Abstract
The football example demonstrated the wisdom of treating tacit assumptions with care: They may be false. Yes, the sum of the probabilities over all events equals one. But when dealing with different individuals, a selective sum of their “implicit probabilities” could differ from unity—which can either hurt or help you.
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- 1.
While readers may know how to compute Taylor series for y = f(x), what about for two variables such as z = f(x, y) or three variables as with u = f(x, y, z)? Answers for multivariable settings can be obtained in the manner described next.
- 2.
At Northwestern University, I would invite Arthur Pancoe, a highly successful investor, to visit my class. (A measure of his success is a headline in a 1988 issue of Money Magazine, “Take Two of Arthur Pancoe’s Drug Stocks and You May Be Rich in the Morning.”) He would ask the students to guess what were his daily readings. To their surprise, the readings included non-finance outlets such as Science and Nature, which gave him insights into what was being discovered. He also declared that he did not accept the EMH. Of course not: Pancoe was a leader, not a follower, so the basic assumption behind the EMH did not apply to him. He was one of those reacting people behind the premise.
- 3.
Thomas Malthus (1786–1834) used an Equation 3.11 type expression to analyze population growth. Similar to Equation 3.4, this approximation is reasonable “in the small” but definitely not in general. Indeed, this incorrect representation led to the several century Malthusian debate. One correction was to replace the linear term with a quadratic expression: Use not just the first, but the first two terms of a Taylor series expansion of aP + bP 2 = P(a + bP). Doing so created a search for explanations of the coefficients, such as carrying capacity, death rates, and the logistic equation. The message: Be careful; compare predictions with data.
- 4.
Rather than \(Y_n=\sum X_i\), the average \( Y^*_n=\frac 1n \sum _{j=1}^n X_j\) is often used. No problem; do the same by finding the associated Z representation, which leads to the same conclusion.
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Saari, D.G. (2019). Modeling. In: Mathematics of Finance. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-25443-8_3
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DOI: https://doi.org/10.1007/978-3-030-25443-8_3
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