 How should we control risk (first post)
 How much risk should we take? (previous post)
 How fast should we trade? (this post)
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 precost 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 midprice, and the actual price traded at.
So for example if a market is 100 bid, 101 offer, then the midprice 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 stoplosses, 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 backtest to see what effect they have on your cost estimates.
Linear and nonlinear
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 precost 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 COVID19 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 stoploss 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 3.10.8 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)
Total cost per year = 0.026 + (0.0058 *5.4) = 0.058
Precost returns: Theory
Let us now turn our attention to precost 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.
Precost 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?
XAxis: Holding Period. YAxis: 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.
Xaxis: Moving average rule N,4N. Yaxis Sharpe Ratio precosts 
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 precost 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 stoplosses 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)
XAxis: Moving average rule, Yaxis 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 precost 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:
XAxis: Moving average rule, Yaxis 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 precost 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 precost returns and costs as equally predictable. But this isn't the case. Precost 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:
Xaxis: Moving average variations. Yaxis: Actual Sharpe Ratio precosts, 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:

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 PRECOST 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
(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:
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. Yaxis: 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
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
0.3 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.
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!