Jekyll2020-07-07T01:38:04+00:00https://smabie.github.io/feed.xmlSturm MabieQuantitative developer interested in finance, machine learning, and functional programming.Sturm MabieThe Low Beta Anomaly2020-06-28T00:00:00+00:002020-06-28T00:00:00+00:00https://smabie.github.io/posts/2020/06/28/beta<p>Perhaps more than any other development, finance was ushered into the
modern era with the development of the <em>Capital Asset Pricing Model</em>
(CAPM) by William Sharpe in the early 60s. Though commonly criticized as
too simple and reductionist, the model is still used today as an easy
way to determine a stock's exposure to the market:</p>
<script type="math/tex; mode=display">r_i - r_f = r_f + \beta_i (r_m - r_f)</script>
<p>$\beta_i$ was originally formulated as
$\frac{\text{Cov}(r_m,r_i)}{\text{Var}(r_m)}$, but it is more commonly
calculated by taking the slope of a linear regression between the asset
and the market. $r_m$ is the expected market return, $r_f$ is the
expected risk-free rate, and $r_i$ is the expected return of the asset.</p>
<p>Perhaps the model is so popular because the interpretation of $\beta$ is
so concrete and easy to understand: a stock's $\beta$ is simply a
multiplier on the market's return. Stocks with high betas are more
volatile than the market, and those with smaller betas less so. In
tumultuous times, investors try and cut their volatility by rotating
into low beta stocks; while in bull markets, investors clamor to those
with the highest betas.</p>
<p>There's been a problem with CAPM, and indeed, the very concept of beta,
for a long time now: it's called the low beta anomaly. Academics
noticed that there was systematic discrepancies between high and low
beta stocks: it seemed like higher beta stocks were under performing and
low beta stocks were outperforming. Consider the <em>Security Market Line</em>
(SML):</p>
<p><img src="/assets/sml.png" alt="Security Market Line" /></p>
<p>According to CAPM, the returns of low beta stocks were supposed to
linearly scaled by their beta exposure, likewise with higher beta
stocks. Instead, it was noticed that there was an unexpected curve:</p>
<p><img src="/assets/lba.png" alt="Low beta anomaly" /></p>
<p>As we can see, high volatility (and thus, high beta) portfolios are
returning significantly less than what is expected, thus throwing a
wrench into the very concept of beta. After seeing the chart, a natural
question is: can we exploit this mispricing while not being exposed to
the market? In this post, we'll look into whether it's possible to
profit off of this effect and also discuss the potential structural and
behavioral reasons for this anomaly.</p>
<h2 id="a-simple-approach">A Simple Approach</h2>
<p>Of course, the simplest way to try and capture the excess return of the
anomaly is to construct a portfolio of low beta equities and call it a
day. But what if we don't want to be exposed to the market, even to the
degree that low beta equities are? Like most answers in finance, the
answer is a market neutral long/short portfolio!</p>
<p>For this strategy, we'll be using the Quantopian <em>Q1500US</em> universe,
which consists of the 1500 most liquid US equities on any given day. We
then construct our low beta factor as such:</p>
<script type="math/tex; mode=display">f = \text{Z}[\text{rank}(-\beta_{252})]</script>
<p>This will give us large positive values for stocks that have low beta,
and small negative values for those that have high beta. Before we rank
and z-score, in order to calculate the beta for each equity, we find the
slope of a simple linear regression over a 252 day rolling window. We
then use these factor values as the weights of our long/short portfolio.
Let's look at the total long/short return of our simple strategy
between 2003-01-01 and 2020-01-01:</p>
<p><img src="/assets/betacum.png" alt="Cumulative return" /></p>
<p>Well… this isn't good! Why are we losing money? Does this mean that
the low beta anomaly isn't actually true? Digging into the numbers a
little more, it turns out that the total portfolio beta for this
strategy is actually negative, -38% to be exact. This means that even
though we are market neutral, we have a negative exposure to the market:
when the market goes up, our portfolio loses money, and vice versa.
Since the market goes up most of the time, this isn't a great property
to have. Ideally, we want to have zero beta exposure, not negative
exposure. Let's look at the skew of the distribution of betas over
time:</p>
<p><img src="/assets/skew.png" alt="Skew over time" /></p>
<p>In order for our market neutral portfolio to have zero beta, we would
need the skew of the betas to be zero, like for a standard normal
distribution. To make this a little more concrete, let's look at a
histogram for a single time period of stock betas:</p>
<p><img src="/assets/hist.png" alt="Histogram" /></p>
<p>Now, the problem becomes evident: even though we are market neutral, we
end up taking on negative beta exposure because the distribution of the
beta of stocks has a long left-side tail. Almost no stocks actually have
negative beta exposure, while many have high beta exposures! This means
that when longing and shorting in equal proportion, we end up with a
large negative beta since our long positions don't have a low enough
beta in order to cancel out our high beta short positions.</p>
<h2 id="fixing-the-problem">Fixing the Problem</h2>
<p>So, how do we proceed? We need to find some way to adjust the weights of
our portfolio so that we end up with zero beta exposure. Fortunately,
calculating the beta exposure is relatively straightforward. Given a
column vector of weights:</p>
<script type="math/tex; mode=display">\mathbf{X} = \begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n \\
\end{bmatrix}</script>
<p>And a vector of beta exposures:</p>
<script type="math/tex; mode=display">% <![CDATA[
\mathbf{B} = \begin{bmatrix}
\beta_1 & \beta_2 & \dots & \beta_n
\end{bmatrix} %]]></script>
<p>We can easily find the beta of portfolio:</p>
<script type="math/tex; mode=display">\beta_p = \mathbf{BX}</script>
<p>Traditionally, we would solve this equation by using an optimizer with a
constraint that $\beta_p =0$. But because we only have one factor, we
can find an analytical solution. What we want to do is to try and solve
for a $\beta_p$ of zero. To do this, we first need to rewrite the
equation in terms of the weights that are positive (the low beta side)
and those that are negative (the high beta side):</p>
<script type="math/tex; mode=display">\mathbf{X_\alpha} = \begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_j
\end{bmatrix}</script>
<script type="math/tex; mode=display">\mathbf{X_\beta} = \begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_k
\end{bmatrix}</script>
<p>Likewise, we split up the betas as well:</p>
<script type="math/tex; mode=display">% <![CDATA[
\mathbf{B_\alpha} =
\begin{bmatrix}
\beta_1 & \beta_2 & \dots & \beta_j
\end{bmatrix} %]]></script>
<script type="math/tex; mode=display">% <![CDATA[
\mathbf{B_\beta} =
\begin{bmatrix}
\beta_1 & \beta_2 & \dots & \beta_k
\end{bmatrix} %]]></script>
<p>Where:</p>
<script type="math/tex; mode=display">n = j + k</script>
<p>Now, our equation becomes:</p>
<script type="math/tex; mode=display">\beta_p = \mathbf{B_\alpha X_\alpha}+\mathbf{B_\beta X_\beta}</script>
<p>Where the $\alpha$ vectors contain the positive weights and the $\beta$
vectors, the negative weights. We want to scale up the negative weights
(make them larger, though still negative), so we introduce a scaling
factor, $\lambda$, and set $\beta_p=0$:</p>
<script type="math/tex; mode=display">\mathbf{B_\alpha X_\alpha}+ \lambda \mathbf{B_\beta X_\beta}=0</script>
<p>We now solve for $\lambda$:</p>
<script type="math/tex; mode=display">\lambda = - \frac{\mathbf{B_\alpha X_\alpha}}{\mathbf{B_\beta X_\beta}}</script>
<p>Assuming a leverage ratio of one, i.e:</p>
<script type="math/tex; mode=display">\sum_{i=1}^j |x_{\alpha,i}| + \sum_{i=1}^k |x_{\beta,i}| = 1</script>
<p>We can calculate the net long exposure, using $\lambda$ as a parameter:</p>
<script type="math/tex; mode=display">l = \frac{1-\lambda}{1+\lambda}</script>
<p>Below is the graph of net long exposure over time:</p>
<p><img src="/assets/ne.png" alt="Net long exposure" /></p>
<p>And the cumulative return:</p>
<p><img src="/assets/betacum2.png" alt="Cumulative return" /></p>
<p>Wow, Looks a lot better! Ignoring the Great Recession, the return stream
seems very solid, with limited volatility. Below is a table of some
additional strategy information:</p>
<table>
<thead>
<tr>
<th>Metric</th>
<th>Value</th>
</tr>
</thead>
<tbody>
<tr>
<td>Beta</td>
<td>1.8%</td>
</tr>
<tr>
<td>Ann. Ret</td>
<td>4.0%</td>
</tr>
<tr>
<td>Ann. Vol</td>
<td>4.6%</td>
</tr>
<tr>
<td>Ret/Vol</td>
<td>0.84</td>
</tr>
</tbody>
</table>
<p>With a beta of only 1.8%, it's clear that our method for reducing beta
exposure works quite well. Still, it's surprising that a net long
exposure of such magnitude can still be taken with little to no
correlation to the market. Evidently, there is indeed an excess return
associated with low-beta equities, and quite a large one, at that.</p>
<h2 id="explanation">Explanation</h2>
<p>Many academics have tried to explain the structural and behavioral
reasons for the low beta anomaly. The most common explanation put forth
is that due to leverage constraints and borrowing costs, investors seek
out high beta securities in order to achieve a higher natural return. If
high leverage ratios were available to all investors, one might expect
that this mispricing of high beta stocks might go away, as the beta of
any given stock would become less important.</p>
<p>Another possible explanation is that the conflict of interest between
money managers and clients creates an incentive for fund managers to
take excess risk through high beta equities. In good times, managers
collect a performance fee and a management fee while in bad times, only
a management fee. This asymmetric payoff incentives managers to try and
score a big "win," while limited liability prevents managers from ever
losing money on losses.</p>
<p>A third explanation is that high volatility (and thus high beta) stocks
receive more attention from the financial community as they are simply
more interesting to discuss. This interest and attention encourages
increased buying, thus pushing down the expected return of said stocks.</p>
<h2 id="conclusion">Conclusion</h2>
<p>In this post we have shown that there is indeed a significant abnormal
return associated with low beta stocks. This abnormal return can not
only be captured with a long only (and beta exposed) portfolio, but also
a beta neutral one. Despite the less than stellar risk-adjusted return
of the strategy, perhaps the Sharpe ratio can be improved by controlling
sector and style risks in addition to beta exposure. Perhaps one could
also overlay the low beta factor on top of an existing factor strategy
in order to reduce volatility and boost the risk-adjusted return.</p>
<p>That's all for now, and thanks for reading! If you're interested in
the code and want to play around with it, check it out
<a href="https://www.quantopian.com/posts/low-beta-anomaly">here</a>.</p>["Sturm Mabie"]Perhaps more than any other development, finance was ushered into the modern era with the development of the Capital Asset Pricing Model (CAPM) by William Sharpe in the early 60s. Though commonly criticized as too simple and reductionist, the model is still used today as an easy way to determine a stock's exposure to the market:Analysis of CS:GO Win-rates2020-06-12T00:00:00+00:002020-06-12T00:00:00+00:00https://smabie.github.io/posts/2020/06/12/csgo<p>This post is going to be a little different from usual; instead of
markets, we're going to look at a video game, namely,<a href="https://en.wikipedia.org/wiki/Counter-Strike:_Global_Offensive"> Counter-Strike:
Global
Offensive</a>
(CS:GO). CS:GO, like most great games, is easy to learn but deceptively
hard to master. For those at are unfamiliar with the game, we'll give a
quick overview below.</p>
<p>CS:GO is a competitive zero-sum game in which two teams of 5 players
each try to win rounds. The first team to 16 points wins the game. The
game is asymmetric as there are two distinct sides: the terrorists
(T-side) and counter-terrorists (CT-side). After 15 rounds, each team
switches sides. The goal of the T-side is to either eliminate all CT
players or to plant a bomb and have it explode before the CT-side can
defuse it; the CT-side wins the round if kill all the players on the
T-side before the bomb is planted, if they defuse the bomb, or if time
runs out on the round. this asymmetric rule-set implies that the CT-side
can lose a round even if they kill all the opposing players, while the T
side cannot lose if they eliminate all CT players.</p>
<p>In order to get a better idea of the structure of the game, let's look
at a top-down perspective of one of the most famous and iconic maps,
Dust2:</p>
<p><img src="/assets/dust2.png" alt="Dust2" /></p>
<p>T-side starts the game in T-spawn and must plant their bomb at one of
two locations: B-site or A-site, designated by the red areas on the map.
the CTs start off in CT spawn and must try and defend these two sites.
If T-side manages to break the CT defenses and plant the bomb, the
remaining players on CT try and retake the bombsite (either A or B) and
defuse the bomb before it explodes.</p>
<p>Because of the asymmetric nature of the game, I thought it would be
interesting to analyze how much a kill effects the game for each side.
To start off with, we'll first look at the situation where the number
of players on each side is equal.</p>
<h2 id="even-match-ups">Even Match-ups</h2>
<p>First up, let's look at the so-called even match-up, where there are an
equal number of players on each team: 5v5, 4v4, etc. Using over 400,000
rounds of match data from mid-2018 (click
<a href="https://www.kaggle.com/skihikingkevin/csgo-matchmaking-damage">here</a>
for the original dataset), we aggregate all rounds with even match-ups
and the side that wins. From this, we can calculate a cumulative win
probability for each match-up:</p>
<p><img src="/assets/evencs.png" alt="Even Match-ups" /></p>
<p>From the above graph, it's seems that the T-side enjoys a significant
advantage. Even when the match starts, the CT-side has a less than 50%
of winning the round and as trades are made (a situation where each team
loses a player), the advantage the T-side has only goes up. But is this
significant? Let's look at a table of CT-side win rates and their
associated p-value:</p>
<table>
<thead>
<tr>
<th>CT win rate</th>
<th>p-value</th>
<th>players</th>
</tr>
</thead>
<tbody>
<tr>
<td>0.48729</td>
<td>2.67482e-55</td>
<td>5</td>
</tr>
<tr>
<td>0.469768</td>
<td>1.17974e-151</td>
<td>4</td>
</tr>
<tr>
<td>0.455098</td>
<td>3.84482e-237</td>
<td>3</td>
</tr>
<tr>
<td>0.439982</td>
<td>1.52963e-319</td>
<td>2</td>
</tr>
<tr>
<td>0.430299</td>
<td>2.35914e-282</td>
<td>1</td>
</tr>
</tbody>
</table>
<p>It's clear from the minuscule magnitude of the p-values that we can
reject the null hypothesis, namely that each even match-up is fair: a
50% chance of each side winning the round.</p>
<h2 id="all-match-ups">All Match-ups</h2>
<p>Now that we've established that even when each side has the same number
of players the T-side has an advantage, let's consider all the possible
combinations. Clearly a situation when 5 CTs are up against 4 Ts is not
a fair fight (We might assume that CTs have the advantage), but unfair
is it? After crunching the number for each permutation, we get the
following graph:</p>
<p><img src="/assets/csperm.png" alt="All Permutations" /></p>
<p>Interesting! When the CTs have close to the number of players as T-side,
the first kills make the most difference. A 5v5 for the CT side gives
them a 48% chance of victory, but netting the first kill shifts the odds
considerably to 68%. When the difference is large, the final kills have
the highest percentage chance, as the chance of the CT-side winning is
so low to start off with. Below is a graph of the exact figures:</p>
<table>
<thead>
<tr>
<th>1 CT</th>
<th>2 CT</th>
<th>3 CT</th>
<th>4 CT</th>
<th>5 CT</th>
<th># T alive</th>
</tr>
</thead>
<tbody>
<tr>
<td>0.00218907</td>
<td>0.0227307</td>
<td>0.106122</td>
<td>0.274956</td>
<td>0.487288</td>
<td>5</td>
</tr>
<tr>
<td>0.00689667</td>
<td>0.0682988</td>
<td>0.235994</td>
<td>0.469768</td>
<td>0.683482</td>
<td>4</td>
</tr>
<tr>
<td>0.0287596</td>
<td>0.187184</td>
<td>0.455091</td>
<td>0.70026</td>
<td>0.858241</td>
<td>3</td>
</tr>
<tr>
<td>0.123364</td>
<td>0.439972</td>
<td>0.732706</td>
<td>0.893306</td>
<td>0.961373</td>
<td>2</td>
</tr>
<tr>
<td>0.430299</td>
<td>0.7915</td>
<td>0.941975</td>
<td>0.985024</td>
<td>0.9967</td>
<td>1</td>
</tr>
<tr>
<td>0.906434</td>
<td>0.989945</td>
<td>0.998562</td>
<td>0.999731</td>
<td>0.999873</td>
<td>0</td>
</tr>
</tbody>
</table>
<h2 id="conclusions">Conclusions</h2>
<p>I know little about professional CS:GO strategy and the data isn't
taken from professional games, but we can quickly infer a couple things
from the data:</p>
<ol>
<li>T-side wants to trade down as much as possible, CT never wants to
trade.</li>
<li>CT-side needs to be play very conservative, maximizing the number of
players on their team that are alive.</li>
<li>T-side wants to play in a very aggressive style in order to take map
control and trade.</li>
</ol>
<p>Anyways, this has been a fun little post to write, I hope you enjoyed
it! Click <a href="https://github.com/smabie/csgo-kill-value">here</a> to view the
GitHub project.</p>["Sturm Mabie"]This post is going to be a little different from usual; instead of markets, we're going to look at a video game, namely, Counter-Strike: Global Offensive (CS:GO). CS:GO, like most great games, is easy to learn but deceptively hard to master. For those at are unfamiliar with the game, we'll give a quick overview below.ETFs, Volatility and Leverage: Towards a New Leveraged ETF Part 32020-06-09T00:00:00+00:002020-06-09T00:00:00+00:00https://smabie.github.io/posts/2020/06/09/vol3<p>In the final installment of this three part series, we are going to use
our results from the previous two posts to construct a fully automated
variable leverage ETF. In <a href="https://cryptm.org/posts/2019/10/04/vol.html">part
one</a>, we derived the
optimal leverage ratio for maximizing returns and in <a href="https://cryptm.org/posts/2020/05/28/vol2.html">part
two</a> we applied the ARMA
and GARCH models to forecast returns and volatility, respectively.</p>
<p>The ETF we are going to build will take variable leverage with a minimum
leverage of 1 and a maximum leverage of 3. This means that the leverage
ratio we will be assuming will deviate considerably from what is
optimal. The reason for this is twofold: 1) many investors don't want
to short the market, regardless of what our model says, preferring a
baseline of pure beta exposure, and 2) the SEC rejected an application
for the creation of a 4x S&P 500 ETF, so it would be unlikely that a
higher leverage ratio could be assumed, at least in an ETF wrapper. If
one was instead obtaining leverage directly through futures, -20x to 20x
leverage could be taken.</p>
<p>Right now, we have two components, $\text{E}(r_m)$, the expected monthly
return generated from our ARMA model, and $\text{E}(\sigma_d)$, the
expected daily volatility from our GARCH model. First off, we need to
forward fill our monthly return data, in order to generate return data
for everyday. We forward fill instead of back fill in order to avoid
lookahead bias. Also, because our ARMA model is forecasted the future
one month expected return, we convert our daily volatility into monthly
variance: $21\text{E}(\sigma_d)^2$. So our raw leverage ratio becomes:</p>
<script type="math/tex; mode=display">l = \frac{\text{E}(r_m^{\text{fill}})}{21\text{E}(\sigma_d)^2}</script>
<p>Let's look at graph of our leverage ratio over time to get a sense of
what we're dealing with:</p>
<p><img src="/assets/olev.png" alt="Optimal leverage" /></p>
<p>Oof, most investors won't be happy with that! Not only is our optimal
leverage calculation taking huge short and long positions, but it's
changing the direction of the portfolio very frequently. We want to
smooth out the changes in leverage as well as constrain the amount taken
between 1 and 3. In order to do this, we apply the following
transformations on the time-series:</p>
<ol>
<li>
<p>First, if the leverage ratio is less than 1 at a time-point, we set
it to 0.</p>
</li>
<li>
<p>In order to constrain the values, we then add $e$ and logscale it.
This should give us values approximately between 1 and 3.</p>
</li>
<li>
<p>To smooth the leverage, we apply an <a href="https://pandas.pydata.org/pandas-docs/stable/user_guide/computation.html#exponentially-weighted-windows">exponential rolling
window</a>,
with an $\alpha$ of 0.05.</p>
</li>
<li>
<p>If any values are above 3, we set them to 3.</p>
</li>
</ol>
<p>Below is the full transformation:</p>
<script type="math/tex; mode=display">% <![CDATA[
f(x) =\begin{cases}
x &\text{if }x > 1\\
0 &\text{else}\\
\end{cases} %]]></script>
<script type="math/tex; mode=display">\sum_{t=1}^n \min(\text{EWM}_{\alpha=0.05}[\log(f(x_t)+e)],\,3)</script>
<p>Consider the graph of adjusted leverage:</p>
<p><img src="/assets/alev.png" alt="Adjusted leverage" /></p>
<p>Though we're significantly deviated from what the optimal leverage is,
our adjusted leverage looks a lot more reasonable. The transitions are
smoother and the values are bounded between 1 and 3, as desired. Now we
simply multiple the leverage ratio by the returns of the S&P 500 and we
have our strategy! This is the moment of truth, let's look at a graph
of the returns of the S&P 500, the returns of our ETF, and the leverage
ratio:</p>
<p><img src="/assets/etf.png" alt="variable leverage S&P 500 vs S&P 500 vs leverage
ratio" /></p>
<p>Not bad! Our ETF is behaving as expected: taking on more and more
leverage during bull runs and reducing exposure when the market drops.
The leverage ratio fluctuates between 1 and 2 due to our smoothing
factor, though more risk could be taken by choosing a greater $\alpha$
value.</p>
<h2 id="conclusion">Conclusion</h2>
<p>At long last, we've reached the end of this series. We've talked about
investors, mostly irrational, aversion to holding leveraged ETFs over a
long period of time, derived the optimal leverage ratio to maximize
returns, created models to forecast returns and volatility, and used all
of this to create a variable leverage ETF. Could such a product be
brought to market, and would investors be interested? Can investors be
convinced to hold any product that isn't just vanilla beta? I'm not
sure, but it is certainly an interesting opportunity. No product on the
market fulfills this niche, if there is even one to exploit. I hope you
enjoyed this series as much as I enjoyed researching and writing it.
Though we did succeed in creating a rudimentary model, so much more work
could be done on it. I believe that a variable leverage product could
have great potential in the retail space, especially now that vanilla
beta has been completely commoditized. New ETF ideas that are "beta
plus" not only have the potential to deliver value to investors, but
also could command much higher fees.</p>
<p>You can check out the notebook
<a href="https://github.com/smabie/towards-a-new-etf-part3">here</a>. Feel free to
play around with any and all parameters.</p>["Sturm Mabie"]In the final installment of this three part series, we are going to use our results from the previous two posts to construct a fully automated variable leverage ETF. In part one, we derived the optimal leverage ratio for maximizing returns and in part two we applied the ARMA and GARCH models to forecast returns and volatility, respectively.ETFs, Volatility and Leverage: Towards a New Leveraged ETF Part 22020-05-28T00:00:00+00:002020-05-28T00:00:00+00:00https://smabie.github.io/posts/2020/05/28/vol2<p>In <a href="https://cryptm.org/posts/2019/10/04/vol.html">part one</a>, we looked
into the relationship between volatility, returns, and leverage and
derived an equation for the optimal leverage ratio that maximizes the
expected return of a portfolio. This leverage ratio is dependent on two
principle components, expected variance and expected returns:</p>
<script type="math/tex; mode=display">l = \frac{r_b}{\sigma^2_b}</script>
<p>Where $r_b$ is the unlevered return and $\sigma^2_b$, the variance of
the portfolio. Our next task, though clear, is hardly straightforward:
we must forecast the future expected return and volatility of the
portfolio in order to set an appropriate leverage ratio. In Part three,
we'll use our forecasts developed in this post, with modifications and
extensions, to realize a complete trading strategy that attempts to
deliver a higher return than the market through the application of
variable leverage.</p>
<h2 id="autoregressive-model">Autoregressive Model</h2>
<p>There are an unaccountably numerous number of modeling techniques that
attempt to do time-series forecasting, each of varying complexity and
sophistication. To start off with, let's consider a moving average (AR)
model:</p>
<script type="math/tex; mode=display">\text{AR}(p): x_t = \alpha + B_1x_{t-1}+B_2x_{t-2}+\cdots+B_px_{t-p}+\epsilon_t</script>
<p>Simply put, an AR model is simply a multiple regression on previous
observed values with the addition of a white noise component. Values of
$x$ come from previous observed values in the series. For example, a
model that simply used today's returns as a predictor for tomorrow's
returns could be approximately formalized as a AR(1) model with
$\alpha=0$ and $B_1=1$:</p>
<script type="math/tex; mode=display">\text{AR}(1): x_t = x_{t-1} + \epsilon_t</script>
<p>Note that $\epsilon_t\sim N(0, \sigma^2)$, which means it is a white
noise term with a mean of zero and a standard deviation that equals the
volatility of the time series in question.</p>
<p>There are various ways to find an appropriate $p$, but perhaps the most
straightforward is to apply the autocorrelation function, also called
ACF. The ACF takes in a number that represents the number of previous
values we are interested in and calculates the correlations over the
different lagged values. For example, if we wanted to find the ACF with
a lag of one, we would simply find the correlation between all values
(except the first one), and the previous value in the time series. By
calculating the ACF over a number of lags, we can discover if there is a
serial correlation between values or if they are independent. For
example, if we found that there is a correlation, positive or negative,
between today's market returns and the returns of yesterday, but not
the day before yesterday, we could use an AR(1) model to capture this
relationship.</p>
<h2 id="moving-average-model">Moving Average Model</h2>
<p>Another type of simple model often used to forecast time series is an
autoregressive or AR model:</p>
<script type="math/tex; mode=display">\text{MA}(q):x_t = \mu + \epsilon_t + \theta_1\epsilon_{t-1}+\cdots+\theta_q \epsilon_{t-q}</script>
<p>A MA model is defined by its lag, $q$, the number of previous terms to
consider when generating a forecast. Each future forecast in
parameterized by the mean, which must remain constant and
$\theta_1\cdots\theta_q$, the lag exposures.
$\epsilon_{t-q}\cdots\epsilon_t$ are lagged white noise terms generated
from previous observations. These epsilon error terms are unobservable,
independent of each other, and respect a normal distribution. It is
important to note that because $\mu$ is constant, the time-series in
question must be stationary: the mean and variance must not change over
time.</p>
<p>Instead of using the ACF to discover the serial correlations and the $q$
parameter, we use the partial autocorrelation function (PACF) instead.
The PACF is similar to the ACF, except that it calculates the partial
correlation instead of a regular correlation over lagged values. Partial
correlation differs in that it controls for some specified variable
before the correlation function is run. In the case of PACF, we control
for the linear dependence of the non-lagged values with the values in
between. For example, let's say we're looking at the PACF of lag 2 and
have three variables to consider: $x_t$, $x_{t-1}$, and $x_{t-2}$. The
partial autocorrelation is the same as normal correlation between $x_t$
and $x_{t-1}$. But when considering $x_t$ and $x_{t-2}$, we first remove
the dependence that $x_t$ has on $x_{t-1}$ before running the
correlation.</p>
<h2 id="arima-model">ARIMA Model</h2>
<p>A common approach is to combine both of these models, forming an ARMA
model:</p>
<script type="math/tex; mode=display">\text{ARMA}(p,\,q): \text{AR}(p) + \text{MA}(q)</script>
<p>Note that an ARMA model is only appropriate when the series is
stationary, the mean and variance must not change over time. Because of
this, such a model is inappropriate for the forecasting of price data,
instead returns must be used. What we're implicitly doing is
differencing the price series, also called integration of order one. An
ARIMA (autoregressive integrated moving average) model captures this
differencing:</p>
<script type="math/tex; mode=display">\text{ARIMA}(p,\,d\,,q)</script>
<p>Where $d$ is the number of times we differenced. This implies that
ARIMA(1, 1, 1) on price data is identical to ARMA(1, 1) on return data:</p>
<script type="math/tex; mode=display">\text{ARIMA}(p,\,1,\,q) p_t = \text{ARMA}(p,\,q) r_t</script>
<p>Also note that:</p>
<script type="math/tex; mode=display">\text{ARIMA}(p,\,0,\,q) = \text{ARMA}(p,\,q)</script>
<p>In order to find an appropriate $d$, wee can use the Augmented Dickey
Fuller (ADF) test to iteratively determine the integration order
necessary. The ADF test produces a p-value indicating the probability
that the series is not stationary. The process for determining the
integration order and therefore $d$ for an ARIMA model might look as
follows:</p>
<ol>
<li>Set $d=0$</li>
<li>Apply the ADF test on the series</li>
<li>If $p\leq0.05$, return $d$</li>
<li>Otherwise, take the difference between each successive value and
increment $d$.</li>
<li>Go to step 2</li>
</ol>
<h2 id="forecasting-returns">Forecasting Returns</h2>
<p>Now that we've covered some background on time series forecasting,
let's jump into it: predicting the future return. We will be using an
ARMA instead of an ARIMA model, as in general, returns are stationary
over long periods of time. First, we should look at the ACF output so we
can find a good $p$. Different return windows might exhibit different
serial correlations, so let's look at the ACF of one day, one week, and
one month returns between 2003-01-01 and 2020-01-01:</p>
<p><img src="/assets/acf.png" alt="ACF" /></p>
<p>It definitely seems that both weekly and daily data do not exhibit very
much, if any, serial correlation between subsequent periods. Monthly
returns, on the other hand, exhibit a weak correlation between adjacent
months. From this graph, a $p$ of 1 is probably the most appropriate
value.</p>
<p>Moving on to the PACF, we observe much the same results as before:</p>
<p><img src="/assets/pacf.png" alt="PACF" /></p>
<p>This isn't surprising, as over short lag periods with low correlation
to each other, the PACF should look relatively similar to the ACF. As
before, we'll decide on a $q$ of 1.</p>
<p>In order to avoid lookahead bias, we will use a starting window value of
96 months, or 8 years. For each subsequent month, we will add the
observation and then refit our ARMA parameters. This means that our
window increases by one every single iteration: on our second to last
observation, we are using almost all of the available data to train the
model. Another possible approach would be to use a constant rolling
window, though we would have to take care to ensure that each window was
stationary. A third approach would be to use some percentage of the data
for fitting parameters and the rest for validation, but an increasing
window ensures that new data is always used for fitting as it is made
available. Below is the graph of the forecasted versus actualized return
for the S&P 500:</p>
<p><img src="/assets/armaret.png" alt="Actual vs forecasted S&P 500 return" /></p>
<p>Though not exceptional, the model does get the direction of the return
(either up or down) right 55% of the time: not terrible for a simple and
straightforward model. It's also evident that the model significantly
underestimates the volatility of the market, with the magnitude of
predicted returns being quite modest. This is unsurprising for several
reasons: market returns exhibit excess kurtosis compared to a normal
distribution, and the residuals must also follow a normal distribution.
All in all, this is a decent start, especially considering that
predicting returns with a high accuracy is a notoriously difficult, and
perhaps intractable, problem.</p>
<h2 id="vix-index">VIX Index</h2>
<p>Now that we have the foundations of the return model out of the way, we
can move onto to the volatility component. Two principle methods of
forecasting volatility are commonly used: the VIX and GARCH. Let's
first discuss the VIX.</p>
<p>By using the implied volatility of S&P 500 calls and puts, the VIX index
aims to predict the one month volatility of the market. By using the
current price of puts and calls for the S&P 500, we can use the Black
Scholes options model to solve for the implied volatility, or the future
volatility necessary to justify the current prices. At the time of
writing, the VIX index currently is at 28.59. This means that the future
expected one month annualized volatility is 9.9% or the square root of
12 multiplied by 2.859%. Unfortunately, using the VIX as a measure of
future volatility has some problems: because options are primarily used
to hedge downside risk, the VIX often underestimates the future upside
volatility and overestimates the future downside volatility.</p>
<p>In a strong bull market, call options are often undervalued, as there
are other, more common ways to express a long view on the market, such
as futures or equities. During bear markets, put options become
overpriced as they are one of the only ways to hedge a portfolio without
assuming unlimited price risk (such a shorting or selling futures). The
consequence of this is that the VIX often serves as nothing more than an
index that represents the inverse of market returns: when the market
goes up, the VIX goes down, and vice-versa. Below are two graphs of the
correlation between the VIX and future one month volatility and previous
volatility and future volatility:</p>
<p><img src="/assets/vix.png" alt="VIX vs lagged Volatility" /></p>
<p>Perhaps surprisingly, the VIX has practically the same predictive power
as simply using last month's volatility.</p>
<h2 id="garch-model">GARCH Model</h2>
<p>Instead of using the VIX, we are going to use an Generalized
Autoregressive Conditional Heteroskedasticity (GARCH) model. GARCH is
more complicated than ARIMA, so we won't get into the mathematics of
how it works, but, in short, GARCH allows for the volatility to
experience "shocks" through time. The output of a GARCH model is the
conditional volatility: the instantaneous volatility with respect to
some model. One can think of the conditional volatility as the
unobservable latent volatility that changes over time. Unlike
traditional volatility, which must be calculated over some time window,
conditional volatility exists at an instantaneous point of time. Though
we're not going to talk about GARCH parameters in this post, for those
that are already familiar with GARCH, note that we are using the
standard GARCH(1, 1) model, the most common parameters for forecasting
the volatility of returns.</p>
<h2 id="forecasting-volatility">Forecasting Volatility</h2>
<p>Unlike with our ARMA model, we are going to using the daily instead of
monthly returns of the S&P 500 for our GARCH model. Much like before, we
are going to take an ever expanding window, starting at 63 days or 3
months. Also like with returns, by the penultimate observation, we will
be using almost all available data for training our model. The following
graph is of annualized conditional volatility vs the annualized 21 day
(one month) volatility:</p>
<p><img src="/assets/vol.png" alt="Conditional volatility vs 21d volatility" /></p>
<p>Just from the graph, the accuracy of our model looks <em>very</em> good. And in
fact it is, with a correlation of 97.6% and a Mean Absolute Percentage
Error (MAPE) of only 14.8%. All things considered, this is quite good
and highlights the incredible predictive power of GARCH models in
forecasting volatility.</p>
<h2 id="conclusion">Conclusion</h2>
<p>Now that we have a monthly model for forecasting returns and a daily
modal for forecasting volatility, we conclude this blog post.
Modifications will need to made to turn this into a real, long-only S&P
500 ETF, but a good foundation has been laid. I hope you enjoyed this
post! Click <a href="https://github.com/smabie/towards-a-new-etf-part2">here</a> to
check out the notebook. If you install
<a href="https://www.anaconda.com/">anaconda</a>, no additional Python libraries
should be necessary to execute the notebook.</p>["Sturm Mabie"]In part one, we looked into the relationship between volatility, returns, and leverage and derived an equation for the optimal leverage ratio that maximizes the expected return of a portfolio. This leverage ratio is dependent on two principle components, expected variance and expected returns:One Weird Trick to Profit off of a Global Market Meltdown2020-05-18T00:00:00+00:002020-05-18T00:00:00+00:00https://smabie.github.io/posts/2020/05/18/owt<p>Did you know there's one weird trick that Wall Street doesn't want you
to know? For only one payment of $39.99, you can get access to this
limited time only exclusive video that will show you how to fight that
bear market! Click here before Wall Street makes it illegal… Or you
could just read this post, I guess.</p>
<p>Anyways, when backtesting data I came across an interesting and
surprisingly simple strategy that has performed very well during the
coronavirus period. Simply put, most of the volatility during this
period has occured overnight, that is, between market close and the
opening auction the next day. This volatility has primarily been
negative, with large downward price differences between the closing
price and the opening price the next day. The intraday market on the
otherhand has mostly stayed flat or gone up modestly. Let's take a look
at three different portfolios: one where we just hold the S&P 500, the
other where we buy the S&P 500 at market open and sell at market close,
and a third where we do the same but also short the S&P 500 between the
close and market open:</p>
<p><img src="/assets/owtr.png" alt="" /></p>
<p>We see that before coronavirus significantly impacts the market, the
returns of all three strategies are relatively similar: little
dispersion is observed in the overnight market and large moves are
primarily contained to the regular intraday session. Around the end of
February, things begin to change: global coronavirus news begins to
dramatically increase the difference between the closing and opening
prices. By executing a strategy as simple as only holding the index
intraday, we convert a 10% loss into a 10% profit, a pretty incredible
difference. Taken one step further, we can even short the overnight
market and net an eye-popping 25% gain over this five and a half month
period!</p>
<p>While the mechanisms for this trend aren't entirely clear, I suspect
this pattern occurs when global markets start to exhibit a very high
correlation to each other. Since coronavirus is a global problem, most
markets across the world are being affected by the same fundamental
factors. Because of this, the price difference between the close and
open in the US has been a reflection of moves in the European and Asian
markets: when the foreign markets crash, it's highly likely that the
American market will open down in the morning. If, on the otherhand,
coronavirus caused a local recession (imagine if coronavirus was only a
US phenomenon), I suspect that we wouldn't see the same overnight
high-volatility trend.</p>
<p>Now, let's look at the previous global recession, here's the same
graph as before but between 2007-06-01 and 2009-06-01:</p>
<p><img src="/assets/owtr2.png" alt="" /></p>
<p>We observe the exact same phenomenon, with large moves in the overnight
markets. Even so, much more volatilty was present in the intraday
markets than now: even when using the long intraday, short overnight
strategy we would have still ended up down about 10% over the two year
period. Using our aforementioned hypothesis, this makes sense at the
overnight difference isn't as pronounced: the US was the epicenter of
the Great Recession, so the overnight markets are affected comparatively
less than during the current period.</p>
<p>Finally, for contrast, the same strategies during the raging bull market
between 2016-01-01 and 2018-01-01:</p>
<p><img src="/assets/owtr3.png" alt="" /></p>
<p>Unsurprisingly, both the long intraday and long intraday, short
overnight strategies leave money on the table compared to the vanilla
portfolio. Even so, both net a not so unsubstantial return during this
two year period: the long intraday of around 23% and the long intraday,
short overnight of around 8%.</p>
<h2 id="conclusion">Conclusion</h2>
<p>One last thing to keep in mind is that even with zero comissions, the
act of buying after the open and selling before the close incurs large
trading costs: even assuming you're buying large cap stocks with low
spreads (let's assume around 4 bips or 0.04%), the annualized cost
comes to around 10% alone!</p>
<p>Even so, shorting or zeroing out overnight exposure can be a powerful
tactic during tumulteous markets: especially if markets around the world
are moving in lockstep. In addition, a small return can be made on top
from the lending of overnight money (though it doesn't help that
interest rates are often lowered close to zero during these periods).</p>
<p>I hope you enjoyed this post! If you want to check out the Quantopian
notebook used to generate the above graphs, click
<a href="https://www.quantopian.com/posts/coronavirus-intraday">here</a>. Feel free
to play around with the dates, though unfortunately, data isn't
available before 2002.</p>["Sturm Mabie"]Did you know there's one weird trick that Wall Street doesn't want you to know? For only one payment of $39.99, you can get access to this limited time only exclusive video that will show you how to fight that bear market! Click here before Wall Street makes it illegal… Or you could just read this post, I guess.Prediction Markets: Strategies for Alpha Generation2020-05-13T00:00:00+00:002020-05-13T00:00:00+00:00https://smabie.github.io/posts/2020/05/13/pr<p>Much like financial markets, prediction markets (PM) have existed in
some informal capacity for hundreds of years; the earliest known
records, dating back to 1503, describe the practice of betting upon
papal succession. Much like traditional financial markets, the process
of centralization took many hundreds of years, as the value of
centralized exchanges slowly began to impress itself upon the
participants. The start of the contemporary era of PM and political
betting, perhaps the foremost type of PM, was heralded by the <em>Iowa
Electronic Markets</em>, established during the 1988 presidential election
by the University of Iowa. Since then, many sites and markets have come
and gone in the relatively unregulated and nascent field. Currently,
<a href="https://predictit.org">PredictIt</a>, run by the <em>Victoria University of
Wellington</em>, dominates the space in both trade volume and number of
independent markets. Other markets include <em>Betfair</em> and <em>Augur</em>.</p>
<p>In this post we will give an overview of how PredictIt markets are
structured and a non-exhaustive list of commonly used strategies for
alpha generation in order to give the reader a better understanding of
how these markets work and how to potentially profit from them.</p>
<h2 id="overview-of-predictit">Overview of PredictIt</h2>
<p>The basic building block of PredictIt is a market, with each market
containing at least two brackets and an expiration; where each bracket
represents a distinct outcome and the expiration setting the date of
when the owners of shares are paid out and the market settled. For each
bracket, traders can buy or sell only two types of shares: <em>yes</em> shares
or <em>no</em> shares. By purchasing a <em>yes</em> share, the buyer is acquiring the
right to be paid $1 if the event happens; conversely, a <em>no</em> share
gives the buyer the right to be paid $1 if the event does not occur.
For every bracket, two distinct order books are used: one for the <em>yes</em>
shares and one for the <em>no</em> shares. When placing the order, only one
order type can be used, akin to a <em>limit</em> order in financial markets
that is good until cancelled or the market expires. Traders can either
wait for their contract to expire or list it on the order book for any
price between 1 and 99 cents, exclusive. If a contract expires or is
executed on the order book for a profit, a 10% tax is taken by PredictIt
in order to fund the site's operations. When buying into or selling out
of multiple brackets that are not mutally exclusive with each other, the
trader's total exposure to the market is taken into account and their
cash balance is credited or debited appropriately. This is often
surprising for new PredictIt traders, as it is possible to actually
receive money for the purchase of new shares (and conversely, to pay
money when selling shares); this usually happens when <em>no</em> shares are
bought on subsequent brackets in the same market as the trader's total
market exposure is often reduced the more <em>no</em> shares are bought across
different brackets in the same markets (click HERE for more information
about the rules). Take the example market described in the table below:</p>
<table>
<thead>
<tr>
<th>Bracket</th>
<th>Best No Ask</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>50¢</td>
</tr>
<tr>
<td>B</td>
<td>50¢</td>
</tr>
</tbody>
</table>
<p>When the trader buys one <em>no</em> of bracket <em>A</em>, she will be debited 50¢,
as that is her total exposure to the market: she cannot lose any more
than that. If she then buys one <em>no</em> on <em>B</em>, her account will be
credited 50¢ on execution as she has zeroed out her exposure to the
market, since in both outcomes she will make $1, cancelling out the
combined value of both contracts. In practice, the math is slightly more
complicated as PredictIt accounts for the 10% tax in these credit and
debit calculations.</p>
<p>Savvy traders can use this mechanism to build up very large positions
with little capital and in certain cases, actually realize cash profits
when the shares are bought. Consider the following market:</p>
<table>
<thead>
<tr>
<th>Bracket</th>
<th>Best No Ask</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>1¢</td>
</tr>
<tr>
<td>B</td>
<td>1¢</td>
</tr>
</tbody>
</table>
<p>As before, the trader when buying one <em>no</em> of <em>A</em> will pay the full cost
of the contract, which in this case is 1¢. The interesting part comes
when the trader purchases the second contract: she will immediately get
credited 99¢ (for a profit of 98¢) and her exposure will, as before, be
zeroed out. Of course, this isn't a very realistic example as the
efficient market sharks will quickly devour all the chum in the water
and move the prices until the sum of the <em>no</em> ask prices is equal to or
greater than $1.</p>
<h2 id="index-of-strategies">Index of Strategies</h2>
<p>Below is the list of strategies we will discuss, ordered from low risk
to high risk:</p>
<ol>
<li>Bracket Arbitrage</li>
<li>Yes/No Arbitrage</li>
<li>Information Arbitrage</li>
<li>Carry</li>
<li>Rule Arbitrage</li>
<li>Statistical Arbitrage</li>
<li>Momentum</li>
<li>Mean-Reversion</li>
<li>Expert Knowledge</li>
</ol>
<p>The next sections will describe each strategy in detail, explain the
conceptual background, and advise you on how and when to apply each in
the markets.</p>
<h2 id="bracket-arbitrage">Bracket Arbitrage</h2>
<p>Perhaps the most straightforward and least risky strategy is bracket
arbitrage, which we discussed brefily above. Simple bracket arbitrage
means buying an equal number shares in all brackets if the sum of all
prices is below some amount. For <em>yes</em> shares, this amount is $1, as
the sum of the probabilities of all events for well-formed probability
mass function (PMF) must equal 1. Therefore, if the sum of all <em>yes</em>
prices is less than one, we can make a completely risk-free profit,
provided that we are able to find the opportunity in the first place.
Also note that theoretically we could short the share if the sum is
greater than one and also make a risk-free profit but unfortunately
PredictIt does not allow the shorting of shares. For clarity, consider
the following market:</p>
<table>
<thead>
<tr>
<th>Bracket</th>
<th>Yes Ask</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>52¢</td>
</tr>
<tr>
<td>B</td>
<td>17¢</td>
</tr>
<tr>
<td>C</td>
<td>29¢</td>
</tr>
</tbody>
</table>
<p>Since the sum of all brackets is less than $1 (98¢), we can buy an
equal amount of shares of each bracket and realize a ~2% return
(pre-tax).</p>
<p>The same general idea works for the <em>no</em> side of the market as well,
except that the sum of the shares must equal the number of brackets (in
dollars) minus $1. For example:</p>
<table>
<thead>
<tr>
<th>Bracket</th>
<th>No Ask</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>98¢</td>
</tr>
<tr>
<td>B</td>
<td>70¢</td>
</tr>
<tr>
<td>C</td>
<td>29¢</td>
</tr>
</tbody>
</table>
<p>For no arbitrage to be possible, the sum of the <em>no</em> prices must add up
to $2 (instead they add up to $1.97), allowing us to buy an equal
amount of each bracket and net a ~1.5% return.</p>
<p>While bracket arbitrage is a risk-free strategy, finding opportunities
to exploit is exceedingly difficult. In fact, it's possible that
you'll never find such an opportunity by just perusing different
markets manually. Instead, one should use PredictIt's market data API
(described HERE) to receive automated alerts on arbitrage opportunities.
Alternatively, one could write a fully automated bot to exploit these
inefficiencies; take caution though, as this is both difficult to do and
against PredictIt's TOS.</p>
<p>A slightly riskier version of arbitrage involves excluding outcomes that
are clearly foreclosed before performing the summation. This strategy
involves some judgement sense but can be profitable. Arbitrage
opportunities that span all outcomes are easier to recognize
mathematically, and so they are usually fixed quickly. Bracket exclusion
is sometimes quite recognizable and can net triple the profits with
little additional risk. For example, take this market from March 7,
2020, "Who will be the Democratic Nominee for President?"</p>
<table>
<thead>
<tr>
<th>Bracket</th>
<th>Yes Ask</th>
</tr>
</thead>
<tbody>
<tr>
<td>Biden</td>
<td>84¢</td>
</tr>
<tr>
<td>Sanders</td>
<td>11¢</td>
</tr>
<tr>
<td>Clinton</td>
<td>5¢</td>
</tr>
</tbody>
</table>
<p>At a glance, there is no arbitrage opportunity here, as the brackets sum
to 1. Clearly Hilllary Clinton is not a candidate in the 2020 primary,
and it is impossible for her to enter the race at this point. Filing
deadlines for all states have closed. Theoretically, the democrats could
settle on her after a contested convention but any casual observer would
agree the probability of this outcome is essentially 0. A good rule of
thumb is that if any bracket has a last transaction price is 99 cents or
1 cent and no offers available on the corresponding side of the order
book, it is safe to count it as $1 or 0 for the purposes of bracket
arbitrage. In the case of the market above, the unusual price of 5 cents
for Hillary Clinton is probably related to liquidity issues, as holders
of no shares created months ago struggle to get their money out at 96
cents in order to reinvest it elsewhere for better returns before the
market resolves. Mispricings for nearly certain outcomes like this are
common when expiration dates are far in the future or rules of
resolution are unclear. Traders prefer not to be exposed to the time and
risk, so they accept unfair prices to convert their investments back
into (liquid) cash.</p>
<h2 id="yesno-arbitrage">Yes/No Arbitrage</h2>
<p>Despite my inability to come up with a better name, yes/no arbitrage is
a reasonable way to make risk-free profits on PredictIt. Implementing
this strategy requires multiple accounts which represents the only risk
associated with it: only one account per individual is allowed as per
PredictIt's TOS so successfully using this strategy could lead to the
banning of your account. Fortunately, if PredictIt decides to ban your
account, they will first liquidate all of your positions and then wire
the total balance to an account of your choosing. Instead of arbitraging
between multiple brackets of the same type (<em>yes</em> or <em>no</em>) as with
bracket arbitrage, we arbitrage between the <em>yes</em> and <em>no</em> side for each
bracket. If the sum of the <em>yes</em> and <em>no</em> ask is less than $1, we can
take advantage of the opportunity by purchasing an equal number of each
kind of share. For example, consider the following market:</p>
<table>
<thead>
<tr>
<th>Bracket</th>
<th>Yes Ask</th>
<th>No Ask</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>63¢</td>
<td>40¢</td>
</tr>
<tr>
<td>B</td>
<td>32¢</td>
<td>64¢</td>
</tr>
</tbody>
</table>
<p>We are unable to take advantage of the mispricing of bracket <em>A</em> (since
we are unable to short shares), but we can purchase both sides for <em>B</em>
and make a risk-free profit. In this case, we purchase one share of each
for a total cost of 96¢ and receive a profit of 4¢, thus netting a
pre-tax return of ~4.1%. Note that the reason this strategy requires
multiple accounts is that PredictIt does not allow traders to buy both
<em>yes</em> and <em>no</em> shares on the same bracket under a single account, one
kind of shares must be sold prior to the purchase of the other.</p>
<p>Opportunities for this strategy are easier to find than for bracket
arbitrage since it necessitates multiple accounts, but even so,
automated and constant monitoring of the market data feed is necessary
to reliably find the mispricing. Caution is needed because of the TOS
and as such, it's advisable to create a pair of accounts purely for
this strategy in order to minimize the downside if the accounts are
frozen.</p>
<h2 id="information-arbitrage">Information Arbitrage</h2>
<p>Information arbitrage is the bread and butter of many PredictIt traders
and perhaps the most commonly used strategy on the site. The mechanics
are straightforward but the amount of work and time required can be
prohibitive: simply read and trade on new information faster than anyone
else. Personally I find this strategy to be time-consuming and boring,
but for those who are either are already reading the news or diligant
about monitoring settlement sources, this strategy can be incredibly
profitable. Beware though that since most markets incorporate new
information reasonably quickly (under 30 minutes), it's necessary to
react fast, preferably under 5 minutes.</p>
<h2 id="carry">Carry</h2>
<p>Another classic and popular PredictIt strategy, exploiting a positive
carry is an ideal way to earn a return on capital while waiting for new
intermittant opportunities. The basic idea is to buy shares (either
<em>yes</em>'s or <em>no</em>'s) in markets and brackets that have a positive carry.
For those unfamiliar with the concept, this means that whatever you're
buying becomes more valuable over time, thus allowing you to hold it and
make a profit day after day. In the context of PredictIt, this usually
manifests in one of two ways:</p>
<ol>
<li>The market expires at a certain date and the event that you are
buying shares of has not yet occured. The trader then buys whichever
bracket and type of share that represents the status quo. If the
status quo is not disrupted, the time decay of the non-status quo
bracket or share will push up the price of the status uo bracket or
share. An example might be: "Will John Smith run for office before
the end of 2020." By purchasing <em>no</em> shares the trader is betting
on the status quo and the price will rise consistently until
expiration, assuming John Smith does not, in fact, ever run for
office. Of course, the risk associated with this trade is that John
Smith may run for office before 2020; even so, it's possible to use
this strategy in combination with information arbitrage to make
money on the carry and quickly get out (or switch the trade to the
other side) before much profit is lost.</li>
<li>The event has already occured and the price has risen, usually to
99¢. Many traders sell contracts at 99¢ before settlement in order
to deploy the capital to other opportunities that have a higher
expected value. By buying these shares, a trader can net a risk=free
1% return over the remaining duration. Note that even when a share
is priced at 99¢ it does not necessarily mean the event has already
occured, reading the actual settlement rules of the market is
critical in order to make well informed trades.</li>
</ol>
<p>An important criteria for identifying opportunities is the ratio between
the total potential profit and the expiration date of the contract: a
99¢ share on a market that expires in one month is significantly more
valuable than one that expires in a year.</p>
<p>Mispricings can sometimes occur because only one market exists to
speculate on a particular eventuality, but the deadline for resolution
is approaching. When new information emerges making such an eventuality
more likely, buyers look for a way to act on the information and the
prices move dramatically. However they may fail to notice that the
resolution date is too soon. For example the market "Will the World
Health Organization Declare Coronavirus a Pandemic by March 6 2020?"
moved dramatically in early March as more cases were reported. However
by March 3, each passing day had such a high carry that the news was
irrelevant.</p>
<h2 id="rule-arbitrage">Rule Arbitrage</h2>
<p>Though not something that a trader can reliably exploit, the potential
windfalls from this strategy can be absolutely massive. By reading the
settlement rules for each market <em>very</em> carefully, a trader can profit
off of those who have not (the majority). I've seen cases where a
contract is priced at 97¢ after an event happens that most traders think
qualifies as a <em>yes</em> but actually, according to a close reading of the
settlement rules, is a <em>no</em>. These events are rare, but a 10-50x profit
is not out of the question. Beware though, in many markets the
setttlement rules stipulate that PredictIt has discretion to decide the
outcome in certain situations, usually pertaining to the unavailability
of settlement sources (if the settlement website goes down, for
example).</p>
<h2 id="statistical-arbitrage">Statistical Arbitrage</h2>
<p>Statistical arbitrage in the context of prediction markets is a type of
strategy that aims to profit off of price deviations relative to some
reference model. Instead of developing their own models, a time
consuming and difficult task, many traders arbitrage prices with
publically posted predictions, such as those on Nate Silver's
<a href="https://fivethirtyeight.com/">538</a>, a political forecasting website.
While small profits can be made consistently using public predictions, a
proprietary model can be much more profitable.</p>
<h2 id="momentum">Momentum</h2>
<p>As with financial markets, momentum trading simply involves trading on a
trend with the hope that the trend continues. This strategy carries the
risk of the trend ending, but can often be used as a proxy for an
information arbitrage strategy. A common way for traders to generate a
monetum signal is for them to buy a single share in all interested
markets and put a sell order at a pre-determined price, maybe 5-10%
greater than the current price. When the trade executes, the trader
receives a notification which signals that there has been a significant
price move. The trader can now manually place a bigger order and hope
for the trend to continue.</p>
<h2 id="mean-reversion">Mean-Reversion</h2>
<p>The opposite of momentum, the mean-reversion strategy involves betting
on the reversion of price. It's common for traders to overbuy or
oversell on new information, allowing mean-reversion traders to buy up
the relatively cheap shares and profit after sentiment becomes more
measured.</p>
<h2 id="expert-knowledge">Expert Knowledge</h2>
<p>As in much of our politics nowadays, traders sometimes seem to use
PredictIt less as a way to make money and more as a way to express their
identity/personal beliefs. This might be analogous to the way wishful
thinking can create unreasonable odds of a famous horse or a well-loved
home team. Readers will not be surprised to learn that these fan-boy
traders tend to skew left, with a few exceptions.</p>
<p>Inside information can be recognized from drastic and unexplained price
swings. It is very risky and difficult to make money by joining these
careening bandwagons, but dramatic price moves should be a reason to
consider downsizing bets you already have open in such markets.</p>
<p>Identifying these irrational prices is hard to quantify, but intuition
and common sense can help to identify these opportunities.</p>
<h2 id="conclusion">Conclusion</h2>
<p>Hopefully this post has gotten you thinking about the standard
strategies in use and how and when to apply them. The best trading
results often occur when multiple signals or strategies are used to
together to inform the buying or selling of shares. Good luck and hang
in there! Prediction markets are often frustrating, but can be quite
profitable and satisfying given a certain level of dedication!</p>["Sturm Mabie"]Much like financial markets, prediction markets (PM) have existed in some informal capacity for hundreds of years; the earliest known records, dating back to 1503, describe the practice of betting upon papal succession. Much like traditional financial markets, the process of centralization took many hundreds of years, as the value of centralized exchanges slowly began to impress itself upon the participants. The start of the contemporary era of PM and political betting, perhaps the foremost type of PM, was heralded by the Iowa Electronic Markets, established during the 1988 presidential election by the University of Iowa. Since then, many sites and markets have come and gone in the relatively unregulated and nascent field. Currently, PredictIt, run by the Victoria University of Wellington, dominates the space in both trade volume and number of independent markets. Other markets include Betfair and Augur.Analyst Recommendations: Got Alpha?2020-01-14T00:00:00+00:002020-01-14T00:00:00+00:00https://smabie.github.io/posts/2020/01/14/rec<p>In today's post, we will dig into the nature and value of analyst
recommendations in order to try to answer an old but controversial
question: do sell-side analyst recommendations hold any predictive
power; and if so, when and why? The data we will use for this analysis
is the <em>FactSet Estimates - Broker Recommendations</em> dataset, available
free of charge on Quantopian.</p>
<h2 id="background">Background</h2>
<p>Analyst recommendations and letters are as old as Wall Street itself:
traditionally provided as free reports for a broker or investment
bank's buy-side institutional clients. A bank might employ many
different equity analysts, each focusing and covering a basket of
stocks, usually all in the same sector. In addition to publishing
qualitative reports, a quantitative numerical score is also provided.
While different firms have adopted different rating systems, all ratings
in the FactSet dataset have been normalized according to the following
table:</p>
<table>
<thead>
<tr>
<th>Rating</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>Buy</td>
</tr>
<tr>
<td>1.5</td>
<td>Overweight</td>
</tr>
<tr>
<td>2</td>
<td>Hold</td>
</tr>
<tr>
<td>2.5</td>
<td>Underweight</td>
</tr>
<tr>
<td>3</td>
<td>Sell</td>
</tr>
</tbody>
</table>
<p>As analyst ratings and reports are provided as a free auxiliary service,
they act as a value-add for the firm's brokerage and trading desks
instead of a primary and monetizable product. In the best light, the
recommendations act to inform or supplement the firm's clients' views
of a given stock instead of defining it. The more contrarian and
critical view is that the recommendations are little more than
entertainment and exist entirely to drive trading activity for the firm.
After all, how much predictive power could these ratings actually have?
If the equity analyst did indeed have any alpha, why would they be
working at a sell-side firm publishing reports instead of trading with
their own money or at a buy-side investment firm? A third possible
explanation is that the very existence of the ratings, especially from
so called all-star analysts, manage to move the market, in effect
becoming a self-fulfilling prophecy.</p>
<p>From the theoretical viewpoint of the <em>Efficient Market Hypothesis</em>
(EMH), the value of analyst ratings is clear: since analysts are not
incorporating any genuinely new or unique information into their reports
and ratings, they should not have any predictive power. In this view,
reading public filings and listening in on earning calls cannot possibly
help, since this information gets quickly absorbed into the market
price, thus rapidly rendering the ratings useless. In addition, the very
existence of the ratings themselves cannot affect the value since no
rational investor would trade on information already incorporated into
the price. Despite the intellectual appeal, this strong EMH has become
niche over the years as the mania of the market has become more apparent
with the advent of the dotcom boom/bust and the great recession.</p>
<p>While the academic and professional consensus is still unclear, there
does seem to be evidence in the literature that analyst recommendations
do hold some predictive power. In <a href="https://www.tandfonline.com/doi/abs/10.2469/faj.v56.n3.2357">Hemang Desai, Bing Liang & Ajai K.
Singh
(2019)</a>,
the authors find a number of interesting effects:</p>
<ol>
<li>All-star analysts outperformed their peers.</li>
<li>There seems to be major herding behavior among analysts.</li>
<li>Analysts who only cover one sector have better predictive power than
those who are responsible for covering multiple sectors.</li>
</ol>
<p>In <a href="https://onlinelibrary.wiley.com/doi/abs/10.1111/0022-1082.00336">Brad Barber, Reuven Lehavy, Maureen McNichols and Brett Trueman
(2002)</a>,
the authors construct a long/short market-neutral portfolio based on
analysts' recommendations and realize a 4% annualized return on the
portfolio.</p>
<p>As suggested by the aforementioned papers, it is possible that <em>some</em>
analysts have <em>some</em> predictive power, but the caveats are long and
many: How do we tell the good analysts from the bad? In what market
conditions are the ratings likely to be correct? With such strong
herding behavior, how independent are each rating? What does the actual
scale mean, and how do we know, for example, that "hold" means the
same thing for all analysts? The questions go on and on, muddying the
waters significantly. In the next section, we will construct a
long/short market neutral portfolio in an attempt to try to answer the
fundamental and basic question on everyone's mind: can I make money
from analyst recommendations?</p>
<h2 id="using-analyst-recommendations-the-simple-way">Using Analyst Recommendations, the Simple Way</h2>
<p>Unfortunately, the FactSet data available on Quantopian is somewhat
limited, containing only 7 relevant fields:</p>
<table>
<thead>
<tr>
<th>Name</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code class="language-plaintext highlighter-rouge">buy</code></td>
<td>number of buy recommendations</td>
</tr>
<tr>
<td><code class="language-plaintext highlighter-rouge">over</code></td>
<td>number of overweight recommendations</td>
</tr>
<tr>
<td><code class="language-plaintext highlighter-rouge">under</code></td>
<td>number of underweight recommendations</td>
</tr>
<tr>
<td><code class="language-plaintext highlighter-rouge">sell</code></td>
<td>number of sell recommendations</td>
</tr>
<tr>
<td><code class="language-plaintext highlighter-rouge">total</code></td>
<td>total number of recommendations</td>
</tr>
<tr>
<td><code class="language-plaintext highlighter-rouge">mark</code></td>
<td>average recommendation</td>
</tr>
<tr>
<td><code class="language-plaintext highlighter-rouge">no_rec</code></td>
<td>number of missing recommendations</td>
</tr>
</tbody>
</table>
<p>We don't have access to individual analyst recommendation in this data
sets, and therefore will not be able to pick up when an individual
analyst changes her recommendation. To start off with, we will just
simply be using the <code class="language-plaintext highlighter-rouge">mark</code> field to construct our alpha factor. We
construct our market-neutral factor based on the following equation:</p>
<script type="math/tex; mode=display">f=\text{Z}(\text{rank}[-mark])</script>
<p>We take the negative of the <code class="language-plaintext highlighter-rouge">mark</code> field since 1 is a buy and we want to
long positive ratings and short negative ones. We then rank from 1 to
$n$, with $n$ being the most highly recommended stock. After that we
Z-score in order to fit the data into the standard normal distribution.
For our universe we'll use <code class="language-plaintext highlighter-rouge">Q500US</code>, which is comprised of the 500 most
liquid securities using a 200-day average dollar volume, capped at a max
of 30% of equities in any single sector. We set our start date at
2003-01-01 with an end date of 2019-01-01, though, ideally, we would
save more data for out-of-sample testing.</p>
<p>To start out, let's break up our data into quantiles and look at the 1
day, 5 day, and 10 day holding periods to get a better sense of the
distribution of returns between quantiles and the persistence or decay
of alpha over these periods:</p>
<p><img src="/assets/recq.png" alt="Mean Return by Quantile" /></p>
<p>Immediately from this graph, we can see that things are not looking so
good for the analyst recommendations. Ideally we would see a
monotonically increasing return stream as we move from the lowest to the
highest quantile. Since we are shorting all equities in the lower half
and longing in the upper half, we are losing money in every single
quantile across all holding periods. Digging deeper, let's look at the
cumulative return by quantile for the 1 day holding period:</p>
<p><img src="/assets/reccret.png" alt="Cumulative Return by Quantile" /></p>
<p>In general, we want the 4th and 5th quantiles to be steadily climbing
upward while quantiles 1 and 2 steadily decline. Except for a brief
period around the financial crisis, we see the opposite. Putting it all
together, let's finally look at the entire long/short portfolio's
cumulative return stream:</p>
<p><img src="/assets/recret.png" alt="Cumulative Return" /></p>
<p>Unsurprisingly (after seeing the previous graphs), the portfolio does
not perform exceptionally well over this time period, only making
significant gains during the financial crisis and then giving them up
(and more) shortly after the start of 2009. While reading time-series
graphs of this nature is often akin to reading tea-leaves, we can
tentatively hazard a few explanations for the apparent lack of
predictive power in the analyst ratings:</p>
<ol>
<li>
<p>Perhaps during turbulent market conditions, analysts ratings exhibit
higher predictive value than in less volatile environments. In these
market regimes, analysts might exhibit less herding behavior as the
consequence of being objectively wrong (instead of merely
contrarian) becomes more severe. Or put more simply, analysts may
get fired for being wrong in bad times while fired for being
contrarian in good times.</p>
</li>
<li>
<p>In <a href="https://www.cambridge.org/core/journals/journal-of-financial-and-quantitative-analysis/article/value-of-client-access-to-analyst-recommendations/AEE68658A8841D8A2862BDC462A38B2C">T. Clifton
Green (2009)</a>,
Green finds systematic mispricing for only two hours after a new
recommendation is released. Quantopian only started storing the
FactSet dataset in a point-in-time fashion starting November 2018;
for data before then, the timestamp is approximated, thus
potentially leading to the situation where all the alpha has been
traded away before the backtester has access to it.</p>
</li>
<li>
<p>In <a href="https://www.researchgate.net/publication/46489709_Informed_Trading_Before_Analyst_Downgrades_Evidence_From_Short_Sellers">Christophe, Stephen & Ferri, Michael & Hsieh,
Jim (2010)</a>,
the authors uncover evidence of abnormal and unusual levels of
short-selling in the three days before analyst downgrades are
announced. This might suggest that individuals and firms are trading
on inside information obtained from analyst recommendations before
their official public release, reducing or eliminating any alpha our
strategy can capture.</p>
</li>
</ol>
<p>Whatever the case, using the data available, there unfortunately does
not seem to be a persistent abnormal return associated with the
portfolio. There is however one hopeful statistic that indicates there
could possibly be some value in analyst ratings, the <em>Information
Coefficient</em> (IC) value:</p>
<p><img src="/assets/ic.png" alt="Information Coefficient Distribution" /></p>
<p>The IC is a measure of predictive power, or the correlation between the
expected or predicted return and the realized forward return bounded
between -1 and 1. An IC of 0 indicates no edge, that is, there is no
relationship between the expected return and the forward return. A value
of 1 indicates a perfect correlation between the two, while -1 means
that the prediction is always incorrect. Despite the strategy yielding
no cumulative abnormal return, the mean IC is, albeit slightly,
positive; thus implying that the predictive power of analyst
recommendations is better than random.</p>
<h2 id="using-analyst-recommendations-20">Using Analyst Recommendations 2.0</h2>
<p>It is clear that using just analyst ratings is not sufficient to
generate and sustain abnormal returns. It might be helpful to go back
and remember what <em>exactly</em> we are trying to do by using analyst
ratings: we are trying to find predictive information that has not yet
been incorporated into the price of the equity. Put a slightly different
but similar way, we are looking for an expert minority view not shared
by the rest of the market. It might stand to reason, for example, that
if the stock is going up and the average analyst recommendation is a
buy, then either the buy rating has already been priced in or analysts
don't possess a significantly different outlook on the stock than the
rest of the market. On the other-hand, if the stock has an average
rating of buy but the stock is declining in price then it stands to
reason that the analysts hold a (presumably) expert minority opinion
that is not shared by the rest of the market yet. Below is a table
illustrating the four primary conditions and the corresponding actions
based on the aforementioned hypothesis:</p>
<table>
<thead>
<tr>
<th>Recommendation</th>
<th>Direction</th>
<th>Action</th>
</tr>
</thead>
<tbody>
<tr>
<td>Buy</td>
<td>Up</td>
<td>No Action</td>
</tr>
<tr>
<td>Sell</td>
<td>Down</td>
<td>No Action</td>
</tr>
<tr>
<td>Buy</td>
<td>Down</td>
<td>Long</td>
</tr>
<tr>
<td>Sell</td>
<td>Up</td>
<td>Short</td>
</tr>
</tbody>
</table>
<p>We can easily construct the corresponding factor as follows:</p>
<script type="math/tex; mode=display">f=\text{Z}(\text{rank}[-\text{SMA}_{20}r_{daily}]\cdot\text{rank}[-mark])</script>
<p>Where SMA is the 20 day simple moving average and $r_{daily}$ the daily
(close-to-close) returns. This factor will allows us to create a
market-neutral portfolio that longs stocks that have had poor last month
returns but positive analyst ratings while shorting stocks that have had
good returns and negative ratings. As like before, let's start off by
looking at the daily mean return by quantile:</p>
<p><img src="/assets/arecret.png" alt="Mean Return by Quantile" /></p>
<p>Not bad! As desired, we have a nice monotonic progression from the
first quantile to the fifth. From this graph, it's clear that our
alpha decays rather quickly with time, so let's focus on the one day
holding period's cumulative return by quantile:</p>
<p><img src="/assets/areccret.png" alt="Cumulative Return by Quantile" /></p>
<p>Again, not terrible: we see the upper quantiles move steadily upward
while the lower quantiles move steadily downward. Now for the total
long/short cumulative return chart and IC:</p>
<p><img src="/assets/arectret.png" alt="Cumulative Return" /></p>
<p><img src="/assets/aic.png" alt="Information Coefficient Distribution" /></p>
<p>From the IC and cumulative returns, it is clear that the portfolio
constructed from the above factor possesses abnormal return (for a a
market-neutral portfolio, any return greater than zero could be
considered abnormal). Even though this is an absolute return strategy
(i.e it has no comparable benchmark), it's illustrative to compare with
the S&P 500 as many investors will implicitly benchmark all strategies
to vanilla market returns:</p>
<table>
<thead>
<tr>
<th>Metric</th>
<th>L/S</th>
<th>S&P 500</th>
</tr>
</thead>
<tbody>
<tr>
<td>Beta</td>
<td>4.6%</td>
<td>100%</td>
</tr>
<tr>
<td>Ann. Ret</td>
<td>1.7%</td>
<td>8.4%</td>
</tr>
<tr>
<td>Ann. Vol</td>
<td>6.2%</td>
<td>17.9%</td>
</tr>
<tr>
<td>Ret/Vol</td>
<td>0.31</td>
<td>1.34</td>
</tr>
</tbody>
</table>
<p>Unfortunately, after looking at the above table, it is clear that even
though our strategy demonstrates abnormal returns, investors would not
find it an attractive investment. The unlevered return is below the
usual cost-of-capital (i.e risk-free rate) so it would be difficult to
lever up the strategy in order to increase returns. Even if we could
borrow at a zero percent interest rate, it is unclear why would even
want to as the S&P 500 has a better return to volatility ratio over
this time-period; in other words, an investor could just invest a
third of their money in the S&P 500 and receive both better returns
and volatility than our strategy. Perhaps the only saving grace is the
low market (beta) exposure, but with so many other uncorrelated asset
classes that have much better risk/reward profiles, the strategy is
functionally useless.</p>
<h2 id="conclusion">Conclusion</h2>
<p>Though one can reasonably conclude that there is some predictive value
in analyst recommendations, using the recommendations alone does not
constitute a viable strategy. However, when combined with other
conditional factors such as price, sentiment, or fundamentals data,
analyst recommendations might very well boost the IC and Sharpe ratio
of the strategy. Moreover, perhaps alpha can be found by utilizing
non-mean aggregate characteristics of the recommendations, such as
skew or standard deviation.</p>
<p>Thanks for reading and I hope you enjoyed this post! If you want to
check out the Quantopian notebook, click
<a href="https://www.quantopian.com/posts/analyst-recommendations-got-alpha">here</a>.</p>["Sturm Mabie"]In today's post, we will dig into the nature and value of analyst recommendations in order to try to answer an old but controversial question: do sell-side analyst recommendations hold any predictive power; and if so, when and why? The data we will use for this analysis is the FactSet Estimates - Broker Recommendations dataset, available free of charge on Quantopian.Diversification, Risk and Leverage2019-11-28T00:00:00+00:002019-11-28T00:00:00+00:00https://smabie.github.io/posts/2019/11/28/div<p>In this post, we will look into the relationship between
diversification, risk, and leverage: first covering the history of
diversification and risk and then expounding upon the mathematical link
between leverage and diversification. Much of the material in this post
is related to another post of mine which you can view here: <a href="https://smabie.github.io/posts/2019/10/04/vol.html">ETFs, Volatility and Leverage: Towards a New Leveraged ETF Part 1</a>.</p>
<p>While this article doesn't constitute the next post in the series (stay
tuned!), it covers some important ideas that I feel to have tangency to
the ideas put forth in the earlier post. So without further ado, let's
jump into the origins of diversification!</p>
<h2 id="background">Background</h2>
<p>Diversification, the strategy of investing in multiple unique return
streams or investments simultaneously, has been a portfolio construction
technique used to mitigate risk for thousands of years. The concept is
mentioned in several ancient religious texts, including the Bible and
Talmud, and came to prominence under the Roman Empire: investors would
form investment partnerships for particularly risky business endeavors,
such as financing the voyages of merchant vessels. While the original
purpose of these partnerships was to provide access to lucrative
investment vehicles for investors lacking sufficient capital to fund
such a voyage alone, they quickly became popular as risk-pooling
strategies; for even the richest Romans would often buy shares of many
different voyages in order to diversify the significant risk of any
single voyage.</p>
<p>The modern concept of diversification was popularized by Harry Markowitz
in the 1950s through his concept of a <em>Markowitz-efficient portfolio</em>
(MEP) and <em>Modern Portfolio Theory</em> (MPT). A MEP is a portfolio that
lowers the portfolio's volatility through diversification for a given
level of expected return. Or alternatively, the portfolio with the
lowest volatility possible (achieved primarily through diversification)
that still satisfies a minimum level of expected return. MPT and MEP
were the first formalizations of diversification put forth, creating a
framework for viewing the essence of a portfolio through its return and
volatility. of a portfolio through the return and volatility of the
portfolio. Using the example of a discretionary equity investor, the MEP
would consist of as many stocks as the investor could find or research
that would pass his investment criteria. Once the investor started
adding less desirable stocks (stocks with a lower expected return) that
culminate in a portfolio with lower return than the desired or expected
return, he no longer would hold a MEP.</p>
<p>More generally, Markowitz in MPT introduced the concept of the <em>Capital
Market Line</em> (CML):</p>
<p><img src="/assets/cml.png" alt="Capital Market Line" /></p>
<p>The efficient frontier is any portfolio that possesses the highest
risk-adjusted return: the portfolio that had the highest expected return
as a function of volatility. While any portfolio on the efficient
frontier is superior to any risky portfolio in the achievable region, by
using the risk-free rate to find the super-efficient portfolio, also
called the tangency or market portfolio, we can create a portfolio that
is a linear combination of risk-free assets and risky assets. By either
going long or short the risk-free rate (lending or borrowing), we can
achieve any level of return as a function of volatility or vice versa.
While this is not strictly true for a compounding portfolio due to
volatility drag (see my aforementioned post), the intuition behind it is
powerful. We can then see risk as a multiple on the volatility of the
super-efficient portfolio.</p>
<p>Later with the advent of the <em>Capital Asset Pricing Model</em> (CAPM) and
multi-factors, the concept of risk was partitioned into two primary
categories: systematic and specific risk. Unlike specific risk which can
be diversified away through the holding of a MEP or similar portfolio,
systematic risk cannot be easily diversified as it represents the latent
factors in which all assets have some exposure to. In the original
formulation of CAPM, specific risk was assumed to be zero and not
included in the equation:</p>
<script type="math/tex; mode=display">r_i - r_f = r_f + \beta_i (r_m - r_f)</script>
<p>Where $\beta_i$ is $\frac{\text{Cov}(r_m,r_i)}{\text{Var}(r_m)}$, $r_f$
is the risk free rate, $r_i$ is the expected return of the asset $i$,
and $r_m$ the expected return of the market. The introduction of CAPM by
William Sharpe and others was the next pivotal step after MPT in
quantifying risk and transformation finance into a legitimate scientific
discipline. In CAPM, a portfolio's risk is defined as some multiple of
the market's risk, generally assumed to be the volatility of the
market. This important idea is best visualized by the Security Market
Line (SML):</p>
<p><img src="/assets/sml.png" alt="Security Market Line" /></p>
<p>Though under the CAPM model, investors may choose where on the line best
fulfills their investment goals ($\beta$) through portfolio
construction, they are not easily able to diversify away their market
risk since, by definition, their $\beta$ will converge to 1 as more and
more assets are added to the portfolio; nor can investors capture excess
returns relative to the market risk, as per the <em>Efficient Market
Hypothesis</em> both overvalued and undervalued stocks mispricings will be
arbitraged away. In it's essence, CAPM posits that investors are
compensated in the exact same proportion to the market risk assumed,
revealing an interesting property of specific risk: since specific risk
can easily be hedged away through the construction of a MEP, investors
are only compensated for systematic or market risk assumed, not specific
risk. A consequence of this is that investors are more likely to judge
an asset by its systematic risk versus its specific risk: an investment
with low sensitivity to the market but high specific risk may be judged
to have much less risk than a naive observer might expect.</p>
<p>In the next section, we look at the mathematics of diversification of
random variables.</p>
<h2 id="portfolio-variance-and-diversification">Portfolio Variance and Diversification</h2>
<p>An interesting proprety of diversification is that mathematically
speaking, with every new security we add to our portfolio, the
volatility of the portfolio monotonically decreases as long as two
conditions are met: the correlation of the new asset to any other asset
is less than 1, and that the sum of portfolio weights never increases.
Let's define a portfolio $P$ that is a linear combination of two
variables (assets):</p>
<script type="math/tex; mode=display">P = x_A X_A + x_B X_B</script>
<p>where $x_A$ and $x_B$ are the proportion or weights of the assets in the
portfolio. While the returns of the portfolio are a linear combination
of the two return streams of the assets, the variance is:</p>
<script type="math/tex; mode=display">\sigma_P^2 = x_A^2 \text{Cov}(r_A,r_A) + x_B^2\text{Cov}(r_B,r_B) + 2x_Ax_B\text{Cov}(r_A,r_B)</script>
<p>Though it may not be immediately obvious, the variance of the portfolio
will always be lower the more assets we add, as long as the
aforementioned conditions are met. With three or more assets, the
equation starts to become too unwieldy so instead, we often represent
the variance of a portfolio using matrices. Using the same example of
two assets, $X_A$ and $X_B$, we first define the covariance matrix:</p>
<script type="math/tex; mode=display">% <![CDATA[
\textbf{P}=\begin{bmatrix}
\text{Cov}(r_A,r_A) & \text{Cov}(r_A,r_B) \\
\text{Cov}(r_B,r_A) & \text{Cov}(r_B,r_B)
\end{bmatrix} %]]></script>
<p>Next we define our vector weights or proportions:</p>
<script type="math/tex; mode=display">\textbf{x}=
\begin{bmatrix}
x_A \\
x_B
\end{bmatrix}</script>
<p>Now we can define our variance in terms of the covariance matrix and the
vector weights:</p>
<script type="math/tex; mode=display">\sigma_P^2 = \mathbf{x}^T\mathbf{Px}</script>
<p>This is often called the quad or quadratic form of $\mathbf{x}$ and
$\mathbf{P}$.</p>
<p>In the special case where:</p>
<script type="math/tex; mode=display">x_A + x_B = 1</script>
<script type="math/tex; mode=display">x = x_A = x_B</script>
<script type="math/tex; mode=display">\sigma = \sigma_A = \sigma_B</script>
<script type="math/tex; mode=display">\rho_{AB} = 0</script>
<p>Then:</p>
<script type="math/tex; mode=display">\textbf{x}=
\begin{bmatrix}
x \\
x
\end{bmatrix}</script>
<script type="math/tex; mode=display">% <![CDATA[
\textbf{P}=\begin{bmatrix}
\sigma^2 & 0 \\
0 & \sigma^2
\end{bmatrix} %]]></script>
<script type="math/tex; mode=display">\sigma_P^2 = \mathbf{x}^T\mathbf{Px} = \frac{\sigma}{\sqrt{2}}</script>
<p>In the general case of $n$ assets:</p>
<script type="math/tex; mode=display">\sigma_n = \frac{\sigma}{\sqrt{n}}</script>
<p>The graph below visualizes the portfolio volatility as a function of
assets each with a standard deviation of 10%:</p>
<p><img src="/assets/sqrtdev.png" alt="Volatility vs Number of Assets" /></p>
<p>This graph makes it very clear on why investors are not compensated for
bearing specific risk: is is relatively easy to reduce much specific
risk by holding a relatively small number of assets in a portfolio. Even
though for equities in particular the volatility cannot be reduced as
much due to most equities being highly correlated to each other through
their common systematic risk factors, as we will see in the next
section, we can still get most of the way there with only 20-30
different correlated assets.</p>
<p>In the next section, we will look at some market data and run some
simulations in order to better understand the effect of diversification
on a theoretical portfolio under real market conditions.</p>
<h2 id="diversification-and-historical-sp-500-returns">Diversification and Historical S&P 500 Returns</h2>
<p>To illustrate how diversification affects volatility and returns in a
real market environment, we ran a Monte Carlo simulation that generated
1000 random portfolios that long $n$ number of random stocks, where $n$
is all even numbers between 2 and 100, inclusively. All stocks were
chosen from the S&P 500 and the daily portfolio returns were constructed
between 2017-01-01 and 2019-01-01. Below is a graph of average
annualized volatility and return as a function of assets held:</p>
<p><img src="/assets/div.png" alt="Volatility and Return as a Function of Number of
Assets" /></p>
<p>In contrast to the essentially constant annual return as a function of
assets, the volatility exhibits a sharp downward trend before
stabilizing, reminiscent of the previous graph of perfectly uncorrelated
assets all sharing the same volatility. Since we are picking stocks
randomly, it is unsurprising that we trend toward the MEP with the same
expected level of return regardless of how many assets are in the
portfolio. Though not shown on the graph, the average annualized
volatility of our portfolio unsurprisingly trends quickly toward the
annualized volatility of the S&P 500 during that same period.</p>
<h2 id="leverage">Leverage</h2>
<p>Despite the prior graph implying that the volatility and return of our
portfolio exist orthogonal to one another, in reality for a compounding
portfolio, this is not actually the case. The essence of the problem is
in the <em>AM-GM Inequality</em>, also called the inequality of arithmetic and
geometric means:</p>
<script type="math/tex; mode=display">\frac{1}{n} \sum_{i=1}^{n}x_i \geq \left(\prod_{i=1}^n x_i \right)^{1 \over n}</script>
<p>This equation implies that the arithmetic average of our returns will
only be accurate if and only if the return stays constant. Otherwise,
our true geometric returns will always lower than the arithmetic return.
Thus, due to the nature of compounding returns and geometric sums, the
volatility of our portfolio has a very real negative affect on our
long-term returns. This phenomenon is coined <em>volatility drag</em> and
increasingly penalizes portfolios with higher and higher volatility:</p>
<script type="math/tex; mode=display">r_a = r_p - \frac{\sigma^2_p}{2}</script>
<p>Where $\sigma_p$ is the standard deviation of the portfolio, $r_p$ the
return of the portfolio and $r_a$ the actualized return of the portfolio
after accounting for volatility drag (for a more in-depth dive into
volatility drag see the link at the beginning of this post). Volatility
drag implies that not only should we aim to reduce volatility for the
classically given reasons, but that reducing volatility actually lets us
boost our returns, which means that diversification provides even more
benefit than we might otherwise think.</p>
<h2 id="leverage-and-diversification">Leverage and Diversification</h2>
<p>A key result of my linked post is the relationship between volatility,
return, and leverage. After a little bit of math we arrived at the
conclusion that in order to maximize our return, we should lever up our
portfolio as per the following equation:</p>
<script type="math/tex; mode=display">l = \frac{r_b}{\sigma_b^2}</script>
<p>Some of the readers might notice the similarity between the ideal
leverage ratio and the fractional Kelly:</p>
<script type="math/tex; mode=display">f^* = \frac{\mu-r}{\sigma^2}</script>
<p>Regardless, we can look at the ideal leverage ratio as a function of the
number of assets in our portfolio:</p>
<p><img src="/assets/l.png" alt="Leverage vs Number of Assets" /></p>
<p>We can see that the optimal leverage for maximizing the geometric growth
of our portfolio increases quickly relative to the number of assets in
our portfolio but also rapidly stabilizes at a leverage ratio between
4.5 and 5 between 2017-01-01 and 2019-01-01.</p>
<p>It is now evident that diversification can be used in one of two ways:
to reduce the uncompensated specific risk of a portfolio, or to increase
the amount of risk or leverage we as investors can take in order to
maximize the long-term geometric growth rate of our portfolio. In
general, we should always aim to maximize our risk while satisfying our
risk controls in order to generate the greatest possible return.
Diversification allows us to minimize specific risk, in which we are not
compensated for, giving us the opportunity to obtain risk for which we
<em>are</em> compensated for, namely, systematic risk.</p>
<h2 id="conclusion">Conclusion</h2>
<p>In this post we arrived at a couple important conclusions. We can easily
reduce our volatility through diversification, even when picking random
assets for our portfolio. Not only does this reduce our risk, but it has
the potential to boost our returns as well. Taking it a step further, we
can actually lever up our portfolio in order to maintain risk parity if
we so choose. Thus we can use diversification in one of two ways: to
either greatly reduce risk and modestly boost the compounded returns of
our portfolio, or to lever up the less risky diversified portfolio in
order to maximize the long-term geometric growth. Either way,
diversification is a powerful tool in an investor's toolbox and should
be exploited to its fullest potential. Furthermore, because
diversification reduces volatility drag and boosts our returns, in
addition to allowing us to take more leverage, it very well might make
sense to diversify past the point of a MEP, as the long-term compounded
savings could outweigh the decrease in single-period expected return.</p>
<p>Thanks for reading, I hope this was entertaining and informative! If you
want to check out the Quantopian notebook I made for this post, click
<a href="https://www.quantopian.com/posts/diversification-risk-and-leverage">here</a>!
Feel free to experiment with different start and end dates, number of
assets, or trials. If you clone the notebook and play with it, you must
also put a file called "SP500.csv" in your data directory. Simply copy
the data off of Wikipedia or some other source and format it to be a one
column CSV file with the header "Ticker".</p>
<p>(1) Check out my deep dive into volatility and leverage: <a href="https://smabie.github.io/posts/2019/10/04/vol.html">ETFs,
Volatility and Leverage: Towards a New Leveraged ETF Part
1</a>. It clarifies
some of the concepts in this post and derives the equation for leverage
that we used in the previous section.</p>["Sturm Mabie"]In this post, we will look into the relationship between diversification, risk, and leverage: first covering the history of diversification and risk and then expounding upon the mathematical link between leverage and diversification. Much of the material in this post is related to another post of mine which you can view here: ETFs, Volatility and Leverage: Towards a New Leveraged ETF Part 1.ETFs, Volatility and Leverage: Towards a New Leveraged ETF Part 12019-10-04T00:00:00+00:002019-10-04T00:00:00+00:00https://smabie.github.io/posts/2019/10/04/vol<p>In part one of this three part series, we will explore the concept of
levered ETFs, common misconceptions, the effect of volatility on the
returns of a portfolio, and the compounded returns of the S&P 500
utilizing different leverage ratios. We will also touch on the basic
mathematical underpinnings of volatility drag and ideal leverage ratio.</p>
<p>In part two, we will look into different ways to forecast future market
regimes and their associated optimal leverage ratios.</p>
<p>In part three, we will construct a fully automated ETF that seeks to
obtain variable leveraged exposure to the S&P 500 conditioned on the
future forecasted market return and volatility regime.</p>
<h2 id="background">Background</h2>
<p>The recent ascent of ETFs as one of the most popular trading vehicles
for both retail and institutional investors has dramatically affected
the business of all funds seeking retail flows, and even many who do
not. ETFs and ETNs serve as wrappers for a diverse array of strategies
that run the gamut from simple and transparent to complex and
proprietary. ETFs (and ETNs though for brevity's sake we will just
refer to all securities employing this legal structure as "ETFs") can
be placed on a two dimensional spectrum: simple to complex on one axis,
and transparent to proprietary on another. These products are united by
several distinguishing features: placement on public equity markets with
tickers alongside equities and a pricing mechanism that, also much like
equities, utilizes arbitrage and a bid-ask mechanism that is used by
market makers to provide liquidity. ETF shares are created and destroyed
in blocks as needed in order ensure that the value of the wrapper is
inline with the value of the underlying securities or strategy that the
ETF conceptually represents.</p>
<p>By far the most popular type of ETF in terms of total asset value and
flows are ETFs that provide exposure to popular indices such as the S&P
500, the Russell 2000, and many others. These ETFs simply aim to match
the relative daily returns of their respective index and occupy the
bottom left spot on the aforementioned two dimensional spectrum. A
closely related (albeit far less popular) type of product that occupies
the bottom middle is the leveraged ETF. A leveraged ETF seeks to obtain
a daily exposure on an underlying index scaled by a constant $l$; which,
at this time, is somewhere between -3 and 3 for products currently on
the market. If $l$ is less than zero, the ETF provides short exposure to
the index and are often called "bear" ETFs; conversely, if $l$ is
greater than zero, the ETF provides long exposure to the index, commonly
referred to as "bull" ETFs. These securities are usually implemented
by means of a rolling futures strategy. By rebalancing everyday, the
ETFs eliminate the risk of ruin (in this case, losing more money than
the fund's total value) and obviate the need for margin payments.</p>
<h2 id="the-popular-argument-against-leverage">The Popular Argument Against Leverage</h2>
<p>At face value, a retail investor might assume that if the S&P 500
returned 10% in a given year, a 3x leveraged ETF would return 30%.
Fortunately or unfortunately depending on your perspective, this is not
the case as the ETF seeks to maintain a 3x multiple on the daily return
of the S&P 500 instead of the annualized return. As touched on in the
previous section, by rebalancing daily the ETF can easily handle the
inflows and outflows of the fund while also eliminating the need for
capital to be held in margin. This strategy also greatly reduces the
risk of ruin, as the S&P 500 would need to lose at least 1/3 of its
total value in a single day for the fund to be wiped out. Though
certainly imaginable, this event is unlikely as the biggest single day
loss in the history of the S&P 500 was Black Monday in 1987, in which
around 20% of value evaporated from the S&P 500 in a single day.</p>
<p>The common wisdom about leveraged ETFs is that they don't so much fill
a roll as an investment vehicle, but merely as a day-trading instrument
that allows a trader to easily obtain short-term tactical Beta exposure
to the market in an efficient and simple fashion. In numerous articles
scattered across the internet, the author always drives home the point
again and again that leveraged ETFs are ill-suited to long-term
buy-and-hold investing and should only be purchased by those who are
savvy enough to lay on short-term trades. Instead of cementing this
advice on a solid mathematical foundation, the author usually cherry
picks some example time window for the S&P 500 of around two months
where the market is mostly sideways and volatility is high. He then
usually remarks on the prescient insight that the levered fund made less
or even lost money whilst the regular fund came out ahead. The
conclusion to be drawn from this I suppose is that levered funds are
deceptive, don't actually multiply your returns by the amount claimed,
and are a bad investment. In the next section, we will discuss the
mathematical foundation of this claim and reason about its validity.</p>
<h2 id="a-little-bit-of-math">A Little Bit of Math</h2>
<p>We can see the immediate effect of volatility from a simple example.
What happens to a $100 portfolio that gains and then loses 10% of its
value versus a portfolio that gains and then losses 50%?</p>
<script type="math/tex; mode=display">\$100(1.10)(0.90) = \$99</script>
<script type="math/tex; mode=display">\$100(1.50)(0.50) = \$75</script>
<p>As we can see, the difference between the arithmetic mean (which for
both examples is 1) and the geometric mean (appropriate for compounded
returns) can be quite significant. In order to calculate the effect that
volatility has on a portfolio we can use the volatility drag formula:</p>
<script type="math/tex; mode=display">r_a = r_p - \frac{\sigma^2_p}{2}</script>
<p>Where $\sigma_p$ is the standard deviation of the portfolio, $r_p$ the
return of the portfolio and $r_a$ the actualized return of the portfolio
after deducting for volatility.</p>
<p>It's important to understand that the value of the volatility drag
equation is when we are changing the volatility of a known return
stream. If we are using past realized returns to forecast future
returns, the volatility equation isn't necessary or applicable, since
the past returns are already reflective of the post-drag return. The
volatility drag equation becomes useful when we are asking ourselves
about the returns of a levered portfolio in terms of the unlevered
returns and volatility. Alternatively, the equation also comes in handy
if we would like to answer questions such as: If we reduced the
volatility our of current portfolio by 25%, how would that affect the
mean return?</p>
<p>We need to rewrite the volatility drag equation in terms of leverage.
Assuming normality, we see that rescaling a normal distribution by a
constant affects the mean linearly and the variance non-linearly (thus
affecting the standard deviation linearly):</p>
<script type="math/tex; mode=display">Y = lX \sim N(l\mu, l^2\sigma^2)</script>
<p>In other words, a 2x levered portfolio with a mean return of 5% and a
volatility of 5% is the same as an unlevered fund with a mean return of
10% and a volatility of 10%. As such we can rewrite $r_p$ in terms of
$r_b$ (the base unlevered return of the portfolio) and the leverage
amount ($l$):</p>
<script type="math/tex; mode=display">r_p = r_bl</script>
<p>As well as $\sigma_p$ in terms of $\sigma_b$ and $l$:</p>
<script type="math/tex; mode=display">\sigma_p = \sigma_bl</script>
<p>Substituting into our drag equation:</p>
<script type="math/tex; mode=display">r_a = r_bl - \frac{\sigma_b^2}{2}l^2</script>
<p>In order to find the maximum leverage that will result in the greatest
mean return, we then take the derivative in terms of the leverage ratio
and set to zero and solve:</p>
<script type="math/tex; mode=display">\frac{\text{d}(r_a)}{\text{d}l} = r_b - \sigma^2_bl = 0</script>
<script type="math/tex; mode=display">l = \frac{r_b}{\sigma^2_b}</script>
<p>Also, our adjusted Sharpe ratio is:</p>
<script type="math/tex; mode=display">S_a = \frac{r_b}{\sigma_b} - \frac{\sigma_b}{2}l - \frac{r_f}{\sigma_bl}</script>
<p>Below is a graph of the adjusted Sharpe ratio, volatility and mean
return of a portfolio given $r_b= \sigma_b=10\%$. We can see that the
returns form a concave quadratic, the Sharpe ratio a negative linear
function, and the volatility a positive linear function. Note that in
this graph we are using log returns:</p>
<p><img src="/assets/rvs.png" alt="Returns vs Sharpe" /></p>
<p>An interesting and important property of volatility drag is that even if
two portfolios look the same given a specific leverage ratio, after the
drag calculation, they often end up quite different. Below is a table of
different example portfolios and leverage ratios that illustrate this
point:</p>
<table>
<thead>
<tr>
<th>Portfolio</th>
<th>Return</th>
<th>Volatility</th>
<th>Leverage</th>
<th>Pre-drag Return</th>
<th>Post-drag Return</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>1%</td>
<td>1%</td>
<td>5</td>
<td>5.025%</td>
<td>4.9%</td>
</tr>
<tr>
<td>B</td>
<td>5%</td>
<td>5%</td>
<td>1</td>
<td>5.125%</td>
<td>5%</td>
</tr>
<tr>
<td>C</td>
<td>10%</td>
<td>10%</td>
<td>0.5</td>
<td>5.25%</td>
<td>5.125%</td>
</tr>
</tbody>
</table>
<p>We found the pre-drag return values by first calculating the natural
unlevered return given no volatility and then scaling it by the leverage
ratio:</p>
<script type="math/tex; mode=display">r_{pre} = l \left( r_p + \frac{\sigma^2_p}{2} \right)</script>
<p>To calculate the post-drag return values in the table, we use the
regular volatility drag equation but use the pre-drag return and levered
volatility as inputs:</p>
<script type="math/tex; mode=display">r_{post} = r_{pre} - \frac{(\sigma_pl)^2}{2}</script>
<p>As we can see, if we are choosing between multiple portfolios with the
same Sharpe ratio, we should always prefer the portfolio that has the
highest natural return. For portfolio A, we pay a significant cost in
volatility. Portfolio B pays no cost, as the historical return has
already taken into account the realized volatility. Portfolio C actually
outperforms the natural portfolio B, since we get some volatility
"lift" from deleveraging. The takeaway is that the Sharpe ratio is not
a sufficient distillation of a portfolio or strategy due to the concave
quadratic nature of volatility drag. Instead, we need to also look at
the unlevered mean return of the strategy, as well as the Sharpe, in
order to determine the actual quality of the strategy. In some cases, it
would actually be preferable to choose a strategy with a lower Sharpe
ratio than one with a higher ratio if the mean unlevered return of the
former is sufficiently greater than the latter.</p>
<h2 id="historical-sp-500-returns-and-leverage">Historical S&P 500 Returns and Leverage</h2>
<p>While the leverage ratio formula is correct if we are sampling from a
normal distribution, market returns often exhibit excess kurtosis and
skew, resulting in the decreased accuracy of our formula. The greater
the skew or kurtosis, the less accurate the model becomes. In periods of
positive skew, the model underestimates the magnitude of mean return,
while in negative skew, the model overestimates the mean return. Below
is a density plot of returns between 2018-01-01 and 2019-09-01:</p>
<p><img src="/assets/kde.png" alt="Density of Returns 2018-01-01 to 2019-09-01" /></p>
<p>For this periods volatility and returns, the model equation suggests a
leverage ratio of approximately 3 in order to maximize returns. Looking
at the cumulative return streams of several different leverage ratios,
we see that despite the non-normality of the data, the prediction is
solid:</p>
<p><img src="/assets/lev1.png" alt="Leveraged S&P 500 Returns" /></p>
<p>Unfortunately, the use of variable leverage conditioned on volatility
and returns does not yet constitute a viable trading strategy: during
actual trading, future volatility and return information is not
available. Without a model to estimate the future returns and volatility
of the market, we will be unable to effectively calculate the optimal
leverage ratio for our portfolio. There are several remediations of
varying complexity and accuracy that we could use to work around this
problem. The most basic model we could employ would be one that uses a
trailing window of returns and volatility in order to predict the future
returns and volatility. Other options would be to use statistical time
series models such as ARIMA (Auto-Regressive Integrated Moving Average)
in order to forecast returns and GARCH (Generalized Auto-Regressive
Conditional Heteroskedasticity) to predict volatility. Other approaches
might include looking at the VIX (Volatility Index) or even constructing
RNNs (Recurrent Neural Networks) to help forecast an ideal leverage
ratio.</p>
<h2 id="conclusion">Conclusion</h2>
<p>While the Sharpe ratio of a levered index ETF will indeed get worse as
leverage is applied, the use of leverage and the associated volatility
drag does not constitute a separate and distinct issue aside from
volatility alone. For all investors, return and volatility are
intimately related through a portfolio's Sharpe ratio. Ultimately
though, investors cannot eat risk-adjusted returns and must instead try
to maximize the Sharpe ratio of their respective portfolios in order to
always assume the greatest amount of risk in line with their investment
objectives. For most investors, alpha generation through security
selection and trading is a lofty and unattainable strategy. Instead of
chasing elusive alpha, these investors adjust the lever of risk through
the management of asset class and factor exposures. When market
volatility is higher than personally tolerable, investors cycle into
lower volatility investments such as value stocks, bonds, and metals.
When volatility and the associated risk-premium is too low, investors
rotate into growth stocks, emerging markets, and real estate.</p>
<p>In this post we touched on a different and arguably simpler way to
manage volatility and risk-premiums: through the conditional application
of leverage. In part two of this three part series, we will look at ways
to forecast the ideal leverage ratio as a function of three parameters:
future returns, volatility, and personal risk limits.</p>
<p>Thanks for reading, I hope you enjoyed this piece! If you want to play
around with the Quantopian notebook, click
<a href="https://www.quantopian.com/posts/etfs-volatility-and-leverage-towards-a-new-leveraged-etf-part-1">here</a>!
Possible things to change would be the start and end dates, reference
leverage ratios, and the ticker to analyze.</p>["Sturm Mabie"]In part one of this three part series, we will explore the concept of levered ETFs, common misconceptions, the effect of volatility on the returns of a portfolio, and the compounded returns of the S&P 500 utilizing different leverage ratios. We will also touch on the basic mathematical underpinnings of volatility drag and ideal leverage ratio.