In AxelCode-R/Master-Thesis: Does not matter.

Mathematical Foundations {#mathfundations}

This chapter provides an overview of the mathematical calculations and conventions used in this thesis. It is important to note that most mathematical formulas are written in matrix notation and vectors are interpreted as one-dimensional matrices. In most cases, this will result in a direct translation to R code. All necessary assumptions required for the modeled return structure are listed in this chapter so that any reader can understand the formulas given. It is important to note that reality is too complex and can only be partially modeled. Simple, basic models are used that do not stand up to reality, but these models or variations of them are commonly used in the financial world and have proven to be helpful. The complexity of solving advanced and basic models does not differ for the PSO because the dimension of the objective function is based on the number of elements that can be selected, see chapter \@ref(challenges).

Basic Operators

The programming language R makes it possible to formulate almost all computations in a matrix notation, since the usual matrix operators and element-wise operators are already efficiently implemented in base R. Element-wise operators are not considered standard in mathematics, so they are defined in more detail in this section. The representation of calculations in a matrix notation helps the user to keep the overview, because the formulas are more compact and in addition, these operators are already implemented efficiently in base R, which makes them more performant than simple for-loops. The operators used here can be found in the table below and are intended to establish the relationship between the notation in the text, in the R code and their meaning:

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    "Meaning" = c("Matrix product", "Outer product (special case of the matrix product)", "Transpose of matrix or vector A", "Scalar or element-wise matrix multiplication", "Scalar or element-wise matrix division", "Scalar or element-wise matrix addition", "Scalar or element-wise matrix subtraction")
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  html_save(., expand = c(-40,-20,-45,-20))

For a better understanding of the operators listed, the following examples are intended to illustrate the resulting dimensions and provide insight into the use of these operators.

The matrix product is defined by the following mapping:

$$\times: \ \ \mathbb{R}^{x \times y} \times \mathbb{R}^{y \times z} \rightarrow \mathbb{R}^{x \times z},$$

with an example:

$$ \begin{bmatrix}a_{1,1} &\cdots & a_{1,y} \ \vdots & \ddots & \vdots \ a_{x,1} & \cdots & a_{x,y} \end{bmatrix} \times \begin{bmatrix}b_{1,1} &\cdots & b_{1,z} \ \vdots & \ddots & \vdots \ b_{y,1} & \cdots & b_{y,z} \end{bmatrix} = \begin{bmatrix} \sum_{i=1}^y a_{1, i} \cdot b_{i,1} &\cdots & \sum_{i=1}^y a_{1, i} \cdot b_{i,z} \ \vdots & \ddots & \vdots \ \sum_{i=1}^y a_{x, i} \cdot b_{i,1} & \cdots & \sum_{i=1}^y a_{x, i} \cdot b_{i,z} \end{bmatrix}. $$

In the case of vectors, the matrix product has the following mapping:

$$\times: \ \ \mathbb{R}^{1 \times N} \times \mathbb{R}^{N \times 1} \rightarrow \mathbb{R},$$

with an example:

$$ \begin{bmatrix}a_{1} &\cdots & a_{N}\end{bmatrix} \times \begin{bmatrix}b_{1}\ \vdots \ b_{N}\end{bmatrix} = \sum_{i=1}^N a_i \cdot b_i. $$

The outer product is defined by the following mapping:

$$\otimes: \ \ \mathbb{R}^{x \times 1} \times \mathbb{R}^{1 \times y} \rightarrow \mathbb{R}^{x \times y},$$

with an example:

$$ \begin{bmatrix}a_{1}\ \vdots \ a_{x}\end{bmatrix} \otimes \begin{bmatrix}b_{1} &\cdots & b_{y}\end{bmatrix} = \begin{bmatrix}a_{1} \cdot b_{1} &\cdots & a_{1} \cdot b_{y} \ \vdots & \ddots & \vdots \ a_{x} \cdot b_{1} & \cdots & a_{x} \cdot b_{y} \end{bmatrix}. $$

This thesis is specified for the use of R, so element-wise operators are very important to make code comparable with formulas. In mathematics, these operators are not common. For this reason, they must be explicitly specified. All element-wise operators work in the same way. Assuming that $\square$ is one of the four element-wise operators, then the mapping is defined as follows:

\begin{align} \square &: \ \ R^{x \times y} \times R^{x \times y} \rightarrow R^{x \times y}\ \text{or }\ \square &: \ \ R^{x} \times R^{x \times y} \rightarrow R^{x \times y}\ \text{or }\ \square &: \ \ R \times R^{x \times y} \rightarrow R^{x \times y}, \end{align}

with examples respectively:

$$ \begin{bmatrix}a_{11} &\cdots & a_{1y} \ \vdots & \ddots & \vdots \ a_{x1} & \cdots & a_{xy} \end{bmatrix} \square \begin{bmatrix}b_{11} &\cdots & b_{1y} \ \vdots & \ddots & \vdots \ b_{x1} & \cdots & b_{xy} \end{bmatrix} = \begin{bmatrix}a_{11} \square b_{11} &\cdots & a_{1y} \square b_{1y} \ \vdots & \ddots & \vdots \ a_{x1} \square b_{x1} & \cdots & a_{xy} \square b_{xy} \end{bmatrix} $$

or \vspace{0.2cm} $$ \begin{bmatrix}a_{1} \ \vdots \ a_{x} \end{bmatrix} \square \begin{bmatrix}b_{11} &\cdots & b_{1y} \ \vdots & \ddots & \vdots \ b_{x1} & \cdots & b_{xy} \end{bmatrix} = \begin{bmatrix}a_{1} \square b_{11} &\cdots & a_{1} \square b_{1y} \ \vdots & \ddots & \vdots \ a_{x} \square b_{x1} & \cdots & a_{x} \square b_{xy} \end{bmatrix} $$

or \vspace{0.2cm} $$ a \ \square \begin{bmatrix}b_{11} &\cdots & b_{1y} \ \vdots & \ddots & \vdots \ b_{x1} & \cdots & b_{xy} \end{bmatrix} = \begin{bmatrix}a \square b_{11} &\cdots & a \square b_{1y} \ \vdots & \ddots & \vdots \ a \square b_{x1} & \cdots & a \square b_{xy} \end{bmatrix}. $$

The operators on the right side of the above example equations corresponds to the common operators in scalar.

Formula Conventions

In mathematics, random variables are written in capital letters, which also applies to matrices. In order to allow an unambiguous assignment, the random variables are written in bold and in capital letters. For example, $A$ should represent a matrix and $\pmb{A}$ should represent a random variable.

Return Calculation

Any portfolio optimization strategy based on historical data must start with returns. These returns are calculated using adjusted closing prices, which show the percentage change over time. Adjusted closing prices reflect dividends and are adjusted for stock splits and rights offerings. These returns are essential for comparing assets and analyzing dependencies.

Simple Returns

The default time frame for all raw data in this thesis is a working day, prices are always measured in the evening, and only simple returns are used. Assuming that there is an asset with price $p_{t_i}$ on working day $t_i$ and price $p_{t_{i+1}}$ on the following working day $t_{i+1}$, the simple rate of return for $t_{i+1}$ can be calculated as follows:

$$ r_{t_{i+1}} = \frac{p_{t_{i+1}}}{p_{t_i}}-1. $$

Markowitz Modern Portfolio Theory

In 1952, Harry Markowitz published his first seminal paper, which had a significant impact on modern finance, primarily by outlining the implications of diversification and efficient portfolios in [@Mark1952]. He later published a monograph on a more in-depth analysis of his findings in [@Mark1959]. The definition of an efficient portfolio is a portfolio that has either the maximum expected return for a given risk target or the minimum risk for a given expected return target. A simple quote to define diversification might be, "A portfolio has the same return but less variance than the sum of its parts." This is the case when assets are not perfectly correlated, as poor and good performances can offset each other, reducing the likelihood of extreme events. For more information, see [@Mari2005].

Assumptions of Markowitz Portfolio Theory

This thesis focuses on problems derived from Markowitz's portfolio theory, without closed form solutions, which are studied in [@Mari2005] by excluding short selling. The following list contains these types of Markowitz assumptions according to [@Mari2005] and [@Mark1952]:

• Perfect market without taxes or transaction costs
• Assets are infinitely divisible
• Short sales are disallowed
• Expected returns, variances and covariances contain all information
• Investors are risk-averse, they will only accept greater risk if they are compensated with a higher expected return

The assumption that the returns are normally distributed is not required, but is assumed in this case to simplify the problem. It is obvious that these assumptions are unrealistic in reality. More details on the requirements for using other distributions can be found in [@Mari2005].

Portfolio Math

Proofs of the basic calculations required for portfolio optimization, as shown in [@Eric2021], are provided in this section. Returns are presented differently than in most sources, as this is the most common data format used in practice. Suppose there are $N$ assets described by a return vector $\pmb{R}$ of random variables, and the relative contribution to the net asset value of the portfolio is described by the weight vector $w$, which typically satisfies $\sum w = 1$:

$$ \pmb{R} = \begin{bmatrix} \pmb{R}{1} & \pmb{R}{2} & \cdots & \pmb{R}{N}
\end{bmatrix} , \ \ w = \begin{bmatrix} w{1} \ w_{2} \ \vdots \ w_{N}
\end{bmatrix}. $$

In this thesis, each return is simplified as being normally distributed with $\pmb{R}_i = \mathbb{N}(\mu_i, \sigma_i^2)$. As a result, linear combinations of normally distributed random variables are jointly normally distributed and have a mean, variance, and covariance that can be used to fully describe them.

Expected Returns

The following formula can be used to get the expected returns of a vector with normally distributed random variables $\pmb{R} \in \mathbb{R}^{1\times N}$:

\begin{align} E[\pmb{R}] &= \begin{bmatrix} E[\pmb{R}{1}] & E[\pmb{R}{2}] & \cdots & E[\pmb{R}{N}]
\end{bmatrix}\ &= \begin{bmatrix} \mu{1} & \mu_{2} & \cdots & \mu_{N} \end{bmatrix} = \mu \end{align}

and $\mu_i$ can be estimated in R using historical data and the formula for the geometric mean of returns (also called compound returns). The function to calculate the geometric mean of returns from an xts object can be found in section \@ref(geomeanret).

Expected Portfolio Returns

The following equation can be used to obtain the expected portfolio return $E[\pmb{R}_p]$ using the formulations from the section above and a weighting vector $w$ (e.g. portfolio weights):

\begin{align} E[\pmb{R}p] &= E[\pmb{R} \times w] = E[\pmb{R}] \times w \ &= \begin{bmatrix} E[\pmb{R}{1}] & E[\pmb{R}{2}] & \cdots & E[\pmb{R}{N}] \end{bmatrix} \times \begin{bmatrix} w_{1} \ w_{2} \ \vdots \ w_{N}
\end{bmatrix} \ &= \begin{bmatrix} \mu_{1} & \mu_{2} & \cdots & \mu_{N} \end{bmatrix} \times \begin{bmatrix} w_{1} \ w_{2} \ \vdots \ w_{N}
\end{bmatrix} \ &= \mu_{1} \cdot w_1 + \mu_{2} \cdot w_2 + \cdots + \mu_{N} \cdot w_{N} = \mu_P \end{align}

Covariance

The general formula of the covariance matrix $\textstyle\sum$ of a random vector $\pmb{R}$ with $N$ normally distributed elements and $\sigma_{i,j}$ as correlation of two distinct values is described as follows:

\begin{align} Cov(\pmb{R}) &= E[(\pmb{R}-\mu)^T \otimes (\pmb{R}-\mu)] \ &= \begin{bmatrix} \sigma_1^2 & \sigma_{1,2} & \cdots & \sigma_{1,N} \ \sigma_{2, 1} & \sigma_2^2 & \cdots & \sigma_{2, N} \ \vdots & \vdots & \ddots & \vdots \ \sigma_{N, 1} & \sigma_{N, 2} & \cdots & \sigma_N^2 \ \end{bmatrix}\ &=\textstyle\sum \end{align}

and can be estimated in R using the base function cov() and historical data. The function cov_() explained in section \@ref(covfun) can be used to calculate the covariance using the geometric mean to estimate the expected returns.

Portfolio Variance {#portvar}

Let $\pmb{R}$ be a random vector with $N$ normally distributed elements and $w$ a weight vector. Assuming that the covariance matrix $\sum$ of $\pmb{R}$ is known, the variance of the linear combination of $\pmb{R}$ can be calculated as follows:

\begin{align} Var(\pmb{R} \times w) &= E[(\pmb{R} \times w - \mu \times w)^2] \ &= E[((\pmb{R} - \mu) \times w)^2] \end{align}

Since $(\pmb{R} - \mu) \times w$ is a scalar, it can be transformed from $((\pmb{R} - \mu) \times w)^2$ to \mbox{$((\pmb{R} - \mu) \times w)^T \cdot ((\pmb{R} - \mu) \times w)$} and results in:

\begin{align} Var(\pmb{R} \times w) &= E[((\pmb{R} - \mu) \times w)^T \cdot ((\pmb{R} - \mu) \times w)]\ &= E[(w^T \times (\pmb{R} - \mu)^T) \cdot ((\pmb{R} - \mu) \times w)]\ &= w^T \times E[(\pmb{R} - \mu)^T \otimes (\pmb{R} - \mu)] \times w \ &= w^T \times Cov(\pmb{R}) \times w \ &= w^T \times \textstyle\sum \times w. \end{align}

The same is true for an estimate of $Var(\pmb{R} \times w)$ by using the estimate of $\textstyle\sum$.

Portfolio Returns {#portfolioreturns}

Suppose there are $N$ assets that form a portfolio with weights $w_{t_0}$ at time step $t_0$, and the portfolio is to go through several time steps until $t_T$ without rebalancing. What are the portfolio returns at each time step $t_i$? Clearly, assets with higher performance at the current time step will have a higher weight at the next time step. This can be done by adjusting the weights after each time step as a function of returns. Suppose we have a complete portfolio $\textstyle\sum w_{t_0} = 1$ with a return matrix $R \in \mathbb{R}^{T \times N}$ and we want to calculate the portfolio returns $R_{t_1}^P$. This can be done with the following two steps:

\begin{align} Z_{t_1} &= (1+R_{t_1}) \cdot w_{t_0}^T\ R_{t_1}^P &= \sum_{n = 1}^N Z_{t_1, n} - 1. \end{align}

And since the portfolio was full in $t_0$, the adjusted weights in $t_1$ are $w_{t_1} = Z_{t_1}/\sum_{n = 1}^N Z_{t_1, n}$ (element-wise division). These new weights can be used in the next time step to replace the weights $w_{t_0}$ in the above formula. The recursive formula for holding a portfolio with weights $\textstyle\sum w_{t_0} = 1$ in $t_0$ and return matrix $R \in \mathbb{R}^{T \times N}$, has portfolio return $R_{t_i}^P = \textstyle\sum_{n = 1}^N Z_{t_i, n} - 1$ in $t_i$ with $i=1, 2, \cdots, T$ for:

$$ Z_{t_i} = \begin{cases} (1+R_{t_i})\cdot w_{t_0}^T &\text{ if }i=1\ (1+R_{t_i})\cdot \frac{Z_{t_{i-1}}}{\sum_{n = 1}^N Z_{t_{i-1, n}}} &\text{ if }i>1. \end{cases} $$

The requirement that the portfolio is full can be achieved by adding an additional weight to $w_{t_0}$, which includes the residual $1-\textstyle\sum w_{t_0}$ and an additional zero return vector to $R$. This calculation of portfolio returns is implemented in the function calc_portfolio_returns() below.

R Functions

The following functions serve the purpose to encapsulate mathematical calculations and to make the calculations reusable. All functions are located in the dedicated GitHub repository inside the R/ folder.

cov_() {#covfun}

This function extends the base R function cov() by allowing the user to pass an expected value vector. This is used to calculate the covariance with expected returns based on the geometric mean instead of the arithmetic mean.

pri_to_ret() {#pritoret}

This function calculates the simple returns of a given xts object containing prices. Before that, missing prices are replaced by the previous value in the respective column.

ret_to_cumret()

This function is used to calculate cumulative returns normalized to 100 from a given xts object containing returns. This function is often used before plotting return time series.

ret_to_geomeanret() {#geomeanret}

The geometric mean of returns is a better estimator than the arithmetic mean of returns because it captures the exact mean price changes over a period of time. The variance estimated from the daily returns is a daily variance, so the returns must have the same time base. This can be done by calculating the geometric mean of the returns from multiple daily returns. Assuming there is an asset with returns $r_1 = 0.01$, $r_2=-0.03$, and $r_3=0.02$, it follows that the geometric mean return $r^{geom}$ can be calculated as:

$$ r^{geom} = ((1+r_1) \cdot (1+r_2) \cdot (1+r_3))^{1/3}-1 = -0.0002353887 $$

And the advantage is that it is a daily average return that gives exactly the same result as the real returns, that is:

$$ (1+r^{geom})^3 = (1+r_1) \cdot (1+r_2) \cdot (1+r_3) $$

This is not the case with the arithmetic mean of the returns. The general formula for calculating the geometric mean return of $n$ days is:

$$ r^{geom} = (\prod_{i=1}^n (1+r_i))^{\frac{1}{n}}-1 $$

and as R code:

ret_to_geomeanret <- function(xts_ret){
  sapply((1+xts_ret), prod)^(1/nrow(xts_ret))-1
}

calc_portfolio_returns()

This is the implementation of a vectorial calculation of portfolio returns over multiple periods with a weighting vector weights at $t_0$ and no re-balancing:

calc_portfolio_returns <- 
  function(xts_returns, weights, name="portfolio"){
  if(sum(weights)!=1){
    xts_returns$temp___X1 <- 0
    weights <- c(weights, 1-sum(weights))
  }
  res <- cumprod((1+xts_returns)) * matrix(
    rep(weights, nrow(xts_returns)), ncol=length(weights), byrow=T)
  res <- xts(
    rowSums(res/c(1, rowSums(res[-nrow(xts_returns),])))-1, 
    order.by=index(xts_returns)) %>% 
    setNames(., name)
  return(res)
}

This function has the same results as the Return.portfolio() function from the PortfolioAnalytics package.

AxelCode-R/Master-Thesis documentation built on Feb. 25, 2023, 7:57 p.m.