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\title{Kelly bet sizing\thanks{as discussed in p.\@ 74 of \cite{Brown} }}
\author{---Humility Consulting}
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\begin{document}
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\begin{abstract}
\noindent This document derives Kelly's rule for optimal bet sizing.
\end{abstract}
%\printclassoptions
\newthought{Here's the game.} You have a \$10,000 stake. You face a series of 60 wins and 40 losses. Their order is chosen by me. You choose each bet's size. After you tell me the size, I get to spend, or not, one of my 40 kills---if I have any left. Your win is my loss, and vice-versa. After 100 rounds, we're done.
%\bigskip \bigskip
If you always bet the same fixed percentage of your stake, then\sidenote{since multiplication is commutative} I have no strategic role to play in determining your overall takeaway. I can't affect your outcome if you play that way.
\bigskip
\section{The Math}
I'll denote by $a$ the aggressiveness of your bets. If you win a bet then $\mathtt{stake}_{t+1} = \mathtt{stake}_t \cdot (1+a)$ and if you lose a bet then $\mathtt{stake}_{t+1} = \mathtt{stake}_t \cdot (1-a)$. A win, then a loss, turns out the same as a loss, then a win:
\vspace{-1em}
\begin{align*}
\mathtt{stake}_{t+2}
&= (1+a) \cdot (1-a) \cdot \mathtt{stake}_t \leavevmode\\
&= (1-a) \cdot (1+a) \cdot \mathtt{stake}_t.
\end{align*}
\noindent
In these terms, Kelly's question is:
what is the optimal%
\sidenote[][-1cm]{assuming, to remove my strategic input, that it's the same fraction $a$ every time}
aggressiveness $a^*$ to \textbf{maximise your final takeaway}%
\sidenote{i.e., $\displaystyle \arg \max_{\{a\}}
\mathtt{stake}_{100} \ \overset{def}{=} \ a^*$}?
\bigskip
\subsection{The Answer}
\begin{quotation}
Twenty percent is the perfect amount to bet. Betting a higher or a lower fraction means doing worse. %This is not probably or only theoretically true. %This is mathematical fact. It really works, in practice as well as theory.
\ldots\
While there are people who dislike the Kelly criterion for various reasons, no intelligent person disputes this aspect of the result.
---Brown, p.\@ 76
\end{quotation}
\begin{table*}
\footnotesize{
\url{http://books.google.com/books?id=eu925DG2xeYC&lpg=PP1&dq=red-blooded\%20risk&pg=PT60#v=snippet&q=mathematical&f=false}
}
\end{table*}
\bigskip \bigskip
\subsection{Solving for optimal aggressiveness}
% \begin{figure}[h]%
% \includegraphics[width=\linewidth]{tumblr_inline_ml2smif1GK1qz4rgp.jpg}
% \caption{The product rule says $$D(f \cdot g) = D(f) \cdot g + f \cdot D(g).$$ \textbf{More:} \url{http://isomorphism.es/tagged/product+rule}
% }
% \label{fig:product rule}
% \end{figure}
This can be solved with calculus 101.
Since the order doesn't matter and there's a fixed number of wins and losses,
we can write the following formula for your total winnings at 60/40 odds:
\[
\label{eqn:main}
\$1000 \cdot (1+a)^{60} \cdot (1-a)^{40}\tag{take}
\].
To find the optimal aggressiveness,
set the derivative of equation \eqref{eqn:main},
with respect to $a$, equal to zero.%
%
\sidenote[][-1 cm]{(There are further conditions to make sure this works, which I'm leaving out.)}
%
The symbolic derivative of $$(1+a)^{60} \cdot (1-a)^{40}$$ with respect to $a$ is:
\[\label{eqn:D}
60 \cdot (1+a)^{59} \cdot (1-a)^{40} \ \ -
\ \ \ 40 \cdot (1+a)^{60} \cdot (1-a)^{39}
\tag{D}
\].
Setting \eqref{eqn:D} = 0
tells me a property that is true
of the optimal $a^*$. Moving things around, that property can be restated as:
$$60 \cdot { (1-a^*)^{40} \over (1-a^*)^{39} }
\ \ = \
40 \cdot { (1+a^*)^{60} \over (1+a^*)^{59} }$$
which reduces to the much nicer $$60 \cdot (1-a^*) \ \ = \ 40 \cdot (1+a^*).$$
\marginnote[-5.5 cm]{The product rule says $D(f \cdot g) = D(f) \cdot g + f \cdot D(g).$}
% \textbf{More:} \url{http://isomorphism.es/tagged/product+rule}
\bigskip
\noindent
Solving then for the optimal aggressiveness $a^*$: \marginnote[+1.4 cm]{Thanks, Artemy!}
%
%
\begin{align}
60-60a^* &= 40 + 40 a^* \\
20 &= 100 a^*
\end{align}
\noindent
which is what Brown gets: 20\% = $1 \over 5$ for the optimal $a^*$ against these odds.
\section{Brown's conclusion}
The ``Ed Thorp takeaway'' is that risk management does not mean taking no risks. You can't sit on your hands. Bill Gross says to ``avoid portfolio mush'', which is similar \& related.
Contrast this to ``lazy CAPM'' style thinking.
When you have an edge (like a 60/40 edge) you need to exploit it. Betting less than 20\% of your stake against 60/40 odds is\sidenote{In this model.} suboptimal. Less aggressive betting does not move you along an optimal frontier; it moves you \emph{off} the optimal frontier.
%\newpage
% The Tufte-\LaTeX\ document classes define a style similar to the style Edward Tufte uses in his books and handouts. Tufte's style is known for its extensive use of sidenotes, tight integration of graphics with text, and well-set typography. This document aims to be at once a demonstration of the features of the Tufte-\LaTeX\ document classes and a style guide to their use.
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