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Improving Efficient Frontiers with Volatility
- 1. Project 2: Volatility as an Asset Class —improving the mean-variance efficient frontier using volatility as an asset class MS&E 445 Projects in Wealth Management Professor Peter Woehrmann Ian Schultz, Linda He Yi, Hai Wei, J.R. Riggs, Andrew Tsai, Vicky Wang, Henry Chen, Erica Jiang
- 2. Project 2: Volatility as an Asset Class Introduction Part I: Portfolio Optimization & Role of Volatility Construct the efficient frontier for universe of 30 (DJIA), 100 stocks (S&P100), 500 stocks (S&P500) & 2600+ stocks (NASDAQ) Include volatility using ETFs tracking the VIX (VXX, VXZ) Part II: Volatility Forecasting & Estimation Volatility forecasting using AR(1) Model Minimize estimation errors by using the Shrinkage Approach to estimate the covariance matrix Part III: Trading & Implementation Comparing selective long/short volatility trading strategies
- 3. Part I: Portfolio Optimization & Role of Volatility
- 4. Part I: Portfolio Optimization & Role of Volatility Efficient Frontier with Market Indices S&P 500 NASDAQ S&P 100 DJIA Efficient Frontier - Markowitz Formula to find two efficient portfolio; i.e. minimum variance for a given return - Two-Fund Theorem to construct the efficient frontier
- 5. Part I: Portfolio Optimization & Role of Volatility Volatility: Negative Correlation S&P 500 Volatility S&P 500 (VIX)
- 6. Part I: Portfolio Optimization & Role of Volatility Price Today Maturity Expected Future Spot Price Forward Price in Normal Backwardation Forward Price in Contango Volatility: Contango Effect of VIX Futures - Each subsequent expiration month of VIX futures prices are traded higher than the closer month's VIX futures prices and the spot VIX overall How to take advantage of the decay effect that is very consistent and significant over time?
- 7. Part I: Portfolio Optimization & Role of Volatility Choice of Volatility Vehicles iPath S&P 500 VIX Short Term Futures TM ETN (VXX) iPath S&P 500 VIX Mid-Term Futures ETN (VXZ)
- 8. Part I: Portfolio Optimization & Role of Volatility Comparing the Effects
- 9. Part I: Portfolio Optimization & Role of Volatility Strategy: Short VXX to Speculate, Long VXZ to Hedge
- 10. Part II: Volatility Forecasting & Estimation
- 11. Part II: Volatility Forecasting & Estimation -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Annualized Return Annualized Standard Deviation Using Calculated Cov S&P500 S&P100 DJIA NASDAQ VXX VXZ After Cov Shrinkage -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Annualized Return Annualized Standard Deviation With Calculated Cov S&P500 S&P100 DJIA NASDAQ After Cov Shrinkage Traditional covariance estimation methods based on historical data incur lots of error and therefore degrade the results through mean-variance optimization Covariance Matrix Shrinkage gives better estimation of covariance coefficients [Ledoit & Wolf 2003] Covariance Matrix Shrinkage
- 12. Part II: Volatility Forecasting & Estimation AR(1): Auto-regressive model of order 1 Key assumptions: 1) Only t-1 information is used to predict the result at t 2) Error term ɛt is independent of time and X AR(1) Model
- 13. Part II: Volatility Forecasting & Estimation For the whole data set of size n, half of the data points are used as training data. Parameters c, ɛt, ϕ are estimated from data points 1, 2, … n/2 AR(1) Model Training
- 14. Part II: Volatility Forecasting & Estimation We use the model to simulate the response from the 1st data point and compare the simulated value with the original data For each method we run the simulation 100 times and calculate the RMS value from the simulated data for each run X difference between real and simulated data AR(1) Model Testing
- 15. Part II: Volatility Forecasting & Estimation C = 3.3802 ϕ = 0.8424 Residue Histogram Mean RMS: 10.5836 AR(1) Model Verification Accurate in the short term
- 16. Part II: Volatility Forecasting & Estimation AR(1) Model Verification – Prediction
- 17. Part III: Trading and Implementation
- 18. Part III: Trading and Implementation Active Volatility Trading It is clear positions on volatility products can enhance overall portfolio performance But… • Does active volatility trading outperform static volatility positions? • How can we actually compare performance of different trading strategies? • What is the best way for a long/short volatility hedge fund to operate? Here we will try to address these issues: 1. Examine simple momentum strategies 2. Incorporate AR(1) volatility predictions to increase returns 3. See how the best returns in active volatility trading can increase efficient portfolios
- 19. Part III: Trading and Implementation 2010 15 20 25 30 35 40 45 50 55 Year Price,$ VIX Future Closing Price 10-Period MA +/- 5% of MA Moving Average Momentum Trading Rules: 1. Enter long VXX position when VIX futures cross n% above N-day moving average 2. Close long VXX position when VIX futures cross N-day moving average 3. Enter short VXX position when VIX futures cross n% below N-day moving average 4. Close short VXX position when VIX futures cross N-day moving average First attempt of a simple momentum strategy Short entry Short close Long entry Long close
- 20. Part III: Trading and Implementation Sharpe Ratios to Compare Strategies Trading Strategy Sharpe Ratio Static short VXX 0.181 Static VXX+VXZ 0.012 PPO & RSI -0.14 Momentum trading 0.524 ARCH Model 0.403 Sharpe Ratio is used to compare excess return and variance against a benchmark SR = E[Ra - Rb ] cov Ra - Rb( ) Ra = Asset return vector Rb = Benchmark return vector Expected Excess Return Variance of Excess Return Here we use the static short VXX as our benchmark portfolio, except for VXX itself
- 21. Part III: Trading and Implementation Exploration of Strategy Variations 2004 2006 2008 2010 2012 2014 0 10 20 30 40 50 60 Year Price,$ VIX Future Closing Price 10-Period MA +/- 5% of MA 2004 2006 2008 2010 2012 2014 0 10 20 30 40 50 60 Year Price,$ VIX Future Closing Price 5-Period MA +/- 10% of MA ν = 73.8% σ = 44.5% SR = 0.501 2004 2006 2008 2010 2012 2014 0 10 20 30 40 50 60 Year Price,$ VIX Future Closing Price 10-Period MA +/- 10% of MA ν = 78.1% σ = 45.5% SR = 0.516 2004 2006 2008 2010 2012 2014 0 10 20 30 40 50 60 Year Price,$ VIX Future Closing Price 10-Period MA +/- 15% of MA ν = 79.8% σ = 45.4% SR = 0.524 ν = 63.3% σ = 44.4% SR = 0.453
- 22. Part III: Trading and Implementation Trading on the AR(1) Model • Recall AR(1) Model of the form: • Fit the AR(1) over the first half of historical VIX (1993-2003) • Test the returns of this strategy on the second half of data (2003-2013) • These results look promising, but perhaps we can simulate many times to eliminate εt noise 1995 2000 2005 2010 0 10 20 30 40 50 60 Year VIXLevel Actual VIX Price Movement ARCH Prediction
- 23. Part III: Trading and Implementation Tomorrows Volatility Today • At each time step we use the today’s VIX in the AR(1) Model to predict tomorrow’s VIX level • Repeat this at each step 30 times and take the average to get tomorrows volatility prediction • When tomorrow’s volatility is higher than today’s, buy the VIX • Tomorrow lower, vice versa Annualized Return = 167% Standard Deviation = 59.9% Sharpe Ratio = 0.403 Qualitative performance of this method looks exceptional. 1995 2000 2005 2010 10 15 20 25 30 35 40 45 50 55 60 Year VIXLevel Actual VIX Price Movement ARCH Prediction Model Fitting Backtesting
- 24. Part III: Trading and Implementation 10 20 30 40 50 60 70 80 90 -40 -20 0 20 40 60 80 100 120 140 160 Annualized Standard Deviation, % AnnualizedReturn,% no vol w/ VXX w/ ARCH-based active trading Excess Returns Expand Efficiency Massive expansion in efficiency frontier due to excess returns from trading on AR(1) predictions of VIX What’s the catch? • The VIX itself is not a tradable product • Can generate VIX sensitivity, however through: - Volatility Swaps - Options Positions - Volatility Futures
- 25. Part III: Trading and Implementation Using The VIX for VXX Timing Three criteria using the VIX to generate a short signal in VXX (and sell long position in VXX): 1.The monthly low is above its 10-month moving average 2.The monthly close is at least 10% above its 10-month moving average (PPO more than 10) - Use the PPO (Percent Price Oscillator) - PPO = (1-day EMA – 10-day EMA)/10-day EMA 3.The monthly close is above the monthly open (Filled Candle Stick) Three criteria using the VIX to generate a cover signal in VXX (and enter long position in VXX): 1. The high of the VIX is below the 10-day moving average (candlestick must be below the 10- day moving average) 2. The monthly close is at least 10% below the 10-month moving average 3. The close is below the open (Hollow Candlestick)
- 26. Part III: Trading and Implementation
- 27. Part III: Trading and Implementation Returns Using the PPO Indicator ν = 18.6% σ = 51.3% SR = -0.1418
- 28. Part III: Trading and Implementation Another Method Using RSI Indicator • When RSI is above 70, VIX is overbought Cover VXX position • When RSI is below 30, VIX is oversold Initiate short VXX position • RSI = 100 – 100/(1+RS) RS = (Average Gain / Average Loss ) in a 5 period (5 month) setting
- 29. Part III: Trading and Implementation Results Using the RSI indicator ν = 15.8% σ = 35.6% SR = -0.1406
- 30. Part III: Trading and Implementation Conclusions Use of volatility ETFs significantly expands efficient frontier • Short position on VXX to speculate • Long position on VXZ to hedge Covariance Matrix Shrinkage technique gives us more reliable covariance estimations and a more accurate efficient frontier AR(1) model helps us predict future movement in VIX Simple momentum trading strategies and trading using AR(1) predictions exhibit promising excess returns above benchmark returns RSI and PPO trading strategies are less viable, returning negative Sharpe ratios
- 31. Project 2: Volatility as an Asset Class —improving the mean-variance efficient frontier using volatility as an asset class Q & A
Editor's Notes
- Hello everyone! We are going to present our work on the project we did as a team on volatility as an asset class. I am Vicky, and on our team we have here Henry, Linda, Hai, JR, Andrew, Ian and Erica.
- A quick introduction of the project. We carried out the project in three parts: First, we constructed the efficient frontier using major market indices, and then included volatility using ETFs that track the Volatility Index to assess the effects on the portfolio optimization. In the second part, we compared different methods and chose AR(1) model to forecast volatility; and also used the shrinkage approach to better estimate the covariance matrix for indices. Lastly, we compared selective long-short volatility trading strategies and explored different ways to leverage the volatility in practice.
- Here shows the efficient frontier derived for a portfolio of four market indices: S&P 500, S&P 100, DJIA and NASDAQ. The way to draw this frontier is to first use Markowitz formula to find two efficient portfolio, which have the minimum variance for a given returnWe then apply the two-fund theorem – which states that any efficient portfolio can be duplicated, in terms of mean and variance, as a combination of two efficient funds.
- Volatility has a negative correlation with the market statistically, as shown in the chart on the left. Intuition leads us to include Volatility Index as an asset and see how it changes the look of our efficient frontier. Not surprisingly, it hedges the market portfolio pretty well as shown in the chart on the right.
- Since volatility are usually overly estimated, the VIX futures are almost in perpetual contango. The S&P 500 VIX Short-Term Futures Index TR is designed to provide access to equity market volatility through CBOE Volatility Index (the “VIX Index (^VIX)”) futures. Specifically, the S&P 500 VIX Short-Term Futures Index TR offers exposure to a daily rolling long position in the first and second month VIX futures contracts and reflects the implied volatility of the S&P 500 Index at various points along the volatility forward curve. The index futures roll continuously throughout each month from the first month VIX futures contract into the second month VIX futures contract.
- What are readily available for us in the market since we cannot trade VIX? Two things that we used, and later on proved to be quite effective, are the S&P 500 VIX Short Term Futures (VXX) and the S&P 500 Mid-Term Futures (VXZ)One issue with these two indices are that they only trace back to 2009, and therefore does not include the whole recession period in 2007-2008. In order to use a more comprehensive data set in our analysis, we produced synthetic historical data, using VIX as the baseline, for these indices to be back to 2004.
- It is interesting to see the results when including each of these volatility futures in the porfolioWith a negative expected return and with the fact that it’s in perpetual contango, VXX seems a very good vehicle for speculation if you hold it in the short position. VXZ, on the other hand, while it is also negatively correlated with the market (same as VXX), but it has a closed to zero expected return, which makes it perfect for hedging.
- When we combine these two and add both in the portfolio, the results look rather encouraging. Both mean and variance have been improved significantly as the efficient frontier expands. By shorting the VXX and longing VXZ, we successfully hedged the portfolio thanks to the fact that the volatility is negatively correlated with the market, and also utilized the contango effect to speculate.
- In the first part of this talk, we showed how including positions on volatility in your portfolio can help expand the efficiency frontier and improve overall performance. Those were all static positions on volatility, however, and now we’d like to examine whether volatility can be actively traded (long/short) to improve performance even more. The questions we have then are: (1) Does active volatility trading provide excess risk-adjusted returns compared to static positions on volatility?, (2) What’s the best way to actually compare the performance of various volatility trading strategies?, and (3) What are the best positions for a hypothetical long/short volatility hedge fund to take?We’ll try to address these issues with the following sections:Using Sharpe ratios to compare back-tested strategiesExamining the performance of simple momentum strategies based on crossing moving average linesIncorporating the auto-regressive model to predict and trade on future movements of the VIXAdvanced momentum-trading strategies based on “technical indictators”
- Sharpe ratio is an accepted standard metric for comparing the excess return of some trading strategy versus a benchmark strategy and comparing it to the standard deviation of that excess return. For volatility trading, we would like to know how much better a potential active trading strategy might be than a static positions. Therefore we use one of the simplest static volatility positions we can take: constantly shorting the VXX (essentially the same as selling VIX futures on a rolling basis). For each strategy we examine, we will compute the corresponding Sharpe ratio benchmarked against this strategy. However, note that the first entry in the slide here shows the static short VXX. This calculation was done benchmarked against a nominal risk-free rate of 2%, and is for reference only. Any position benchmarked against itself would of course have a Sharpe ratio of 0. So, of the strategies we have examined, those with positive Sharpe ratios offer excess returns, and those with negative sharpe ratios do not offer excess returns compared to our benchmark.
- Here we see that by perturbing the conditions on which we trade using the momentum strategy outlined previously, we can achieve different Sharpe ratios, returns, and variances, based on the moving average we use and how far we allow the price to drift from the moving average before we enter a position. Of the four cases shown here, the best results are found when we use a 10-month exponential moving average and enter positions when price moves more than 5% off the moving average value. From the period of March 2004 to Feb. 2013 this strategy would have yielded annualized returns of nearly 80% with about 45% annual standard deviation.
- We also tried a different strategy based on the AR model presented previously. Recall the AR model was of the form shown above. As before, we fit the model parameters C and phi based on the first half of the historical data for the VIX. Then, we can plug in the actual VIX value each day and the model gives the predicted VIX value for the next day. As shown in the plot, this alone does a pretty good job of predicting VIX price movements in the future, but the results are clouded by the noise term epsilon. Therefore we next made multiple predictions and averaged over them in order to help eliminate the noise terms from the model.
- We see that the results of this exercise produce remarkably accurate estimates of future VIX closing prices. From here, trading on the prediction becomes trivial. When the AR model says tomorrows volatility will be higher than today’s price, we buy volatility. When the model suggests tomorrows volatility will be lower than today’s, we sell volatility. Since the model coefficients are based only the period of 1993-2003, we can back-test this strategy without bias over the period of 2004-2013. Over this period it produces remarkable annualized returns of 167%, with only 60% standard deviation and a Sharpe ratio of 0.403. At first glance, these look like pretty compelling results, but lets dive a little deeper and try to see how this strategy might expand the efficient frontier for an entire portfolio.
- Up at the very top of the efficient frontier diagram, you can see the AR model risk-return data point. Now, even though this strategy is very risky, allocating even a small amount of the overall portfolio allows one to capture huge gains in annualized returns while actually decreasing risk! Basically, everything is great! So, by now you should be asking yourself what the catch is. Because there’s got to be a catch, right? And there is. The issue here is that all of this analysis is based on perfectly trading the VIX. Of course, the VIX is not actively traded and so we can’t truly capture all of these gains. There are ways of generating sensitivity to VIX price movements, however. One way is to enter into volatility swaps. Another would be to take straddle-like option positions (or even vanilla option positions if you also have opinion on price direction). Finally, volatility futures, which we’ve already discussed at length here, are another great avenue to get volatility exposure.
- So now lets move back for a few minutes to a couple of more advanced momentum trading strategies. Here we’ll address the issue raised previously and, while the entry and exit of a trade will be based on the VIX levels, the positions will be taken on the VXX and those price movements will account for the P&L over the duration we hold any position.The first of these trading strategies will be based on a technical indicator called a percent price oscillator. And we trade based on the rules outlined above. Percent price oscillators are typically used by day traders who use the VIX to trade the SPX. The idea of using VIX to generate a signal is the same approach.
- This plot shows both the VIX as well as the PPO indicator of the VIX. Cyan arrows show points where we would like to sell short the VXX, and purple arrows show the where we would cover the short position. You can see over the period of 2004 to 2013 we enter several trades but still only a limited number. Yellow Arrows indicate where PPO is above 10, but a short signal is not confirmed because monthly lows are not above the 10 month moving average.
- Results for this strategy on an efficient frontier are shown here. We see that it offers modest returns, however it has significant standard deviation. The Sharpe ratio is negative because this strategy does not offer excess returns compared to selling the VXX short.However, we argue that if we used daily data instead of monthly to determine our trading entry and exits, our numbers would have been more accurate. This is because we would be able to capture more opportunities within the course of 10 years.
- Next we will look at returns based on trades using the Relative strength indicator. This indicator works exactly like it sounds: When RSI is high, the underlying is considered overbought and should be sold. When RSI is relatively low, the underlying is considered oversold and therefore likely to rise.However, the strategy is counterintuitive. Typically when the VIX is overbought (RSI >70), we anticipate that the VIX is going to move up, and VXX in the same direction. Instead, results have shown that we should cover any VXX position we had previously. This works the same way when VIX’s RSI is below 30 – we would initiate a short position.
- We incorporate results from using the RSI indicator here, shown in the left plot as the Green line. This strategy has a slightly smaller annualized return than the PPO strategy, however it also has significantly smaller standard deviation. Nevertheless, the Sharpe ratio remains similar to the trading strategy based on the RSI indicator. However, again, using daily data could yield a more accurate result.On the right hand plot we summarize all the results using technical indicators. We can also verify some of the difference between the RSI and PPO active trading strategies based on the expansion of the efficiency frontier when both are considered for a Markowitz-efficient portfolio. As with active trading based on AR model, both methods do a reasonable job of expanding the high-return portion of the efficiency frontier. However, if one only desires to reduce volatility while maintaining standard-security returns, using the VXX as a portfolio hedge remains optimal.