Tag: quant

Fat Tails, Sampling

While developing a model, historical data alone may not be sufficient to test its robustness. One way to generate test data is to re-sample historical data. This “re-arrangement” of past time-series can then be fed to the model to see how it behaves.

The problem with sampling historical market data is that it may not sufficiently account for fat-tails. Typically, a uniform sample is taken. The problem with this is it under-represents the tails. This leads to models that work on average but blow up on occasion. Something you’d like to avoid.

One way to overcome this problem is through stratified sampling. You chop the data into intervals and use their frequencies to probability weight the sample. This preserves the original distribution in the sample.

Notice the skew and the tails in the “STRAT” densities for both NIFTY and MIDCAP indices. This distribution is far more likely to result in a robust model compared to the one that just uses uniform sampling.

You can check out the R-code here.

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90 days of Minimum Volatility

We had discussed portfolios optimized for minimum volatility back in January (see: Low Volatility: Stock vs. Portfolio) and had setup Themes that track such strategies. Broadly, these fall into ETL (Expected Tail Loss) and Min-Var (Minimum Variance) optimized portfolios that either take in the entire universe of stocks or only those that have a high momentum score. So, we have Minimum Expected Tail Loss, Minimum Variance, Momentum (Min-ETL) and Momentum (Min-Variance).

We expect optimized portfolios of momentum stocks to perform better during market up-trends. During bears, we expect them to have lower drawdowns than the market. The Corona Virus Panic put these portfolios in through the wringer. Glad to report that they came out largely unscathed.

Minimum Volatility Portfolios vs. NIFTY 50

Our back-tests showed that optimized momentum portfolio would under-perform “raw” momentum during up-trends but should have lower drawdowns during down-trends.

Momentum: Optimized vs. raw

Optimized momentum portfolios saved the investor about 3-4% in drawdowns compared to the “raw” momentum portfolio. May not sound like much in this instance but think about the cumulative effect over multiple market corrections when you invest for the long-term.

Overall, optimized portfolios delivered what they promised.

WhatsApp us at +91-80-26650232 if you are interested in knowing more about these strategies.

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Low Volatility: Stock vs. Portfolio

Lower the Risk, Higher the Returns?

Typically, investors expect higher returns from high risk investments compared to low risk ones. However, realized returns are the opposite of what they expect.

High Beta vs. Low Vol

This anomaly, where lower risk systematically results in higher returns, has spawned a number of “betting against beta” strategies. A common approach is to rank a universe of stocks by volatility and create a portfolio off the top-N. However, this approach could lead to a highly volatile portfolio if relative correlations are not considered.

1+1 = 0

Here are two stocks with their volatility plotted against time:

correlated low-vol stocks

If you create a portfolio off these two stocks, what happens to portfolio volatility?

A portfolio made off low-vol stocks can be high vol

In the worst case scenario, where all the volatilities are correlated, portfolio volatility can end up being a sum of all component volatility.

If low-vol stocks can create a high-vol portfolio, can high-vol stocks create a low-vol portfolio? Yes! It all depends on how the volatilities are correlated.

inversely correlated high-vol stocks

In the best case scenario, portfolio volatility can be a very low constant value if the components are inversely correlated.

a low-vol portfolio created off volatile components

It doesn’t matter if individual volatilities are high or low. What matters is the correlation of volatilities.

Portfolio Optimization

A simple ranking of stocks will not help in creating a low-vol portfolio. What we need is a holistic approach that considers the correlation of volatilities and optimizes the entire portfolio.

One way to go about this is to use gradient descent. Start with a random portfolio and go in the direction that minimizes variance (min-var) or expected tail loss (min-ETL)

min-var and min-ETL backtest

With a monthly rebalance, the chances of the portfolio getting trapped in a local-minima are low. And the backtest looks promising.

Investing in Low-Volatility Portfolios

Equity investors can map our Minimum Variance and Minimum ETL Themes to their portfolios to gain exposure to these low-vol strategies.

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Factor Momentum Everywhere

Earlier this year, AQR had published a paper that showed momentum behavior also exists in equity factors (like value, quality, etc.) and not just in vanilla equities.

In this article, the authors document robust momentum behavior in a large collection of 65 widely-studied, characteristic-based equity factors around the globe. They show that, in general, individual factors can be reliably timed based on their own recent performance. A time series “factor momentum” portfolio that combines timing strategies of all factors earns an annual Sharpe ratio of 0.84. Factor momentum adds significant incremental performance to investment strategies that employ traditional momentum, industry momentum, value, and other commonly studied factors. Their results demonstrate that the momentum phenomenon is driven in large part by persistence in common return factors and not solely by persistence in idiosyncratic stock performance.

Factor Momentum Everywhere – Tarun Gupta, Bryan T. Kelly (AQR)

We put this idea to the test by constructing a long-only portfolio with five of the strongest factors – Momentum, Quality, Low-volatility, Value and Small-cap. The strategy was to go long whatever factor had the best returns over the last 12-months. We also looked at going long the best factor from the previous month. In both cases, the portfolio was re-balanced every month.

The representative indices and ETFs used for this back-test can be perused from the code: factor-momentum-india.ipynb and factor-momentum-US.ipynb

Results

The strategy using a 12-month formation period was a disappointment. There was no discernible improvement over a buy-and-hold of the large-cap index.

India (12-month Formation)
US (12-month Formation)

However, the one-month formation period widely out-performed the large-cap benchmark.

India (1-month Formation)
US (1-month Formation)

Needless to say, shorter the look-back period, larger the number of trades. So we added another back-test that averaged factor returns over 6-through-12 months to check if there was an acceptable middle-ground. Turns out, there is.

India (6..12-month Formation)
US (6..12-month Formation)

Forward Test

We setup US portfolios back in May this year when we first got to know about this paper. The 12-month and the 6…12-month formation period portfolios can be found here and here. They both seem to have out-performed SPY and MTUM so far.

We constructed the Factor Momentum 6-12 Theme for Indian equities that tracks the last strategy outlined above but given the lack of liquidity in factor ETFs, it trades the underlying stocks directly.

Investing in Factor Momentum

Indian investors can use brokers like TD Ameritrade or Interactive Brokers to invest in US stocks. The trades are posted on the slack channel mentioned on the pages linked above. You can execute the trades yourself by monitoring the messages on the channel.

If you are interested in executing this strategy on Indian equities, talk to us!

Related:
Factor Holding Periods for Excess Returns
Funding Your Dollar Dreams

Questions? Slack me!

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Does Momentum Trend or Mean-Revert?

Outline

We consider long-only momentum returns to be composed of market returns plus excess returns.

If excess returns over a specified period, n-days, (say, 5- or 10-days) either trends or mean-reverts, then that can be used to trade the momentum portfolio.

One way to check if a time-series is trending is to calculate the Hurst exponent (H) over a rolling window (say, 5-years) of n-day excess returns . If H < 0.5, then the time-series is mean-reverting; if H > 0.5, then it is trending, else it is random.

A simple strategy would be, for H < 0.5 (mean-reverting), if excess returns is greater than its median, then exit or if excess returns is less than its median, then enter.

The problem boils down to specifying the excess-return calculation periods (n-days) and the Hurst exponent rolling windows so that it makes sense (avoid data-mining.)

Setup

We use the Barclays Euro-zone, UK, Japan and US momentum index data-sets to run our experiment. Since they provide both an excess-return index and a total-return index, we can use the former to time entries and exits in the latter.

We ran for two n-day configurations: 5 and 10. We set the Hurst (and median) rolling window at 5-years.

We expected to find H to be either consistently above or below 0.5. My personal expectation was that H would be above 0.5 (trend.)

Results

Using the Hurst exponent did not improve momentum returns. In both 5-day and 10-day configurations, a strategy that went long if n-day returns were less than their median out-performed those that incorporated H.

The back-test using 5-day returns mostly worked on Euro-zone and US momentum indices. So we are skeptical that this approach can be generalized and it will likely fall prey to data-mining.

The back-test using 10-day returns saw buy-and-hold emerge a consistent winner.

The 5-day H toggled between trending and mean-reverting but spent most of its time trending. The 10-day H was consistently trending. For the specifications that we tested, momentum excess returns trends.

Code and Charts

The R code is here: 5-day and 10-day config. You can login to pluto and play around with the lookback and statWindow variables to see how H, median and back-test results change.

Questions? Ask them over at our slack workspace.

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