Jonathan, Author at QUANTITATIVE RESEARCH AND TRADING

July 9, 2019

Is Internal Bar Strength A Random Walk? The Case of Exxon-Mobil

For those who prefer a little more rigor in their quantitative research, I can offer more a somewhat more substantive statistical argument in favor of the IBS indicator discussed in my previous post.

Specifically, we can show quite convincingly that the IBS process is stationary, a highly desirable property much sought-after in, for example, the construction of statistical arbitrage strategies. Of course, by construction, the IBS is constrained to lie between the values of 0 and 1, so non-stationarity in the mean is highly unlikely. But, conceivably, there could be some time dependency in the process or in its variance, for instance. Then there is the further question as to whether the IBS indicator is mean-reverting, which would indicate that the underlying price process likewise has a tendency to mean revert.

Let’s take the IBS series for Exxon-Mobil (XOM) as an example to work with. I have computed the series from the beginning of 1990, and the first 100 values are shown in the plot below.

Autocorrelation and Unit Root Tests

There appears to be little patterning in the process autocorrelations, and this is confirmed by formal statistical tests which fail to reject the null hypothesis that the first 20 autocorrelations are not, collectively, statistically significant.

Next we test for the presence of a unit root in the IBS process (highly unlikely, given its construction) and indeed, unsurprisingly, the null hypothesis of a unit root is roundly rejected by the Dickey-Fuller and Phillips-Perron tests.

Variance Ratio Tests

We next conduct a formal test to determine whether the IBS series follows a random walk.

The variance ratio test assesses the null hypothesis that a univariate time series y is a random walk. The null model is

y(t) = c + y(t–1) + e(t),

where c is a drift constant (assumed zero for the IBS series) and e(t) are uncorrelated innovations with zero mean.

When IID is false, the alternative is that the e(t) are correlated.
When IID is true, the alternative is that the e(t) are either dependent or not identically distributed (for example, heteroscedastic).

We test whether the XOM IBS series is a random walk using various step sizes and perform the test with and without the assumption that the innovations are independent and identically distributed.

Switching to Matlab, we proceed as follows:

q = [2 4 8 2 4 8];
flag = logical([1 1 1 0 0 0]);
[h,pValue,stat,cValue,ratio] = vratiotest(XOMIBS,’period’,q,’IID’,flag)

Here h is a vector of Boolean decisions for the tests, with length equal to the number of tests. Values of h equal to 1 indicate rejection of the random-walk null in favor of the alternative. Values of h equal to 0 indicate a failure to reject the random-walk null.

The variable ratio is a vector of variance ratios, with length equal to the number of tests. Each ratio is the ratio of:

The variance of the q-fold overlapping return horizon
q times the variance of the return series

For a random walk, these ratios are asymptotically equal to one. For a mean-reverting series, the ratios are less than one. For a mean-averting series, the ratios are greater than one.

For the XOM IBS process we obtain the following results:

h = 1 1 1 1 1 1
pValue = 1.0e-51 * [0.0000 0.0000 0.0000 0.0000 0.0000 0.1027]
stat = -27.9267 -21.7401 -15.9374 -25.1412 -20.2611 -15.2808
cValue = 1.9600 1.9600 1.9600 1.9600 1.9600 1.9600
ratio = 0.4787 0.2405 0.1191 0.4787 0.2405 0.1191

The random walk hypothesis is convincingly rejected for both IID and non-IID error terms. The very low ratio values indicate that the IBS process is strongly mean reverting.

Conclusion

While standard statistical tests fail to find evidence of any non-stationarity in the Internal Bar Strength signal for Exxon-Mobil, the hypothesis that the series follows a random walk (with zero drift) is roundly rejected by variance ratio tests. These tests also confirm that the IBS series is strongly mean reverting, as we previously discovered empirically.

This represents an ideal scenario for trading purposes: a signal with the highly desirable properties that is both stationary and mean reverting. In the case of Exxon-Mobil, there appears to be clear evidence from both statistical tests and empirical trading strategies using the Internal Bar Strength indicator that the tendency of the price series to mean-revert is economically as well as statistically significant.

July 2, 2019

The Internal Bar Strength Indicator

Internal Bar Strength (IBS) is an idea that has been around for some time. IBS is based on the position of the day’s close in relation to the day’s range: it takes a value of 0 if the closing price is the lowest price of the day, and 1 if the closing price is the highest price of the day.

More formally:

IBS = (Close – Low) / (High – Low)

The IBS effect may be related to intraday over-reaction to news or market movements, which are then ”corrected” the next day. It serves as a measure of the tendency of a price series to mean-revert over daily horizons. I use the term “daily” advisedly: so far as I am aware, there has been no research (including my own) demonstrating the existence of an IBS effect at time horizons shorter, or longer, than one day. Indeed, there has been very little in the way of academic research into the concept of any kind, which is strange considering how compelling are the results it is capable of producing. Practitioners have been happy enough with that state of affairs, content to deploy this neglected indicator in their trading strategies, where it has often proved to be extremely useful (we use IBS in one of our volatility strategies). Since 2013, however, the cat has been let out of the bag, thanks to an excellent research paper by Alexander Pagonidis, who writes an interesting quantitative finance blog.

The essence of the idea is that stocks that close in the lowest part of the daily range, with an IBS of below, say, 0.2, will tend to rally the next day, while stocks that close in the highest quintile will often decline in value in the following session. In his paper “The IBS Effect: Mean Reversion in Equity ETFs” (2013), Pagonidis researches the IBS effect in equity index ETFs in the US and several international markets. He confirms that low IBS values in these assets are associated with high returns in the following day session, while high IBS values are associated with low returns. Average returns when IBS is below 0.20 are .35% ,while average returns when IBS is above 0.80 are -0.13%. According to his research, this effect has been present in equity ETFs since the early 90s and has been highly consistent through time.

IBS Strategy Performance

To give the reader some idea of the potential of the IBS effect, I have reproduced below equity curves for the IBS strategy for the SPDR S&P 500 ETF Trust (SPY) and iShares MSCI Singapore ETF (EWS) index ETFs over the period from 1999 to 2016. The strategy buys at the close when IBS is below 0.2, and sells at the close when IBS exceeds 0.8, liquidating the position at the following market close. Strategy CAGR over the period has been of the order of 13% for SPY and as high as 40% for EWS, ignoring transaction costs.

Note that in both cases strategy returns for SPY and EWS have diminished in recent years, turning negative in 2015 and 2016 YTD and this is true for ETFs in general. It remains to be seen whether this deterioration in strategy performance is temporary or permanent. There are some indications that the latter holds true, but the evidence is not quite definitive. For example, the chart below shows daily equity curve for the SPY IBS strategy, with 95% confidence intervals for the latest 100 trades (up to the end of May 2016), constructed using Monte-Carlo bootstrap. The equity curve appears to have penetrated the lower bound, indicating a statistically significant deterioration in the performance of the IBS strategy for SPY over the last year or so (EWS is similar). That said, the equity curve does fall inside the boundaries of the 99% confidence interval, so those looking for greater certainty about the possible breakdown of the effect will need to wait a little longer for confirmation.

Whatever the outcome may be for SPY and other ETFs going forward, it is certainly true that IBS effects persist strongly for some individual equities, Exxon-Mobil Corp. (XOM) being a case in point (see below). It’s worth taking note of the exceptional performance of the XOM IBS strategy during the latter quarter of 2008. I will have much more to say on the application of the IBS indicator for individual equities in a future blog post.

The Role of Range, Volume, Bull/Bear Markets, Volatility and Seasonality

Pagonidis goes on to detail several further important findings in relation to IBS. It is clear from his research that high volatility is related to increased predictability of returns and a more powerful IBS effect, in particular the high IBS-negative return aspect. As might be expected, the effect is also larger after days with high range, both for high and low IBS extremes.

Volume turns out to be especially important for U.S. index ETFs: in fact, the IBS effect only appears to work on high-volume days.

Pagonidis also separates the data into bull and bear market environments, based on whether 200-day returns are positive or not. The size of the effect is roughly similar in each environment (slightly larger in bear markets), but it is greater in the direction of the overall trend: high IBS readings are followed by larger negative returns during bear markets, and vice versa.

Day of Week Effect

The IBS effect is also strongly seasonal, having the greatest impact on returns from Monday’s close to Tuesday’s close, as illustrated for the SPY ETF in the chart below. This accounts for the phenomenon known popularly as “Turnaround Tuesday”, i.e. the tendency for the market to recover strongly from losses on a Monday. The day-of-week effect is weakest for Fridays.

The mean of the returns distribution is not the only aspect that IBS can predict. Skewness also varies significantly between IBS buckets, with low IBS readings being followed by highly skewed returns, and vice versa. Close-to-close returns after a bottom-bucket IBS day have average skewness of 0.65 across Equity Index ETF products, while top-bucket IBS days are followed by returns with skewness of 0.03. This finding has very useful risk management applications for investors concerned with tail risk.

IBS as a Filter for a Swing Trading Strategy in QQQ

The returns to an IBS-only strategy are both statistically and economically significant. However, commissions will greatly decrease the returns and increase the maximum drawdowns, however, making such an approach challenging in the real world. One alternative is to combine the IBS effect with mean reversion on longer timescales and only take trades when they align.

Pagonidis offers a simple demonstration using the Cutler’s RSI indicator that shows how the IBS effect can be used to boost returns of a swing trading strategy while significantly decreasing the number of trades needed.

Cutler’s RSI at time t is calculated as follows:

Pagonidis tests a simple, long-only strategy that trades the PowerShares QQQ Trust, Series 1 (QQQ) ETF using the Cutler’s RSI(3) indicator:

• Go long at the close if RSI(3) < 10

• Maintain the position while RSI(3) ≤ 40

filter these returns by adding an additional rule based on the value of IBS:

• Enter or maintain long position only if IBS ≤ 0.5

Pangonis claims that the strategy produces rather promising results that “easily beats commissions”; however, my own rendition of the strategy, assuming commissions of $0.005 per share and slippage of a further $0.02 per share produces results that are distinctly less encouraging:

Strategy Code

For those interested, the code is as follows:

Inputs:
RSILen(3),
RSI_Entry(10),
RSI_Exit(40),
IBS_Threshold(0.5),
Initial_Capital(100000);
Vars:
nShares(100),
RSIval(0),
IBS(0);
RSIval=RSI(C,RSILen);
IBS = (C-L)/(H-L);

nShares = Round(Initial_Capital / Close,0);

If Marketposition = 0 and RSIval > RSI_Entry and IBS < IBS_Threshold then begin
Buy nShares contracts next bar at market;
end;
If Marketposition > 0 and ((RSIval > RSI_Exit) or (IBS_Threshold > IBS_Threshold)) then begin
Sell next bar at market;
end;

Strategy Optimization and Robustness Testing

One can further improve performance by optimizing the trading system parameters, using Tradestation’s excellent Walk Forward Optimization (WFO) module. This allows us to examine the effect of re-calibrating the strategy parameters are regular intervals, testing the optimized model on out-of-sample data sets of various sizes. WFO can be used, not only optimize a strategy, but also to examine the sensitivity of its performance to changes in the levels of key parameters. For example, in the case of the QQQ swing trading strategy, we find that profitability increases monotonically with the length of the RSI indicator, and this effect is especially marked when an IBS threshold level of 0.2 is used:

Likewise we can test the consistency of the day-of-the-week effect over several OS data sets of varying size and these tests are consistent with the pattern seen earlier for the IBS indicator, confirming its role as a filter rule in enhancing system profitability:

A model that is regularly re-calibrated using WFO is subjected to a series of tests designed to ensure its robustness and consistency in live trading. The tests include the following:

In order to achieve an overall pass rating, the system is required to pass all five tests of its out-of-sample performance, from which Tradestation deems it likely that the system will continue to perform well in live trading. The results from this procedure appear much more promising than the strategy in its original form, as can be seen from the performance table and equity curve chart shown below.

However, these results include both in-sample and out-of-sample periods. An examination of the results from the WFO indicate that the overall efficiency of the strategy is around 55%, meaning that the P&L produced by the system in out-of-sample periods amounts to a little over one half of the rate of profit produced during in-sample periods. Going forward, therefore, we might expect the performance of the system in live trading to be only around half as good as shown here. While this is still superior to the original system, it may not be considered good enough. Nonetheless, for the purpose of illustrating the benefits of the IBS indicator as a trade filter, it makes the point.

Another interesting example of an IBS-based trading strategy in the QQQ and SPY ETFs can be found in the following blog post.

Conclusion

Internal Bar Strength is a powerful mean-reversion indicator for equity products traded at daily frequencies, with a consistent effect that has continued from the 1990s through to the current decade. IBS can be used on its own in mean-reversion strategies that have worked well for both US equities and US and International equity index ETFs, or used as a trade filter when combined with other alpha signals.

While there is evidence of a weakening of the IBS effect since around 2013 this is not yet confirmed statistically (at the 99% confidence level) and may simply be the result of normal statistical variation in its efficacy.

June 25, 2019

Seasonal Effects in Equity Markets

There are a plethora of seasonal anomalies documented in academic research. For equities these include the Halloween effect (“Sell in May”), January effect, turn-of-the-month effect, weekend effect and holiday effect. For example, Bouman and Jacobsen (2002) and Jacobsen and Visaltanachoti (2009) provide empirical evidence on the Halloween effect, Haug and Hirschey (2006) on the January effect, Lakonishok and Smidt (1988) on the turn-of-the-month (TOM) effect, Cross (1973) on the weekend effect, and Ariel (1990) on the holiday effect.

An excellent paper entitled An Anatomy of Calendar Effects in the Journal of Asset Management (13(4), 2012, pp. 271-286) by Laurens Swinkels of Erasmus University Rotterdam and Pim van Vliet of Robeco Asset Management gives a very good account of the various phenomena and their relative importance. Using daily returns data on the US value-weighted equity market over the period from July 1963 to December 2008, the researchers find that Halloween and turn-of-the-month (TOM) are the strongest effects, fully diminishing the other three effects to zero. The equity premium over the sample 1963-2008 is 7.2% if there is a Halloween or TOM effect, and -2.8% in all other cases. These findings are robust with respect to transactions costs, across different samples, market segments, and international stock markets. Their empirical research narrows down the number of calendar effects from five to two, leading to a more powerful and puzzling summary of seasonal effects.

The two principal effects are illustrated here with reference to daily returns in the S&P 500 Index, using data from 1980-2016.

Halloween Effect

The Halloween effect refers to the tendency of markets to perform better in the six month period from November to April, compared to the half year from May to October. In fact, for the S&P 500 index itself, performance during the months of May and October has historically been above the monthly average, as you can see in the chart below. According to this analysis, the period to avoid spans the four months from June to September, with September being the “cruelest month”, by far. Note that, between them, the months of November, December and April account for over 50% of the average annual return in the index since 1980.

Turn-of-the-Month Effect

The TOM effect refers to the finding that above average returns tend to occur on the last trading day of the month and (up to) the first four trading days of the new calendar month. For the S&P 500 index the TOM effect spans a shorter period comprising the last trading day of the month and the first two trading days of the new month. It is worth noting also the anomalous positive returns arising on 16th – 18th of the month and negative returns around the 19th and 20th of the month. My speculative guess is that these mid-month effects arise from futures/option expiration.

Seasonal Tactical Allocation

Let’s assume we allocate to equities (in the form of the S&P 500 Index) only during the period from October to May, or on the last or first two trading days of each month. How do the returns from that seasonal portfolio compare to the benchmark buy and hold portfolio? If we ignore transaction costs (and income from riskless Treasury investments when we are out of the market), the seasonal portfolio outperforms the buy and hold benchmark over the 36 year period since 1980 by around 88bp per annum (continuously compounded), and with an annual volatility that is 258bp lower. The outperformance of the seasonal portfolio becomes particularly noticeable after the 2000/2001 crash.

A much more rigorous analysis of the performance characteristics of the seasonal portfolio is given in the research paper, taking account of transaction costs, with summary results as follows:

Conclusion

There is a sizable body of credible academic research demonstrating the importance of calendar effects and this paper suggests that investors’ focus should be on the Halloween and TOM effects in particular. A tactical allocation program that increases the allocation to equities towards the end of the month and first few trading days of the new month, and during the November to April calendar months is likely to significantly outperform a buy-and-hold portfolio, according to these findings.

There remain unaccounted-for seasonal effects in the mid-section of the month that may arise from the expiration of futures and option contracts. These are worthy of further investigation.

June 18, 2019

Trading With Indices

In this post I want to discuss ways to make use of signals from relevant market indices in your trading. These signals can add value regardless of whether you trade algorithmically or manually. The techniques described here are one of the most widely applicable in the quantitative analyst’s arsenal.

Let’s motivate the discussion by looking an example of a simple trading system trading the VIX on weekly bars. Performance results for the system are summarized in the chart and table below. The system outperforms the buy and hold return by a substantial margin, with a profit factor of over 3 and a win rate exceeding 82%. What’s not to like?

Well, for one thing, this isn’t really a trading system – because the VIX Index itself isn’t tradable. So the performance results are purely notional (and, if you didn’t already notice, no slippage or commission is included).

It is very easy to build high-performing trading system in indices – because they are not traded products, index prices are often stale and tend to “follow” the price action in the equivalent traded market.

This particular system for the VIX Index took me less than ten minutes to develop and comprises only a few lines of code. The system makes use of a simple RSI indicator to decide when to buy or sell the index. I optimized the indicator parameters (separately for long and short) over the period to 2012, and tested it out-of-sample on the data from 2013-2016.

inputs:
Price( Close ) ,
Length( 14 ) ,
OverSold( 30 ) ;

variables:
RSIValue( 0 );

RSIValue = RSI( Price, Length );
if CurrentBar > 1 and RSIValue crosses over OverSold then
Buy ( !( “RsiLE” ) ) next bar at market;

The daily system I built for the S&P 500 Index is a little more sophisticated than the VIX model, and produces the following results.

Using Index Trading Systems

We have seen that its trivially easy to build profitable trading systems for index products. But since they can’t be traded, what’s the point?

The analyst might be tempted by the idea of using the signals generated by an index trading system to trade a corresponding market, such as VIX or eMini futures. However, this approach is certain to fail. Index prices lag the prices of equivalent futures products, where traders first monetize their view on the market. So using an index strategy directly to trade a cash or futures market would be like trying to trade using prices delayed by a few seconds, or minutes – a recipe for losing money.

Nor is it likely that a trading system developed for an index product will generalize to a traded market. What I mean by this is that if you were to take an index strategy, such as the VIX RSI strategy, transfer it to VIX futures and tweak the parameters in the hope of producing a profitable system, you are likely to be disappointed. As I have shown, you can produce a profitable index trading system using the simplest and most antiquated trading concepts (such as the RSI index) that long ago ceased to offer any predictive value in actual traded markets. Index markets are actually inefficient – the prices of index products often fail to fully reflect all relevant, available information in a timely way. Such simple inefficiencies are easily revealed by indicators such as moving averages. Traded markets, by contrast, are highly efficient and, with the exception of HFT, it is going to take a great deal more than a simple moving average to provide insight into the few inefficiencies that do arise.

Strategies in index products are best thought of, not as trading strategies, but rather as a means of providing broad guidance as to the general condition of the market and its likely direction over the longer term. To take the VIX index strategy as an example, you can see that each “trade” spans several weeks. So one might regard a “buy” signal from the VIX index system as an indication that volatility is expected to rise over the next month or two. A trader might use that information to lean on the side of being long volatility, perhaps even avoiding any short volatility positions altogether for the next several weeks. Following the model’s guidance in that way would would certainly have helped many equity and volatility traders during the market sell off during August 2015, for example:

The S&P 500 Index model is one I use to provide guidance as to market conditions for the current trading day. It is a useful input to my thinking as to how aggressive I want my trading models to be during the upcoming session. If the index model suggests a positive tone to the market, with muted volatility, I might be inclined to take a more aggressive stance. If the model starts trading to the short side, however, I am likely to want to be much more cautious. Yesterday (May 16, 2016), for example, the index model took an early long trade, providing confirmation of the positive tenor to the market and encouraging me to trade volatility to the short side more aggressively.

In general, I would tend to classify index trading systems as “decision support” tools that provide a means of shading opinion on the market, or perhaps providing a means of calibrating trading models to the anticipated market conditions. However, they can be used in a more direct way, short of actual trading. For example, one of our volatility trading systems uses the trading signals from a trading system designed for the VVIX volatility-of-volatility index. Another approach is to use the signals from an index trading system as an indicator of the market regime in a regime switching model.

Designing Index Trading Models

Whereas it is profitability that is typically the primary design criterion for an actual trading system, given the purpose of an index trading system there are other criteria that are at least as important.

It should be obvious from these few illustrations that you want to design your index model to trade less frequently than the system you are intending to trade live: if you are swing-trading the eminis on daily bars, it doesn’t help to see 50 trades a day from your index system. What you want is an indication as to whether the market action over the next several days is likely to be positive or negative. This means that, typically, you will design your index system using bar frequencies at least as long as for your live system.

Another way to slow down the signals coming from your index trading system is to design it for very high accuracy – a win rate of 70%, or higher. It is actually quite easy to do this: I have systems that trade the eminis on daily bars that have win rates of over 90%. The trick is simply that you have to be prepared to wait a long time for the trade to come good. For a live system that can often be a problem – no-one like to nurse an underwater position for days or weeks on end. But for an index trading system it matters far less and, in fact, it helps: because you want trading signals over longer horizons than the time intervals you are using in your live trading system.

Since the index system doesn’t have to trade live, it means of course that the usual trading costs and frictions do not apply. The advantage here is that you can come up with concepts for trading systems that would be uneconomic in the real world, but which work perfectly well in the frictionless world of index trading. The downside, however, is that this might lead you to develop index systems that trade far too frequently. So, even though they should not apply, you might seek to introduce trading costs in order to penalize higher frequency trading systems and benefit systems that trade less frequently.

Designing index trading systems in an area in which genetic programming algorithms excel. There are two main reasons for this. Firstly, as I have previously discussed, simple technical indicators of the kind employed by GP modeling systems work well in index markets. Secondly, and more importantly, you can use the GP system to tailor an index trading system to meet the precise criteria you have in mind, such as the % win rate, trading frequency, etc.

An outstanding product that I can highly recommend in this context is Mike Bryant’s Adaptrade Builder. Builder is a superb piece of software whose power and ease of use reflects Mike’s engineering background and systems development expertise.

June 11, 2019

Some Further Notes on Market Timing

Almost at the very moment I published a post featuring some interesting research by Glabadanidis (“Market Timing With Moving Averages” (2015), International Review of Finance, Volume 15, Number 13, Pages 387-425 – see Yes, You Can Time the Market. How it Works, And Why), several readers wrote to point out a recently published paper by Valeriy Zakamulin, (dubbed the “Moving Average Research King” by Alpha Architect, the source for our fetching cover shot) debunking Glabadanidis’s findings in no uncertain terms:

We demonstrate that “too good to be true” reported performance of the moving average strategy is due to simulating the trading with look-ahead bias. We perform the simulations without look-ahead bias and report the true performance of the moving average strategy. We find that at best the performance of the moving average strategy is only marginally better than that of the corresponding buy-and-hold strategy.

So far, no response from Glabadanidis – from which one is tempted to conclude that Zakamulin is correct.

I can’t recall the last time a paper published in a leading academic journal turned out to be so fundamentally flawed. That’s why papers are supposed to be peer reviewed. But, I guess, it can happen. Still, it’s rather alarming to think that a respected journal could accept a piece of research as shoddy as Zakamulin claims it to be.

What Glabadanidis had done, according to Zakamulin, was to use the current month closing price to compute the moving average that was used to decide whether to exit the market (or remain invested) at the start of the same month. An elementary error that introduces look-ahead bias that profoundly impacts the results.

Following this revelation I hastily checked my calculations for the SPY marketing timing strategy illustrated in my blog post and, to my relief, confirmed that I had avoided the look-ahead trap that Glabadanidis has fallen into. As the reader can see from the following extract from the Excel spreadsheet I used for the calculations, the decision to assume the returns for the SPY ETF or T-Bills for the current month rests on the value of the 24 month MA computed using prices up to the end of the prior month. In other words, my own findings are sound, even if Glabadanidis’s are not, as the reader can easily check for himself.

Nonetheless, despite my relief at having avoided Glabadanidis’s blunder, the apparent refutation of his findings comes as a disappointment. And my own research on the SPY market timing strategy, while sound as far as it goes, cannot by itself rehabilitate the concept of market timing using moving averages. The reason is given in the earlier post. There is a hidden penalty involved in using the market timing strategy to synthetically replicate an Asian put option, namely the costs incurred in exiting and rebuilding the portfolio as the market declines below the moving average, or later overtakes it. In a single instance, such as the case of SPY, it might easily transpire simply by random chance that the cost of replication are far lower than the fair value of the put. But the whole point of Glabadanidis’s research was that the same was true, not only for a single ETF or stock, but for many thousands of them. Absent that critical finding, the SPY case is no more than an interesting anomaly.

Finally, one reader pointed out that the effect of combining a put option with a stock (or ETF) long position was to create synthetically a call option in the stock (ETF). He is quite correct. The key point, however, is that when the stock trades down below its moving average, the value of the long synthetic call position and the market timing portfolio are equivalent.

June 4, 2019

The Information Content of the Pre- and Post-Market Trading Sessions

I apologize in advance for this rather “wonkish” post, which is aimed chiefly at the high frequency fraternity, or those at least who trade intra-day, in the equity markets. Such minutiae are the lot of those engaged in high frequency trading. I promise that my next post will be of more general interest.

Pre- and Post Market Sessions

The pre-market session in US equities runs from 8:00 AM ET, while the post-market session runs until 8:00 PM ET. The question arises whether these sessions are worth trading, or at the very least, offer a source of data (quotes, trades) that might be relevant to trading the regular session, which of course runs from 9:30 AM to 4:00 PM ET. Even if liquidity is thin and trades infrequent, and opportunities in the pre- and post-market very limited, it might be that we can improve our trading models by taking into account such information as these sessions do provide, even if we only ever plan to trade during regular trading hours.

It is somewhat challenging to discuss this in great detail, because HFT equity trading is very much in the core competencies of my firm, Systematic Strategies. However, I hope to offer some ideas, at least, that some readers may find useful.

A Tale of Two Pharmaceutical Stocks

In what follows I am going to make use of two examples from the pharmaceutical industry: Alexion Pharmaceuticals, Inc. (ALXN), which has a market cap of $35Bn and trades around 800,000 shares daily, and Pfizer Inc. (PFE), which has a market cap of over $200Bn and trades close to 50M shares a day.

Let’s start by looking at a system trading ALXN during regular market hours. The system isn’t high frequency, but trades around 1-2 times a day, on average. The strategy equity curve from 2015 to April 2016 is not at all impressive.

ALXN – Regular Session Only

But look at the equity curve for the same strategy when we allow it to run on the pre- and post-market sessions, in addition to regular trading hours. Clearly the change in the trading hours utilized by the strategy has made a huge improvement in the total gain and risk-adjusted returns.

ALXN – with Pre- and Post-Market Sessions

The PFE system trades much more frequently, around 4 times a day, but the story is somewhat similar in terms of how including the pre- and post-market sessions appears to improve its performance.

PFE – Regular Session Only

PFE – with Pre- and Post-Market Sessions

Improving Trading Performance

In both cases, clearly, the trading performance of the strategies has improved significantly with the inclusion of the out-of-hours sessions. In the case of ALXN, we see a modest increase of around 10% in the total number of trades, but in the case of PFE the increase in trading activity is much more marked – around 30%, or more.

The first important question to ask is when these additional trades are occurring. Assuming that most of them take place during the pre- or post-market, our concern might be whether there is likely to be sufficient liquidity to facilitate trades of the frequency and size we wish to execute. Of various possible hypotheses, some negative, other positive, we might consider the following:

(a) Bad ticks in the market data feed during out-of-hours sessions give rise to apparently highly profitable “phantom” trades

(b) The market data is valid, but the trades are done in such low volume as to be insignificant for practical purposes (i.e. trades were done for a few hundred lots and additional liquidity is unlikely to be available)

(c) Out-of-hours sessions enable the system to improve profitability by entering or exiting positions in a more timely manner than by trading the regular session alone

(d) Out-of-hours market data improves the accuracy of model forecasts, facilitating a larger number of trades, and/or more profitable trades, during regular market hours

An analysis of the trading activity for the two systems provides important insight as to which of the possible explanations might be correct.

(click to enlarge)

Dealing first with ALXN, we that, indeed, an additional 11% of trades are entered or exited out-of-hours. However, these additional trades account for somewhere between 17% (on exit) and 20% (on entry) of the total profits. Furthermore, the size of the average entry trade during the post-market session and of the average exit trade in the pre-market session is more than double that of the average trade entered or exited during regular market hours. That gives concerns that some of the apparent increase in profits may be due to bad ticks at prices away from the market, allowing the system enter or exit trades at unrealistically low or high prices. Even if many of the trades are good, we will have concerns about the scalability of the strategy in out-of-hours trading, given the relatively poor liquidity in the stock. On the other hand, at least some of the uplift in profits arises from new trades occurring during the regular session. This suggests that, even if we are unable to execute many of the trading opportunities seen during pre- or post-market, the trades from those sessions provides useful additional data points for our model, enabling it to increase the number and/or profitability of trades in the regular session.

Next we turn to PFE. We can see straight away that, while the proportion of trades occurring during out-of-hours sessions is around 23%, those trades now account for over 50% of the total profits. Furthermore, the average PL for trades executed on entry post-market, and on exit pre-market, is more than 4x the average for trades entered or exited during normal market hours. Despite the much better liquidity in PFE compared to ALXN, this is a huge concern – we might expect to see significant discrepancies occurring between theoretical and actual performance of the strategy, due to the very high dependency on out-of-hours trading.

(click to enlarge)

As we dig further into the analysis, we do indeed find evidence that bad data ticks play a disproportionate role. For example, this trade in PFE which apparently occurred at around 16:10 on 4/6 was almost certainly a phantom trade resulting from a bad data point. It turns out that, for whatever reason, such bad ticks are a common occurrence in the stock and account for a large proportion of the apparent profitability of out-of-hours trading in PFE.

CONCLUSION

We are, of course, only skimming the surface of the analysis that is typically carried out. One would want to dig more deeply into ways in which the market data feed could be cleaned up and bad data ticks filtered out so as to generate fewer phantom trades. One would also want to look at liquidity across the various venues where the stocks trade, including dark pools, in order to appraise the scalability of the strategies.

For now, the main message that I am seeking to communicate is that it is often well worthwhile considering trading in the pre- and post-market sessions, not only with a view to generating additional, profitable trading opportunities, but also to gather additional data points that can enhance trading profitability during regular market hours.

May 28, 2019May 28, 2019

A High Frequency Scalping Strategy on Collective2

Scalping vs. Market Making

A market-making strategy is one in which the system continually quotes on the bid and offer and looks to make money from the bid-offer spread (and also, in the case of equities, rebates). During a typical trading day, inventories will build up on the long or short side of the book as the market trades up and down. There is no intent to take a market view as such, but most sophisticated market making strategies will use microstructure models to help decide whether to “lean” on the bid or offer at any given moment. Market makers may also shade their quotes to reduce the buildup of inventory, or even pull quotes altogether if they suspect that informed traders are trading against them (a situation referred to as “toxic flow”). They can cover short positions through the repo desk and use derivatives to hedge out the risk of an accumulated inventory position.

A scalping strategy shares some of the characteristics of a market making strategy: it will typically be mean reverting, seeking to enter passively on the bid or offer and the average PL per trade is often in the region of a single tick. But where a scalping strategy differs from market making is that it does take a view as to when to get long or short the market, although that view may change many times over the course of a trading session. Consequently, a scalping strategy will only ever operate on one side of the market at a time, working the bid or offer; and it will typically never build inventory, since will it usually reverse and later try to sell for a profit the inventory it has previously purchased, hopefully at a lower price.

In terms of performance characteristics, a market making strategy will often have a double-digit Sharpe Ratio, which means that it may go for many days, weeks, or months, without taking a loss. Scalping is inherently riskier, since it is taking directional bets, albeit over short time horizons. With a Sharpe Ratio in the region of 3 to 5, a scalping strategy will often experience losing days and even losing months.

So why prefer scalping to market making? It’s really a question of capability. Competitive advantage in scalping derives from the successful exploitation of identified sources of alpha, whereas market making depends primarily on speed and execution capability. Market making requires HFT infrastructure with latency measured in microseconds, the ability to layer orders up and down the book and manage order priority. Scalping algos are generally much less demanding in terms of trading platform requirements: depending on the specifics of the system, they can be implemented successfully on many third party networks.

Developing HFT Futures Strategies

Some time ago my firm Systematic Strategies began research and development on a number of HFT strategies in futures markets. Our primary focus has always been HFT equity strategies, so this was something of a departure for us, one that has entailed a significant technological obstacles (more on this in due course). Amongst the strategies we developed were several very profitable scalping algorithms in fixed income futures. The majority trade at high frequency, with short holding periods measured in seconds or minutes, trading tens or even hundreds of times a day.

The next challenge we faced was what to do with our research product. As a proprietary trading firm our first instinct was to trade the strategies ourselves; but the original intent had been to develop strategies that could provide the basis of a hedge fund or CTA offering. Many HFT strategies are unsuitable for that purpose, since the technical requirements exceed the capabilities of the great majority of standard trading platforms typically used by managed account investors. Besides, HFT strategies typically offer too limited capacity to be interesting to larger, institutional investors.

In the end we arrived at a compromise solution, keeping the highest frequency strategies in-house, while offering the lower frequency strategies to outside investors. This enabled us to keep the limited capacity of the highest frequency strategies for our own trading, while offering investors significant capacity in strategies that trade at lower frequencies, but still with very high performance characteristics.

HFT Bond Scalping

A typical example is the following scalping strategy in US Bond Futures. The strategy combines two of the lower frequency algorithms we developed for bond futures that scalp around 10 times per session. The strategy attempts to take around 8 ticks out of the market on each trade and averages around 1 tick per trade. With a Sharpe Ratio of over 3, the strategy has produced net profits of approximately $50,000 per contract per year, since 2008. A pleasing characteristic of this and other scalping strategies is their consistency: There have been only 10 losing months since January 2008, the last being a loss of $7,100 in Dec 2015 (the prior loss being $472 in July 2013!)

Annual P&L

Strategy Performance

Offering The Strategy to Investors on Collective2

The next challenge for us to solve was how best to introduce the program to potential investors. Systematic Strategies is not a CTA and our investors are typically interested in equity strategies. It takes a great deal of hard work to persuade investors that we are able to transfer our expertise in equity markets to the very different world of futures trading. While those efforts are continuing with my colleagues in Chicago, I decided to conduct an experiment: what if we were to offer a scalping strategy through an online service like Collective2? For those who are unfamiliar, Collective2 is an automated trading-system platform that allowed the tracking, verification, and auto-trading of multiple systems. The platform keeps track of the system profit and loss, margin requirements, and performance statistics. It then allows investors to follow the system in live trading, entering the system’s trading signals either manually or automatically.

Offering a scalping strategy on a platform like this certainly creates visibility (and a credible track record) with investors; but it also poses new challenges. For example, the platform assumes trading cost of around $14 per round turn, which is at least 2x more expensive than most retail platforms and perhaps 3x-5x more expensive than the cost a HFT firm might pay. For most scalping strategies that are designed to take a tick out of the market such high fees would eviscerate the returns. This motivated our choice of US Bond Futures, since the tick size and average trade are sufficiently large to overcome even this level of trading friction. After a couple of false starts, during which we played around with the algorithms and boosted strategy profitability with a couple of low frequency trades, the system is now happily humming along and demonstrating the kind of performance it should (see below).

For those who are interested in following the strategy’s performance, the link on collective2 is here.

Disclaimer

About the results you see on this Web site

Past results are not necessarily indicative of future results.

These results are based on simulated or hypothetical performance results that have certain inherent limitations. Unlike the results shown in an actual performance record, these results do not represent actual trading. Also, because these trades have not actually been executed, these results may have under-or over-compensated for the impact, if any, of certain market factors, such as lack of liquidity. Simulated or hypothetical trading programs in general are also subject to the fact that they are designed with the benefit of hindsight. No representation is being made that any account will or is likely to achieve profits or losses similar to these being shown.

In addition, hypothetical trading does not involve financial risk, and no hypothetical trading record can completely account for the impact of financial risk in actual trading. For example, the ability to withstand losses or to adhere to a particular trading program in spite of trading losses are material points which can also adversely affect actual trading results. There are numerous other factors related to the markets in general or to the implementation of any specific trading program, which cannot be fully accounted for in the preparation of hypothetical performance results and all of which can adversely affect actual trading results.

Material assumptions and methods used when calculating results

The following are material assumptions used when calculating any hypothetical monthly results that appear on our web site.

Profits are reinvested. We assume profits (when there are profits) are reinvested in the trading strategy.
Starting investment size. For any trading strategy on our site, hypothetical results are based on the assumption that you invested the starting amount shown on the strategy’s performance chart. In some cases, nominal dollar amounts on the equity chart have been re-scaled downward to make current go-forward trading sizes more manageable. In these cases, it may not have been possible to trade the strategy historically at the equity levels shown on the chart, and a higher minimum capital was required in the past.
All fees are included. When calculating cumulative returns, we try to estimate and include all the fees a typical trader incurs when AutoTrading using AutoTrade technology. This includes the subscription cost of the strategy, plus any per-trade AutoTrade fees, plus estimated broker commissions if any.
“Max Drawdown” Calculation Method. We calculate the Max Drawdown statistic as follows. Our computer software looks at the equity chart of the system in question and finds the largest percentage amount that the equity chart ever declines from a local “peak” to a subsequent point in time (thus this is formally called “Maximum Peak to Valley Drawdown.”) While this is useful information when evaluating trading systems, you should keep in mind that past performance does not guarantee future results. Therefore, future drawdowns may be larger than the historical maximum drawdowns you see here.

Trading is risky

There is a substantial risk of loss in futures and forex trading. Online trading of stocks and options is extremely risky. Assume you will lose money. Don’t trade with money you cannot afford to lose.

May 21, 2019

High Frequency Trading: Equities vs. Futures

A talented young system developer I know recently reached out to me with an interesting-looking equity curve for a high frequency strategy he had designed in E-mini futures:

Pretty obviously, he had been making creative use of the “money management” techniques so beloved by futures systems designers. I invited him to consider how it would feel to be trading a 1,000-lot E-mini position when the market took a 20 point dive. A $100,000 intra-day drawdown might make the strategy look a little less appealing. On the other hand, if you had already made millions of dollars in the strategy, you might no longer care so much.

A more important criticism of money management techniques is that they are typically highly path-dependent: if you had started your strategy slightly closer to one of the drawdown periods that are almost unnoticeable on the chart, it could have catastrophic consequences for your trading account. The only way to properly evaluate this, I advised, was to backtest the strategy over many hundreds of thousands of test-runs using Monte Carlo simulation. That would reveal all too clearly that the risk of ruin was far larger than might appear from a single backtest.

Next, I asked him whether the strategy was entering and exiting passively, by posting bids and offers, or aggressively, by crossing the spread to sell at the bid and buy at the offer. I had a pretty good idea what his answer would be, given the volume of trades in the strategy and, sure enough he confirmed the strategy was using passive entries and exits. Leaving to one side the challenge of executing a trade for 1,000 contracts in this way, I instead ask him to show me the equity curve for a single contract in the underlying strategy, without the money-management enhancement. It was still very impressive.

The Critical Fill Assumptions For Passive Strategies

But there is an underlying assumption built into these results, one that I have written about in previous posts: the fill rate. Typically in a retail trading platform like Tradestation the assumption is made that your orders will be filled if a trade occurs at the limit price at which the system is attempting to execute. This default assumption of a 100% fill rate is highly unrealistic. The system’s orders have to compete for priority in the limit order book with the orders of many thousands of other traders, including HFT firms who are likely to beat you to the punch every time. As a consequence, the actual fill rate is likely to be much lower: 10% to 20%, if you are lucky. And many of those fills will be “toxic”: buy orders will be the last to be filled just before the market moves lower and sell orders will be the last to get filled just as the market moves higher. As a result, the actual performance of the strategy will be a very long way from the pretty picture shown in the chart of the hypothetical equity curve.

One way to get a handle on the problem is to make a much more conservative assumption, that your limit orders will only get filled when the market moves through them. This can easily be achieved in a product like Tradestation by selecting the appropriate backtest option:

The strategy performance results often look very different when this much more conservative fill assumption is applied. The outcome for this system was not at all unusual:

Of course, the more conservative assumption applied here is also unrealistic: many of the trading system’s sell orders would be filled at the limit price, even if the market failed to move higher (or lower in the case of a buy order). Furthermore, even if they were not filled during the bar-interval in which they were issued, many limit orders posted by the system would be filled in subsequent bars. But the reality is likely to be much closer to the outcome assuming a conservative fill-assumption than an optimistic one. Put another way: if the strategy demonstrates good performance under both pessimistic and optimistic fill assumptions there is a reasonable chance that it will perform well in practice, other considerations aside.

An Example of a HFT Equity Strategy

Let’s contrast the futures strategy with an example of a similar HFT strategy in equities. Under the optimistic fill assumption the equity curve looks as follows:

Under the more conservative fill assumption, the equity curve is obviously worse, but the strategy continues to produce excellent returns. In other words, even if the market moves against the system on every single order, trading higher after a sell order is filled, or lower after a buy order is filled, the strategy continues to make money.

Market Microstructure

There is a fundamental reason for the discrepancy in the behavior of the two strategies under different fill scenarios, which relates to the very different microstructure of futures vs. equity markets. In the case of the E-mini strategy the average trade might be, say, $50, which is equivalent to only 4 ticks (each tick is worth $12.50). So the average trade: tick size ratio is around 4:1, at best. In an equity strategy with similar average trade the tick size might be as little as 1 cent. For a futures strategy, crossing the spread to enter or exit a trade more than a handful of times (or missing several limit order entries or exits) will quickly eviscerate the profitability of the system. A HFT system in equities, by contrast, will typically prove more robust, because of the smaller tick size.

Of course, there are many other challenges to high frequency equity trading that futures do not suffer from, such as the multiplicity of trading destinations. This means that, for instance, in a consolidated market data feed your system is likely to see trading opportunities that simply won’t arise in practice due to latency effects in the feed. So the profitability of HFT equity strategies is often overstated, when measured using a consolidated feed. Futures, which are traded on a single exchange, don’t suffer from such difficulties. And there are a host of other differences in the microstructure of futures vs equity markets that the analyst must take account of. But, all that understood, in general I would counsel that equities make an easier starting point for HFT system development, compared to futures.

May 14, 2019May 14, 2019

ETFs vs. Hedge Funds – Why Not Combine Both?

Grace Kim, Brand Director at DarcMatter, does a good job of setting out the pros and cons of ETFs vs hedge funds for the family office investor in her LinkedIn post.

She points out that ETFs now offer as much liquidity as hedge funds, both now having around $2.96 trillion in assets. So, too, are her points well made about the low cost, diversification and ease of investing in ETFs compared to hedge funds.

But, of course, the point of ETF investing is to mimic the return in some underlying market – to gain beta exposure, in the jargon – whereas hedge fund investing is all about alpha – the incremental return that is achieved over and above the return attributable to market risk factors.

But should an investor be forced to choose between the advantages of diversification and liquidity of ETFs on the one hand and the (supposedly) higher risk-adjusted returns of hedge funds, on the other? Why not both?

Diversified Long/Short ETF Strategies

In fact, there is nothing whatever to prevent an investment strategist from constructing a hedge fund strategy using ETFs. Just as one can enjoy the hedging advantages of a long/short equity hedge fund portfolio, so, too, can one employ the same techniques to construct long/short ETF portfolios. Compared to a standard equity L/S portfolio, an ETF L/S strategy can offer the added benefit of exposure to (or hedge against) additional risk factors, including currency, commodity or interest rate.

For an example of this approach ETF long/short portfolio construction, see my post on Developing Long/Short ETF Strategies. As I wrote in that article:

My preference for ETFs is due primarily to the fact that it is easier to achieve a wide diversification in the portfolio with a more limited number of securities: trading just a handful of ETFs one can easily gain exposure, not only to the US equity market, but also international equity markets, currencies, real estate, metals and commodities.

More Exotic Hedge Fund Strategies with ETFs

But why stop at vanilla long/short strategies? ETFs are so varied in terms of the underlying index, leverage and directional bias that one can easily construct much more sophisticated strategies capable of tapping the most obscure sources of alpha.

Take our very own Volatility ETF strategy for example. The strategy constructs hedged positions, not by being long/short, but by being short/short or long/long volatility and inverse volatility products, like SVXY and UVXY, or VXX and XIV. The strategy combines not only strategic sources of alpha that arise from factors such as convexity in the levered ETF products, but also short term alpha signals arising from temporary misalignments in the relative value of comparable ETF products. These can be exploited by tactical, daytrading algorithms of a kind more commonly applied in the context of high frequency trading.

For more on this see for example Investing in Levered ETFs – Theory and Practice.

Does the approach work? On the basis that a picture is worth a thousand words, let me answer that question as follows:

Systematic Strategies Volatility ETF Strategy

Conclusion

There is no reason why, in considering the menu of ETF and hedge fund strategies, it should be a case of either-or. Investors can combine the liquidity, cost and diversification advantages of ETFs with the alpha generation capabilities of well-constructed hedge fund strategies.

May 7, 2019

A New Approach to Equity Valuation

How Analysts Traditionally Value Equity

I learned the traditional method for producing equity valuations in the 1980’s, from Chase bank’s excellent credit training program. The standard technique was to develop several years of projected financial statements, and then discount the cash flows and terminal value to arrive at an NPV. I’m guessing the basic approach hasn’t changed all that much over the last 30-40 years and probably continues to serve as the fundamental building block for M&A transactions and PE deals.

Amongst several excellent texts on the topic I can recommend, for example, Aswath Damodaran’s book on valuation.

Arguably the weakest point in the methodology are the assumptions made about the long term growth rate of the business and the rate used to discount the cash flows to produce the PV. Since we are dealing with long term projections, small variations in these rates can make a considerable difference to the outcome.

The Monte Carlo Approach

Around 20 years ago I wrote a paper titled “A New Approach to Equity Valuation”, in which I attempted to define a new methodology for equity valuation. The idea was simple enough: instead of guessing an appropriate rate to discount the projected cash flows generated by the company, you embed the riskiness into the cash flows themselves, using probability distributions. That allows you to model the cash flows using Monte Carlo simulation and discount them using the risk-free rate, which is much easier to determine. In a similar vein, the model can allow for stochastic growth rates, perhaps also taking into account the arrival of potential new entrants, or disruptive technologies.

I recall taking the idea to an acquaintance of mine who at the time was head of M&A at a prestigious boutique bank in London. About five minutes into the conversation I realized I had lost him at “Monte Carlo”. It was yet another instance of the gulf between the fundamental and quantitative approach to investment finance, something I have always regarded as rather artificial. The line has blurred in several places over the last few decades – option theory of the firm and factor models, to name but two examples – but remains largely intact. I have met very few equity analysts who have the slightest clue about quantitative research and vice-versa, for that matter. This is a pity in my view, as there is much to be gained by blending knowledge of the two disciplines.

The basic idea of the Monte Carlo approach is to formulate probability distributions for key variables that drive the business, such as sales, gross margin, cost of goods, etc., as well as related growth rates. You then determine the outcome in terms of P&L and cash flows over a large number of simulations, from which you can derive a probability distribution for the firm/equity value.

There are two potential sources of data one can use to build a Monte Carlo model: the historical distributions of the variables and information from line management. It is the latter that is likely to be especially useful, because you can embed management’s expertise and understanding of the business and its competitive environment directly into the model variables, rather than relying upon a single discount rate to account for all the possible sources of variation in the cash flows.

It can get a little complicated, of course: one cannot simply assume that all the variables evolve independently – COGS is likely to fall as a % of sales as sales increase, for example, due to economies of scale. Such interactive effects are critically important and it is necessary to dig deep into the inner workings of the business to model them successfully. But to those who may view such a task as overwhelmingly complicated I can offer several counter examples. For instance, in the 1970’s I worked on large scale simulation models of the North Sea oil fields that incorporated volumes of information from geology to engineering to financial markets. Another large scale simulation was built to assess how best to manage tanker traffic at one of the world’s busiest sea ports.

Creating a simulation model of the financials of a single firm is a simple task, by comparison. And, after you have built the model it will typically remain fundamentally unchanged in basic form for many years making the task of producing valuation estimates much easier in future.

Applications of Monte Carlo Methods in Equity Valuation

Ok, so what’s the point? At the end of the day, don’t you just end up with the same result as from traditional methods, i.e. an estimate of the equity or firm value? Actually no – what you have instead is an estimate of the probability distribution of the value, something decidedly more useful.

For example:

Contract Negotiation

Monte Carlo methods have been applied successfully to model contract negotiation scenarios, for instance for management consulting projects, where several rounds of negotiation are often involved in reaching an agreed pricing structure.

Stock Selection

You might build a portfolio of value stocks whose share price is below the median value, in the expectation that the majority of the universe will prove to be undervalued, over the long term. Or you might embed information about the expected value of the equities in your universe (and their cashflow volatilities) into you portfolio construction model.

Private Equity / Mergers & Acquisitions

In a PE or M&A negotiation your model provides a range of values to select from, each of which is associated with an estimated “probability of overpayment”. For example, your opening bid might be a little below the median value, where it is likely that you are under-bidding for the projected cash flows. That allows some headroom to increase the bid, if necessary, without incurring too great a risk of over-paying.

Recent Research

A survey of recent research in the field yields some interesting results, amongst them a paper by Magnus Pedersen entitled Monte Carlo Simulation in Financial Valuation (2014). Pedersen takes a rather different approach to applying Monte Carlo methods to equity valuation. Specifically, he uses the historical distribution of the price/book ratio to derive the empirical distribution of the equity value rather than modeling the individual cash flows. This is a sensible compromise for someone who, unlike an analyst at a major sell-side firm, may not have access to management information necessary to build a more sophisticated model. Nevertheless, Pedersen is able to demonstrate quite interesting results using MC methods to construct equity portfolios (weighted according to the Kelly criterion), in an accompanying paper Portfolio Optimization & Monte Carlo Simulation (2014).

For those who find the subject interesting, Pedersen offers several free books on his web site, which are worth reviewing.