Thursday, 2 April 2020

How fast should we trade?

This is the final post in a series aimed at answering three fundamental questions in trading:


  • How fast should we trade? (this post)
Understanding these questions will allow you to avoid the two main mistakes made when trading: taking on too much risk and trading too frequently. Incidentally, systematic traders can add another list to that sin: overfitting. But that is a topic too large to be covered in a single post, and I've written about it enough elsewhere in the blog.

As with the other two posts this topic is covered in much more detail in my two books on trading: "Systematic Trading" and "Leveraged Trading"; although there is plenty of new material in this post as well. If you want more information, then it might be worth investing in one of those books. "Leveraged Trading" is an easier read if you're relatively new to trading.

The timing of my posts about risk has turned out to be perfect, with the Coronavirus currently responsible for severe market movements as well as thousands of deaths. It's less obvious why trading too frequently is a problem. The reason is costs. Taking on too much risk will lead to a fast blowup in your account. Trading too often will result in high costs being paid, which means your account will gradually bleed to zero. As I write this, I notice for the first time how often we use metaphors about losing money which relate to death. For obvious reasons I will try and avoid these for the rest of the post.

Incidentally, I'm not going to post anything about 'trading and investing through the Coronavirus'. I have put a few bits and pieces on twitter, but I don't feel in the mood for writing a long post about exploiting this tragedy for financial gain.

Neithier will I be writing anything about the likely future path of markets from here. As you know, I don't feel that making predictions about price movements is something I'm especially good at. I leave that to my trading systems. Finally, I won't be doing any analysis of the models used for predicting Coronavirus deaths. I leave that to epidemiologists.

I will however be posting my normal annual update on performance after the UK tax year ends in a few days time. And I will probably, at some point in the future, write a post reviewing what has happened. But not yet.



Overview


How fast should we trade? We want to maximise our expected returns after costs. That's the difference between two things:


  • Our pre-cost returns
  • Our costs

The structure of this post is as follows: Firstly I'll discuss the measurement and forecasting of trading costs. Then I will discuss how expected returns are affected by trading speed. Finally I will talk about the interaction between these two quantities, and how you can use them to decide how quickly to trade.



Types of costs


There are many different kinds of costs involved in trading. However there are two key categories:


  • Holding costs
  • Trading costs

Holding costs are costs you pay the whole time you have a position on, regardless of whether you are trading it. Examples of holding costs include brokerage account fees, the costs of rolling futures or similar derivatives, interest payments on borrowing to buy shares, funding costs for FX positions, and management fees on ETFs or other funds.

Trading costs are paid every time you trade. Trading costs include brokerage commissions, taxes like UK stamp duty, and the execution costs (which I will define in more detail below). 

Some large traders also pay exchange fees, although these are normally included as part of the brokerage commission. Other traders may receive rebates from exchanges if they provide liquidity.

The basic formula for calculating costs then is:

Total cost per year = Holding cost + (Trading cost * Number of trades)




Execution costs


Most types of costs are pretty easy to define and forecast, but execution costs are a little different. Firstly a definition: the execution cost for a trade is the difference between the cost of trading at the mid-price, and the actual price traded at.

So for example if a market is 100 bid, 101 offer, then the mid-price is just the average of those: 100.5

Some people calculate the mid price as a weighted average, using the volume on each side as the weight. Another term for this cost is market impact.

If we do a market order, and our trade is small enough to be met by the volume at the bid and offer, then our execution cost will be exactly half the spread. If our order is too large, then our execution cost will be larger.

Who actually earns the execution cost you pay? Judging by his smile, it's this guy

Broadly speaking, we can estimate execution costs or measure them from actual trading. You can estimate costs by looking at the spreads in the markets you trade, or using someone elses estimates.

A nice paper with estimates for larger traders is this paper by Frazzini et al, check out figure 4. You can see that someone trading 0.1% of the market volume in a day will pay about 5bp (0.05%) in execution costs. Someone trading 0.2% of the volume will pay 50% more, 7.5bp (0.075%).

When estimating costs, there are a few factors you need to bear in mind. Firstly, the kind of trading you are doing. Secondly, the size of trading.

  • Smaller traders using market orders: Assume you pay half the spread
  • Smaller traders using limit orders or execution algos: You can pay less, but  (I pay about a quarter of the spread on average, using a simple execution algo)
  • Larger traders: Will pay more than half the spread, and will need to acccount for their trading volume.
You can use execution algos (which mix limit and market orders) if you are trading reasonably slowly. You can use limit orders if you're trading a mean reversion type strategy of any speed, with the limits placed around your estimate of fair value (though you may want to implement stop-losses, using market orders). If you are trading a fast trend following strategy, then you're going to have to use market orders.

If you're trading very quickly, then assuming a constant cost of trading is probably unrealistic since the market will react to your order flow and this will significantly change your costs. In this case I'd suggest only using figures from actual trades.

There are other ways to reduce costs, such as smoothing your position or using buffering. If you are trading systematically you can incorporate these into your back-test to see what effect they have on your cost estimates.


Linear and non-linear



An important point here is that smaller traders, to all intents and purposes, face fixed execution costs per trade. If they double the number of trades they do, then their trading costs will also double. Smaller traders have linear trading costs. 

Holding costs will be unaffected by trading, and other costs eg commissions may not increase linearly with trade size and frequency, but this is a reasonable approximation to make.

But larger traders face increasing trading costs per trade. If they do larger size or or more trades, their costs per trade will increase (eg from 5bp to 7.5bp in the figures given in the Frazzini paper above). If they double the number of trades they do their execution costs will more than double; using the figures above they will increase be a factor of 3: twice because they are doing double the number of trades, and then by another 50% as the cost per trade is increasing. Larger traders have non linear trading costs.

Normalisation of costs

What units should we measure costs in? Should it be in pips or basis points? Dollars or as a percentage of our acount value?

For many different reasons I think the best way to measure costs is as a return adjusted for risk. Risk is measured, as in previous posts, as the expected annualised standard deviation of returns.

Suppose for example that we are buying and selling 100 block shares priced at $100 each. The value of each trade is $10,000. We work out our trading costs at $10 per trade, which is 0.1%. The shares have a standard deviation of 20% a year. So each trade will cost us 0.1 / 20 = 0.005 units of risk adjusted return. Notice how similar this is to the usual measure of risk adjusted returns, the Sharpe Ratio. We are effectively measuring costs as a negative Sharpe Ratio.

We don't include a risk free rate in this calculation, as otherwise we'd end up cancelling it out when we subtract costs as a Sharpe Ratio from pre-cost returns measured in the same units.

Why does this make sense? Well, it makes it easier to compare trading costs across different instruments, account sizes, and time periods. Trading costs measured in dollar terms look very high for a large futures contract like the S&P 500, but they're actually quite low. Because of the COVID-19 crisis, spreads in most markets are pretty wide at the moment, but this means costs in risk adjusted terms are actually pretty similar. 

It also relates to how we scale positions in the second post of this series. Since we scale positions according to the risk of an instrument, it makes sense to scale costs accordingly.



Estimating the number of trades


Let's return to the basic formula above: 

Total cost per year = Holding cost + (Trading cost * Number of trades)

We're going to need to calculate the expected number of trades. How to do this?

  • We can infer it from the size of our stop-loss relative to volatility, defined in the first post as X (this works no matter what kind of trader you are)
  • Systematic Traders: We can get it from a backtest
  • Systematic Traders: We can use some heuristics based on the kind of trading system we are running

You can find heuristics for different trading systems in both of my books on trading; in this post I'm going to focus on the stop loss method as it's simpler, applies to all traders, and is consistent with the methodology I'm using in the other posts.

Here's the table you need:

Fraction of volatility 'X'    Average trades per year

0.025                                97.5
0.05                                 76.5
0.1                                  46.9
0.2                                  21.4
0.3                                  11.9
0.4                                   7.8
0.5                                   5.4
0.6                                   4.0
0.7                                   3.1
0.8                                   2.4
0.9                                   2.1
1.0                                   1.7



We will use the data in this table later when we try and work out how fast we should be trading.


Trading cost calculations: example


We know have enough information to work out how the trading costs for a given instrument and stop loss fraction.

In my book, "Leveraged Trading", I include examples for all the main types of traded instruments (futures, spot FX, spread bets, CFDs and stock/ETF trading). Here however there isn't really enough space, so I'm just going to focus on my favourite: futures.

As I started out life as a fixed income trader, let's consider the costs of the Eurodollar future. Eurodollars are relatively pricey to trade for a future, but still cheaper than the products most retail investors prefer like CFDs, spread bets and spot FX.

Each contract index point is worth $2500 and the current price of the June 2023 I hold is $99.45 (but that may change!). So each contract has a current notional value of 2500*99.45 = $248,625. My broker charges $1 per contract in commission, and the spread is 0.005 of a point wide (except on the front contract: but don't trade that!).

To trade one contract as a small trader with a market order will cost half of the spread: 0.5*0.005*$2500 = $6.25 plus the commission of a $1 = $7.25. That is 0.0029% of the notional value. There are no taxes or further fees due. It doesn't matter how many contracts we trade, it will always cost 0.0029% of the notional value per trade.

What about holding costs? Each contract has to be rolled quarterly. It's usually possible to do the roll as a calendar spread rather than two seperate trades. This reduces risk, but also means it will cost the same as a regular trade in execution cost (though we will pay two lots of commission). So each roll trade will cost $6.25 plus $2 = $8.25, or 0.00332% of the notional value. Four lots of that per year adds up to 0.0132% in holding costs.

Let's convert these into risk adjusted terms. The risk of Eurodollars is currently elevated, but in more normal times it averages about 0.5% a year. So the execution cost will be 0.0029/0.5 = 0.0058 and the holding cost is 0.0132/0.5 = 0.026. Both in units of Sharpe Ratio.

Here's our formula again:

Total cost per year = Holding cost + (Trading cost * Number of trades)

Total cost per year = 0.026 + (0.0058 * Number of trades)

We could now plug in a value of X into the table above, for example if we used X=0.5 -> 5.4 trades per yer:

Total cost per year = 0.026 + (0.0058 *5.4) = 0.058


Pre-cost returns: Theory


Let us now turn our attention to pre-cost returns. How are these affected by trading speed? Naively, if we double the number of trades we do in a given timeframe, can we double our profits?

We can't double our profits, but they should increase. Theoretically if we double the number of trades we do we will increase our profits  by the square root of 2: 1.414 and so on. This is down to something called The Law Of Active Management. This states that your 'information ratio' will be proportional to the square root of the number of uncorrelated bets that you make. If we make some assumptions then we can boil this down to your return (or Sharpe Ratio) being proportional to the square root of the number of trades you make in a given time frame.


Pre-cost returns: Practice


LAM is a theory, and effectively represents an upper bound on what is possible. In practice it's extremely unlikely that LAM will always hold. Take for example, the Sage of Omaha.
Ladies and Gentlemen, I give you Mr Warren Buffet.

His information ratio is around 0.7 (which is exceptionally good for a long term buy and hold investor), and his average holding period is... well a long time but let's say it's around 5 years. Now under the Law of Active Management what will Warren's IR be if he shortens his holding period and trades more?

X-Axis: Holding Period. Y-Axis: Information ratio


Shortening it to 2 years pushes it up to just over 1.0; pretty good and probably achievable. Then things get silly and we need a log scale to show what's going on. By the time Warren is down to a one week holding period his IR of over 10 put's him amongst the best high frequency trading firms on the planet, despite holding positions for much longer.

When the graph finishes with a holding period of one second, still well short of HFT territory, Warren has a four figure IR. Nice, but very unlikely.

This is a silly example, so let's take a more realistic (and relevant) one. The average Sharpe Ratio (SR) for an arbitrary instrument achieved by the slowest moving average crossover rule I use, MAV 64,256, is around 0.28. It does 1.1 trades per year. What if I speed it up by using shorter moving averages, MAV 32,128 and so on? What does the LAM say will happen to my SR, and what actually happens.

X-axis: Moving average rule N,4N. Y-axis Sharpe Ratio pre-costs

If I turn the dial all the way and start trading a MAC 2,8 (far left off the graph) the LAM says the Sharpe should be a stonking 1.68. The reality is a very poor 0.07. Momentum just doesn't work so well at shorter timeframes, although it does consistently well between MAC8 and MAC64. You can't just increase the speed of a trading rule and expect to make more money; indeed you may well make less.


Net returns


We are now finally ready to put pre-cost returns together with costs and see what they can tell us about optimal trading speeds. For now, I will stick with using a set of moving average rules and the costs of trading Eurodollar futures. Later in the post I'll discuss how you can set stop-losses correctly in the presence of trading costs.

Let's take the graph above, but now subtract the costs of trading Eurodollar futures using the formula from earlier:

Total cost per year = 0.026 + (0.0058 * Number of trades)

The number of trades for each trading rule will come from backtests, but there are also values in both of my trading books that you can use.

X-Axis: Moving average rule, Y-axis Sharpe Ratio before and after costs


The faster rules look even worse now and actually lose money. For this particular trading rule the question of how fast we can trade is clear: as slow as possible. I recommend keeping at least 3 variations of moving average in your system for adequate diversification, but the fastest two variations are clearly a waste of money.

Important: I am comparing the average SR pre-cost across all instruments with the costs just for Eurodollar. I am not using the backtested Sharpe Ratios for Eurodollar by itself, which as it happens are much higher than the average due to secular trends in interest rates. This avoids overfitting.

These results are valid for smaller traders with linear costs. Just for fun, let's apply an institutional level of non linear costs. We assume that costs per trade increase by 50% when trading volume is doubled:

X-Axis: Moving average rule, Y-axis Sharpe Ratio with LAM holding before and after costs for larger traders

I'm only showing the LAM here; the actual figures are much worse. Even if we assume that LAM is possible (which it isn't!), then speeding up will stop working at some point (here it's at around MAC16). This is because pre-cost returns are improving with square root of frequency, but costs are increasing more than linearly.


Net returns when returns are uncertain


So far I've treated pre-cost returns and costs as equally predictable. But this isn't the case. Pre-cost returns are actually very hard to predict for a number of reasons. Regular readers will know that I live to quantify this issue by looking at the amount of statistical variation in my estimates of Sharpe Ratio or returns.

Let's look at the SR for the various speeds of trading rules, but this time add some confidence intervals. We won't use the normal 95% interval, but instead I'll use 60%. That means I can be 20% confident that the SR estimate is above the lower confidence line. I also assume we have 20 years of data to estimate the SR:

X-axis: Moving average variations. Y-axis: Actual Sharpe Ratio pre-costs, with 60% confidence bounds applied

Notice that although the faster crossovers are kind of rubbish, the confidence intervals still overlap fairly heavily, so we can't actually be sure that they are rubbish.

Now let's add costs. We can treat these as perfectly forecastable with zero sampling variance, and compared to returns they certainly are:

X-axis: Moving average variations. Y-axis: Actual Sharpe Ratio net of costs, with 80% confidence bounds applied

Once we apply costs there is much clearer evidence that the fastest crossover is significantly worse than the slowest. It also looks like we can be reasonably confident (80% confident to be precise) that all the slower crossovers have an expected SR of at least zero.



A rule of thumb


All of the above stuff is interesting in the abstract, but it's clearly going to be quite a lot of work to apply it in practice. Don't panic. I have a heuristic; I call it my speed limit:



SPEED LIMIT: DO NOT SPEND MORE THAN ONE THIRD OF YOUR EXPECTED PRE-COST RETURNS ON COSTS

How can we use this in practice? Let's rearrange:


Total cost per year = Holding cost + (Trading cost * Number of trades)


(speed limit) Max cost per year = Expected SR / 3

Expected SR / 3 =  Holding cost + (Trading cost * Max number of trades)

Max number of trades = [(Expected SR / 3) - Holding cost] / Trading cost

Specifically for Eurodollar:

Total cost per year = 0.026 + (0.0058 * Number of trades)
Max number of trades = [(Expected SR / 3) - 0.026] / 0.0058

The expected SR varies for different trading rules, but if I plug it into the above formula I get the red line in the plot below:

X axis: Trading rule variation. Y-axis: Blue line: Actual trades per year, Red line: Maximum possible trades per year under speed limit

The blue line shows the actual trades per year. When the blue line is above the red we are breaking the speed limit. Our budget for trading costs and thus trades per year is being exceeded, given the expected SR. Notice that for the very fastest rule the speed limit is actually negative; this is because holding costs alone are more than a third of the expected SR for MAC2.

Using this heuristic we'd abandon the two fastest variations; whilst MAC8 just sneaks in under the wire.  This gives us identical results to the more complicated analysis above.


Closing the circle: what value of X should I use?!


The speed limit heuristic is awfully useful for systematic traders who can accurately measure their expected number of trades and . But what about traders who are using a trading strategy that they can't or won't backtest? All is not lost! If you're using the stoploss method I recommended in the first post of this series, then you can use the table I included earlier to imply what value of X you should have, based on how often you can trade given the speed limit.

For trading a single instrument I would recommend using a value for expected Sharpe Ratio of around 0.24 (roughly in line with the slower MAC rules). 

Max number of trades = [(Expected SR / 3) - Holding cost] / Trading cost
Max number of trades = [0.08 - Holding cost] / Trading cost

Let's look at an example for Eurodollars:

Max number of trades = [0.08 - 0.026] / 0.0058 = 9.3

From the table above:

Fraction of volatility 'X'    Average trades per year

...                                   ...
0.3                                  11.9
0.4                                   7.8
0.5                                   5.4
...                                   ...


This implies that the maximum value for 'X' in our stop loss is somewhere between 0.3 and 0.4; I suggest using 0.4 to be conservative. That equates to 7.8 trades a year, with a holding period of about 6 to 7 weeks.

Important: You also need to make sure your stop loss is consistent with your forecast horizon. For discretionary traders, if you're expecting to trade once a month make sure your trading is based on expected price movements over the next few weeks. For systematic traders, make sure you use a trading rule that has an expected holding period which matches the stoploss holding period.




Summary


I've gone through a lot in the last few posts, so let's quickly summarise what you now know how to do:


  • The correct way to control risk using stop losses: trailing stops as a fraction of annualised volatility ('X')
  • How to calculate the correct position size using current volatility, expected performance, account size, strength of forecast and number of positions.
  • The correct value of 'X' given your trading costs


Knowing all this won't guarantee you will be a profitable trader, but it will make it much more likely that you won't lose money doing something stupid!


Thursday, 5 March 2020

How much risk should we take?

This is the second of three posts aimed at answering three fundamental questions in trading:

  • How should we control risk (previous post)
  • How much risk should we take? (this post)
  • How fast should we trade? (next post)

These questions are extremely important, IMHO much more important than the question of which funky indicator to use.

I won't be able to discuss all the finer details of position scaling here, as the post would be extremely long. If you find my approach interesting, you may want to read some more about it in my latest book, Leveraged Trading.

I will also be making some references to my first book, Systematic Trading. That's a more advanced piece of work, and if you're relatively new to trading you're probably going to want to read Leveraged Trading first.

Inevitably then this post will partly read like a long advert for both books. But there is still enough here for you to do basic position scaling without needing to spend money.


Recap


In the previous post I explained why you should set stop losses at a multiple of the annualised standard deviation of price changes, and ignore position sizing and account size. Instead, I promised that I'd address the issue of position sizing in this current post.


Overview 


Broadly speaking, the optimal position size will depend on the following factors:


  • The size of your account ('account size')
  • How risky the instrument you are trading is ('instrument risk')
  • How much risk you are willing / able / should to take ('risk target')
  • How confident you are about the position ('forecast scaling')
  • How many positions you are currently / likely to hold ('portfolio size')
  • How to translate position size into units of what you're actually trading ('position scaling')

Account size

Positions should always be calculated as a proportion of the 'amount of capital you have at risk' (capital for short). What on earth does 'capital at risk' mean? It's the money you're prepared to lose trading. Usually this would be the current value of your trading account: the current value of any positions held, plus any cash in your account.

Some exceptions to this rule might include:

  • Your account is worth more than your capital at risk. For example, in my account about 85% of my account value is at risk, the rest is 'spare'
  • Your account is worth less than your capital at risk. You have other funds outside of your account that you're prepared to lose trading. 

If your account value is not equal to your capital at risk, you need to make sure that your capital at risk is adjusted for trading profits and losses:

  • If you lose money, then deduct any losses. 
  • If you make money, then add any profits to your account value
  • You may decide not to increase your capital at risk beyond a certain point (this is what I do). You won't benefit from compounding, but you will ensure you can't lose more than a certain amount (slightly technical discussion here).
This can get complicated, which is why it's easier to use your current account value. In particular, I really don't recommend keeping money outside of your account, which you intend to use to replenish it in case of losses. Once you get used to shovelling money into your account to replace losses, you may find it hard to get out of the habit.

Steve had a $5,000 bankroll. He had originally intended to put another $5,000 into his account if he lost money.
That was $250,000 ago.



Instrument risk

There are as many ways of measuring risk as there are people who care about risk (okay slight exaggeration); none are perfect, but I'm going to opt for a relatively simple measure: the expected annualised standard deviation of returns.

To give you a feel, the relevant statistic for S&P 500 stocks is ~16% a year (although at the time of writing, amongst the turmoil of the Coronavirus sell off, it's considerably higher). Government bonds with a maturity of around 10 years are about half that. Something like Bitcoin is more like ~100% a year.

You can use a spreadsheet to find this figure given some data, like this one. If you prefer to measure risk using the well known ATR, then as a rule of thumb multiplying the daily ATR by 14 will give you the annual standard deviation.


Risk targeting


Once you have your capital you need to work out what risk target you are going to have on it (again measured as the expected annualised standard deviation of returns). Your risk target will depend on a few different factors

  • Your appetite for risk
  • How much leverage your broker will give you
  • How much leverage is safe
  • What your expected performance is like

Risk appetite


How much risk can you handle?

Would you be happy running your system at the same kind of level of risk as stocks (around 15% to 25% a year)? So, for example, on a really bad day like the ones we had as I write this in early March 2020, you might lose 5% in a single day? What about even higher than that?


Broker leverage


Your broker will limit the amount of leverage you can use (in practice it might be the regulator, or the exchange that is setting the limits). They will do this by setting a certain amount of minimum margin that is required to hold a given instrument.

In practice broker leverage limits are still pretty generous, and will rarely be a constraint for a sensible trader.

To work out what risk target is implied by a particular leverage limit, simply multiply the possible leverage by the instrument risk. So, for example, if you're limited to 5 times leverage and the instrument risk is 10%, then the maimum risk target allowed by your broker is 5*10% = 50%.


Safe leverage




Heard of fat tails AKA Black Swans AKA bad stuff happening? Occasionally the market just pukes. October 1987 for example. In a market where the average annual standard deviation is ~16% it ought to be impossible for the market to crash by 23% in a day, but it did.

If a 1987 style crash happened, how much pain could you take? Would you be happy to lose say half your account? Then the maximum leverage you should use is 2.17 times: with 2.17 times leverage your loss would have been 23%*2.17 = 49.9%.

Again you'd need to translate this into a risk target by multiplying by the instrument risk. If for example the instrument risk is 15%, then with a safe leverage of 2.17 your maximum risk target would be 15%*2.17 = 32.6%


Expected performance


The better your trading system is, the more risk you can take. If for example you had a system that always made money, then you could take infinite risk. If your system always lost money then the correct risk target is 0%.

In between these two extremes there is a neat theoretical formula called the Kelly Criteria which basically says this:

Optimal risk target = Expected Sharpe Ratio

If for example your Sharpe Ratio was 0.5, then your optimal risk target would be 50%. Most people think the Kelly formula is too aggressive. A better rule of thumb is to use half the optimal risk target. In this case we'd use a risk target of 25%.

What kind of Sharpe Ratio should we expect?

Most amateur traders don't know their risk target. From reading about the kind of systems many so called 'experts' on the internet are running, risk targets of 100% or even higher are not uncommon. This is madness.

We can see from this list that some muppet 'guru' who punts FX and has a YouTube channel is unlikely to be justified in using a 100% implicit risk target which would imply a Sharpe Ratio of at least 2.0.

Back in 2012 Alex Hope was a 23-year-old self-proclaimed currency trading expert who received a wave of publicity after reportedly spending £125,000 on a single bottle of champagne, here seen with some Z list reality TV Star in some dodgy nightclub.
It turned out he was a very naughty boy and he got 7 years in jail.

It's also possible to measure your trading performance, and infer from that how much it is safe to increase your expected Sharpe Ratio. So a new trader might start assuming a SR of 0.24 (risk target 12%), and eventually scale up to 0.5 or higher if they are very profitable over several years.


So... what risk target should I use?


You should use the most conservative risk target. If for example:

  • Your tolerance for risk is 25% annualised standard deviation of returns
  • Your broker will allow a leverage limit which translates to 50% annualised standard deviation of returns
  • A safe leverage limit to use is equivalent to 32.6% annualised standard deviation of returns
  • The Kelly criteria suggests you should use 12% annualised standard deviation of returns

... then you should use 12% as your annualised risk target. In my book Leveraged Trading I recommend using a risk target of 12% if you're running the simplest 'one indicator / one instrument' system. More complex systems can have higher risk targets. This is a good starting point for most traders.

Most professional managers have risk targets of between 10% and 30%. I myself run at 25%.



Forecast scaling

Have you ever had a trade that you thought "This is a slam dunk. I'm going to go all in". A trade so good that it made you mix your metaphors until the cows came home?

You've probably also had so-so trades that you put on because you were bored and were waiting for the "big one".

Should these two types of trades have the same risk? Probably not. You should probably put more money into the slam dunks than the so-sos. To do this I like to calculate what I call a forecast.

This is a number between -20 and +20 reflecting how confident we are about our forecast, scaled to have an average absolute value of 10. Slam dunks would be -20 (max short) or +20 (max long). So-Sos would be like -3, or +2.5. And the average long trade would be +10.

If you're running a mechanical trading system you can design it so that it will automatically produce a number between -20 and +20 (more in Systematic Trading). Otherwise you can use gut feel.


Portfolio size

How does portfolio size affect position sizing?

The short answer: Bigger portfolios mean proportionally less risk per position. If you have two positions on, then you should have roughly half the average risk per position. Three positions, a third of the risk. And so on.

The slightly longer answer: An interesting question is 'how many positions do you have'. You're unlikely to have exactly 5 positions all the time. What if you usually have 5 positions on, but sometimes 10? I won't deal with this here, but it's covered in Systematic Trading.

The much longer answer: A diversified portfolio can take on more risk per position. It's also possible to allocate capital in such a way to increase the diversification. Again, I explain how to handle this in both Systematic Trading and Leveraged Trading.


Position scaling: putting it all together


Let's recap what we have:

  • The size of our capital at risk: a £,$ or other number: C
  • Our risk target, measured as an annualised standard deviation of risk: T
  • The risk of the instrument we are trading, measured as an annualised standard deviation of risk: V 
  • A scaled forecast: A number between -20 and +20 reflecting how confident we are about our forecast, scaled to have an average absolute value of 10: F
  • Portfolio size: A number indicating how many positions we expect to hold at any one time: N

We can now calculate the amount of exposure we want to take, which will be:

(1/N)*C*(T/V)*(F/10)

Some intuition around this formula:

  • The more positions we have, the less we can put into each. Here our capital is equally split (1/N), with no account for diversification. More complex methods can be found in my first or third book.
  • The more capital we have, the more we can bet
  • We want to scale our bet according to the ratio between our risk target, and the risk of the instrument
  • We want to scale our bet according to the strength of the forecast, where an average forecast is 10

So we now know that we want to take the equivalent of £1,500 of exposure in BP shares, or $123,456 in Crude oil futures. What do we actually do now?


  1. For BP shares that's easy; we just divide the exposure by the share price (about £4) and buy or short the required number of shares (about 375 shares). 
  2. For futures it's a bit more complicated; the price of Crude is about $50 but each contract price point has a value of $1,000 so the required number of contracts is $123456/($50*1000) = 2.46, which you would round to 2. 
  3. For FX you may need to convert the exposure into a different currency and then do some rounding if you're limited to a certain lot size.



Summary


You know now how to set your stop losses, and how big your positions should be. In the final post in this series I'll discuss how fast you should trade, since this determines the exact calibration of our stop loss.


Thursday, 6 February 2020

What is the right way to set stop losses?

Stop losses are the most common method used by traders to control risk. However, they're often used inappropriately. In this post I'll quickly bust some of the myths around them, and explain how to use them properly.

This is the first of three posts aimed at answering three fundamental questions in trading:


  • How should we control risk (this post)
  • How much risk should we take? (next post)
  • How fast should we trade? (final post)
If you find my approach interesting, you may want to read some more about it in my latest book, "Leveraged Trading".

(Note to regular readers of the blog or any of my books: you won't find much new here. But I'm writing these posts in the hope that they will start appearing in Google searches to stop people making silly mistakes. Think of these three posts as the 'gateway drug' to the world of trading professionally).



What is a stop loss?



To make sure we're all on the same page, I'm defining a stop loss in the following way. I use trailing stop losses. 

Assuming you have a long position, you should sell when the price has fallen by more than X below it's high watermark. The high watermark is the highest price that a stock (or future, or whatever) has reached since you purchased it. Of course if the thing turned out to be a complete lemon, then the high watermark will be the purchase price.

If you have a short position, then you should buy (closing your short) when the price has risen by more than X above the low watermark. The low watermark, as you've probably guessed, is the lowest price that the stock has reached since you bet on it going down. I'll mostly frame the examples here in terms of long positions, since they're easier to get your head around, but everything I say is applicable to shorts once you've flipped the language around (long->short, fallen-> risen, high watermarket-> low watermark, highest price->lowest price).

Here's a real example. I bought shares in Go-Ahead group (a UK bus company) in September 2017. Go-ahead group has the ticker GOOG, and everyone I know that isn't a finance geek assumes this is the ticker for Google. But as any fool knows, Google nowadays prefers to be identified as Alphabet and hence has the ticker.... GOOGL. So no danger at all of confusing a modestly sized UK bus company with a global tech titan.

Let's have a look at the chart:

https://www.hl.co.uk/shares/shares-search-results/g/go-ahead-group-ordinary-10p/share-charts

My acquisition price was £15.65 and I set my stop loss at 30% below that:

 (1-0.3)*15.65 = £10.96

(The whys and wherefores of where to set the stop loss will follow later in the post. For now, just take 30% as given).

If the price had subsequently fallen below £10.96 then I would have sold GOOG. But it didn't! In fact it went up, hitting a high water market (HWM) of just under £18 in November 2017. I would have adjusted my stop loss on every high, and it would have reached:


 (1-0.3)*17.89 = £12.52


Then some bad ju-ju hit GOOG and the price fell. Fell below £17, £16... kept falling until it was below the price I had acquired it at (£15.65, if you have a short attention span).

But it never reached my stop loss level of £12.52, so I hung on. In February 2018 the price bottomed out at £13.38. And then it rallied. And rallied some more. And by April it was making new highs, hitting a HWM of £18.10. So I adjusted my stop loss to:


 (1-0.3)*18.10 = £12.67

In fact it continued rallying some more all the way up to £19.64, which meant my stop loss was now £13.75.

I won't continue to bore you with the ups and downs of this stock, but to cut a long story short the most recent HWM was £22.78 set in November last year, and thus my current stop is £15.95. Notice that if the price falls to that level then, assuming I can sell exactly at my stop, I will 'lock in' a profit of £15.95 - £15.65 = £0.30. Not amazing after over 3 years, but better than nothing.


Some other kinds of stop losses



Notice some key features of the trailing stop loss:
  • The stop 'trails' the price, ratcheting upwards with every new HWM
  • Eventually, we're 'guaranteed'* not to lose more than our initial investment, and even to lock in a profit.
  • The most* we can lose at any time is 30% of our profits*
* Caveats! This assumes we can get out exactly at our stop

We can contrast this with a fixed stop loss, where I would have set the stop at £10.96 and left it there. A few scenarios can play out here. The price could plummet to £10.96, and you'd close: exactly the same as the trailing stop. If the price follows the path we've seen here, well it would still be holding the position, so again now change. 

But importantly, if the price now falls down to £10 then the trailing stop will prevent us losing more than 30% off the best price, whilst the fixed stop will mean we always lock in a loss on our initial investment when we close. 

Even if the price never gets down that far we could end up owning stocks forever when there are better opportunities out there; with a trailing stop set at the right level you will eventually close your position and have a chance to redeploy your capital.

A special case of a fixed stop is the breakeven stop. You initially set a stop below your purchase price, but once it's risen a bit your reset it permanently at your entry price. Guess what, there is nothing special about your entry level as far as the market is concerned. This is no better than any other kind of fixed stop.

Some people like to use stop profits eithier instead of, or in addition to, stop losses. A stop profit or profit target is a target price at which you will sell the stock. Clearly this doesn't provide you with any protection against losses, so you need to have a stop loss as well. Even then I have two issues with using profit targets. Firstly, they add unneccessary complexity to your trading. Secondly, I don't know where a stock might end up, so why set a target? 

Consider Intermediate Capital Group (ICP), which I bought at £4.26 in July 2016:

https://www.hl.co.uk/shares/shares-search-results/i/intermediate-capital-group-ord-gbp0.2625/share-charts
A typical setting for profit target is twice your stop loss, i.e. 60% of the purchase price or £6.82. ICP is now over £17, a rise of 300%! That's an awful lot of profit I would have missed out on. 

Note: For some kinds of mean reversion systems profit targets make sense, but they should be set using a proper analysis of the mean reversion process rather than a single figure. They make no sense at all if you're looking for trends.

Another stop loss strategy I am not keen on is the time based stop. You hold your position for 3 days or 3 months, and then close it if it hasn't reached some kind of profit target. Unless you're trading options where timing is inherent you shouldn't normally try and predict exactly when something is going up. It's hard enough predicting if it will go up. 

Of course any kind of stop loss is better than no stop at all. Without a stop loss I may well have sold GOOG at £18 to lock in my profits, missing out on the additional profits I've made subsequently. Or I could have panicked when it dropped below £13, and sold at a loss.


Where should we set our stop?


The figure of 30% above came out of thin air. But how in reality do we calculate this number? 

Here's an interesting quote:

"Each system has its unique and optimal betting percentage."

Is this really true? Do we have to risk calibrating/fitting the optimal percentage using a back-test? Or can we set the relevant figure with some simple rules? (Spoiler- the latter)

Let's review some facts:

Capital loss: A wider stop (more than 30%) means we'll lose more money. For example if we've invested half our account in a stock, then we're risking 30%*50% = 15% of our capital. Widening our stop will increase the likely damage to our account when the position is closed, and vice versa.

Time: The longer we hold a trade for, the more likely a stop will eventually be hit. GOOG is unusual; most of the trades I've done using this stop-loss last for about 6 months to a year.
The tighter a stop is, the quicker we'll hit it. For example if I set my stop for GOOG at 10% (i.e. initially at £14.09, then ratcheting up to 10% below £17.89=£16.10) I would have been stopped out in mid June 2018. Conversely, if I widen the stop enough then it might never be hit (a 99% stop will only be hit if a company is completely wiped out).

Volatility: If something is more volatile, it is more likely to hit a stop at a given level. Consider Bitcoin, here represented by an ETF:

https://www.hl.co.uk/shares/shares-search-results/x/xbt-provider-ab-bitcoin-tracker-eur/share-charts
It takes a few months -or even years - for most shares to move 30%, but 30% moves happen in a few days for this crazy POS asset.

Magic beans price level: We want to set our stop at the level where, if the price breaches it, it is guaranteed to keep falling. You may notice from the subheading that I am somewhat skeptical of this. Things like using fibbonaci numbers as 'key resistance' levels are pretty bogus. You may want to use a rule to tell you when the market is likely to fall, but it's better to drop the stop-loss entirely if you're going to use such a rule (this is discussed in chapter nine of "Leveraged Trading").

Let's start with the issue of capital loss. A common method for setting stops is to do so such that N% of your capital is at risk, eg N=1%. So for example if you bought 100 shares of Apple, that would be a position of $32,000 based on the current price of ~$320. If you had invested all your capital in Apple, then that would correspond to a 1% price move: $0.32. You're likely to be in the trade for less than a day.

This leads to traders saying stuff like this:

Yes, but I am trading low float stocks not something like forex where that 1% actually makes sense. If I put a stop loss at 1% on stocks 9/10 times it will probably just stop me out.

This trader appears to have a hobsons choice! They can set their stop appropriate to the volatility of the market at say 30% for low float stocks... but then they will be risking too much of their capital. Or they can set their stop appropriate to their capital, and risk being stopped out too quickly.

This makes no sense! Instead what you should do is:
  1. Set the size of your position according to your capital, and the ratio of your risk appetite and the risk of the underlying market.
  2.  Then set your stop according to volatility and time horizon.

Let's explore that. Suppose for example that you are happy to run your account at about the same level of risk as Apple shares, that you typically own two shares, and that you have $30,000 in capital. This implies that you should risk half your capital: $15,000 on your Apple stock position. At the current price of $320 that equates to 46 shares. 

Now you are free to set the stop-loss at a level which makes sense for the market. You should set the stop-loss at X * volatility of the market. Where does X come from? A larger X will mean wider stops, so you will hold positions for longer. A smaller value of X will result in holding positions for less time. You should set X according to how long you want to hold positions for.  I'll come back to that decision in in a future post. 

For now, let's use X=0.5, and assume we measure volatility as the annual standard deviation of the market which for Apple is around 20%.

Note: In subsequent posts I'll explain how you can use different measures of risk, such as the daily ATR, to calibrate your stop losses.

0.5 of 20% is 10%, so the stop loss for Apple should be set at 10% below your entry price. What proportion of our capital is at risk if we hit our stop loss? Well, it will be 10% of $15,000; $1500. Which is 5% of our capital.

Notice that the stop loss will have nothing to do with the size of your account. A small trader will have the same stop loss as a large trader, but a smaller position. A trader who has more appetite for risk will have a larger position, but the same stop loss. You can set your stop loss differently for FX and stocks without worrying about using too much capital. 

To see why this is a superior method, consider what will happen if you decide to invest in a Bitcoin ETF. Or maybe that should be 'invest'. Or to be accurate, invest in gamble on.

Suppose that Bitcoin is four times as volatile as Apple. We had a risk target for our account which was the same as Apple, 20% a year standard deviation of returns. But Bitcoin has four times the risk, 80% a year.

That means your position in Bitcoin will be a quarter of the size to compensate for the extra risk: $3,750. 

The stop loss will be set at 0.5 * 80% = 40% for Bitcoin. Much wider, to reflect the higher chance that Bitcoin will make a big move. 

How much of our capital is at risk? A 40% move on a $3750 position will cost us $1500. That's the same as for the Apple trade! Setting our position size and stop loss independently means we can target a particular % of our account at risk, whilst setting our stop differently according to market conditions.

Notice also that the same stop-loss can be used for different instruments, and it will adjust automatically if the volatlity of the market changes. A stop that made sense in the quiet days of 2006 would have been far too tight in the madness of late 2008!

Another benefit of this method is that you can switch to a trading system that doesn't have an explicit stop loss (like my automated futures trading account), and your risk will still be accurately sized.


Some unanswered questions


In this post we've learned that position size and stop levels should be set independently, and that stop levels should only depend on your expected holding period and the volatility of the market. 

But! There are hefty two elephants left in this particularly crowded room, namely:

  • How big should our positions be: how much risk should we take on our account?
  • What should the stop loss multiplier 'X' be: How fast should we trade
I will answer these questions in the next couple of posts (number two and number three).