We consider the equilibrium drawdown distribution for a biased random walk — in the context of a repeated investment game, the drawdown at a given time is how much has been lost relative to the maximum capital held up to that time. We show that in the tail, this is exponential. Further, when mean drift is small, this has an exponent that is universal in form, depending only on the mean and standard deviation of the step distribution. We give simulation examples in python consistent with the results.

# Finance

## Compounding benefits of tax protected accounts

Here, we highlight one of the most important benefits of tax protected accounts (eg Traditional and Roth IRAs and 401ks). Specifically, we review the fact that not having to pay taxes on any investment growth that occurs while the money is held in the account results in compounding / exponential growth with a larger exponent than would be obtained in a traditional account.

## Martingales

Here, I give a quick review of the concept of a Martingale. A Martingale is a sequence of random variables satisfying a specific expectation conservation law. If one can identify a Martingale relating to some other sequence of random variables, its use can sometimes make quick work of certain expectation value evaluations.

This note is adapted from Chapter 2 of Stochastic Calculus and Financial Applications, by Steele.

## Stochastic geometric series

Let $a_1, a_2, \ldots$ be an infinite set of non-negative samples taken from a distribution $P_0(a)$, and write

$$\tag{1} \label{problem}

S = 1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \ldots.

$$

Notice that if the $a_i$ were all the same, $S$ would be a regular geometric series, with value $S = \frac{1}{1-a}$. How will the introduction of $a_i$ randomness change this sum? Will $S$ necessarily converge? How is $S$ distributed? In this post, we discuss some simple techniques to answer these questions.

Note: This post covers work done in collaboration with my aged p, S. Landy.