The Math Merc Andries Steenkamp

Let’s suppose that you have managed, agianst all the expectation, to climb out of debt and the hand-to-mouth lifestyle. You listened to Minority Mindset, settled your high interest loans and built up a cash safety cushion. You have a few pennies spare and decide to invest in stocks. Fire up the old online brokerage, scan for the highest returns stock you can recognize and sink all your funds into it. A few days pass, you check up on your investment only to see red. You bought high and the price is down along with your will to exist.

Assuming you don’t quit, you learn of something called “diversification”. The idea is to buy stocks that don’t rise and, more importantly, don’t fall all at the same time. But how do I know if stocks are casually linked in such a manner? You don’t, it’s too hard to answer, so we answer an easier question, hoping that it has bearing on the original. We check to see to what extent the returns on our various stocks correlate. The idea now is to get the right mixture of stocks in your portfolio to minimize the total variance of the mix while getting some nice returns.

Enter the Markowitz model for portfolio optimization. Back in the 1950’s a guy called Harry Markowitz modeled the problem of portfolio optimization as

Where

• models your risk aversion. 0 if you don’t care about returns you just want to minimize variability. 1 if you are all in on returns, variance be damned. Note if you choose this path you end up where we began in the first paragraph, sinking all your money into a single stock.
• where is the returns data matrix. Thus, , is just the average returns for each of your stocks.
• is the covariance matrix of your portfolio.
• is the the weight allocation of your portfolio, i.e. is the percentage of your current total portfolio value that comes from the stock.

The model is predicated on a few assumptions that have been shown not to hold in reality but don’t let that stop you from throwing your hard earned money away. I highlight only two assumptions:

1. The underlying distribution of your stock returns is Gaussian.
2. The variance of your portfolio is a proxy for risk and minimizing it should make you safer.

If you want to know more about the topic of risk but you don’t want to touch math. I refer you to Nicholas Nassim Taleb and his many great books.

Assuming you still want to go through with this how do you solve (MV)? Simple just solve the optimization problems and recover the maximizers w. It’s not that simple. Quadratic programs contain the max-cut problem and are hence NP-hard to solve. Which is the math way of saying you may have to wait a while for your computer to solve the problem. However, there are techniques and computational approaches around this but that is for another video. Let us assume you get through all of that. Are you done now? nope.

The Markowitz model consider exceptionally bad (or good) returns more unlikely than they really are. There is no account for fat tails. As a result your portfolio is at risk that you don’t know about.

So what are “fat-tails”? First, in case you missed it, the tapering ends of the Gaussian distribution are called tails, don’t ask me why. When either of them is fat it indicates that the likelihood of events occurring there is higher than expected using only the pure Gaussian. What does this mean to you. It means that you could experience heavy losses more often than you had planned for. In the world of finance this could ruin you on all levels of existence.

Don’t despair, we can extend the Markowitz model by including what are called higher order moments namely skewness and kurtosis. The skewness of can be seen as a leaning to one side or the other, see the gif below. Kurtosis is a bit harder to visualize but it is contributes fatter tails. In general investors want more positive skewness and no kurtosis.

So the optimization problem becomes.

Where :

• is the centralized skewness matrix.
• is the centralized kurtosis matrix.

This is called the Mean-Variance-Skewness-Kurtosis model and it is a polynomial goal programming problem. It is not any easier than the (MV) model. It does account for fatter tails. Next times I’ll show you how to solve this (approximately) and use it to “optimize” a portfolio. Until then stay well and keep improving.

P.S. If you have a math topic with practical relevance that you would like me to address please send me a mail.

Obsidian is a a great tool for taking notes. One of its best features is that it supports LaTeX via MathJax. Essentially LaTeX is a type setting program that is often used in academia to create documents with many funny symbols and equations. MathJax is a JavaScript display engine that allows you to see Latex inside Obsidian.

Suppose you are doing research, you stumble upon an equation and you would like to add it to your notes. The quick and dirty approach would be to take a screenshot and just link it as an image. Like this:

This is unsatisfactory for at least two reasons. First, you have to keep that screenshot around adding to the file clutter. Second, you can’t modify the equation. So how to add the equation in properly? You can use the inbuilt LateX of Obsidian as follows:

\begin{align*} 
    \xi_t^{sep}(\rho) := \inf \Big\{ L(1) ~|~ &L: \mathbb{C}
    [\mathbf{x},\bar{\mathbf{x}},\mathbf{y},\bar{\mathbf{y}}]_{2t}
     \ \text{ s.t. }  \\ 
    & L(\bar{p}) = \overline{L(p)} ~~~\forall~~ p \in \mathbb{C}
    [\mathbf{x},\bar{\mathbf{x}},\mathbf{y},\bar{\mathbf{y}}]\\
    & L(\mathbf{x}\bar{\mathbf{x}}^*\otimes \mathbf{y}\bar{\mathbf{y}}^*) = \rho,  \\
    & L \geq 0 \text{ on }  \mathcal{M}_{2t}(S_\rho),  \\
    & M_{t-2}(G\otimes L)=L(G \otimes 
    [\mathbf{x},\bar{\mathbf{x}},\mathbf{y},\bar{\mathbf{y}}]_{t-2} 
    [\mathbf{x},\bar{\mathbf{x}},\mathbf{y},\bar{\mathbf{y}}]_{t-2}^*) \succeq 0\Big\}.
\end{align*}

This looks good. However, the code is bloated and it is a pain to write \mathbb{R} every time you invoke the reals. LaTeX addresses this with macro’s. The idea is that you encode your own LeTeX commands that are local to your document. Like encoding \mathbb{R} as \R. Doing this for the equation we get:

\newcommand{\C}{\mathbb C}                   
\newcommand{\xidsep}[1]{\xi_{#1}^{\mathrm{sep}}(\rho)}
\newcommand{\bx}{\mathbf{x}}                    
\newcommand{\bbx}{\bar{\mathbf{x}}}             
\newcommand{\by}{\mathbf{y}} 
\newcommand{\bby}{\bar{\mathbf{y}}}             
\newcommand{\monoV}{[\bx,\by,\bbx,\bby]}       
\newcommand{\polyS}{\C\monoV}                    
\newcommand{\calM}{\mathcal {M}}                 

\begin{align*} 
    \xidsep{t} := \inf \Big\{ L(1) ~|~ &L: \polyS_{2t}\to\C \ \text{ s.t. }  \\ 
    & L(\bar{p}) = \overline{L(p)} ~~~\forall~~ p \in \polyS\\
    & L(\bx\bx^*\otimes \by\by^*) = \rho,  \\
    & L \geq 0 \text{ on }  \calM_{2t}(S_\rho),  \\
    & M_{t-2}(G\otimes L)=L(G \otimes \monoV_{t-2} \monoV_{t-2}^*) \succeq 0\Big\}.
\end{align*}

We cleaned up the equation at the cost of those 9 extra lines where we define new LaTeX commands. To get the effect of the macros throughout your vault I have found a trick. You make a note that has all of the macros you want, (mine is called “THE GRAND TEX”). Then you just open it and close it once and then carry on as usual. That is all. I suppose behind the scenes all the macros are loaded into MathJax and you just use them in the notes without having to redefine the macros. This lasts for the duration of the Obsidian session. You can then view and print your pretty math symbols.

To conclude we have seen how to define LateX macros globally in an Obsidian vault.

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