Edge, Expectancy, R-Multiples & Links

Risk management, probability and statistics are one of those "nod and wink" topics discussed by most traders.  They know they should rigorously implement these three topics into their business plan but if most traders fail, I doubt that any new trader is probably doing so from day one.

Summary

1.  Define your edge and risk reward profile before putting on a trade.
2.  Edge can be expressed in terms of expectancy per dollar risked or  Expectancy = (Winning Percentage * Average Win) - (Losing Percentage * Average Loss).
3.  Think in terms of probabilities over a large sample size and realize that each trade is unique.
4.  An R-Multiple is really just a simple expression of Reward to Risk for a given trade. An R-Multiple of 2 means that for that trade, you are rewarded with $2 for every $1 risked.
5.  R-Multiple is Expected Gain - Expected Loss or (Exit Price - Entry Price) - (Entry Price - Stop Loss).
6.  Often times you will see 1R calculated which is just the Expected Loss or (Entry Price - Stop Loss).
7.  R-Multiple Distributions are an easy way to express the expectancy of a given trading system based on the system's average risk/reward ratio per trade.
8.  An R-Multiple distribution is the average or sample mean of R-Multiples of a number of sample trades.

Define Your Edge

When I first got into trading, I wasn't sure what an "Edge" was.  How can someone know what will happen in the future?  It is still something that I struggle with - simply knowing how to calculate an edge is different than internalizing what your edge truly is.  Stated another way, readiness to start trading is different than knowing how to trade.  The best way I have found to look at edge is by thinking about edge in terms of the law of large numbers.

Definition:  In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.  From:  Wikipedia, Law of Large Numbers.

Let's look at an example.  Take a six sided dice and let's calculate what the odds are that you will roll a 5.  The statistical edge is easy to calculate because the outcome of any roll is determined by a mechanical probability system (ie, the six sided dice).  The exact edge can be calculated by thinking through all possible outcomes of the game.  The die has six sides so the odds that a 5 will be rolled on a given roll is 1 in 6 or 16.66%.  However, five of the six sides will not roll a 5 so the odds of not rolling a 5 are 5 in 6 of 83.33%.

Let's say that you offer a friend the following bet:  he wins $4 for every time a 5 is rolled and you win $1 every time any other number but 5 is rolled.  If you roll the dice one time, the odds of him winning $4 is 16.66% and the odds of him losing $1 is 83.33%.  But let's say you change the rules of the game and require the dice to be rolled 100 times before you will pay him.  At the end of 100 rolls, you would expect to have won the $1 wager 83 times for a total profit of $83.  You friend would have won about 17 times, delivering a loss of $68.  This is what can be referred to as expectancy.

Expectancy = (Winning Percentage * Average Win) - (Losing Percentage * Average Loss)

So our Expectancy from the dice roll bet over 100 rolls is:

Expectancy = (.8333 * $1) - (.1667 * $1) = $15 or 15 percent profit for each dollar wagered.

According to the law of large numbers, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

So how does this relate back to trading?  It is important to internalize that you can never predict the outcome of a single event or trade.  When you put a trade on, you don't care to predict what the outcome of that specific trade will be because you cannot predict the outcome.

Mark Douglas, in Trading in the Zone, defines two levels of thinking in probabilities:

1.  Micro Level - Each Trade is Unique - The outcome of each individual trade is statistically independent of every other trade and there is a random distribution between wins and losses in any given string of trades.  You have to believe in the uncertainty and unpredictability of the outcome of each individual trade.

2.  Macro Level - Think in Terms of Probabilities - You have to believe that the outcome over a series of trades is relatively certain and predictable. The degree of certainty is based on the fixed or constant variables that are known in advance and specifically designed to give an advantage (edge) to one side or the other.

Risk to Reward Ratios

So how does edge and expectancy interact with risk to reward ratios.  Remember, edge or expectancy is a function of the probability of a losing event versus its positive risk to reward ratio.

Van Tharp uses something called an R-Multiple.  An R-Multiple is the expected gain on a trade minus the expected loss on a trade. An R-Multiple can be expressed as the amount risked per unit or the total risk.  In it's most basic form, an R-Multiple is simply an expression of Risk to Reward.

R-Multiple = Expected Gain on a Trade - Expected Loss on a Trade


R-Multiple = (Expected Exit Price - Entry Price) - (Entry Price - Stop Loss)

Gains are positive R-Multiples.  Losses are negative R-Multiples.

Let's look at two examples.

Trade #1:  Say you buy a stock for $20 and you enter a stop loss order for $15.  Your profit target on the trade is $30.  So your initial R-Multiple for this trade is:

R-Multiple = ($30 Exit Price - $20 Entry Price)/($20 Entry Price - $15 Stop Loss) = 2R

Trade#2:  But say now you buy a stock for $20 and enter a stop loss order for $10.  Your profit target is $40 but the stock fails to rally and you are stopped out at $10.  You R-Multiple for this trade is:

R-Multiple = ($10 Exit Price - $20 Entry Price)/($20 Entry Price - $15 Stop Loss) = -2R

To make the calculation a little easier, you will often see R-Multiples expressed as 1R.  1R is simply defined as the risk of any one trade or:

1R = Entry Price - Stop Loss


For Trade #1, 1R is equal to $5.  For Trade #2, 1R is equal to $10.

Okay, so now that your mind is swimming with R-Multiples and such, how do we relate R- Multiple, which is basically an expression of our risk to reward on a given trade to expectancy, which we learned was how much you can expect to make on the average, per dollar risked, over a number of trades.

Van Tharp directs you to look at R-Multiple Distributions to determine your expectancy for a given trading system.

Let's go back to the dice problem again.  The law of large numbers states that as the sample size grows, the closer the average of the past outcomes approaches the statistical prediction of the probable outcome.  So what is the expected value of a single dice roll?

The formula is the sum of each individual outcome divided by the total number of outcomes or:



According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the accuracy increasing as more dice are rolled. From:  Wikipedia, Law of Large Numbers.

As you roll the dice more and more times and take the average of these rolls, the sample mean should approach 3.5.  

Back to R-Multiple distributions, you can determine the expectancy or probable outcome of any trading system by taking the average of you R-Multiples.  As more and more trades are taken, the average or "sample mean" of all of the trades should approach the probable outcome of the trading system, just like with the dice.

Let's look at an example of 5 Trades to calculate the R-Multiple distribution for a given system.

Trade #1:  Entry: $100, Stop: $90, 1R = 10, Exit Price: 120.  R-Multiple = 2R
Trade #2:  Entry: $50, Stop: $45, 1R = 5, Exit Price: 43.  R-Multiple = -1.4
Trade #3:  Entry: $120, Stop: $90, 1R = 30, Exit Price: 150.  R-Multiple = 1R
Trade #4:  Entry: $100, Stop: $80, 1R = 20, Exit Price: 300.  R-Multiple = 10R
Trade #5:  Entry: $78, Stop: $70, 1R = 8, Exit Price: 95.  R-Multiple = 2.13R

An R-Multiple distribution is found by calculating the R-Multiple for each trade and then taking the average or sample mean for all trades.  For the 5 trades above, this system has an average R-Multiple or expectancy of 2.745R.

R-Multiple Distribution = Sum of All R-Multiples for Each Trade / No. of Trades

Over 100 trades, we can expect to make 274.5R.  So if we keep R at $20 for all trades, we can expect to make 274.5 * 20 = $5490.  5 trades is far to small of a sample size to determine if we have a profitable system (we will get into sample size requirements later) but you can see that how R works.

Putting it All Together

So now you should be able to take a given trading system's results and determine the expectancy of the system using the R-Multiple Distribution calculation.  A couple of points to remember/rules of thumb:

1.  R-Multiple is an easy way to determine the Risk to Reward ratio of a given system and compare systems across the board.  If a trader approaches you and says he has a system that makes $10,000 across a population of 500 trades that sounds great.  But what if the R-Multiple Distribution of the system is 0.1R.  That is, for every $1 risked, you can expect $0.1 in return.  That is a damn risky system in my opinion!

2.  Generally, a rule of thumb is to only use a system that gives you a "2 to 1" or "3 to 1" risk reward profile.  This can be expressed as a 2R or 3R system.

3.  Don't focus on winning percentages alone!  Focusing only on winning percentages of a system ignores the risk reward profile and expectancy (edge) we just talked about.  I made this mistake early on and paid dearly for it.

Let's run the numbers.  For these examples, assume you have a $10,000 account, you trade 100 times and $100 is risked on each trade.  We will need these formulas:

Winning Percentage = % of Profitable Trades
Number of Winners = Total Number of Trades Taken * Winning Percentage
Total Profit = Number of Winners * Avg Profit Per Trade


Losing Percentage = (1 - % of Profitable Trades)
Number of Losers = Total Number of Trades Taken * Losing Percentage
Total Loss = Number of Losers * Average Loss


Net Profit = Total Profit + Total Loss

Assume System #1 has a winning percentage of 70% and System #2 has a winning percentage of 25%.  What does this tell us?  Really nothing because we don't know what the average win, average loss and expectancy are.  Assume System #1 has a $30 average win and a $100 average loss.  Assume System #2 has a $500 average win and a $100 average loss.

System #1 Analysis


What is the expectancy for this system?

Expectancy = (Winning Percentage * Average Win) - (Losing Percentage * Average Loss)
Expectancy = (.7*$30) - (.3*$100) = -9 percent expectancy or a loss of $9 for every $100 risked on the trade.


What is the Net Profit of this System?

No. Winners = 100 Trades * 70% Winners = 70 Winners
Total Profit  = 70 Winners * $30 Average Profit Per Trade = $2100

No. Losers = 100 Trades * 30% Losers = 30 Losers
Total Loss = 30 Losers * $100 Average Loss Per Trade  = $3000

Net Profit = Total Profit - Total Loss = $2100 - $3000 = -$900.

You could have easily found the Net Profit doing this:

Expectancy * Total # Trades * Total Risked Per Trade = -.09 * 100 * $100 = -$900

System #2 Analysis



What is the expectancy for this system?

Expectancy = (Winning Percentage * Average Win) - (Losing Percentage * Average Loss)
Expectancy = (.25*$500) - (.75*$100) = 50 percent expectancy or $50 for every $100 risked.



What is the Net Profit of this System?

No. Winners = 100 Trades * 25% Winners = 25 Winners
Total Profit  = 25 Winners * $500 Average Profit Per Trade = $12,500

No. Losers = 100 Trades * 75% Losers = 75 Losers
Total Loss = 75 Losers * $100 Average Loss = $7,500

Net Profit = Total Profit - Total Loss = $12,500 - $7,500 = $5,000.

Again Net Profit was .5 * 100 * $100 = $5,000.

As you can see, a high winning percentage does not guarantee a high rate of return for a trading system.  System #1 had a -9% expectancy even with a 70% win rate!

Links:

A good article explaining expectancy and R-Multiples from BackTestingBlog.com.

Van Tharp's article on R-Multiples and R-Multiple Distributions.  Van Tharp Institute.

A note to the R-Multiple Haters.  TraderMike.

A mathematical approach to eliminating emotions.  FX360.