues and the company has not reported a profit, the power-law tells us that the rate of increase of costs, with increasing revenues, is actually decreasing. Also, if we examine the data carefully, we see that the data at the higher revenues begins to deviate from the power-law. The power-exponential law, with the exponential term exp (-ax), then becomes a more appropriate model.

    I have also found the value of the parameter "a", holding m and n constant (to match with 2012 cumulative quarterly data) so that Tesla Motors could start making a profit once they exceed sales (or deliveries as they put it, a car is made only after there is a commitment by the customer by advanced reservations) of 15,000 units per year.

    Current production capability is 20,000 units per year. This means the last 5000 units produce a profit. The first 15,000 units help them breakeven (recover fixed costs and the variable costs which increase as N, the number of units sold, increases). After that it is all profits. So, one plant can produce a profit of 5000 times $100,000 per unit (the current average price for Tesla's Model S) or $500 million in profits. The profits per plant could potentially even go up to $ 1 billion --- at least on my computer!

    That's good news Tesla Motors can use, I think, after some of the beating they have received from other experts here who have been dealing with such problems for much longer than I have been.

    (click to enlarge)

    Figure 1: The dramatic change in the slope of the cumulative costs versus cumulative revenues in 2012 for Tesla Motors, Inc. The rate of increase of costs, as measured by the slope of the graph has decreased.

    Finally, the meaning of the term "costs" is clarified in both the articles above. The "breakeven" analysis can only be performed if all the costs of operation are properly accounted for. The reader is urged to review this point. Also, although this might all seem academic to many here who are interested in more practical matters such as making money, even the ideas such as fixed costs and variable costs, and the three constants (a, b, p) which appear in the classical breakeven analysis (see discussion in Ref.[1]) are academic concepts. To date, I am not aware of any attempts to quantify the three constants (a, b, p) in the classical breakeven analysis. As discussed in the articles cited, Tesla Motors offers a unique test case to study the meaning of these real "fundamentals" about a company. As new operational data on revenues, profits, and costs become available in the near future, we will be able to gain deeper insights into the meanings of these theoretical concepts. More importantly, we might actually start "designing" companies, like scientists and engineers design our cars, airplanes, rockets, cell phones, computers, washing machines, refrigerators, etc. Call me a fool, if you like. Cheers. :)

    Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

    Stocks: TSLA
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  • vlaxmanan
    , contributor
    Comments (9) | Send Message
    Author’s reply » The 2012 quarterly data presented graphically in Figure 1 shows that costs are increasing with increasing revenues but at a decreasing rate. Hence, the data can be analyzed using a nonlinear model. The intersection point of the costs-revenues curve with the breakeven line on such a diagram then gives the breakeven revenue. The breakeven line has the equation y = x. Hence, the breakeven point is determined by solving the equation y = x = m*(x^n)*exp(-ax). It is easier to just prepare a graph, as illustrated in the references cited above. This is done very easily using modern computer programs such as Microsoft Excel.
    2 Mar 2013, 06:03 AM Reply Like
  • vlaxmanan
    , contributor
    Comments (9) | Send Message
    Author’s reply » The simpler power-law, y = m*(x^n), can be rewritten, after taking logarithms, as logy = n logx + logm. Hence, the graph of log x versus log y, should be a straight line, with a slope n and intercept log m. Thus, we can fix m and n and fit a nice smooth curve to the data in Figure1. This is illustrated in Figure 7, on page 22 of Ref. [2]. The constant n = 0.70, or roughly 2/3 (this value of n is commonly observed when we study the behavior of many complex systems, including materials processing experiments that I did during my doctoral thesis work at MIT). The decreasing slope (since n < 1) means there is an intersection point with the breakeven line y = x. However, if we look carefully at the data, we see that at the highest revenue, there is a deviation from the power law curve. Hence, we fit the exponential term, and this allows the determination of the constant "a". If these extrapolations hold, we can predict the breakeven point, see Figure 9. The validity of this line of reasoning is supported by other calculations presented in Refs.[1] and [2]. For completeness, it is worth noting that the experimental observations on the radiation spectrum of heated bodies (also called blackbody radiation) were analyzed using the logic just presented, before Max Planck developed his quantum theory, see additional discussion of this point in Ref. [2].
    2 Mar 2013, 06:45 AM Reply Like
  • Asim Srivastava
    , contributor
    Comments (2) | Send Message


    I had a quick read through your article on scribd, and I must say, your application of intuition and principles from Physics to Finance is really quite interesting. I was wondering if you could provide some further insight into what the parameters m,n, and a mean, or what their physical interpretation should be. Also, is there any reason to explain why the value of a is so small, at -2.25*10^-4, especially since you mention it may be related to fixed cost?
    I've always believed the descriptions for systems we obtain from Physics can be applied to a variety of other practical situations, and so its good to see that happening!


    I also have interests in applying mathematical techniques to financial situations, and I've found Physics to be a great place to draw inspiration from, needless to say it was my favorite subject in school and still continues to amaze me as I learn more about further concepts in Quantum Physics, much in the way that it did when I read about concepts like electromagnetism and em-waves for the first time in school!


    In fact, I was doing some research into the Capital Asset Pricing Model (CAPM) in finance and looking at ways to provide improvements. I found that it was possible to use a function that had a similar equation to the Maxwell-Boltzmann distribution of Kinetic Energy for gas molecules, to provide a more general model. Perhaps I could send you the paper so you could provide your thoughts and insight on it?
    2 Mar 2013, 09:34 AM Reply Like
  • vlaxmanan
    , contributor
    Comments (9) | Send Message
    Author’s reply » Thanks, Asim for your comments and sorry for the late response. I am NOT a frequent visitor here -- mainly because I am still so frustrated with being unable to upload documents properly. So, I post them at scribd and leave a short summary here.


    The nonlinear equation, called Wien's law in blackbody radiation physics, and Maxwell-Boltzmann distribution, are both identical mathematically speaking although their physics is very different. Of all models from physics that can be applied to financial and economic data analysis, the nonlinear law called Wien's law, also used by Einstein in 1905, is the most useful. In particular, I think the idea of a work function that Einstein develops in his photoelectricity paper, if fully understood and applied outside physics, will change the so-called soft sciences. That is what I have tried to focus on. Getting distracted with too many other models, and getting fascinated with nonlinear models, is IMHO, not a very good idea. A few years ago, I tried to present the same ideas in another forum and was mocked and called a fool. Now, the same model that I tried to describe then has been proved to be true beyond my wildest dreams. In particular, financial data for several companies (unfortunately due to the financial crisis) has now revealed a maximum point. GM's profits-revenues graph shows a maximum point and it was forced into bankruptcy. Other companies are operating past their maximum point and struggling. Unfortunately, most business leaders, even the CEOs of the companies (like Ford Motor Company) do not know that their company is operating past the maximum point.


    Now, that brings me to your most important question. What is the meaning of the parameters m, n, and a in the model and why is "a" so small? I wish I could answer you fully. But, here is a partial answer. The nonlinear model is a generalization of the linear model, y = hx + c, where it is very clear what the slope h and the intercept c mean. This is discussed in almost every article that I have written. Both "h" and "c" can be related to the fixed cost, unit variable cost, and the unit price. Please read the article on Tesla, or on Google, or Apple, or Microsoft.


    Since the constants h and c in the linear law can be both positive or negative, we have three cases and this then leads to the nonlinear law as well. So far, I have NOT related the parameters m, n, and a in the nonlinear law to the three constants - fixed constant, unit variable cost and unit price - in the financial model. For the linear model it is possible but not yet for the nonlinear model.


    Finally, why is "a" in the nonlinear model so small? At this stage, it is simply a fitting parameter. I have generally used value of the exponent "n" as 0.6667 or 2/3 based on what we see in physical processes. Once n is fixed, m can be fixed using just a few points and least squares analysis. Very rarely we see deviations from the "pure" power law (case of a = 0 with n < 1 of n > 1). Tesla is one rare example where we see deviation from the power law. Hence, all I did was to "fit" the model to the observed value.


    Without appearing to flatter myself, if all this is accepted by the large majority in the financial and business world, we would where physics was when Wien proposed his law, purely as a curve-fitting model. There was no deeper understanding. Although I have been able to show that Planck's law can be generalized to financial world, I have NOT yet given a meaning to each of the parameters. That, may not be what I have to focus on.


    I strongly believe that I must simply focus on getting the idea of the linear law and the work function accepted by the financial and economic world. If people start seeing the problem with the use of the y/x ratio when the law is y = hx +c, that would be a BIG BIG step. The rest that you, Asim, are interested in, will follow. Thanks for your thoughtful comments. I hope I have added some clarity here.


    Do look at the baseball analogy that I have posted today.
    26 May 2013, 06:19 AM Reply Like
  • vlaxmanan
    , contributor
    Comments (9) | Send Message
    Author’s reply » The following may also be of interest Here I have continued to use baseball statistics to illustrate some fundamental concepts in data analysis, not generally used in the "soft sciences" like financial, business, social and political sciences and even economics. Here's the link


    29 May 2013, 02:47 AM Reply Like
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