Fixed-Fractional Position Sizing, and What an R-Multiple Is

A fraction of what, exactly?

Say you have decided to risk a fixed share of your account on each trade. You have done the arithmetic from the first article, and it works.

Then you lose three trades in a row. The account is smaller now.

So here is the question nobody asks at the start. Is the share you risk a share of the account you began with, or of the account you have today?

The answer decides how a losing streak plays out. It is the whole difference between two methods that look almost identical on day one.

Two ways to size the same bet

The first way keeps the rupee amount fixed. You risk the same number of rupees on every trade, whatever has happened to the account since.

The second way keeps the fraction fixed. Before every trade you look at the balance as it stands now, and risk the same slice of that.

On the first trade the two methods give exactly the same number. They only part ways after the account has moved, and the direction they part in is the part worth understanding.

The two percent is there because it divides neatly. It is an example, and this article does not suggest a figure for you.

What happens in a losing streak

Every method meets a run of losses sooner or later. So the useful question is what the rule does to you while it is happening.

Both start with an identical first loss. On the left every step is the same size, so the staircase runs into the floor. On the right each loss is a slice of a smaller account, so every step is a little shorter than the last and the staircase flattens out.

With a fixed rupee amount, every loss takes the same bite. The account falls in a straight line and eventually meets zero.

With a fixed fraction, every loss takes a smaller bite than the one before, because it is a slice of a smaller account. The fall slows as it goes. In the pure arithmetic it never reaches zero at all.

Here is the same streak in invented numbers.

Fifty losses in a row is extreme, and almost nobody would keep trading through it. The row is there to show the shape. One method runs out. The other keeps cutting its own bet until the losses become small.

The catch: the climb back is slower

The same property works against you on the way up. After a losing run, the fixed-fractional account is betting less, so each win adds less.

In the invented example, after ten losses the fixed-rupee account holds ₹80,000 and the fixed-fraction account holds ₹81,707. Suppose every win after that returns twice the amount risked.

The fixed-rupee account is back above ₹1,00,000 after five wins. The fixed-fraction account needs six, because its bets are smaller the whole way up. It fell less and it climbs slower.

This is the trade-off at the centre of the method. The rule that keeps you in the game during a bad run is the same rule that makes the recovery take longer. You cannot keep one half and drop the other.

Meet R

Everything so far has been measured in rupees. That works for one account, but it makes trades hard to compare. A ₹3,000 loss on a large position and a ₹3,000 loss on a small one are not the same kind of mistake.

R fixes that by measuring every result against the loss you planned for.

Take an invented trade. You buy 100 shares at ₹500 with a stop at ₹480. The gap is ₹20 a share, so R is ₹2,000.

The distance from entry to stop is one R. Everything else is measured with it: close the trade at the stop and you have lost one R, close it twice as far above the entry and you have made two R.

If the price reaches ₹540, that is ₹40 a share, twice the ₹20 you risked, so the trade made plus two R. If it falls to your stop, it lost minus one R. That is the whole idea, and it takes a few minutes to get used to thinking in it.

Why measuring in R is worth the effort

The first reason is that it strips out size. A trade that made plus two R was a good trade whether R was ₹500 or ₹5,000. You are judging the decision, and the position size no longer gets in the way.

The second reason is that it keeps you honest about the stop. R is fixed at the moment you enter. Moving the stop further away during the trade does not change R. It means you are taking more risk than you planned, and a loss then shows up as something worse than minus one R.

It was minus one and a half R. A loss bigger than one R always has a reason behind it. The price may have jumped past the stop, the stop may have been moved, or it was never placed. Each of those needs a different fix, and measuring in R is how you notice them.

The average R: whether a method holds up

One trade tells you almost nothing. Ten tell you a little. What you actually want to know is the average result in R across many trades.

Six of the ten lost. Four won. The method was wrong more often than it was right, and still came out ahead. The four wins averaged 1.75R, and every loss was held to one R.

The arithmetic behind that average is simple. Four in ten trades won an average of 1.75R, and six in ten lost one R each. Four tenths of 1.75 is 0.7, six tenths of one is 0.6, and the difference is the plus 0.1R per trade.

This is the same argument Mark Minervini makes about the size of your losses in Trade Like a Stock Market Wizard. R is simply the unit that lets you check it on your own trades.

Ten trades is still far too few to trust. A run of four lucky wins would produce this table by chance. The average only starts to mean something after many more trades than most people have the patience to record.

What this looks like in India

The arithmetic is the same everywhere. What changes here is how cleanly you can follow it.

On Indian exchanges you buy whole shares. The sizing rule might say 37.6 shares, and you buy 37. Rounding down every time means you usually risk a little less than the fraction says, which is the safe direction to be wrong in.

Futures and options come in fixed lot sizes set by the exchange. You cannot buy part of a lot, so on a smaller account the correct size is often zero lots. That is the method telling you the contract is too big for the account.

Costs sit inside every R. Brokerage, securities transaction tax, stamp duty and exchange charges come out of every trade, winning or losing. On a small position they can be a noticeable slice of the risk. A loss at your stop can then come out a little worse than minus one R.

And a stop is only a plan. A share that opens well below it, or locks at its lower circuit, sells below the level you chose. That is where losses bigger than one R come from in this market.

Where this breaks down

Fixed-fractional sizing controls how much you risk. It says nothing about whether the trades are any good. A method with a negative average R loses money at every fraction. The only thing the fraction changes is how fast.

The fraction itself is a guess. Too large, and a normal losing streak cuts deep enough that you abandon the rules at the worst moment. Too small, and costs eat the gains. No formula hands you the right number.

Several open positions at once break the neat picture. Suppose five trades all depend on the same thing happening in the market, so they move together. Risking two percent on each is then closer to risking ten percent on one idea.

And R depends on the stop being real. If the stop is decided loosely, or moved during the trade, every result measured in R is quietly wrong.

A practical checklist

Things you can do before your next trade

7

All of this is arithmetic you can do on your phone. None of it tells you what to buy or sell.

• Open your broker app and note the total account value today. Is that the number you are sizing from, and not the one you started with?Fixed-fractional sizing always works from what you have now.
• Choose your fraction before the next trade, and write it beside that number. Did you pick it while calm, and not straight after a loss?A fraction picked in the middle of a bad run tends to creep up.
• For your next idea, write down the entry, the stop and the gap between them in rupees. Is the gap written down before you think about quantity?R is fixed at entry. Decided later, it stops meaning anything.
• Divide the rupee risk by that gap, then round down to whole shares or whole lots. Does the answer come out as zero?Zero means the trade is too big for the account, and that is a useful answer.
• Run that quantity through the charges calculator in your broker app. Are the costs a large slice of the rupee risk?Costs come out of every trade, win or lose.
• After each closed trade, write the result in R rather than rupees. Do you have at least twenty before judging the method?A handful of trades says almost nothing about the average.
• Look back at your last few losses. Did any of them go past the stop, beyond minus one R?Losses bigger than one R point to gaps, circuits or a moved stop, and each needs a different fix.

Nothing here says what to buy or sell, or what fraction to risk. Every line is something to work out or write down.

Check yourself

4 questions

Answer in your head first. Guessing and being wrong still helps you remember.

1. 01Why does a fixed-fraction account never quite reach zero in a losing streak?Show

   Because each loss is a fraction of what is left. As the account shrinks, the rupee amount risked shrinks with it, so the losses get smaller and the fall slows instead of continuing in a straight line.

2. 02Why is the recovery slower after a losing run?Show

   The same rule that shrank the bets on the way down keeps them smaller on the way up. Each win is a fraction of a smaller account, so it adds fewer rupees than it would have at the start.

3. 03You planned to risk ₹1,000 and the trade made ₹2,500. What was it in R?Show

   Plus two and a half R, because the result was two and a half times the planned risk.

4. 04Why can a method be wrong most of the time and still come out ahead?Show

   Because the average result in R depends on the size of the wins and losses as well as how often each happens. If losses are held to one R and the wins are larger, fewer wins can outweigh more losses.

Where these ideas come from

The arithmetic of risking a fixed fraction of capital was worked through in detail by Ralph Vince in Portfolio Management Formulas (Wiley, 1990). The idea of measuring every trade in units of its initial risk, and calling that unit R, was popularised by Van K. Tharp in Trade Your Way to Financial Freedom (McGraw-Hill, 1998). Both go a long way further than an article can, and both are worth reading in full.