Portfolio Analysis for Vector Calculus

Abstract

Classic stock portfolio analysis provides an applied context for Lagrange multipliers that undergraduate students appreciate. Although modern methods of portfolio analysis are beyond the scope of vector calculus, classic methods reinforce the utility of this material. This paper discusses how to introduce classic stock portfolio analysis in a vector calculus course including basic assumptions, worked examples, and sources for real data.

Keywords:

• Lagrange multipliers
• optimization
• portfolio analysis

Notes

¹ We compute this annual gain from the fact that (1 + 0.00101)²⁵⁰ − 1 ≈ 0.287.

² The standard deviation is a measure of the spread of data about its average value.

For a list of data denoted , the mean, denoted by and the standard deviation, denoted by is . Since this is vector calculus, we can define the vectors and . Then we can define the standard deviation as .

³ Note that a normal distribution means that the daily growth rates are distributed symmetrically about the average growth rate. Specifically, we are assuming that the data is well-approximated by the graph of where is the average daily return and is the risk.

⁴ This is because we expect the value to grow exponentially. That is, the stock price is modeled as Pe^rt where P is the initial value of the stock and r is the growth rate. The growth rate from time t₁ to t₂ is given by . If, instead, we took the logarithm of the data first, then the growth rate from time t₁ to t₂ is given by . This second expression extracts the growth rate, r, as a linear term. For small r, the first term in the Maclaurin series for is . This approximation is good enough for our application.

⁵ Selling short was restricted in 2008 during the financial crisis.

Additional information

Notes on contributors

Samuel R. Kaplan

Samuel R. Kaplan discovered “chaos” had something to do with math his freshman year at the University of North Carolina, Chapel Hill and never looked back. He attended Boston University for his Ph.D. where his research mixed chaos and celestial mechanics. After graduating in 1996, he took a visiting position at Bowdoin College where he fell in love with teaching in a liberal arts setting. After three years in Maine, he took a position at the University of North Carolina Asheville where he continues to teach, mentor undergraduate research students, and promote math literacy in the community.