Chapter 3
Portfolio Theory and CAPM
Abstract This chapter deals with the compilation of stocks into a portfolio that suits
an investor’s hunger for profit, while minimizing the risk of losing the initial investment. Since the theory is based on variational methods and on Lagrange multipliers,
applications of these concepts in physics are juxtaposed to their use in finance. As
an extension of the portfolio theory, the basic equations of the capital asset pricing
model are derived and used to value companies.
Let us assume that we want to invest a sum of money in shares. So, how do we split the
available money among shares in a way that maximizes our profit, while minimizing
the risk to actually lose money? This question was addressed by Markowitz in the
1950s, which led to his winning a share of the 1990 memorial prize in A. Nobel’s
memory. We start by considering his analysis of choosing a portfolio [1], which relies
heavily on methods we use in physics: Lagrange multipliers and variational methods.
Let us look at different traders having different degrees of eagerness to make
a profit and varying degrees of acceptance to risk. A very risk-averse trader, for
example, would simply place all his available cash in a bank account that provides
the risk-free rate r f , which does not have any volatility σ and poses no risk of losing
any money. A more ambitious investor, a speculator, will request a higher return
rate than the risk-free rate r f , but at the expense of having to tolerate some level
of risk or volatility, which may potentially lead to a reduction of the value of the
original investment. One usually only has a finite amount of money available to start
trading. This poses a constraint on one’s ability to select stocks. Such a constraint
is efficiently handled by Lagrange multipliers. Moreover, optimizing a portfolio is
based on methods from variational calculus. We therefore take the opportunity to
briefly touch upon the use of both concepts in classical mechanics [2, 3].
Electronic supplementary material The online version of this chapter
(https://doi.org/10.1007/978-3-030-63643-2_3) contains supplementary material, which is
available to authorized users.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_3
15
Portfolio Theory and CAPM
Abstract This chapter deals with the compilation of stocks into a portfolio that suits
an investor’s hunger for profit, while minimizing the risk of losing the initial investment. Since the theory is based on variational methods and on Lagrange multipliers,
applications of these concepts in physics are juxtaposed to their use in finance. As
an extension of the portfolio theory, the basic equations of the capital asset pricing
model are derived and used to value companies.
Let us assume that we want to invest a sum of money in shares. So, how do we split the
available money among shares in a way that maximizes our profit, while minimizing
the risk to actually lose money? This question was addressed by Markowitz in the
1950s, which led to his winning a share of the 1990 memorial prize in A. Nobel’s
memory. We start by considering his analysis of choosing a portfolio [1], which relies
heavily on methods we use in physics: Lagrange multipliers and variational methods.
Let us look at different traders having different degrees of eagerness to make
a profit and varying degrees of acceptance to risk. A very risk-averse trader, for
example, would simply place all his available cash in a bank account that provides
the risk-free rate r f , which does not have any volatility σ and poses no risk of losing
any money. A more ambitious investor, a speculator, will request a higher return
rate than the risk-free rate r f , but at the expense of having to tolerate some level
of risk or volatility, which may potentially lead to a reduction of the value of the
original investment. One usually only has a finite amount of money available to start
trading. This poses a constraint on one’s ability to select stocks. Such a constraint
is efficiently handled by Lagrange multipliers. Moreover, optimizing a portfolio is
based on methods from variational calculus. We therefore take the opportunity to
briefly touch upon the use of both concepts in classical mechanics [2, 3].
Electronic supplementary material The online version of this chapter
(https://doi.org/10.1007/978-3-030-63643-2_3) contains supplementary material, which is
available to authorized users.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_3
15