Position Sizing using the Exponentially Weighted Moving Average of Volatility

19th Dec '15 by mpat89

35 comments 15642 reads

I believe it is currently the accepted principle that a portfolio of shares will be bought with each share making up an equal amount of the portfolio.

Therefore, the same amount of capital is at risk in each individual stock.

The benefit of this is that it is a simple way to spread risk fairly evenly across a broad number of shares. This diversification reduces the specific risk of the portfolio, whereby movements in individual stocks affect the overall value of the portfolio. We aim to reduce this specific risk to a level where we are exposed only to more general factors such as movements in the overall market and the QVM premium.

Recently, after reading some books on the matter, I have been inclined to examine alternate methods of position sizing.

One method that I have found I like the idea of is the use of the EWMA (exponentially weighted moving average) of Volatility.

Using this method, position sizes are based on recent volatility of each stock rather than equal sizes based on £ or $ amounts.

The result is that lower volatility stocks are given higher position sizes, while higher volatility stocks are given lower position sizes. This standardises the overall volatility of the portfolio and should result in a much smoother equity curve, such that sudden spikes in value are less frequent and overall up and down moves should happen more smoothly.

For example, if buying a company with a very low valuation (high value rank) with a recent recovery (quality and momentum) but ultimately extremely high gearing, this company is likely to be far more volatile. Any negative sentiment towards its recovery will result in sharper and more sudden downward movements in price because the price is sensitive to the recovery. Under the volatility position sizing method, this stock would have a lower position size to compensate for this.

A much lower volatile stock, such as a company with a good track record, sensible valuation and continuing good fundamentals, represents a far lower risk. It's share price is likely to be much less volatile, i.e. strong steady move in one direction. The volatility method would allocate this stock a larger position size.

In practice, stocks with a much lower position size:
CNC (expected volatility 104.3%)

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Mark Simpson

22nd Dec '15

50 day ATR won't, of course, be the only consideration in determining position size; but it doesn't seem crazy to me to suggest that for many investors it should be a significant consideration.

I'm afraid it does seem crazy to me that someone would underweight a low risk, high return-potential stock because it has a large 50 day ATR (not specifically talking about Moss Bros (LON:MOSB) or French Connection (LON:FCCN) here.) But then you don't get opportunities in the market without people under-weighting good companies for reasons unrelated to the underlying performance of the business. So in reality I'm glad there are people out there for whom it is a 'significant consideration' :-)

TangoDoc

The problem I have with this and other quasi-scientific systems are that they seem to be a more complicated way to shuffle the deckchairs on a ship that, for all you know, is called the Titanic. I suppose we are all looking for a way to overcome what is, for most of us, an emotionally driven "hobby" away from our own individual arenas of real expertise. I do think we need to be careful not to delude ourselves by using systems to cloud the reality; when we part with our money and back a team we know so little about, we are out of control. It's a bit like supporting a football club in some respects but, with the added touch that is sometimes reminiscent of religion.

tomomo

You wouldn't be underweight the volatile stock. You'd be overweight it, since it would have a place in your portfolio. Perhaps it would be 3% of that portfolio, as opposed to 0.000x% of the FTSE All-Share.

But while you'd be overweight that stock, you might reasonably choose to underweight it within your portfolio, in the sense of giving it less than equal weight.

I guess one way of looking at it that might seem less crazy to you is that you'd be giving the low-volatility stocks which you want to contribute to the performance of your portfolio the opportunity to do so: their potential contributions to the portfolio shouldn't be swamped by the contributions of more volatile stocks. If you don't want their contributions to be diminished by what you refer to as "reasons unrelated to the underlying performance of the business" you need to give them greater than equal weight in the portfolio.

giving the low-volatility stocks which you want to contribute to the performance of your portfolio the opportunity to do so: their potential contributions to the portfolio shouldn't be swamped by the contributions of more volatile stocks.

You want to maximise the return and minimise the volatility of your portfolio. This means you need to consider the co-variance not variance of components.

So what you are talking about is developing an expected return model and something like a stochastic expected volatitility/co-variance model and applying these to your portfolio with Markowitz or Kelly.

Don't get me wrong, I am a big fan of Robert Haugen who was one of the people who found the low-volatilty anomoly and developed expected return and variance models but you have to recognise the implementation of these concepts is a quant strategy.

Taking a bad implementation (RTA or EWMA of vol. vs a true stochastic vol. model) of a small part of a quant strategy and assuming it will enhance a small cap stocking picking strategy doesn't make sense. It's like trying to add nitrous injection to a Nissan Leaf - the technology is completely different, it is at best wasted effort and at worst dangerous.

The point of the kind of ATR-based position sizing I referred to is to try to equalise the contributions of stocks with differing levels of volatility.

Suppose, for example, you were using StockRanks to select 10 stocks to be held for 1 year from selection. You could simply invest the same amount in each issue: 10% of the invested cash into each. If you did so, however, the more volatile issues would have a disproportionately large effect on the end of year p&l. So in this case you might decide to use ATR-based position sizing to equalise the contributions of the different stocks in the portfolio.

Trying to equalise the contributions of different stocks is not the same as trying to capture the low volatility anomaly.

Trying to equalise the contributions of different stocks is also not the same as trying to minimise the volatility of a portfolio for a given level of expected return.

I think ATR-based position sizing is a rational way to attempt to equalise the contributions of different stocks. I do not think that ATR-based position sizing is an effective means to capture the low volatility anomaly, or that it is an effective means to minimise the volatility of a portfolio for a given level of expected return. These are quite different objectives and, as you correctly say, require quite different instruments.

Trying to equalise the contributions of different stocks is also not the same as trying to minimise the volatility of a portfolio for a given level of expected return.

Which is exactly why it is pointless.

It doesn't follow that it is pointless, merely that it has a different point.

I guess so but given that you admit it isn't intended to help generate higher risk-adjusted returns then that point is clearly unrelated to investing.

In its most straightforward application a position sizing rule based on ATR will simply help you to adjust the weight of the stocks in your portfolio so that they contribute more equally to p&l, reducing the dollop of randomness that comes from differences in volatility. I think this is an objective which is both worthwhile and related to investing.

You're right that such a rule won't produce an optimal stock allocation, or overall asset allocation, where "optimal" is understood as meaning: having the highest level of expected return for a given level of volatility. That isn't what such a position-sizing rule does or is intended to do. But an ATR-based position-sizing rule doesn't stop you seeking to generate returns, and it doesn't stop you managing risk. It also doesn't stop you from seeking to capitalise on the low volatility anomaly. You can do any or all of these things while implementing this position-sizing rule. It isn't the enemy of higher risk-adjusted returns: it's simply one instrument that may be useful to investors seeking to achieve higher risk-adjusted returns.

Back in post #10 you seemed to repudiate the equation of risk with volatility and to be scornful of the idea that the only aim of portfolio management is reducing volatility. Now in post #23 you seem to be maintaining that anything which isn't a procedure for maximising volatility-adjusted return is "unrelated to investing" because it isn't "intended to help generate higher risk-adjusted returns". I disagreed less with the post #10 version of you.

There's no inconsistency I was very intentional to say 'higher risk-adjusted returns' precisely because volatility does not equal risk and not optimising but any improvement. Volatility can be a component of risk but only if you have career risk or use gearing. But since there is no evidence that your volatility measure even improves volatility-adjusted returns then it certainly doesn't improve risk-adjusted returns. Volatility is at best a subset of risk.

There is no reason that 'weight [ing] of the stocks in your portfolio so that they contribute more equally to p&l has any advantage to the investor unless it reduces portfolio volatility. I'm not sure why you think this has any value to you - it certainly won't put more money in your bank account or help you sleep better at night and in fact it costs you thinking time that could be better spent on more logically sound portfolio constrction.

pka

Tomomo,

Paradoxically, there might be an advantage in over-weighting a volatile stock in your portfolio if it has negative co-variance with the other stocks in your portfolio, because that would reduce the overall volatility of your portfolio.

In reply to dangersimpson #post 25

Thinking time is negligible. This is a very simple position-sizing procedure. It's supposed to be an improvement on equal cash-weighting, not a panacea for all your risk-adjusted return problems. Quick and easy. Better not perfect. Not right for everyone all the time. Right for some people some of the time.

Suppose you want to capture the out-performance which you believe higher StockRank stocks possess.

To do this you need to purchase enough different stocks for the law of large numbers to start to bite: you don't expect the out-performance to be equally distributed among high StockRank stocks, so you need to buy a decent number of these stocks in order to stand a reasonable chance of getting something close to the average out-performance.

But of course you also need to keep a lid on the number of stocks you buy because otherwise you'll just give all the out-performance to your broker in commissions.

Perhaps you end up deciding that 20 stocks is about the right number.

Maybe you then limit the number of stocks you select from any sector to a maximum of 4. The objective would be to protect yourself to some degree from nasty macro impacts that may be around the corner, while not forcing yourself into under-performing sectors.

So now you have your list of 20 high StockRank stocks.

Time elapsed so far: Hardly any.

Why buy more of the lower-volatility ones? Simply because your objective is to get your hands on the outperformance that you think is distributed unequally and unpredictably between different high-StockRank stocks, and you want to do this as effectively as is reasonably practicable.

If you buy equal cash quantities then you'll have to hope that the StockRank outperformance happens to be predominantly realised in the higher volatility stocks, since these will tend to be the ones which affect the portfolio value most. But of course you have no reason to expect this outcome rather than the opposite.

If on the other hand you use an ATR-based position sizing rule, you will somewhat equalise the impacts of lower and higher volatility stocks on the portfolio. This will give you more reliable access to the out-performance you're trying to get hold of. You'll get more mileage from the diversification you paid your broker to access because each stock you've paid to include in the portfolio will have an approximately equal influence on your p&l.

So you look at the ATRs of the 20 stocks, divide these by the current prices of these stocks and put more money in the stocks where the ATR represents a low proportion of the price level than in those where it represents a high proportion.

Time elapsed so far: Still hardly any. Brain power expended: Very little.

Using a more efficient approach to getting hold of out-performance will, other things being equal, put more money in your bank account. It will reduce the risk of you getting poorer. It has relevance to investing. It isn't pointless.

If this is all too simple and you want to spend more time thinking about portfolio construction then by all means do so. Nothing in using ATR-based position sizing prevents you from doing so. If it doesn't suit your purposes, don't use it. If it doesn't meet all your needs, combine it with something else. If you're especially worried about tail risks then limit the application of ATR-based position sizing so that you don't have more than 10% in any particular stock. If you're concerned about market risk, pre-screen for low beta stocks. Then get the ones with the best StockRank and use ATR-based position sizing.

Obviously if you really, really don't believe in the persistence of volatility (crikey) then ATR-based position sizing will be at best pointless. But suppose you look at a few charts and see that actually volatility does persist enough to make it seem sensible to factor it in: Why then not do so using ATR-based position sizing?

I agree. If you want to manage the volatility of the portfolio you need to take co-variance into account and it may have the implication that you should over-weight a higher volatility stock.

I've just done a quick exercise to illustrate what I think the value of an ATR-based position-sizing procedure is. I think the results of this exercise are quite interesting.

I took the Top 10 stocks Ed Croft selected using StockRanks for his new year "NAPS" portfolio on 6th January. (To keep things simple I ignored the 2nd choice "honourable mention" stocks he also identified.)

http://www.stockopedia.com/content/new-year-naps-t...

These 10 stocks have collectively done staggeringly well since 6th January. If you'd put £10,000 into each at the mid closing price on 6th January (a simplification) and had received no dividends (another simplification), you'd have a profit of £34,794 at the end of today, 22nd December.

I looked at the volatility of the stocks Ed selected. This varied considerably. Using 50-day ATR and calculating this as a proportion of the closing price on 6th January, the volatility varied from 4.57% of closing price for Lamprell (LON:LAM) at the top end down to 1.67% for Animalcare (LON:ANCR) at the bottom.

One relevant issue is: How persistent have these differences in volatility been? Supposing it makes sense in theory to underweight more volatile stocks, does this make any sense in practice, or does the variability of volatility over time mean that, unless you're continually re-balancing the portfolio, the initial weightings are soon completely out of whack with the differing levels of volatility which were their initial justification?

The relative volatility levels of these 10 stocks have been pretty persistent over the course of a year (judging by the beginning and end points). The three most volatile stocks at the outset are still the three most volatile (though their relative order has changed). Two of the three least volatile at the outset are still among the three least volatile. And of the 4 with mid-range relative volatility at the outset, 3 of these still lie in the mid-range.

Another relevant issue is: How much of a bearing does the ATR measure of volatility have on the degree to which performance is likely to deviate from the average over a longer period? Judging from these 10 stocks the answer is: A lot. The average (price change) performance of the 10 shares since 6th January has been +34.8%. The two most volatile stocks at the outset, measured by ATR relative to price level, are the two which have deviated the most from this average. The most volatile, Lamprell (LON:LAM), returned -57.6 percentage points relative to the average. The second most volatile, Dart (LON:DTG), returned +60.1 percentage points relative to the average. Conversely, the least volatile, Animalcare (LON:ANCR), gave a return which differed by only -3.4 percentage points from the average, while the second least volatile, Idox (LON:IDOX), gave a return differing by only 6.9 percentage points from the average.

So what happens when you use ATR-based position sizing?

Firstly, you get significant differences in initial position sizes: Animalcare (LON:ANCR) is allocated £14,188 vs. the allocation for £5,185 for Lamprell (LON:LAM). (At least that was how I applied ATR-based position sizing: Animalcare (LON:ANCR) gets 2.74 X as much money because Lamprell (LON:LAM) has a volatility level which is 2.74 X higher.)

Secondly, the contribution of the more volatile stocks to overall p&l is therefore lessened. Lamprell (LON:LAM) contributes a loss of £2,278 to the equal-weighted NAPS portfolio, which is reduced to a loss of only £1,181 in the ATR-weighted NAPS portfolio. Dart (LON:DTG) contributes a £9,487 profit to the equal-weighted portfolio, which is reduced to £6,612 in the ATR-weighted portfolio.

Thirdly, and conversely, the contribution of the less volatile stocks is therefore increased. £ANCR's contribution of £3,144 profit to the equal-weighted NAPS increases to £4,460 in the ATR-weighted portfolio. £IDOX's contribution of £2,790 profit is increased to £3,866.

And returns? Well, for these stocks for this period returns are very close for the two differently-weighted versions of the portfolio. The equal-weighted return is +34.8% (excluding dividends). The return for the ATR-weighted version is +34.4% (again excluding dividends). (If you were looking at total returns, I think the ATR-weighted version would edge it in terms of return, because the highest volatility stocks in the portfolio carry the lowest dividend yield.)

So is the ATR-weighted version preferable, and if so why?

Obviously looking at 10 stocks over a period of less than 1 year is hardly a sufficient basis for seeking to draw a well-founded, empirically grounded conclusion. But I do think the fate of the NAPS portfolio over the past year does indicate why ATR-weighting is at least appealing.

The worry I have with the equal-weighted version of the portfolio is the outsized contributions of Lamprell (LON:LAM) and Dart (LON:DTG), the two stocks with the highest ATR-calculated volatility when the portfolio was set up. They both made contributions around 60 percentage points different from the average for the portfolio as a whole. As it happened, these two contributions cancelled each other out, as one was positive and the other was negative; but while the actual past was reassuring, there are all-too-possible pasts in which the two contributions didn't cancel each other out. The two most volatile stocks could both have made big positive contributions, but equally they could both have made big negative contributions. By applying ATR-based position sizing the size of both of their contributions was significantly reduced, and the chance of the portfolio producing something close to the average StockRank return for 2015 was thereby increased. Luck was given a smaller part in determining the outcome, StockRank stock-selection a bigger part.

23rd Dec '15

If you buy equal cash quantities then you'll have to hope that the StockRank outperformance happens to be predominantly realised in the higher volatility stocks, since these will tend to be the ones which affect the portfolio value most.

The problem is you are confusing volatility with expected return. By saying higher volatility stocks are the ones that are affecting the portfolio most you are saying that they have a higher expected return. You then conclude that in order to avoid this higher expected return you should underweight them in your portfolio. This is simply a bad strategy.

But suppose you look at a few charts and see that actually volatility does persist enough to make it seem sensible to factor it in: Why then not do so using ATR-based position sizing?

The problem is that looking at a few charts is not a scientific way of whether volatility is persistent or not. Before basing a strategy on it a sensible investor asks for evidence. I've not done an exhaustive search of all research but the research papers I have read on the topic show index volatility to be at least weakly serially auto-correlated with the implication that large caps would be even more weaklyserially auto-correlated and small caps unlikley to be statistically significant. But I'm quite open to changing my mind if you have some academic studies that I'm not aware of. I'm just prepared to do so based on someone eyeballing a chart.

Why then not do so using ATR-based position sizing?

Because a good strategy is based on academic evidence or logical deduction, preferably both. This has neither.

Tomomo, you wrote in your latest post:

"Obviously looking at 10 stocks over a period of less than 1 year is hardly a sufficient basis for seeking to draw a well-founded, empirically grounded conclusion."

Although your post is interesting, I would argue that the sample size you used of 10 stocks over a year is far too tiny to draw any statistically valid conclusions from its results.

The expected excess return for all the stocks bought is the same (that is the purchaser's operating hypothesis). It is not higher for the more volatile stocks. The expected variability of return is higher for the more volatile stocks.

The likelihood of achieving something close to the expected excess return is increased by buying more stocks (but then you have to pay for that, and ultimately you'll run out of the top end StockRanks stocks) and by equalising the extent to which each stock purchased contributes to the overall performance of the portfolio.

A volatility-based position sizing rule doesn't have the effect of telling you to buy less where there's higher expected return and more where there's lower expected return. That would indeed be a bad strategy. Rather a volatility-based position sizing rule gets you to buy less where there a higher expected variability of return and more where the expected variability of return is lower. That makes sense to me.

No doubt there is masses that academic papers could tell us about serial auto-correlation of volatility, and volatility of stocks more generally. I haven't read any papers on this at all. But really, I'm not going to wait til I've read a paper on the hydrative effects of H2O before I drink a glass of water.

If you've looked at a lot of charts you're not going to be in any doubt that different stocks have very different volatility characteristics, and that these are somewhat persistent. Even a few charts will give you the idea. But if you refuse to use your eyeballs because it's too amateurish for you then you won't see it. Personally, I don't think such fastidiousness is especially rational -- rather the opposite.

By the way, it isn't the persistence in the level of volatility that's relevant. Its the extent of persistence in the relative levels of volatility of stocks in your portfolio that will make ATR-based position sizing more or less beneficial. You can't reasonably infer that just because the level of volatility varies a lot for indexes there is therefore unlikely to be much persistence in levels of relative volatility between different stocks.

If volatility drops significantly for the market as a whole it will be possible to look at the serial auto-correlation level of volatility readings for a market index and say, "Oh, so not much serial auto-correlation here". OK, but that's not relevant. What matters is whether the more volatile stocks continue to be the more volatile stocks, the less volatile stocks continue to be the less volatile stocks. It's this that will determine whether the levels of relative volatility of the stocks in your portfolio persist or not.

I agree, pka.

But I also think looking at that portfolio over that period illustrates how ATR-based position sizing would function, and why it might be beneficial.

For an amateur investor like me equal cash position-sizing is the most obvious default position sizing approach. It's almost invisible because it's so ubiquitous. It's likely to be simply taken for granted.

Looking at the NAPS portfolio with two different kinds of position sizing hopefully helps to defamiliarise the obvious option of equal position sizing, and to show why ATR-based position sizing might be better.

mpat89

25th Dec '15

I'm glad to see plenty of discussion around the topic.

To clarify, the purpose of the idea mentioned in the thread was to ensure that no one stock would have a larger or smaller impact on profit simply due to volatility. That way, we have less of a chance of our returns being affected by volatility of individual stocks and a better chance of earning the premium we set out to earn.

I would like to stress this is not intended to be a low volatility strategy. The aim is not to reduce volatility of the portfolio. It's simply to normalise the contribution each stock makes to the portfolio with respect to volatility.

People have discussed implementation, and stated EWMA is a poor method of implementation. If anyone could please point us to resources of better models that would be appreciated.

Like I said it's the investors choice whether to do this. I see the arguments beginning to get academic. I don't have training in finance however I can't understand why higher volatility implies higher expected returns. As far as I know higher volatility simply implies higher variability in returns.

I understand that tomomo has provided an example which is not statistically significant, however I don't believe he set out to do this. He's just demonstrating how it works in practice and why it could be beneficial. Obviously there is a chance the portfolio returns less using volatility based position sizing when compared with equal dollar, however I am under the impression that 'risk-adjusted returns' should improve, due to the normalisation of each stock's contribution to the overall volatility of the portfolio.

In practice I am beginning to favour this method. I don't use volatility based position sizing yet but I tend to find my high volatility share holdings eat away at my overall returns. Obviously by reducing my exposure to these I would also be limiting my upside.

Therefore, if anyone is willing to move the discussion towards how a volatility based approach to position sizing could best be implemented (i.e. better alternative to EWMA), I'm sure that would be of interest to those who agree with the idea highlighted in the original post.

gus 1065

Morning Mpat.

Just a non-academic quickie as to your comment "why does higher vol imply higher expected returns...". It's just a simple observation that from a given price a stock can go up to infinity (unlimited gain) but only down to zero (a maximum loss).

If you invest £1 in a single stock and assign classic Black-Scholes probability to every price movement up or down from this point, there are some potential gains (think bell curve distribution about £1 level) which are greater than £1 ie >£2) but no possible situation where the loss is more than £1. The distribution is asymmetric. Accordingly, the more volatile the stock price, the greater the chance of moving into the >£2 area of the curve so the higher the potential profit from the universe of possible outcomes. Lots of reasons why this falls down in the real world and even more theoretical ones too, but intuitively it seems reasonable.

I think this is one of the main arguments rolled out to suggest shorting a stock is potentially very risky because your upside is limited (stock price falls to zero) but loss is unlimited.

Anyway, back to dealing with the turkey.....

Happy Christmas!

Gus.

Position Sizing using the Exponentially Weighted Moving Average of Volatility

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