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an optimal risk allocation across multiple strategies/main.tex

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+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+% Core packages
+\usepackage{amsmath,amssymb}
+\usepackage{tikz-cd}
+\usepackage{multicol}
+
+% Paragraphs
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{1\baselineskip}
+
+\title{An Optimal Risk Allocation Across Multiple Strategies}
+\author{algorembrant}
+\date{\today}
+
+\begin{document}
+\maketitle
+
+Say we have multiple alpha/edge/system/strategy's $Winrate$, in decimal, which came from trading setups, denoted as $W_1,W_2,W_3,\ldots,W_{\alpha}$. The certain Winrate, which is $W_c \in W_{\alpha}$, divided by the sum of all winrates, is the fraction to use to allocate risk from our max risk.
+\begin{align}
+    O_c = C \times R \times \frac{W_c}{\displaystyle\sum_{\text{all }{\alpha}} W_{\alpha}}, \quad O_c \in \mathbb{N}, \quad \sum \frac{W_c}{\displaystyle\sum_{\text{all }{\alpha}} W_{\alpha}} = 1, \quad \displaystyle\sum_{\text{all }{\alpha}} O_{\alpha} = C \times R
+\end{align}
+where
+\begin{align*}
+    O_c &= \text{is the Optimal Bet at certain alpha/edge/system/strategy, a cash, } O_c \in O_{\alpha}, \text{ and a whole number } \mathbb{N} \\
+    C &= \text{is the Capital in cash, or balance with realized PnL} \\
+    R &= \text{is the max Risk\%, in decimal, to expose in one Session or within active trading time}.
+\end{align*}
+For example, suppose we deploy five alpha/edge/system/strategies with corresponding win rates of $\{0.60, 0.63, 0.67, 0.69, 0.89\}$, and our capital $C$ is 100,000 cash, and we are willing to expose $R$ as 0.05 (5\%) for the whole trading day. Therefore:
+\begin{align*}
+    O_c &= C \times R \times \frac{W_c}{\displaystyle\sum_{\text{all }{\alpha}} W_{\alpha}} \Rightarrow
+    O_{0.67} &= 100000 \times 0.05 \times \frac{0.67}{0.60 + 0.63 + 0.67 + 0.69 + 0.89} = 963
+\end{align*}
+It said that the optimal bet of that certain strategy (which had the 0.67 winrate) is 963 cash. Which could be used as proportional to our stop-loss size, but SL size is out of scope in this paper. Moving on, the following rules could be used for safeguard-related pipelines, which are always true.
+\begin{align*}
+     \sum \frac{W_c}{\displaystyle\sum_{\text{all }{\alpha}} W_{\alpha}} &= \underbrace{0.17 + 0.18 + 0.19 + 0.20 + 0.26}_{\text{each was rounded into 2 decimals}} = 1
+\end{align*}
+Where the sum of all fractions of optimal bets must be equal to 1. And,
+\begin{align*}
+    \displaystyle\sum_{\text{all }{\alpha}} O_{\alpha} = \underbrace{862 + 905 + 963 + 991 + 1279}_{\text{each was turned into $\mathbb{N}$}} = 100000 \times 0.05
+\end{align*}
+Where the sum of all optimal bets must be equal to the max risk exposure.
+
+That's all, the end.
+\end{document}