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 \(p\) is the probability of winning and \(q\) is the probability of losing, then \(p + q = 1\). 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 \( \left( p > .5 \right) \), 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 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 \( f^* = \frac{bp-q}{b} \) of the bankroll, where the quantities represent

\(f^*\): fraction of the current bankroll to be invested.

\(b\): net odds received (\( b\) to 1).

\(p\): probability of winning

\(q\): probability of losing

Because \( p+q=1 \), students can rewrite the formula as \( f^* = \frac{p(b+1)-1}{b} \).

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 \(p\) with a slider and graphing \(f^*\) as a function of \(b\). So, for \(p = 0.6\) and \(b = 1\), the Kelly criterion suggests investing 20% of the bankroll (Figure 1). For a fixed value of \(p\), students can explore the relation between \(b\) and \(f^*\). 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 \(p\).

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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 (\(b = 1\), \(p = 0.6\)). 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? (\(0.6 \times 2 = 1.2\)) What is the expected net win for each investment? (\(\$200 \times 0.2 = \$40\)) What is the expected win after 200 investments? (\(200 \times \$40 = \$8,000\))

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 \( p = 0.6\). Figure 2 shows two simulation examples for \(p = 0.6\). 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.

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Figure 2a. Bankroll change with fixed investment amount, Example 1.

Figure 2b. Bankroll change with fixed investment amount, Example 2.

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 (\(b = 1\)). The probability of winning is \(p\) and \(q\) of losing. For \(b = 1\), Kelly’s formula is simply \(f^* = p - q\). 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 \(p = 0.6\). 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 \(p = 0.6\). 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.

Figure 3b. Bankroll growth with Kelly criterion, Example 2.

Students can then run the program for \(p = 0.7\) 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. Retrieved from http://www.corestandards.org/Math/

Enthought (2014). Canopy [Computer software]. Retrieved from https://www.enthought.com/downloads/.

Kelly criterion (n.d.). In Wikipedia. Retrieved February 18, 2019, from https://en.wikipedia.org/wiki/Kelly_criterion

Credits

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.

From soliciting answers to eliciting reasoning: Questioning our questions in digital math tasks

Heather Lynn Johnson, Gary Olson, Amber Gardner, Amy Smith

University of Colorado Denver

Students can interact with digital math tasks in different locations, on different devices, and for different purposes. What kinds of questions do students encounter when interacting with digital math tasks? And why might the kinds of questions matter?

We designed digital math tasks to provide opportunities for students to engage in math reasoning. Questions are a key component of the tasks. With our questions, our goal was to do more than solicit students’ answers. We intended to elicit students’ reasoning.

We share a digital math task and a question from the task. Then we provide three design principles guiding our questions.

The Toy Car Task

We developed the The Toy Car task in collaboration with Dan Meyer and the Desmos team. The task begins with a video of a toy car moving along a curved path (Figure 1). Then students investigate and graph relationships between a toy car’s distance from a shrub, and its total distance traveled.

Figure 1. The toy car and the shrub

The Toy Car task is part of a group of tasks that we call Techtivities. The Techtivities include video animations and dynamically linked, interactive graphs. Students have opportunities to sketch different graphs to represent the same relationship between attributes. Then students reflect on what those graphs represent. To learn more about the Techtivities, see Johnson (2018).

In the Toy Car task, students sketch, then reflect on two different graphs, shown in Figure 2. Each graph represents the toy car’s total distance traveled as a function of the toy car’s distance from the shrub.

Figure 2. Two different graphs in the Toy Car task

Students might wonder how it is possible for two different looking graphs to represent the same function relationship. Furthermore, students might notice that the graph shown at right in Figure 2 does not pass the vertical line test, meaning that a vertical line would intersect the graph at more than one point.

Students can apply the vertical line test based solely on the shape of a graph, and they may miss how graphs can represent relationships between attributes in a situation (Moore, Silverman, Paoletti, & LaForest, 2014). In the Toy Car task, and across the Techtivities, our goal was for students to focus on relationships between different attributes in the situations. We worked to design questions that could help us to achieve our goals.

A Question

We posed this question in the Toy Car task: Val says that both of these graphs represent the toy car’s total distance traveled as a function of the toy car’s distance from the shrub. Do you agree or disagree? Why or why not? (Graphs are shown in Figure 2.)

We purposefully posed this question as person’s (Val’s) claim, rather than as a claim devoid of human connection. Furthermore, we used precise language to clarify Val’s claim. In particular, we used the phrase as a function of, rather than the more general term, function. We did this so that Val’s claim focused on the function relationship that the graphs represented. Overall, we aimed to position Val as a capable doer of mathematics, who made a claim worthy of consideration.

Three Design Principles

1. Provide opportunities for students to consider other students’ claims. Mathematics is a human endeavor (Freudenthal, 1973). In our questions, we decided to have students respond to another student’s claim. We could have asked students: Do both graphs represent the toy car’s total distance traveled as a function of the toy car’s distance from the shrub? By framing our questions as a response to another student, we aimed to humanize students’ interactions with the digital math tasks.

2. Allow for gender ambiguity when incorporating student names into task questions. Students can think that gender identity plays a role in mathematical ability (Boaler, 2002; Leyva, 2017; Rubel, 2016). In our questions, we aimed to use gender ambiguous names, and names we selected were often informal. We could have used a pronoun to assign a gender identity to Val, or selected a more gendered name. Instead, we intended to open possibilities for students to use a variety of pronouns, or no pronouns at all, when responding to the student claims given in the tasks.

3. Elicit sense making, rather than soliciting judgments of correct/incorrect. To promote students’ reasoning, we posed questions to elicit sense making rather than solicit judgments. We could have asked students if Val was right or wrong. Instead of asking students to judge another student’s claim as correct/incorrect, we chose to ask students to explain why they agreed or disagreed. We intended to offer students opportunities to consider possibilities, rather than rushing to judgments.

Closing Remarks

Doing mathematics is so much more than finding answers. With our questions, we can work to create spaces for students to engage in reasoning and sense making. In designing questions for our digital math tasks, we are aiming to do just that.

Acknowledgments. This work was supported by a grant from the National Science Foundation (DUE-1709903). Opinions, findings and conclusions are those of the authors. We thank Dan Meyer and the team at Desmos for their work with us. We are grateful to our colleagues who provided feedback to help us to grow.

References

Boaler, J. (2002). Paying the price for “sugar and spice”: Shifting the analytical lens in equity research. Mathematical Thinking and Learning, 4(2-3), 127–144.

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel Publishing.

Johnson, H. L. (2018). Helping students see how graphs work | Edutopia. Retrieved July 30, 2018, from https://www.edutopia.org/article/helping-students-see-how-graphs-work

Leyva, L. A. (2017). Unpacking the male superiority myth and masculinization of mathematics at the intersections: A review of research on gender in mathematics education. Journal for Research in Mathematics Education, 48(4), 397–452.

Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. (2014). Breaking conventions to support quantitative reasoning. Mathematics Teacher Educator, 2(2), 141–157.

Rubel, L. H. (2016). Speaking up and speaking out about gender in mathematics. The Mathematics Teacher, 109(6), 434–439.