A Class Experiment to Illustrate Multiple-Asset. Arbitrage and Other Important Concepts in Finance. R. Brian Balyeat. *. ABSTRACT. One of the most important ...
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A Class Experiment to Illustrate Multiple-Asset Arbitrage and Other Important Concepts in Finance R. Brian Balyeat* ABSTRACT One of the most important forces in finance is the concept of arbitrage. Arbitrage is the driving force behind the Law of One Price, numerous asset pricing models, the efficient markets hypothesis, and many topics in derivatives. The purpose of this paper is to introduce a simple inclass exercise that transitions students from trivial cases of arbitrage to examples involving multiple assets. Additionally, the exercise reinforces the concepts of basic descriptive statistics, the risk-free rate, mean-variance efficiency, diversification, and short-selling. All that is needed for the exercise is a chalkboard and fifteen minutes of class time.
Introduction Paradoxically, the concept of arbitrage is one of the easiest concepts for students to learn, but one of the most difficult to apply beyond trivial illustrations. Students frequently struggle in applications beyond the traditional example of one asset selling for two different prices in two distinct markets. However, the concept of arbitrage may be the single most important force in finance. For as Stephen A. Ross (1987) noted, referring to a parrot who had already been taught the economic concepts of “supply” and “demand,” “To make the parrot into a learned financial economist, he only need to learn the single word ‘arbitrage.’ ” The concept of arbitrage extends into many areas of finance. Arbitrage is the driving force behind the Law of One Price, the Arbitrage Pricing Theory (APT) and other asset pricing models, the efficient markets hypothesis (which is a lack of arbitrage), implied forward rates, futures pricing, option bounds, option pricing, binomial models, the put-call parity, Modigliani-Miller capital structure propositions, and financial engineering. This paper provides a simple in-class exercise that helps students make the transition from trivial examples of arbitrage to understanding how to identify arbitrage opportunities in multiple-asset settings. In the exercise, students will be presented with a budget and four asset classes. The exercise design allows students to buy or sell each asset for the same price. The payout structure for each asset depends upon which of three states of nature is randomly selected. Purchasing and selling the right combination of assets results in a costless portfolio that is both riskless and guarantees a positive profit. Hence, the portfolio represents a multiple-asset arbitrage opportunity. The design of the exercise is to provide the intuition behind multiple-asset arbitrage without getting lost in either detailed mathematical analysis or technical detail. Additionally, the exercise engages students in an interactive critical thinking activity that both expands upon the notion of arbitrage and reinforces the concepts of basic descriptive statistics, the risk-free rate, mean-variance efficiency, diversification, and short-selling. Lastly, the resources needed for the exercise are minimal. To implement the exercise, all that is needed is either chalkboard or whiteboard and fifteen minutes of class time. Frequently, the best way to learn a new concept is to apply it to a new situation. As noted by Fels (1993), not all students learn in the same manner; thus, repeating a concept in a variety of different contexts can be fundamental to the learning process. In-class exercises have been shown to have a positive effect on the learning process. Cebula and Toma (2002) noted that the use of an in-class investment portfolio management project enhances student learning outcomes. In an in-class options game, Pavlik and Nienhaus (2004) concluded that the exercise helped *
Associate Professor, Department of Finance, Xavier University, Cincinnati, OH 45207-1213. The author would like to thank Julie Cagle, the editor, and the anonymous referee for helpful comments. As usual, all remaining errors are the author’s.
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students to better internalize the material from the textbook and lecture. This paper offers a simple numerical exercise as a supplementary means of exploring the concept of arbitrage beyond lectures and the traditional textbook applications.
Exercise Methodology and Solution The exercise was performed with an undergraduate Investments class and in a MBA Investments class. The undergraduate class is one of the first classes in the major above the required course for all undergraduate business majors and the graduate class is an elective MBA students take after completing an introductory course and a Corporate Finance class. Before the exercise was performed in class, arbitrage had been presented in both classes in the context of the Arbitrage Pricing Model (APT). The basic definition of an arbitrage was presented as a portfolio that simultaneously (1) costs nothing, (2) has no risk, and (3) provides a positive profit. Additionally, simple examples of arbitrage were covered where students could simultaneously buy and sell a single asset at two different prices. Only covering simple examples of single-asset arbitrage, while requiring less class time than the exercise detailed in this paper, did not provide the necessary background for students to easily comprehend examples of multiple-asset arbitrage. Additionally, students frequently confused the concept of arbitrage with the APT model. Without the exercise, the topics of implied forward rates, valuation of callable bonds, and triangular currency arbitrage were more difficult for the students to grasp later in the course. About a week before the exercise is to be performed in class, students are given the following return structure for four assets in a three-state economy.
Return on Asset A (%) Return on Asset B (%) Return on Asset C (%) Return on Asset D (%)
Boom (Pr = 30%) 6 11 4 6
Neutral (Pr = 50%) 5 4 12 6
Bust (Pr = 20%) 10 6 5 6
Please note that as indicated in the table, the probability of the boom, neutral, and bust states are 30%, 50%, and 20%, respectively. In a discussion of what and how financial theory should be taught, Chang (2005) noted that an integral part of financial theory education involves the use of applications with numbers. To this end, students are asked to calculate the expected return and standard deviation for each of the four assets. Additionally, students are asked to calculate all of the pair-wise correlations among the four assets. This assignment is to be completed outside of class time and prior to the day the exercise is to be done in class. The day of the exercise, the students are given the following set of instructions. The class has $300 to invest in any of the assets (A through D). Each asset can be bought or sold at a price of $100. The class’s job is to create a portfolio that maximizes expected return while minimizing risk. This condition is met when the class devises a portfolio for which the professor cannot find a more efficient portfolio.
Typical In-Class Exercise Results The exercise starts by writing the payoff structure for the exercise on the board and reading the instructions to the students. It is important to make sure the students understand that the initial endowment is $300 and that each asset can be bought or sold for $100. Besides the payoff structure, a blank vertical list with the numbers 1 to 6 can be used to track the class’s progress towards reviewing the six concepts of basic descriptive statistics, mean-variance efficiency, the risk-free rate, diversification, short-selling, and arbitrage. The list can be referred to as the “concept list” for the exercise. Inform the class that the exercise concludes when they have satisfied the two conditions of the exercise. First, the students must uncover the six concepts for the exercise. Second, the students must find a portfolio such that the professor cannot find a more efficient alternative. This facilitates a discussion on what it means for a portfolio to be mean-variance efficient. At the conclusion of the mean-variance efficiency discussion, “meanvariance efficiency” can be written beside the first number on the concept list.
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Next, students can fill in the following descriptive statistics table. Given that the students are assigned the descriptive statistics in advance, they can be called upon randomly to fill in the table. It should be noted that since the three states in the exercise are not equally weighted, frequently many of the students will have miscalculated the descriptive statistics.
Asset A Asset B Asset C Asset D
Exp. Ret. (%) 6.3 6.5 8.2 6.0
Stnd. Dev. (%) 1.90 3.04 3.82 0.00
Corr. w/ A 1.00 0.15 -0.62 undef.
Corr. w/ B 0.15 1.00 -0.87 undef.
Corr. w/ C -0.62 -0.87 1.00 undef.
Corr. w/ D undef. undef. undef. 1.00
If necessary, a brief review of expected return, standard deviation, and correlation under the case of unequal probabilities can be undertaken. Special attention needs to be given to Asset D. Asset D has a zero standard deviation and a zero covariance with each of the other three assets. Because the correlation between two assets is defined as covariance between those assets divided by the product of the two standard deviations, the correlations of Asset D with any of the other assets is zero divided by zero, which is undefined. “Basic descriptive statistics” can now be written by the number 2 on the concept list. The exercise proceeds by simply asking the students if they notice anything unusual, if any element of the payoff structure seems important, or if any of the results from the descriptive statistics stands out. Without too much prodding, students usually notice that Asset D is the risk-free asset as it returns the same amount regardless of the state of the economy. The definition of the risk-free asset is an asset that has zero standard deviation. From the descriptive statistics, Asset D has a zero standard deviation. Thus, Asset D is the risk-free asset and yields the riskfree rate. “Risk-free rate” can now be written by the number 3 on the concept list. While it may take some help, a student frequently notices that each of the remaining assets (i.e. A, B, and C) does extremely well in one of the three states. Furthermore, this superior performance is in a different state for each asset. The payoff structure of the exercise visually facilitates this observation in that in each state a different asset has a two-digit return while all of the other returns are a single digit. Additionally, the cross-correlations for the first three assets are either low or negative. These observations can be used to guide the discussion towards assets in a portfolio with low correlations and the idea of diversification. The low correlation among assets A, B, and C can be seen visually in the table. In each state, while one of the three assets performs extremely well, the other two assets perform poorly. “Diversification” can now be written besides the number 4 on the concept list. Next, students frequently notice that for assets A, B, and C the returns summed down the columns always sums to 21. This observation reinforces the benefits of diversification. At this point, students usually suggest buying one unit each of Asset A, Asset B, and Asset C. This is denoted as portfolio ABC and uses the entire $300 initial endowment. The following results table can be put on the board and denotes the portfolio, its return, its risk, and leaves an extra column that will be used later. Because of the arbitrage opportunity inherent in the exercise, it is easier to talk about the return on the portfolios in dollar terms (in excess of the initial investment) rather than as percentages or as an average percentage. Portfolio ABC
Return $21
Risk 0
????? -------
The portfolio ABC returns $21 in all three states and has no risk. Students are usually surprised to learn that the exercise allows for other portfolios that are more mean-variance efficient than portfolio ABC. Further analysis by the students leads to the conclusion that any portfolio that is superior to the ABC portfolio must also be risk-free. At this point, it is frequently necessary to guide the students by reminding them that portfolio DDD was also risk-free. Eventually students realize that the exercise allows them to short-sell DDD to fund ABC. This is similar to analysis of the capital market line where investors must create leverage by borrowing at the risk-free rate to achieve an expected return larger than the market return. Short-selling portfolio DDD to fund portfolio ABC will be denoted as portfolio (ABC)-(DDD) and leads to a net risk-free return of only $3. However, this portfolio also does not use any of the $300 initial endowment. The last column in the results table is revealed to be “money left.”
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Portfolio ABC (ABC)-(DDD)
Return $21 $3
Risk 0 0
Money Left $0 $300
While the initial portfolio ABC costs $300 and thus has no money left, the (ABC)-(DDD) portfolio did not use any of the $300 initial endowment. Additionally, “short-selling” can now be written by the number 5 on the concept list. However, the exercise is not finished because there are other portfolios that offer a higher return for the same risk and the 6th item on the concept list is still blank. Because portfolio (ABC)-(DDD) leaves the initial $300 endowment untouched, frequently students will suggest using the $300 to buy an additional ABC portfolio. This portfolio is denoted portfolio 2(ABC)-(DDD) and returns $24 risk-free. Portfolio ABC (ABC)-(DDD) 8[(ABC)-(DDD)]
Return $21 $3 $24
Risk 0 0 0
Money Left $0 $300 $0
From this point, the class discussion can be guided by asking the question, “Can anyone construct a risk-free portfolio that yields $24 without spending the $300?” Students quickly note that 8 portfolios (ABC)-(DDD) would yield the desired risk-free return of $24 without spending any money up front as illustrated by the last line in the results table. Thus, with portfolio (ABC)-(DDD) we have a portfolio that costs nothing, has no risk, and delivers a positive return. Students are usually quick to point out that portfolio (ABC)-(DDD) is an arbitrage portfolio. Given that this portfolio costs nothing and is infinitely scalable, one can generate any desired return without spending any of the initial endowment. “Arbitrage” can now be written by the number 6 on the concept list. The students have now satisfied the two conditions of the exercise: the concept list is completed and the students have found a portfolio for which the professor cannot find a more efficient alternative.
Student Perception and Assessment of Exercise As a supplement to the exercise, both classes were given a pre-test and post-test. The pre-test was given after the lecture on the APT and directly before the exercise was performed in class. The pre-test consists of three scenarios where the students were asked to identify if the stated scenario provided an arbitrage opportunity. The three scenarios are as follows: 1. You are quoted the following exchange rates: 80 yen per 1 US dollar, 1 US dollar for 0.75 euros, and 100 yen per 1 euro. 2. You have $100. You can invest (i.e. lend) money for 1 year at 5% and then you can re-invest all of the funds for a second year at 6%. Likewise, you could invest the $100 for two years at 5.25% for both years. Please assume that for each rate where you are investing, you could also borrow money at that same rate. 3. You find a convertible bond selling for $900. The coupon on the bond is 10% and the bond can be converted into 20 shares of stock. The stock currently sells for $30 per share. The first and second scenarios involve an arbitrage opportunity whereas the last does not. The students were given a post-test directly after the completion of the exercise. The post-test was identical to the pre-test. Both the pre-test and the post-test were administered anonymously; thus, student performance on the tests did not directly impact their grade in the class. After the post-test, the answers to the tests were revealed and briefly discussed. After the discussion, the students in both classes were asked one final question: “Do you think the exercise increased your ability to identify arbitrage opportunities in multiple-asset settings?” Including the two tests and accompanying discussion adds about 10 to 15 minutes to the exercise.
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The undergraduate class had 23 students. On the pre-test, these students correctly identified 34 of the 69 (49.3%) possible arbitrage opportunities. This result is what one would expect if the students were simply guessing. On the post-test, the undergraduate students correctly identified the arbitrage opportunities in 47 of the 69 cases (68.1%). For the graduate class of 28 students, the arbitrage opportunities were correctly identified in the pre-test 56.0% of the time and improved to 71.4% in the post-test. In both classes, the number of incorrect answers on the tests decreased by over 35% after the exercise. Given that the students did not have time to study the concept of multipleasset arbitrage in detail between the two tests, these results are encouraging and demonstrate the teaching effectiveness of the exercise. Additionally, the exercise is generally well perceived by students. Directly after the completion of the exercise, 87.0% of the undergrads felt that the exercise increased their ability to identify arbitrages and 89.3% of the graduate students responded similarly. Many of the students in both classes commented both during and after the debriefing of the exercise that they now understood why they missed some of the questions on the post-test. Lastly, referring to the exercise proved to be valuable when covering the concepts of implied interest rates, convertible bonds, and the put-call parity later in the semester.
Conclusion The concept of arbitrage is pervasive in finance. Arbitrage is the driving force behind the concepts of the Law of One Price, numerous asset pricing models, the efficient markets hypothesis, implied forward rates, derivatives pricing, option bounds, binomial models, the put-call parity, Modigliani-Miller capital structure propositions, and many examples of financial engineering. The application of arbitrage theory is critical to understanding the development and implementation of many advanced concepts in finance. This paper presents a simple in-class exercise designed to help students transition from trivial examples of arbitrage to identifying arbitrage opportunities in more complicated multiple-asset applications. Students are presented with a budget and four assets that can be bought or sold for the same price. Each asset has a potentially different payout depending upon which of three states of nature are chosen. The correct combination of buying and selling the assets creates an arbitrage portfolio. The exercise also has an optional pre-test and post-test assessment tool that further reinforces the concept of multiple-asset arbitrage. The only resources required to implement the exercise are a blackboard and fifteen minutes of class time. The exercise can be used to either introduce the concept of arbitrage or to lay the groundwork for more complicated examples of arbitrage typically used in asset pricing and derivatives settings. Additionally, the exercise reinforces the concepts of basic descriptive statistics, the risk-free rate, mean-variance efficient portfolios, diversification, and short-selling.
References Cebula, Richard and Michael Toma. 2002. “The Effect of Classroom Games on Student Learning and Instructor Evaluations.” Journal of Economics and Finance Education 1(2)(Winter): 1-10. Chang, S. J. 2005. “A Theoretical Discussion on Financial Theory: What Should We Teach and How?” Journal of Economics and Finance Education 4(2)(Winter): 39-48. Fels, Rendigs. 1993. “This Is What I Do, and I Like It.” The Journal of Economic Education 24(4): 365-370. Pavlik, Robert M. and Brian J. Nienhaus. 2004. “Learning from a Simple Options Trading Game.” Journal of Economics and Finance Education 3(2)(Winter): 21-29. Ross, Stephen A. 1987. “Papers and Proceedings of the Ninety-ninth Annual Meeting of the American Economic Association.” The American Economic Review 77(2)(May): 29-34.
