- 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?

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?

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.

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.

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?

- The average retail trader: -1.0
- Bank acccount: 0.0
- A retail trader using a simple mechanical system, with a single indicator and instrument: 0.24
- An inexperienced discretionary retail trader: 0.1 to 0.3
- The best long term investor in the world: 0.7
- An experienced retail trader using a mechanical system diversified over at least 30 instruments including several asset classes, and many indicators: 1.0
- A world-class systematic hedge fund diversified over hundreds of instruments and using bleeding edge technology: 1.5 to 2.0 before fees
- The best hedge fund in the world: from 2.0 to 4.0
- The best high frequency traders on the planet, trading thousands of instruments with holding periods that can be measured in the microseconds: 10.0+

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.

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

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

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

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

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?

- 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).
- 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.
- 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.

Thank you for all the excellent information. I get a huge amount out of it. In this post and in your books, I believe you calculate the annual standard deviation (V) based on a period equal to the last 25 trading days. At the moment, fixed income is trading in a very narrow band so the V is quite low compared to historical average, so the resulting position size would be quite large compared to the average size. For example, at the moment, with identical risk targets, the number of futures contracts needed for a Eurodollar position using a V based on the last 25 days would be roughly 5 times the number needed using a V based on the last 250 trading days. You would get similar results for a variety of STIR and fixed income futures at the moment. I’d appreciate your views on the tradeoffs between using a longer or shorter period. A few thoughts:

ReplyDelete- “Luck” plays more of a roll in trading results when using a shorter period. When a period of very low V is followed by very high directional V, by the time you reduce position size to account for the higher V, you will be either very happy or very sad. If you calculate V over 250 trading days, you are less likely to have extreme position sizing. In the example of the 2015 EUR/CHF crash, a position based on 25 day V would have been about three times the size of a position based on 250 day V. There are certainly more extreme examples than that. If a trader had reason to believe their system is more likely than not to be positioned correctly when extreme high V follows extreme low V (which I don’t think is realistic), then I suppose having a massive position may make sense, at least over the long term.

- Based on my simplistic research over a variety of instruments, using a shorter period to calculate V is clearly superior to using a much longer period. Since that result incorporates a variety of situation involving what I described above as “luck,” perhaps using a shorter period is warranted, as long as you know what you are getting into and have the stomach for it.

- A not very original solution would be to incorporate a V floor into a system. Another not very original solution is to use common sense when things look wacky, but that is impossible to back-test.

- You’ve spent a great deal of time, I think, researching ways to make more efficient use of capital, especially for futures-based systems with smaller account sizes (keep those ideas coming, by the way). Doesn’t a shorter V window exacerbate the problem, since the band of possible positions sizes is so much wider than with a longer V window?

Thanks again for all the information – it is much appreciated.

- The length of V I use is calibrated to be the best possible forecast of future V over my average holding period. It seems that is the best measure of the correct V, rather than trying to optimise system performance. Using a longer V doesn't help much in reducing extreme returns, or luck, in fact it makes things worse because with a more inaccurate forecast of future vol we end up being exposed to fat tails when vol is in a period of being relatively high (this is something I will post on at some point).

Delete- I do indeed use a V floor in my system, for the reasons you describe. A better forecast of V would be to use a comination of recent historical vol, plus a long term measure of V, since V tends to mean revert in the long run.

- "Doesn’t a shorter V window exacerbate the problem, (minimum capital size)" To be honest I haven't thought about that. But in fact a wider band of possible position sizes is good for those with reduced capital, since it makes it more likely that at times they will be able to hold a position in a given instrument.

Hi Rob

ReplyDeleteWhen you say you use 25% risk target. Is that the account risk before applying a idm and forecast or is that the maximum you allow including those factors?

That's what I'm targeting. I target 25% average on each instrument but it will vary depending on the forecast. And the IDM is there to try and hit 25% on average across the portfolio.

DeleteFor the standard deviation, why not use log returns instead of returns? For example, a 10% up followed by a 10% down does not get you a 0% total return, but if you use simple returns it will appear something like this...

ReplyDeleteI think simple returns make more sense here, and I'm struggling to think how you'd get back from a standard deviation of log returns to a % figure which is what you need for risk scaling.

DeleteHello Rob, how do you define your Sharpe (for use in the exposure calculation, i.e. choosing risk target) when Sharpe depends on the amount of exposure available to the instruments? Do you calculate per-instrument returns assuming 100% allocation separately then combine somehow? Thanks.

ReplyDeleteBasically the portfolio level SR determines the portfolio level risk target; everything is then scaled to that including each individual instrument, and since instruments are given portfolio weights adding up to 100% you'll hit the portfolio risk target (after applying an 'instrument diversification multiplier' to account for diversification which would otherwise result in us undershooting the risk target).

Delete