smabie

Forecasting Market Kurtosis with the Volatility Smile (Poorly)

2020-09-02T00:00:00+00:00

Note: I want to thank Algoseek for providing historical SPY option-chain data at a very reasonable price. Algoseek provides all the market data you could possibly need, including equity, future, option, forex, crypto data, and more! If you're interested in high quality market data at low quality prices, check them out here!

Before the Black Monday market crash of 1987, options of different strike prices for the same expiration had roughly the same implied volatility (IV), inline with the popular Black-Scholes model. After the crash however, sellers of options realized that perhaps they were taking on too much long-tail risk since the underlying equities exhibit excess kurtosis compared to a normal distribution – one of the assumptions of Black-Scholes. Sellers proceeded to start charging a premium for options that were either far in-the-money (ITM) or out-of-the-money (OTM). Let's look at different SPY calls on 2020-05-01 all expiring on 2020-06-01:

While Black-Scholes would suggest a uniform distribution (each option would have the same IV), it's clear from the graph that the far ITM and OTM options are overpriced with respect to Black-Scholes: this phenomenon is dubbed the "volatility smile." But if the volatility smile is due to excess kurtosis of the underlying, can we forecast future market kurtosis using the dispersion of option IVs? In this short post we'll look at SPY call options between 2012-01-03 and 2020-07-16 in order to see if a higher standard deviation of option IVs is a predictor of SPY excess kurtosis.

Methodology

The methodology is simple, we will look at a simple linear regression between the standard deviation of calls with the same maturity date and the excess kurtosis of SPY between the current date and the maturity date:

\[\text{Kurt}[r_{SPY}] \sim \sigma_{IV}\]

We will restrict the data set to only calls that expire before today's date so that the kurtosis of the market between the date of the option and maturity can be fully quantified. If the IVs of each option at each strike is the same in accordance to Black-Scholes, the standard deviation will be zero; if the market returns fit a log-normal distribution, the excess kurtosis will be zero as well. Let's look at the output of the linear regression:

Not a particularly great fit, but it does seem like there's a positive correlation between the standard deviation of IVs and the future market kurtosis. In addition, the R-squared is 4.12%, which means that the linear model predicts only 4.12% of the future kurtosis of the market. Even so, the model does explain some of the future kurtosis of the market and the coefficient is positive, meaning that indeed a greater dispersion of option IVs is positively correlated with future excess kurtosis.

Conclusion

While not a particularly powerful nor predictive model on its own, this result does lend some credence to the origin of and reason for the volatility smile of IVs.

Thanks for reading, hope you liked this short post! Like last time, I won't be sharing the code for this one as the option data isn't publicly available and without the data, the code doesn't have much value. If, however, you are interested in accessing the data, check out Algoseek for all of your option data needs (and more!).

Portfolio Construction with Risk Parity

2020-08-01T00:00:00+00:00

Just like how there's more than one way to skin a cat, there's more than one way to construct a portfolio. The first systematic method of construction was Modern Portfolio Theory, put forth by Harry Markowitz. MPT's approach is simple: choose weights for each asset that maximize the amount of return received for the amount of risk or volatility taken. However, trying to maximize the risk-adjusted return of a portfolio leads to very unstable allocations, as ex-ante estimates of return are notoriously difficult and previous return isn't a good predictor of future performance.

Another approach that came later is the minimum variance approach which concerns itself with solely minimizing volatility, irrespective of return. While this method might converge on sub-optimal portfolios according to MPT, they tend to be a lot more stable as volatility is easier to forecast than returns. Minimum variance optimization too suffers from some problems: without any constraints, the minimum variance portfolio often heavily weights only a few different assets, suggesting minimal allocations in the rest. For example, given two assets, a stock and a bond, the minimum variance method would almost always return a portfolio in which the bond was allocated the vast majority of the capital. Even if it produced reasonable results, the results are highly unstable: the change in portfolio weights year to year or even month to month can be massive.

For more background on both MPT and minimum variance optimization, check out one of my previous posts: The Value of Alternative Investments. In the mid-90s, a third approach was discovered by the hedge fund Bridgewater, founded by Ray Dalio, that tries remedy some of the problems with both of the aforementioned portfolio construction methods: risk parity.

Math of Risk Parity

Risk parity is a conceptually simple approach to portfolio construction in which we try and construct a portfolio in which each asset contributes a commensurate level of risk to the portfolio. In contrast to a traditional 60/40 portfolio in which the equity component is contributing almost all of the risk, risk parity tries to equalize the risk between each asset. In the two asset case of equities and bonds, the risk parity approach would spit out a set of weights so that each asset contributes equally to the volatility of the overall portfolio. To do this, it would most likely heavily weight the bond compared to the equity. After the weights are determined, the portfolio would be leveraged up in order to match a investor defined volatility budget. With this approach, we often end up with a better risk-adjusted return than a 60/40 portfolio and also get the benefits on diversification, unlike with a minimum variance portfolio.

We're going to use the example of a two asset portfolio with allocation weights $x_a$ and $x_b$ and standard deviations $\sigma_a$ and $\sigma_b$. To keep things explicit, we're going to eschew matrix notation. As we showed in the blog post linked above, the variance of the entire portfolio is:

\[\sigma_P = x_a^2\sigma_a^2 + x_b^2\sigma_b^2 + 2 x_a x_b \sigma_a \sigma_b \rho\]

Where $\rho$ is the standard correlation between $x_a$ and $x_b$. So now we want to divide up the risk so that they equal each other. But how do we do that?

To start off, we can use Euler's homogeneous function theorem to help. Euler's theorem states that if a function is homogeneous, i.e:

\[f(tx, ty) = t^n f(x, y)\]

Then:

\[n t^{n-1} f(x, y)= x \frac{\partial f}{\partial (x\,t)} + y\frac{\partial f}{\partial (y\,t)}\]

In the case of $t=1$:

\[n f(x, y) =x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y}\]

Well, it turns out that our equation for $\sigma_P(x_a, x_b)$ is homogeneous. By using Euler's theorem, we can figure out how to split up the risk for each asset. First let's calculate the partial derivative with respect to each variable:

\[\frac{\partial \sigma_P}{\partial x_a} = 2(\rho\sigma_a\sigma_b x_b + \sigma_a^2x_a)\] \[\frac{\partial \sigma_P}{\partial x_b}= 2(\rho \sigma_{a} \sigma_{b} x_{a} +\sigma_{b}^{2} x_{b})\] \[n \sigma_P = 2x_a(\rho\sigma_a\sigma_b x_b + \sigma_a^2x_a) + 2x_b(\rho \sigma_{a} \sigma_{b} x_{a} +\sigma_{b}^{2} x_{b})\]

Simplifying and solving for $n$ we get:

\[n = \frac{2x_a(\rho\sigma_a\sigma_b x_b + \sigma_a^2x_a) + 2x_b(\rho \sigma_{a} \sigma_{b} x_{a} +\sigma_{b}^{2} x_{b})}{\sigma_P} = 2\]

We want each component to be equal:

\[x_a \frac{\partial \sigma_P}{\partial x_a}=x_b \frac{\partial \sigma_P}{\partial x_b}\] \[\rho\sigma_{a}\sigma_{b}x_{a}x_{b}+\sigma_{a}^{2}x_{a}^{2} =\rho \sigma_{a} \sigma_{b} x_{a} x_{b} + \sigma_{b}^{2} x_{b}^{2}\]

Assuming:

\[x_a + x_b = 1\]

We substitute $x_b = x_a - 1$ and then solve for $x_a$. Discarding the negative solution, we get:

\[x_a = \frac{\sigma_b}{\sigma_a + \sigma_b}\]

And solving for $x_b$:

\[x_b = \frac{\sigma_a}{\sigma_a+\sigma_b}\]

Note that the solution does not depend on the correlation $\rho$, which some might find counter-intuitive. While we showed that there exists a closed form solution for the case of two assets, a numerical solution is required when $n>2$.

Risk Parity in Practice

We're going to construct a risk-parity portfolio from two assets: SPY (S&P 500), and LQD, which is an investment grade bonds ETF. Our weights will be calculated as described above, using a 252-day rolling window for the volatility calculations. Below is a graph of the weights over time:

As expected, bonds make up the majority of the portfolio in order to match the risk of the equity component. Let's look at the cumulative returns of our risk parity portfolio compared to a traditional 60/40 portfolio:

Due to the majority of our risk parity portfolio being bonds, the portfolio underperforms a classic 60/40 portfolio that takes on significantly more equity risk. Having a relatively low natural return is expected of a risk parity portfolio, which is why leveraged is applied after portfolio construction. But what leverage ratio to use? In order to have an apples-to-apples comparison between a risk parity portfolio and a 60/40 one, we will leverage up in order to try and match the volatility of a 60/40 portfolio:

\[l = \frac{\sigma_{60/40}}{\sigma_{RP}}\]

For each $\sigma$, we use a 252-day rolling window. We calculate the leverage ratio needed on each day and leverage up accordingly. Ideally, this will ensure that the amount of risk taken by the risk parity portfolio is always inline with the amount of risk a 60/40 portfolio is taking. Let's look at a graph of our leveraged risk parity portfolio, a classic 60/40 portfolio, and the leverage ratio:

Not bad! The amount of leverage taken is reasonable and the performance of the risk parity portfolio looks pretty good. Let's look at a table of metrics:

Metric	Risk Parity	60/40
Beta	0.51	0.68
Ann. Ret	9.9%	7.8%
Ann. Vol	12.8%	12.8%
Ann. Sharpe	0.77	0.6

Our simple dynamic leverage strategy works very well, with the volatility of the two portfolios being identical. But even with the same volatility, the risk parity portfolio significantly outperforms while also having less beta exposure and a better Sharpe ratio as well. Another plus is that the amount of leverage taken is very reasonable, and well within the ability of even retail investors to obtain.

Conclusion

Risk parity is an exciting and effective technique that is a viable alternative to minimum variance portfolios as well as fixed weight portfolios such as 60/40. While risk parity portfolios might not reduce volatility as much as minimum variance portfolios, they tend to be more stable over time, incur less turnover, and provide greater diversification. Compared to a 60/40 portfolio, they are usually superior unless the borrowing costs are too high. This interest rate exposure represents a distinct risk for risk parity strategies; a risk investors should be cognizant of.

Thanks for reading and hope you liked this post! You can check out the Quantopian notebook here. Feel free to change the time periods and assets used to construct the portfolio.

Risk Imparity: A Simple Strategy for Alpha Generation

2020-07-29T00:00:00+00:00

Risk parity is a popular method of investing that aims to outperform the traditional 60/40 equity/bond portfolio by equalizing the risk of the bond portion with that of the equity part of the portfolio and then leveraging up the portfolio to hit the desired risk target. Using the covariance, and volatility of the components, we can generate a portfolio and leverage ratio for a given level of desired risk. This works for two reasons: bonds have a low or inverse correlation to equities and they generally have superior risk-adjusted returns as well, and combining low correlation assets allows us to lower the volatility of the portfolio. Bridgewater and Ray Dalio pioneered this approach in the 90s with the introduction of his now famous All Weather fund.

The appeal of this approach is obvious compared to the 60/40 portfolio: by setting a volatility target, we stabilize our risk throughout different market regimes. This is an attractive property for investors who might otherwise choose a 60/40 or even all equity portfolio; they can sleep well knowing an appropriate level of risk will be taken, regardless of the market. Well, in this post, we're going to take a less sensible approach and instead of equalizing the risk in our portfolio, we're going to leverage up the equity component (representing almost all of the risk of the portfolio) and use the bonds to generate yield.

Risk Imparity

The intuition behind this approach is pretty straightforward: we want to take on leverage with equities and cancel out the leverage by choosing an appropriate weight. For example, if we purchased a 2x bull leveraged S&P 500 ETF such as SSO, we would choose an appropriate weight to cancel out the leverage, 0.5 in this case. If we kept the rest of the money in cash, the return of the portfolio would only be slightly worse than that of the S&P 500, due to the 0.90% expense ratio we pay. If we didn't have to pay an expense ratio, the return streams would look identical besides for perhaps some marginal tracking error. But if we instead invested the rest of the money in a low risk investment, such as investment grade bonds, we would earn yield on the money and diversify our portfolio. As long as the bonds contribute more profit than the cost of the expense ratio, the strategy should beat the vanilla index. Below is the strategy in full:

Our portfolio will consist of two assets, UPRO, a 3x leveraged S&P 500 ETF, and LQD, an investment grade bond ETF.
Each day, we will rebalance so that 1/3 of our portfolio is allocated towards UPRO and 2/3s towards LQD.

Let's look at the returns against the S&P 500 between 2009-06-26 and 2020-07-28 (we choose the beginning date because UPRO was at the beginning of 2009 and the data only became available for it at the start date):

Not bad, let's look at a table of metrics:

Metric	Portfolio	S&P 500
Ann. Rets	18.1%	14.1%
Ann. Vol	18.4%	17.4%
Ann. Sharpe	0.98	0.81
Beta	1.02	1.0

The results are pretty impressive for such a simple strategy: better Sharpe ratio, comparable beta, and a 4% better annualized return, with only slightly higher volatility. One problem however, is apparent: if the traditional negative correlation between bonds and equities starts to break like it did during the start of Coronavirus, the portfolio takes a major hit, performing worse than the S&P 500. In order to confirm this, let's look at LQD alone:

And indeed, even investment grade bonds collapsed during the start of Coronavirus. But what about the Great Recession, how did the strategy perform then? We don't have real data from that time-period, but we can synthetically generate a comparable return stream by just multiplying SPY by 3. This of course won't take into account the expense ratio of UPRO and the tracking error, but the results should still be pretty similar. Let's take a look:

Unlike during Coronavirus, The strategy held up pretty well during the crash of '07-08, returning approximately the same as a pure S&P 500 portfolio. This might suggest a regime change in bond/equity correlation or a temporary blip that isn't indicative of future correlation.

Conclusion

Though there are risks associated with this strategy not present in a pure S&P 500 portfolio, in most market environments, this approach delivers a better risk-adjusted return without taking on any additional market risk. Another benefit of the strategy is its simplicity: by using only two different ETFs, a retail investor can manage the strategy with minimal effort. investor could also adjust the rebalancing period to weekly or monthly in order to reduce the complexity and transaction costs. The strategy can also be modified by adjusting the type of bond for the portfolio: more risk-averse investors could stick with short to medium term Treasury notes while higher octane ones could substitute in junk bonds or other alternative investments.

Thanks for reading, hope you enjoyed this post. Click here for the Quantopian notebook used. Seeing that this strategy is pretty trivial, if anyone could help me in locating prior literature about it, I would be most appreciative. Email me at sturm@cryptm.org if so.

Beta Stability: A New Persistent Factor?

2020-07-23T00:00:00+00:00

The beta exposure of a stock is one of the first and most important statistics any investor looks at. Though perhaps simplistic, it distills a myriad of various properties of a stock into a single, easily digestible number. Using beta, it becomes easy to roughly forecast how an investor's portfolio would perform under different market conditions. Maybe out portfolio has a beta of 1.3 and we expect the market to return 11% next year: great, our portfolio should return around 14.3% (1.3*11%). While there's a lot of problems with this simplistic model, in this post, we're going to focus on just one: the beta of a stock changes significantly over time. Consider this graph of Apple's rolling, 252-day beta:

As we can see, the dispersion is quite high, and the rolling beta rarely stays the same for any reasonable period of time. There could be any number of reasons for why the rolling beta changes, such as changes in the drivers of market movements, the evolution of Apple's business model, geographic changes in sales, and etc.

Clearly, some companies are going to have higher beta stability than others. One might imagine that a mature utility company with stable recurring revenue would have greater beta stability than a recently IPO'd tech company who's revenue model and business units are still in flux. So that brings up the question, when constructing a portfolio, which kind of companies would we prefer: the relatively unstable beta company, or the more stable one? In this post, we're going to form and test the hypothesis that, in the long run, a portfolio that consists of stocks with high beta stability generates excess returns.

The Beta Stable Factor

Our goal is to construct a beta neutral long/short portfolio that tries to capture the abnormal return that we hypothesized might be associated with a beta stable portfolio. To do this, we're going use to the Q500US universe provided by Quantopian. This universe consists of the top 500 most liquid (and therefore, probably largest market cap) equities on any given day. We construct our alpha factor as follows:

\[f = \text{Z}[\text{rank}(-|\beta_{21}-\beta_{63}|)]\]

Where $\beta_{21}$ is the 21-day (one month) rolling beta, and $\beta_{63}$ the 63-day (three month) rolling beta. We then take the absolute value, negate, rank, and Z-score. This factor gives us a portfolio that goes long stocks that have lower changes in beta and shorts those that have greater changes in beta.

Because we suspect that higher beta stocks tend to have more instability, we will zero out the beta exposure by dynamically adjusting our net long exposure. We use the same method that we did in The Low Beta Anomaly post, so go check it out if you want the details. Backtesting over a 10 year period, 2010-01-01 to 2020-01-01, let's first look at our net long exposure:

The reason for our net long exposure being positive is that there is some correlation between beta instability and high beta stocks. Beta generally has a right hand skew, so our unadjusted portfolio would have a higher average beta for the short-side than the long-side. Overall, this would result in a portfolio that had significant negative beta exposure, an undesirable property to have in a world where the market usually goes up. By increasing the long-side of our portfolio, we can offset the high beta short-side and maintain beta neutrality. Even so, the net long exposure is relatively small compared to what we observed for our low beta portfolio (again, to compare, check out The Low Beta Anomaly post): around 10% vs 45%, respectively. This is good, because it means we are capturing an independent effect with our beta stability hypothesis, not just another reformulation of the low beta/volatility anomaly.

Now for the good part, the cumulative return over the same time period:

Wow, that looks pretty good for a beta neutral strategy! Let's decompose the returns into quantile:

This graph shows the cumulative return by quantile if each quantile was held long (remember that the lower quantiles are held short in our portfolio). The profit from our strategy comes from the difference between the upper quantiles and the lower ones. Let's look at a table of metrics to better understand the strategy:

Metric	Value
Beta	-.018
Ann. Ret	3.56%
Ann. Vol	2.7%
Ann. Sharpe	1.28

Not bad at all! Our beta minimization worked well, the return is pretty significant, and our Sharpe ratio is very respectable, at 1.28.

Even so, in an ideal beta neutral strategy, we would want to be making money off the short-side of the portfolio, not merely hedging our market risk with it: i.e, we would want to see the lower quantiles lose money, not merely make less than the greater quantiles. In the next section, we'll consider a long-only variation of the same beta stable strategy.

Long-only Beta Stable

For the long-only variation, we'll use the same factor as before, but instead of investing in the entire Q500US universe, only invest in the top 100 stocks. This will give us a portfolio with higher allocations to stocks with greater beta stability and one with 100% net long exposure. Using the same time period, consider the quantile graph below:

As expected, we only see a small difference of the returns by quantile. This is because we are only investing in the 100 most stable equities, significantly narrowing the gap between the lowest and highest quantile. Still, there is some effect (the first quantile realizes the lowest returns), which bodes well for the robustness of our factor. Now, for the moment of truth, our factor portfolio compared to the S&P 500:

Wow! This is an amazing result: over a 10 year period, our factor portfolio returns a little under a 50% superior return compared to the S&P 500. Here's a table of metrics for comparison:

Metric	Portfolio	S&P 500
Beta	0.99	1.0
Ann. Ret	14.7%	13.37%
Ann. Vol	15.5%	14.6%
Ann. Sharpe	0.94	0.91

With over a 1% of excess return, a higher Sharpe ratio, and an equivalent beta, the long-only beta stable strategy appears to be a huge success!

Conclusion

Based on the results, perhaps we should consider beta stability a new persistent factor, alongside the classics like low-volatility, value, and size? This is a bold claim of course, and much more research needs to be done; but the preliminary results are very positive, especially for the long-only version. It would be very easy to overlay onto an existing beta exposed portfolio, generating a moderate amount of excess return without taking on any more risk. The beta neutral version also has promise, though ideally it would be combined with other factors (such as maybe value or low-beta) in order to boost the Sharpe ratio and unleveraged return.

Thanks for reading, I hope you enjoyed this post! for the source code, check out the Quantopian notebook here. Feel free to play around with the start and end dates, universe, and anything else. If you have any comments or feedback, contact me at sturm@cryptm.org.

The Value of Alternative Investments

2020-07-09T00:00:00+00:00

A retail investor would be forgiven for assuming that there are really only two assets classes one should consider for personal investments: equities (stocks) and fixed income (bonds). Wherever you look, these two asset classes dominate the financial landscape. All robo-advisors and most personal advisors implement a mix of bonds and equities for the portfolios of their clients. The idea behind this mix is simple: bonds are low risk and will earn you a marginal return, while equities are riskier and will be the real engine of returns in good times. In bad times, bonds will rise, mitigating some of the losses of your equity portfolio: it's been long known and accepted that stocks and bonds have a negative correlation. Tying this strategy together is periodic rebalancing, taking money off the table during bull markets (and moving them to the safer bonds), and deploying money to the equity portion during bear markets. This in effect overlays a mean-reversion strategy onto the portfolio, boosting the returns of an otherwise static portfolio. The archetypal allocation is 60% equities, and 40% bonds, though each investor's allocation is going to differ based on his risk tolerance, age, and personal goals.

Though there's nothing wrong with this two asset class mix, the historically negative correlation between stocks and bonds means that in most market environments, bonds are going to drag down the portfolio. Ideally, we would want to mix in uncorrelated assets and investments instead of negatively correlated ones, in order to reduce the volatility of our portfolio without dragging down our returns. Alternative investments such as hedge funds, venture capital, metals, or real estate all serve this purpose. These alternative investments are not only popular for their occasionally spectacular returns, but also for their low correlation to the broader equity market. For this post, we'll use the oldest and possibly most maligned investment, gold, and explore ways to mix it into a pure equity, S&P 500 portfolio.

Gold

Gold and other metals are an interesting and, unfortunately, underappreciated asset class. The much derided "gold-bug" is made out to be paranoid, crazy, and irrational. Despite being so rare and valuable, the percentage of the world's gold supply that is used for industrial uses is comparatively small, creating a phenomenon that untethers the price of gold from other asset classes. Let's look at the returns of gold versus the S&P 500 between 2010-01-01 and 2020-07-08:

It's evident that the returns of gold are unspectacular, to say the least. It doesn't seem to consistently lose or gain value as much as it meanders around, seemingly uncorrelated to the S&P 500. And indeed, it is almost entirely uncorrelated, with a correlation to the S&P 500 of only 1% over this time period. Is it even worth including in our portfolio? And if it is, how would we determine the allocation size?

Monte-Carlo

To clarify, we're trying to determine the weights for the S&P 500 and gold parts of our portfolio, taking on no leverage:

\[|x_a| + |x_b| = 1\]

And maximizing the risk-adjusted return. For every unit of risk we take, we would like to maximize the amount of return we receive. We'll first start off by doing the most straightforward thing, and randomly choosing different weights for our two asset portfolio and then graphing them:

This hyperbola formed is called the Markowitz Bullet, after Harry Markowitz, winner of the Nobel Prize for Economics for his work on Modern Portfolio Theory. The portfolio with the best return vs volatility profile is called the frontier or efficient portfolio. There exists only one such portfolio and it is the portfolio every rational investor should prefer, ignoring borrowings costs and volatility drag (for more information about volatility drag, check out my other post here, or the beginning of my three part series here).

While it's relatively easy to find what would have been the frontier portfolio looking backwards, it is much more difficult to estimate the frontier portfolio over some future time period. We can forecast and then reduce future volatility with modest accuracy, but forecasting expected return is a notoriously difficult problem. So difficult, in fact, that few quantitative investors even try, instead resigning themselves to solely minimize volatility. Likewise, we'll resign ourselves to the same fate.

In the general case of a many asset portfolio, no closed form solution can be found for the minimization of volatility, we must instead use an optimizer or do a Monte-Carlo simulation; but in the two asset case, we can find a symbolic solution. So that's what we'll do next!

Symbolic Solution

Recall that our portfolio return is going to be a function of our two assets and their weights:

\[P = x_a X_a+ x_b X_b\]

Likewise, our expected portfolio mean would look like:

\[\begin{align} E[P] &= \mu_P\\ &= x_a E[X_a] + x_b E[X_b]\\ &= x_a \mu_a + x_b \mu_b\\ \end{align}\]

And the equation for variance:

\[\text{Var}[X] = \sigma^2 = E[(X-\mu)^2]\]

Covariance is similar, except instead of squaring, we multiply each variable after demeaning:

\[\text{Cov}[X,Y] = E[(X-\mu_X)(Y-\mu_Y)]\]

Now we just need to derive the portfolio variance:

\[\text{Var}[P] = \text{Var}[x_aX_a + x_b X_b]\]

First we substitute for variance and rearrange:

\[\begin{align} \text{Var}[x_aX_a + x_b X_b] =& E[(x_aX_a + x_b X_b - E[x_a X_a + x_b X_b])^2]\\ =& E[(x_a X_a - E[x_a X_a] + x_b X_b - E[x_b X_b])^2] \end{align}\]

Now we can pull the constants out of the expectations, substitute, and expand:

\[\begin{align} \text{Var}[x_a X_a + x_b X_b] =& E[(x_a (X_a - \mu_a) + x_b (X_b - \mu_b))^2] \\ =& E[x_a^2(X_a - \mu_a)^2 + x_b^2(X_b - \mu_b)^2 + 2 x_a x_b (X_a - \mu_a)(X_b - \mu_b)] \end{align}\]

Finally, we break up the expectations, and replace:

\[\begin{align} \text{Var}[x_a X_a + x_b X_b] =& x^2_a E[(X_a - \mu_a)^2] + x^2_b E[X_b - \mu_b] + 2x_a x_b E[(X_a-\mu_a)(X_b-\mu_b)]\\ =& x^2_a \sigma_a^2 + x^2_b \sigma_b^2 + 2 x_a x_b \text{Cov}[X_a,X_b]\\ =& x^2_a \sigma_a^2 + x^2_b \sigma_b^2 + 2 x_a x_b \rho_{ab} \sigma_a \sigma_b \end{align}\]

Where $\rho_{ab}$ is the correlation between $X_a$ and $X_b$:

\[\rho_{ab} = \frac{\text{Cov}[X_a,X_b]}{\sigma_a\sigma_b}\]

Now that we have an equation for variance in terms of the weights, mean, volatility, and correlation between the two assets, we can formulate our objective:

\[\min\left( \text{Var}[x_aX_a + x_b X_b]\right)\]

Where:

\[x_a + x_b = 1\]

We perform the substitution $x_b = 1 - x_a$ and then take the derivative with respect to $x_a$:

\[\frac{\mathrm{d} \sigma_P}{\mathrm{d}x_a}=2 \, \rho_{\mathit{ab}} \sigma_{a} \sigma_{b} - 2 \, \sigma_{b}^{2} - 2 \, {\left(2 \, \rho_{\mathit{ab}} \sigma_{a} \sigma_{b} - \sigma_{a}^{2} - \sigma_{b}^{2}\right)} x_{a}\]

Now we set to 0 and solve for $x_a$:

\[x_{a} = \frac{\rho_{\mathit{ab}} \sigma_{a} \sigma_{b} - \sigma_{b}^{2}}{2 \, \rho_{\mathit{ab}} \sigma_{a} \sigma_{b} - \sigma_{a}^{2} - \sigma_{b}^{2}}\]

And $x_b$:

\[x_b = 1 - x_a\]

Note that sometimes this equation will give us values for $x_a$ and $x_b$ that imply leverage, i.e:

\[|x_a| + |x_b| > 1\]

But we can always rescale them by dividing each one by $|x_a| + |x_b|$. In the next section, we'll put this formula to work.

Portfolio Weights

We're going to use a one year or 252 day rolling window to generate the weights. Everyday, we slide our window over one and generate the new weights for that day. Below is the generated graph:

Well that's interesting, the weights change considerably over time. In a real portfolio, not only would we be using an optimizer and a multitude of risk factors to determine the weights, but we would also have constraints on position size, turnover, and the magnitude of deviation from the "ideal" portfolio, and etc. But even in this simple and contrived example, we see that our formula is doing its job: as the volatility of the S&P 500 increases, like at the end of 2018 and during Coronavirus, we start cutting back our equity exposure.

Let's now look at the returns of our portfolio compared to both gold and the S&P 500:

A lot of gains from the market are given up, as we might have expected, but the return stream becomes a lot less volatile. Just looking at a graph of cumulative returns isn't super informative, so here's a table of various metrics:

Metric	Gold	S&P 500	Portfolio
Ann. Vol	15.8%	17.4%	10.8%
Ann. Ret	4.3%	12.4%	8.4%
Beta	0.01	1	0.41
Ann. Sharpe	0.27	0.71	0.77

Wow! Our volatility is lower than either gold or the S&P 500 alone, our beta has been cut more than in half compared to the S&P 500, and our Sharpe ratio is even higher than both! Even with mixing in an asset that had poor returns and high volatility, we've managed to construct a portfolio that, on a risk-adjusted basis, is superior to the S&P 500. And, if we so desired, could be levered up in order to beat the return of the S&P 500, while maintaining comparatively low volatility.

Conclusion

Even with a bad Sharpe ratio and less than stellar returns, gold enhanced our pure equity portfolio. The same also can be said about other, more popular, alternative investments. Hedge funds specifically are often derided for their low returns and frequent meltdowns (such as Long-Term Capital Management); but like with gold, looks can be deceiving. Returns and volatility aren't the whole story. When designing a portfolio, each individual investment is irrelevant, and instead the return stream of the portfolio as a whole is what matters. Considered in isolation, a lot of alternative investments look sub-optimal and irrational. It is only when you zoom out and think about the needs of the investor (their existing allocations, investment goals, monetary needs, etc) does everything come into perspective. When it comes to investing, the whole is certainly greater than the sum of its parts!

I hope you liked the post and if you did, let me know! You can also check out the notebook, developed on Quantopian, here. Possible things you could mess around with are the start and end dates and the two assets to construct a portfolio from.

The Low Beta Anomaly

2020-06-28T00:00:00+00:00

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:

\[r_i - r_f = r_f + \beta_i (r_m - r_f)\]

$\beta_i$ 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.

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.

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 Security Market Line (SML):

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:

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.

A Simple Approach

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!

For this strategy, we'll be using the Quantopian Q1500US universe, which consists of the 1500 most liquid US equities on any given day. We then construct our low beta factor as such:

\[f = \text{Z}[\text{rank}(-\beta_{252})]\]

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:

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:

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:

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 side-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.

Fixing the Problem

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:

\[\mathbf{X} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \\ \end{bmatrix}\]

And a vector of beta exposures:

\[\mathbf{B} = \begin{bmatrix} \beta_1 & \beta_2 & \dots & \beta_n \end{bmatrix}\]

We can easily find the beta of portfolio:

\[\beta_p = \mathbf{BX}\]

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):

\[\mathbf{X_\alpha} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \end{bmatrix}\] \[\mathbf{X_\beta} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_k \end{bmatrix}\]

Likewise, we split up the betas as well:

\[\mathbf{B_\alpha} = \begin{bmatrix} \beta_1 & \beta_2 & \dots & \beta_j \end{bmatrix}\] \[\mathbf{B_\beta} = \begin{bmatrix} \beta_1 & \beta_2 & \dots & \beta_k \end{bmatrix}\]

Where:

\[n = j + k\]

Now, our equation becomes:

\[\beta_p = \mathbf{B_\alpha X_\alpha}+\mathbf{B_\beta X_\beta}\]

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 $\beta_p=0$:

\[\mathbf{B_\alpha X_\alpha}+ \lambda \mathbf{B_\beta X_\beta}=0\]

We now solve for $\lambda$:

\[\lambda = - \frac{\mathbf{B_\alpha X_\alpha}}{\mathbf{B_\beta X_\beta}}\]

Assuming a leverage ratio of one, i.e:

\[\sum_{i=1}^j |x_{\alpha,i}| + \sum_{i=1}^k |x_{\beta,i}| = 1\]

We can calculate the net long exposure, using $\lambda$ as a parameter:

\[l = \frac{1-\lambda}{1+\lambda}\]

Below is the graph of net long exposure over time:

And the cumulative return:

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:

Metric	Value
Beta	1.8%
Ann. Ret	4.0%
Ann. Vol	4.6%
Ret/Vol	0.84

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.

Explanation

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.

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.

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.

Conclusion

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.

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 here.

Analysis of CS:GO Win-rates

2020-06-12T00:00:00+00:00

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.

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.

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:

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.

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.

Even Match-ups

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 here 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:

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:

CT win rate	p-value	players
0.48729	2.67482e-55	5
0.469768	1.17974e-151	4
0.455098	3.84482e-237	3
0.439982	1.52963e-319	2
0.430299	2.35914e-282	1

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.

All Match-ups

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:

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:

1 CT	2 CT	3 CT	4 CT	5 CT	# T alive
0.00218907	0.0227307	0.106122	0.274956	0.487288	5
0.00689667	0.0682988	0.235994	0.469768	0.683482	4
0.0287596	0.187184	0.455091	0.70026	0.858241	3
0.123364	0.439972	0.732706	0.893306	0.961373	2
0.430299	0.7915	0.941975	0.985024	0.9967	1
0.906434	0.989945	0.998562	0.999731	0.999873	0

Conclusions

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:

T-side wants to trade down as much as possible, CT never wants to trade.
CT-side needs to be play very conservative, maximizing the number of players on their team that are alive.
T-side wants to play in a very aggressive style in order to take map control and trade.

Anyways, this has been a fun little post to write, I hope you enjoyed it! Click here to view the GitHub project.

ETFs, Volatility and Leverage: Towards a New Leveraged ETF Part 3

2020-06-09T00:00:00+00:00

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.

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.

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:

\[l = \frac{\text{E}(r_m^{\text{fill}})}{21\text{E}(\sigma_d)^2}\]

Let's look at graph of our leverage ratio over time to get a sense of what we're dealing with:

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:

First, if the leverage ratio is less than 1 at a time-point, we set it to 0.
In order to constrain the values, we then add $e$ and logscale it. This should give us values approximately between 1 and 3.
To smooth the leverage, we apply an exponential rolling window, with an $\alpha$ of 0.05.
If any values are above 3, we set them to 3.

Below is the full transformation:

\[f(x) =\begin{cases} x &\text{if }x > 1\\ 0 &\text{else}\\ \end{cases}\] \[\sum_{t=1}^n \min(\text{EWM}_{\alpha=0.05}[\log(f(x_t)+e)],\,3)\]

Consider the graph of adjusted leverage:

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:

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.

Conclusion

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.

You can check out the notebook here. Feel free to play around with any and all parameters.

ETFs, Volatility and Leverage: Towards a New Leveraged ETF Part 2

2020-05-28T00:00:00+00:00

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:

\[l = \frac{r_b}{\sigma^2_b}\]

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.

Autoregressive Model

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:

\[\text{AR}(p): x_t = \alpha + B_1x_{t-1}+B_2x_{t-2}+\cdots+B_px_{t-p}+\epsilon_t\]

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 $B_1=1$:

\[\text{AR}(1): x_t = x_{t-1} + \epsilon_t\]

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.

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.

Moving Average Model

Another type of simple model often used to forecast time series is an autoregressive or AR model:

\[\text{MA}(q):x_t = \mu + \epsilon_t + \theta_1\epsilon_{t-1}+\cdots+\theta_q \epsilon_{t-q}\]

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.

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.

ARIMA Model

A common approach is to combine both of these models, forming an ARMA model:

\[\text{ARMA}(p,\,q): \text{AR}(p) + \text{MA}(q)\]

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:

\[\text{ARIMA}(p,\,d\,,q)\]

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:

\[\text{ARIMA}(p,\,1,\,q) p_t = \text{ARMA}(p,\,q) r_t\]

Also note that:

\[\text{ARIMA}(p,\,0,\,q) = \text{ARMA}(p,\,q)\]

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:

Set $d=0$
Apply the ADF test on the series
If $p\leq0.05$, return $d$
Otherwise, take the difference between each successive value and increment $d$.
Go to step 2

Forecasting Returns

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:

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.

Moving on to the PACF, we observe much the same results as before:

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.

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:

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.

VIX Index

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.

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.

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:

Perhaps surprisingly, the VIX has practically the same predictive power as simply using last month's volatility.

GARCH Model

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.

Forecasting Volatility

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:

Just from the graph, the accuracy of our model looks very 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.

Conclusion

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 here to check out the notebook. If you install anaconda, no additional Python libraries should be necessary to execute the notebook.

One Weird Trick to Profit off of a Global Market Meltdown

2020-05-18T00:00:00+00:00

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.

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:

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!

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.

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:

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.

Finally, for contrast, the same strategies during the raging bull market between 2016-01-01 and 2018-01-01:

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%.

Conclusion

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!

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).

I hope you enjoyed this post! If you want to check out the Quantopian notebook used to generate the above graphs, click here. Feel free to play around with the dates, though unfortunately, data isn't available before 2002.