In this post I would like to introduce Alpay Kaya, author of

Leveraged ETFs: How Biased Statistics Affect Your Portfolio
**OnlyVIX**:
Before we get to the math, would you tell us about yourself?

**Alpay Kaya**:
Like many others, I came to the world of finance from a technical educational background, Control Systems, which is an applied math field mostly populated by electrical & mechanical engineers. My research from those days has been published in academic journals and presented at conferences.

I was drawn to the investing sector of finance because of its connection to reality and the relation between human behavior & math. I joined Koch Capital, the prop trading unit of Koch Industries, where I developed systematic trading strategies (long/short equities, then FX options and rates futures). My output was a trading protocol implemented by the execution desk at notional values exceeding $100MM. Although the systematic approach appeals to me, the office politics between our unit and corporate became quite silly; to wit, a corporate director came to the office one day and announced that my FX trade would be doubled because it made the most money the previous year. Since portfolio allocation is not to be done by dictate, I left shortly thereafter to join a start-up fund, but whatever was lacking in office politics, the owner made up for with his outsized ego.

I was working on a general-interest investing book a little less than a year ago when I heard someone on CNBC mention that LETFs lost value because of 'compounding'...a term I find to be meaningless. There were no books on the topic, so figuring supply did not meet demand, I wrote one and found ample opportunity to cover important quantitative finance topics also. By defining negative drift in mathematical terms (not by example), my work expands even upon the current academic literature.

**OnlyVIX**: What can readers expect to gain from reading your book?

**Alpay Kaya**:
Since fundamentals serve as the foundation for everything, a step-by-step review of all necessary math - from the definition of return to geometric Brownian motion (GBM) and Ito's lemma - is included. Readers can expect to understand GBM better than most because it is developed under both the arithmetic and logarithmic return models. The conventional derivation confuses many because it is statistically convoluted; that is to say, the parameters are not independent!
Practitioners will also appreciate my use of a unified model (as opposed to long & short variants), which is justified by showing the equivalence of long- and short-leveraged portfolios backed by futures or the assets themselves (when applicable). For traders, my book explains why leveraged ETFs trade the way they do, which will help them avoid basic trading mistakes. The long-term value evolution differences between long- and short-leveraged (inverse) ETFs is derived. For the more mathematically inclined, my book is a compete resource on the quantitative aspects of leveraged ETFs, complete with formulas that can be easily implemented in Excel.

**OnlyVIX**: We have corresponded about leveraged ETFs and disagreed on some points; would you mind sharing them with the reader?

**Alpay Kaya**:
The following take on distributions is directly from that text. For consistency of presentation, some points from

Leveraged ETFs Decay And Symmetric vs Skewed Distributions are quoted below and addressed along the way.

** **
**A Package Deal**
It is important to view the log-normal distribution φ

_{L} as a system of parts:

- a normally distributed INPUT: x ~ φ
_{N}( μ, σ )
- a continuous growth PROCESS function: f(x) = exp( x )
- a growth factor (price ratio) OUTPUT: y
_{i} = p_{i} / p_{i-1} = exp( x_{i} )

for the purpose of correctly attributing any characteristics. For example, anyone faulting φ

_{L} with nonzero skew should place blame with the exponential function. It is ‘guilty’ by virtue of transforming additive inverses to multiplicative inverses.

*Then again* doing so has served all branches of the physical & life sciences very well. Besides, is it not logical that returns over any number of periods should sum to zero

x_{1} + … + x_{n} = 0 <==> p_{n} = p_{0}
if and only if final price equals initial price? If the input driving the process is symmetrically distributed, do not necessarily expect the same of the output distribution.

This furcation is consistent with the notion that a price history is a discrete sampling of a continuously evolving reality. A standard result

E( y_{i} ) = exp( μ + σ^{2}/2 )
is that of the mean output being a function of mean input and volatility. To those who think of period return in terms of percent change, note that percent change is the first-order approximation of log return (i.e., the input x as defined above). Put another way, consider changing your perception of the price evolution process.

**“…downward drift in leveraged ETFs is due to mean-median divergence…”**
I disagree. A log-normal distribution’s mean output differs from the median yet φ

_{L} exhibits no such drift. Given successive log returns x

_{1}, x

_{2} such that x

_{1} + x

_{2} = 0, the two-period growth factor

p_{2} / p_{0} = y_{2} × y_{1} = exp( x_{1} ) × exp( x_{2} ) = 1
shows this to be the case. Since

*every* symmetric input distribution is comprised of such pairs, this accurately reflects the general case for symmetric distributions. This applies to the discussion on LETF drift because it was first noticed in sideways markets (during which log returns sum to zero). Any choice of input leverage will not change this result.

**“Many people have a mental model of ETF prices…some kind of symmetric price distribution.”**
It is true that many subscribe to such a mental model, and, they are wrong. Considering equal-magnitude positive and negative moves ($1 up, $1 down) as symmetric is consistent with the percent change model. As one decreases the period of time over which percent change is calculated (or used to evolve prices), the results over multiple periods tend towards those consistent with the exponential growth process.

Serious study of financial mathematics should not be undertaken with a percent change model. I derive all the results in my book using both return models as a way of exposing all of the issues associated with doing so.

**Don't Mess with Causality**
I have one final point of contention with conventional presentations: distributions parametrized on volatility. Parametric graphs are a valuable visual tool; unfortunately, many are designed backwards. The future distribution of prices is a function of the

*input*, and a parametric visualization should show how changing the input affects the

*output*.

Increasing σ (keeping μ constant) keeps the median output constant and

*increases* the mean output.

It is unfortunate the designers of such graphs often decide that the mean output should be kept constant, thus changing both of the input's parameters. As they increase σ, they decrease μ

*just enough* to keep the mean output constant. This convolutes the intent of parametric graphs. Both make clear the growing difference between mean and median price (and look pretty much the same), but the conventional presentation misleads many into thinking volatility is reducing the mean input.

This detail is no less important for its subtlety. LETF dynamics are mostly driven by the increased volatility that comes with leverage. As a related point, I would like to mention that a fair price is not one equal to the expectation of tomorrow’s price distribution. Today’s fair price (ignoring one day’s interest) is one such that tomorrow’s expected log return is zero.

**Why Do LETFs Exhibit Negative Drift?**
If neither the mean-median divergence nor the increased volatility driven by leverage leads to negative drift in φ

_{L} , then why do LETFs exhibit it? Quite simply, LETFs leverage the percent change implied by the output ( y

_{i} - 1 ) not the distribution’s input ( x

_{i} ). LETFs are not log-normally distributed, but they can be pretty accurately represented by a log-nromal distribution with the correct parameters.

**OnlyVIX**: Any advice to someone who wants to start trading leveraged ETFs?

**Alpay Kaya**:
All leveraged instruments help to increase alpha, but they do so in different ways. This necessarily means their risk profiles are different also. The most important thing is to know the fundamentals so you will not be surprised at how your portfolio responds.

Leveraged ETFs: How Biased Statistics Affect Your Portfolio
is now
available on Amazon.com