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Getting More out of Two Asset Portfolios Tom Arnold University of Richmond, [email protected]

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Getting More out of Two Asset Portfolios Tom Arnold University of Richmond, Tarnold@Richmond.Edu University of Richmond UR Scholarship Repository Finance Faculty Publications Finance Spring 2006 Getting More Out of Two Asset Portfolios Tom Arnold University of Richmond, [email protected] Terry D. Nixon Follow this and additional works at: http://scholarship.richmond.edu/finance-faculty-publications Part of the Applied Mathematics Commons, and the Finance and Financial Management Commons Recommended Citation Arnold, Tom, and Terry D. Nixon. "Getting More Out of Two Asset Portfolios." Journal of Applied Finance 16, no. 1 (Spring/Summer 2006): 72-81. This Article is brought to you for free and open access by the Finance at UR Scholarship Repository. It has been accepted for inclusion in Finance Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. Getting More Out of Two-Asset Portfolios Tom Arnold, Lance A. Nail, and Terry D. Nixon Two-asset portfolio mathematics is a fixture in many introductory finance and investment courses. However, (he actual development ofthe efficient frontier and capital market line are generally left tn a heuristic discussion with diagrams. In this article, the mathematics for calculating these attributes of two-asset portfolios are introduced in a framework intended for the undergraduate classroom. [G 10, Gil] •The use of two-asset portfolios in the classroom is in Section I, the efficient frontier is developed for a very convenient,, as the instructor is able to demonstrate portfolio consisting oftwo risky securities. Pursuant the benefits of risk diversification without introducing to that goal, formulae for determining the minimum much in the way of mathematics. By varying portfolio variance portfolio weights are given and an example weights, it is simple to demonstrate that some portfolio is developed for a two-stock portfolio consisting of weight combinations result in better risk-return McDonald's and Pepsico. In Section II,, the framework tradeoffs than others (i.e.. the efficient frontier). is expanded to include a risk-free security, and formulae Although the portfolio in question is small, the basic are generated to find the portfolio weights ofthe two lessons it demonstrates are applicable to much larger risky assets that comprise the portfolio tangent with portfolios. However, more can be demonstrated with the efficient frontier developed in Section I {using the two-asset portfolio than portfolio mean and McDonald's and Pepsico). Further, a capital allocation portfolio variance calculations. In this article, a line consisting of the risk-free security and the framework is developed that allows the student to tangency portfolio containing the two risky securities calculate the minimum variance portfolio weights and is developed. The dominance ofthe capital allocation the weights of a tangency portfolio when a third line relative to the efficient frontier becomes apparent. "riskless" security is added to the portfolio. This Section III concludes the article. An appendix is then method allows the student to demonstrate how a capital provided to clarify the derivation of the tangency allocation line dominates the efficient frontier, naturally portfolio formulae. leading to a discussion of the Capital Asset Pricing Model (Sharpe, 1964). The remainder of this article is organized as follows; I. A Simple Framework for Determining the Efficient Frontier for a Portfolio of Tom Arnold is a Professor of Finance at the University of Two Risky Stocks Richmond in Richmond, VA 23173. Lance A. Nail is a Professor of Finance at the University of Alabama-Birmingham in Birmingham, AL 35294. Terry D. Nixon is a Professor of The equations for calculating a two-asset portfolio's Finance at Miami University in Oxford. OH 45056-1879. mean and standard deviation are available in any number of basic finance and investment textbooks. The author.^ gratefully acknowledge the helpful comments Let W^ be the proportion ofthe portfolio invested in provided by Mark D. Griffiths. David G. Shrider. and an anonymous referee. Security A and W^ be the proportion ofthe portfolio 72 ARNOLD. NAIL, & NIXON — GETTING MORE OUT OF TWO-ASSET PORTFOLIOS 73 invested in Security B. (Note: W^ + W,, - 100%.) Securities A and B bave associated expected returns (^ and fijj. respectively), associated standard (4) deviations (a^ and a^^, respectively), and an associated covariance (cr^j,)- Equations (1) and (2) demonstrate the calculation of the two-asset portfolio mean and w. standard deviation denoted with P as a subscript; ''^..*^.y (1) Equation (5) is simplified further by defining F: ., _ a, ^ cr. F = Many texts and instructors prefer to use correlation rather than covariance. Because the correlation coefficient is a "standardized" measure ranging between negative one and positive one, it is usually preferred, as it provides information on both the magnitude and the direction of co-movement provided W and Wjj ^^^ are the appropriate security by covariance. Correlation (p^j,) is derived from weights, resulting in the minimum variance or tbe covariance: p^^ ^ <s^^ ^ [CT^ * aJ, and is easily inserted minimum standard deviation for a two-security as the substitute for covariance {o^^ ^ p^^ + (j^ * ^^^ j portfolio. into Equation (2): Let us now further develop the example of McDonald's and Pepsico. The minimum variance portfolio is found at W^,^|,^^^|j,^ = 18.88% and Wp^^^^^^, = (3) 81.12%. These weights result in a mean annual return of 10.00% and an annual portfolio return standard deviation of 24.65%.^ Let us demonstrate these equations through an To demonstrate that these weights do, in fact, investor who invests 50% of the portfolio funds in the generate the minimum variance portfolio, increase the shares of McDonald's and the other 50% in Pepsico portfolio weight in the security (relative to tbe (see Exhibit 1 for tbe latest eleven years of annual minimum variance portfolio weight) with the higher stock prices and returns for these companies).' The expected return and calculate the portfolio mean and investor's mean annual return is 0.50 * 0.1551 + 0.50 * standard deviation. For example, a portfolio with 30% 0.0872 = 12.11% and the annual standard deviation of of its wealth invested in McDonald's and the remaining the portfolio is (0.50- * 0.4229- + 0.50- * 0.2591- + 2 * 70% in Pepsico has an expected mean annual return of 0.50 * 0.50 * 0.3037*0.4229*0.2591)'- - 27.95%. 10.75% and standard deviation of 25.09%. This Given these basic formulae as a starting place, we portfolio's mean and standard deviation are larger than will now graph the efficient frontier for an investor the minimum variance portfolio's mean and standard who holds these two risky securities. Our first goal is deviation. Next, increase tbe portfolio weight (relative to determine tbe minimum variance portfolio (which is to the minimum variance portfolio weight) in the also the portfolio that minimizes the standard security with the lower expected return and calculate deviation). In order to minimize the portfolio variance the portfolio mean and standard deviation. A portfolio (standard deviation), substitute (1 - W^) for W^, take with 10% of its wealth invested in McDonald's and the derivative of the square of Equation (3) relative to the remaining 90% in Pepsico has an expected annual W^, set the derivative equal to zero (in calculus terms return of 9.40%, which is lower than the mean of the this is solving for a local minima, which, in this case is minimum variance portfolio, and a standard deviation also shown to result in a global minima), then solve for of 24.93%, which is greater than tbe standard deviation W. (call it of tbe minimum variance portfolio. Clearly, this portfolio is inefficient; thus, portfolios tbat provide 'Data for this analysis were obtained at Yahoo! Finance using Yahoo's closing prices adjusted for dividends and splits. 'Rounding results in this minimum variance portfolio having a standard deviation equal to that of a portfolio with 20% of the -U is more convenient to solve for weights resulting in the investor's wealth in McDonald's shares and 80% in Pepsico minimum variance porilolio knowing that ihese weights also shares (see Exhibit 2). The difference between the two lead to the minimum portfolio standard deviation. portfolios" standard deviations is 0.000046. ^^ JOURNAL OF APPLIED FINANCE — SPRING/SUMMER 2006 Exhibit 1. Price and Return Data for McDonaid's and Pepsico McDonald's (A) Pepsico (B) Date Adj-Close Annual Return Adj-Close Annual Return 3-Jan-05 $32.39 0.2818 $40.96 -0.1386 2-Jan-04 $25.27 0.8351 $47.55 0.2418 2-Jan-03 $13.77 -0.4685 $38.29 -0.0599 2-Jati-02 $25.91 -0.0653 $40.73 -0.2373 2-Jan-Ol $27.72 -0.2096 $53.40 0.0230 3-Jan-0() $35.07 -0.0423 $52.20 -0.1113 4-Jan-99 $36.62 0.6915 $58.74 0.0171 2-Jan-98 $2L65 0.0494 $57.75 0.1286 2-Jan-97 $20.63 -0.0835 S51.I7 0.5525 2-Jan-% $22.51 0.5621 $32.96 0.4558 3-Jan-95 $14.41 $22.64 Annuatized Statistics McDonald's (A) Pepsico (B) Mean Return: 0.1551 0.0872 Standard Deviation of Returns: 0.4229 0.2591 Covariance of Return: 0.0333 Correlation of Returns: 0.3037 higher expected returns relative to those of tbe of the feasible set including the efficient frontier'' minimum variance portfolio are optimal choices. Exhibit 2 graphs the return-standard deviation II. Extending the Framework to relationship (also known as the feasible set) based Determine the Capital Allocation Line upon different portfolio weightings of McDonald's and Pepsico stock. for a Two-Stock Portfolio Theefficient frontier fora two-stock portfolio is just To generate a capital allocation line, a risk-free a subset of the feasible set.
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