Students do probabilistic models for investment strategies and bankroll management. Students develop a better understanding for the Kelly criterion as a function of paying odds using an interactive graph. Then students conduct computer simulations to compare two investment strategies, first investing a fixed amount, and then using the Kelly criterion

Flores, Alfinio

English

University of Northern Colorado

COLORADO MATHEMATICS TEACHER FEBRUARY 2019, VOLUME 52, ISSUE 1, ARTICLE 1 PUBLISHED BY THE COLORADO COUNCIL OF TEACHERS OF MATHEMATICS HTTP://WWW.CCTMATH.ORG Investing under uncertainty: Mathematical models and computer simulations Alfinio Flores   University of Delaware  INTRODUCTION Mathematical modeling is stressed in the Common Core State Standards (Common Core State Standards Initiative, 2010) both as a content standard for high school and as a standard of mathematical practice for all ages. Modeling investment growth is one example mentioned for high school. Students need to experience mathematical models not only of deterministic situations but also of random phenomena. The following activities have students use computer simulations in the context of investing money with uncertainty. Students need to be able to interpret graphs of rational functions and have basic probability knowledge, such as if 𝑝𝑝 is the probability of winning and 𝑞𝑞 is the probability of losing, then 𝑝𝑝 + 𝑞𝑞 = 1. Students can download and run the simulations without needing any programming knowledge. By running many simulations, and comparing and contrasting the outcomes, students can learn that in probabilistic situations we can predict with some confidence the results for the long term, but that for small numbers of trials the results can vary widely. The Need for Bankroll Management A particular investment may be a win or a loss for an investor. The uncertainty of investments can be better described and understood using probabilistic models rather than deterministic models. Investors, due to skill and experience, may have a favorable probability (𝑝𝑝 > .5), but even so, long losing streaks are not uncommon. Thus, it is important for investors to properly manage their bankroll, the total money that they can Abstract: Students do probabilistic models for investment strategies and bankroll management. Students develop a better understanding for the Kelly criterion as a function of paying odds using an interactive graph. Then students conduct computer simulations to compare two investment strategies, first investing a fixed amount, and then using the Kelly criterion. Keywords: Tools and Technology, Python, financial literacy ARTICLE COLORADO MATHEMATICS TEACHER VOLUME 52 | ISSUE1 | ARTICLE 1 | 2 invest. For a long series of investments, is it better to invest each time a fixed amount, or a variable amount? Kelly developed a criterion that is used nowadays in investment theory and practice (“Kelly criterion,” n.d.). Students can use GeoGebra to develop a better understanding of the Kelly criterion, a real life long term optimal investment model, and Python programs to simulate and contrast two investment strategies. The Kelly Criterion Kelly’s formula determines the optimal size of a series of investments to maximize the expected utility over long trials (“Kelly criterion,” n.d.). Considering simple investments with two possible outcomes, losing all the invested money, or winning with the corresponding payoff odds, the Kelly criterion indicates to invest the fraction 𝑓𝑓∗ = 𝑏𝑏𝑏𝑏−𝑞𝑞𝑏𝑏 of the bankroll, where the quantities represent  𝑓𝑓∗: fraction of the current bankroll to be invested  𝑏𝑏: net odds received (𝑏𝑏 to 1)  𝑝𝑝: probability of winning  𝑞𝑞: probability of losing Because 𝑝𝑝 + 𝑞𝑞 = 1, students can rewrite the formula as 𝑓𝑓∗ = 𝑏𝑏(𝑏𝑏+1)−1𝑏𝑏. An Interactive Graph for the Kelly Criterion Using GeoGebra, students can interact with the graph at https://beta.geogebra.org/m/CkxdBZ9n by seting the value of 𝑝𝑝 with a slider and graphing 𝑓𝑓∗ as a function of 𝑏𝑏. So, for 𝑝𝑝 = 0.6 and 𝑏𝑏 = 1, the Kelly criterion suggests investing 20% of the bankroll (Figure 1). For a fixed value of 𝑝𝑝, students can explore the relation between 𝑏𝑏 and 𝑓𝑓∗. For example, ask students to find a specific value that would suggest investing 35% of a bankroll. Let students discuss and describe how the fraction invested varies for different values of the odds. Students can identify the type of graph (a rational function, a hyperbola) and describe the two asymptotes. By dragging the slider, students can see how the graph changes for different values of 𝑝𝑝. COLORADO MATHEMATICS TEACHER VOLUME 52 | ISSUE1 | ARTICLE 1 | 3  Figure 1. The Kelly criterion for b = 1 and p=0.6. SIMULATING BANKROLL MANAGEMENT WITH PYTHON Python is a free programming language; different versions are available. We used Canopy, which has a friendly user interface (Enthought, 2014). Teachers or students can download Python. The file bankroll.py used in this article runs well with Python 3.x. After students download the file from http://udel.edu/~alfinio/bankroll.py, they need to right click and save it as a Python file (file extension .py). After opening the file with Python, in the “Run” menu select “Run File.” Students can then execute the different functions in the file by typing the name of the function with the desired value for the parameter, for example fixed(0.6), and pressing Enter or Return. Exploration of Investing a Fixed Amount In this activity, students will learn that by investing a fixed amount, in the long term, investors can estimate their wins by using a linear function, but that for short sequences of investments the outcomes can vary widely. Suppose a skilled investor can double the money invested 60% of the time (𝑏𝑏 = 1,𝑝𝑝 = 0.6). The other 40% of the time the investor loses the money. The starting bankroll is $1000 and $200 is invested each time. What is the expectation for this case? (0.6 × 2 = 1.2) What is the expected net win for each investment? ($200 × 0.2 = $40) What is the expected win after 200 investments? (200 × $40 = $8,000) We simulate the situation with the Python function fixed(p). The investor will make 200 investments or stop investing if the bankroll is depleted. Students can type fixed(0.6) to run the function for 𝑝𝑝 = 0.6. COLORADO MATHEMATICS TEACHER VOLUME 52 | ISSUE1 | ARTICLE 1 | 4 Figure 2 shows two simulation examples for 𝑝𝑝 = 0.6. Let students observe the graphs, discuss, and describe the results of different simulations, paying attention to the scale, identify interesting parts of the graphs, such as long losing streaks, and contrasting long-term predictions with short-term variability.  Figure 2a. Bankroll change with fixed investment amount, Example 1.  Figure 2b. Bankroll change with fixed investment amount, Example 2. COLORADO MATHEMATICS TEACHER VOLUME 52 | ISSUE1 | ARTICLE 1 | 5 Exploration of Using the Kelly Criterion In these activities, students will learn that by using the Kelly criterion, in the long term, investors can make their bankroll grow much more than by investing a fixed amount, but that short sequences of investments can vary widely and include considerable losses. We consider again two possible outcomes, one losing the entire investment and the other doubling the money (𝑏𝑏 = 1). The probability of winning is 𝑝𝑝 and 𝑞𝑞 of losing. For 𝑏𝑏 = 1, Kelly’s formula is simply 𝑓𝑓∗ =𝑝𝑝 − 𝑞𝑞. Thus, if an investor has a probability of 60% of winning, Kelly’s formula recommends investing 20% of the bankroll. Students can run the function kelly(p) using 𝑝𝑝 = 0.6. The initial bankroll is $1000, and the investor will wager 200 times or stop investing if the bankroll is depleted. Figure 3 shows the results of two simulations for 𝑝𝑝 = 0.6. When observing and discussing the graphs, students should pay special attention to the scale on the vertical axis.  Figure 3a. Bankroll growth with Kelly criterion, Example 1. COLORADO MATHEMATICS TEACHER VOLUME 52 | ISSUE1 | ARTICLE 1 | 6  Figure 3b. Bankroll growth with Kelly criterion, Example 2. Students can then run the program for 𝑝𝑝 = 0.7 several times. What seems to be a modest increase in the favorable probability (from 0.6 to 0.7) can drastically change the outcomes. FINAL REMARKS AND EXTENSIONS Bankroll management is also important in other contexts where people face uncertainty, such as professional gamblers. In this case too, bankroll management “should eventually reference the Kelly criterion.” (Luis Flores, personal communication, 2014). Other situations involving money and uncertainty can also be simulated using probabilistic models. Students can simulate the effect of different strategies for withdrawing money in retirement when the assets face unpredictable fluctuations, such as in the stock market. REFERENCES Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. http://www.corestandards.org/Math/ Enthought (2014). Canopy [Computer software]. https://www.enthought.com/downloads/ Kelly criterion (n.d.). In Wikipedia. Retrieved February 18, 2019, from https://en.wikipedia.org/wiki/Kelly_criterion COLORADO MATHEMATICS TEACHER VOLUME 52 | ISSUE1 | ARTICLE 1 | 7  Author: Alfinio Flores is the Hollowell Professor of Mathematics Education at the University of Delaware. He teaches mathematics and mathematics methods courses con ganas. His students use hands-on materials, visual representations, computers, and calculators to explore mathematical ideas and to develop a better understanding of concepts and relations. Associate Editor: Gulden Karakok edited this article. © Author(s), 2/21/2019. “Investing under uncertainty: Mathematical models and computer simulations” is distributed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC 4.0) which permits any non-commercial use, reproduction, and distribution of the work without further permission provided the original work is properly attributed. Suggested Citation: Flores, A. (2019). Investing under uncertainty: Mathematical models and computer simulations. Colorado Mathematics Teacher, 52(1), Article 1. https://digscholarship.unco.edu/cmt/vol52/iss1/1 

Investing under uncertainty: Mathematical models and computer simulations

https://digscholarship.unco.edu/cgi/viewcontent.cgi?article=1000&amp;context=cmt

Investing under uncertainty: Mathematical models and computer simulations

Abstract

Similar works

Full text

Available Versions

University of Northern Colorado