Reward-Risk is a term for the ratio of the potential profit to the potential loss of a trade. In using such analysis, the entry price and the prices at which profits or losses will be taken are predetermined Measuring the units of reward per risk in advance aids in deciding whether to enter a position. However, since the outcome of a trade is uncertain, this widely known and used measurement must be revised in order to account for the probability of the trade’s success or failure, which can be done through expected value.

In expected value, a set of future values, or payouts, are adjusted to reflect the probability that each will materialize. For a stock, given a $10 entry with a profit target of $16 and a stop-loss at $8, the reward-risk is $6 to $2, a 3:1 relationship; these are the set of future values.

Additionally, say the probability of prices reaching the target before the stop-loss is 40%. The odds of being stopped out are thus 60% [=1-0.4].To obtain the expected value, one multiplies the chances of a profit by the potential profit, .4*$6 = $2.4, and the chances of loss by the potential loss, .6*$2 = $1.2. Thus, the expected reward-risk (ERR) is $2.4 to $1.2, or 2:1. The expected value is the gain minus the loss, or $1.2 [=$2.4-$1.2]. The unadjusted ratio of $6:$2, or 3:1 will from here on be referred to as the simple reward-risk (SRR). Note how in this example, the SRR is 3:1 but the ERR is 2:1. With a probability of success of .25, instead of .4, the expected reward is $1.5 [=.25*$6], the expected risk is $1.5 [=.75*$2], and the expected value is 0, or breakeven. Calculating an ERR of 1:1 reveals the odds needed to break even on a trade.

To use a non-market and non-monetary example, say a football player makes two-thirds, or roughly 67%, of his field goals. A field goal is worth 3 points if obtained, or 0 points if missed. For this player, his expected value is 2 points [=.67*3 + .33*0]. If in the last play of the game, his team is down by 3 points and he is up to kick a field goal, a strictly rational gambler will bet against the team. If the team is down by 1 point, the gambler will bet for the team. Although non-intuitive, a better description of reality is that this player scores 2 points per field goal rather than 3 points per field goal. Our gambler, by the way, will stay in business.

An SRR accounts for one observation, whereas an ERR accounts for all possible observations. Over the long term, the values of the ERR will materialize, similar to how the football player only scores 2 points per field goal. And like the gambler, a rational trader will base his decisions on ERR profiles that have a positive expected value. A positive expected value, however small, will result in long term success, and vice-versa.

Let us return to the example of the $10 stock with the SRR of 3:1. Given the present market conditions, the trader has assessed that either $16 or $8 will be reached first; whichever target is reached first will close the trade. If the analysis ends there, the trader unwittingly assumes that there is a 50/50 chance of a profit (of $6) or a loss (of $2). The expected return is thus $3 [=.50*$6] and the expected risk is $1 [=.50* $2]. Observe that the $3 profit to $1 loss remains a 3:1 relationship; the SRR equals the ERR. Any and every reward-risk setup that does not define probabilities assumes a 50/50 chance that its respective targets will be met. Therefore, the classical concept of reward-risk does—and has always— incorporated expected value, but as a passive assumption! Determining an expected reward-risk based on trade specific probabilities is not a supplement to the simple reward-risk construction, but rather, a revision of it.

The passive assumption of a 50/50 chance for a profit or loss has exerted a silent influence on the subject of reward-risk. Literature in Technical Analysis often prescribes a minimum reward-risk of 3:1 for one to remain profitable over time. As discussed above, “over time” is when expected values pave over individual observations. If a risk-reward lower than 3:1 is considered to ultimately result in a trader’s ruin, one can extract the assumed probabilities. The breakeven odds for an SRR of 3:1 are 25/75. Therefore, one assumes that successful traders are profitable slightly over 25% of the time, because anything less than 3:1 will result in a negative expected value. Interestingly, this success rate matches the figure of 30%, which is the percentage of time that the market is estimated to be trending. A positive expected return can result from various reward-risk proportions and their respective probabilities of success and failure. For example, assume an SRR of 1:2.5, that is, for every potential profit of 1, there is a potential loss of 2.5. With a probability of success of .8, the ERR is 0.8:0.5, or 1.6:1, and the expected value is .30. Over the long term, this reward-risk profile makes .30 per trade—not per every winning trade, but per trade. That’s success. Now, examine a trade with a large profit potential and a “tight” stop-loss, a temptation particular to futures traders. Even with an SRR of 7:1, a 90% chance of being stopped out yields an ERR of 0.7/0.9, or .77:1, and a negative expected value, -.40. Trading only these 7:1 setups ensures a quick bankruptcy. As you can see, even if the SRR is less than 1, any trade with a positive expected value—adjusting for commissions and other frictions—should be undertaken. In this light, the prescription of trading only 3:1 or higher setups is crude, and most likely a result of an implicitly assumed success rate of around 25-30%, which may be related to the fact that markets trend 30% of the time.

**Estimating the Probability**

Trading is inherently probabilistic. The preceding arguments strongly suggest that if one is to use a reward-risk ratio in assessing trades, an expected value must be calculated and serve as a deciding factor.—Well, that’s the easy part. The real challenge to this model is in determining an accurate measure for the probability that will be used to arrive at expected values. Due to the complex nature of trading in the financial markets, probabilities can only be estimated, not determined. In the examples throughout this article, the probabilities used were assumed to be accurate for the sake of illustration. This section sketches four methods that can be used to estimate the probability of success for a given trade. Note that only one value needs to be calculated, because 1 minus the probability of success equals the probability of failure.

**Historical Success Rate**

For an individual trader with a consistent style, his historical success rate can be used as a proxy for the probably of success. For example, if he finds that 55% of his trades are winners, .55 and .45 will be the probability of success and of failure, respectively. With these probabilities, an SRR of .82:1 yields an ERR of 1:1, or breakeven.

You will see later on in this article, the probability of success can be inversely related with the reward to risk ratio. As higher price targets are attempted, there are fewer successes and vice-versa. The trader can balance the success rate and reward-risk proportion to see what combination yields the highest expected value. In the example above, the SRR must be greater than .82:1 for profitability. Based on the inverse relationship established above, to maintain a 55% success rate the reward may not extend too far beyond .82:1. Note, however, that if these measures of probability are consistent and accurateall one needs is a positive expected value for a trade to be worthwhile; in this example, even an SRR as “low” as 1.05:1 will have an expected value of 0.1275 per trade.

The historical success rate approach has the following advantages and disadvantages:

*Advantages*

- Based on empirical data
- Quick and simple to calculate
- Requires no estimates
- A long enough track record has some reliability

*Disadvantages*

- Past performance is not indicative of future results; the conditions that have allowed that style obtain a given success rate may change
- Not a scientifically rigorous approach

**Technical Analysis Statistics**

Technical traders use various methods to determine buy and sell signals. These include trendlines, support and resistance, continuation and reversal patterns, indicators, and many more. The statistical success rates of these methods of technical analysis can be used for calculating probabilities.

In Charles Kirkpatrick and Julie Dahlquist’s Technical Analysis,the authors summarize the results from Thomas Bulkowski’s Trading Classic Chart Patterns. In his book, Bulkowski documented the success rate of various technical continuation and reversal patterns from a sample of 700 stocks during a span of ten years. I will use his figures for illustrative purposes.

For a rectangle pattern, also known as a trading range, if there is a viable upward breakout, established after the fact in Bulkowski’s research, 80% of the time there will be a gain of 10%. (Technical Analysis, p. 315) That yields an expected reward of 8%. A stop-loss at – 4% [.2*-4% = -8%] is thus the breakeven point. A stop-loss of less than 4% below the entry price is necessary to obtain a positive expected value. Continuing with this example, 75% of rectangle breakouts are false signals, making the ex ante success rate 20% [=.8*(1-.75)]. The expected reward is not 8%, but 2%. A stop-loss at 2.5% yields an expected value of 0. The breakeven SRR is thus 5:1, a 10% gain to a 2.5% loss. Pretty grim. These numbers also assume that the odds already discount the possibility of being stopped-out prior to the 10% gain.

Using technical analysis statistics to determine probabilities has the following advantages and disadvantages:

*Advantages*

- If technical analysis is used to determine the reward and risk targets, this method is the most consistent with the analysis
- Expected values are simple to calculate
- Requires no estimates

*Disadvantages*

- The body of research on the success rate of technical patterns and indicators is limited, an enormous hindrance to this approach
- Based on historical data
- Both the success rates of valid signals and the likely percentage gain prior to a reversal of the signal must be known to properly calculate the probability

**Volatility of Returns**

All else equal, of two price targets in opposing directions, the target closer to the entry price has greater odds of being reached first. To a trader, a stop-loss being reached prior to an upside target is not an inconsequential occurrence. The standard deviation of returns, specified for the expected holding period, aids in assessing the chances of being stopped out of a trade due to the proximity of the profit and loss targets to the entry price.

Assume a trade with an expected four month holding period. The security in question has a historical four month mean return of 1% and a standard deviation (σ) of 3%. The entry price is $10, profits will be taken at $10.70, and a stop-loss is placed at $9.80. Once these reward-risk targets are set, one can determine the distance, in standard deviations, of each from the entry price. In this example, the upside target is 7%, 2σ from the mean of 1% and the stop-loss is at 2%,-1σ from the mean. On a normal distribution, +2σ or higher is reached 2.4% of the time and -1σ or lower is reached 16% of the time. The balance for which neither target is achieved is 81.6%. The image below depicts these zones; red is for the loss target, green is for the profit target, and blue is neither.

Two assumptions needed to simplify the example is that the trader will not readjust the stop-loss or the price target and that after the four month holding period, the trade will be exited at the middle of the blue target range if neither target has been met. Since there are three probabilities involved, there must be a third step in calculating expected value.

Expected reward: .024*7% = .168%

Expected risk: .16*-2% = -.32%

Expected exit if neither target achieved (middle of blue range): .816*2.5% = 2.04%; since the result is positive, must be added to expectedreward

Expected reward-risk: 2.208% to .32% = 6.9:1

Expected value = .168% + -.32% + 2.4% = 1.89% profit

**The simple reward-risk of 7% to 2%, or 3.5:1, is not a meaningful number**

Let’s examine the reverse setup, in which the profit target is 1σ, 4%, and the stop loss is 2σ, -5%:

**EXPECTED REWARD: .16*4% = .64%**

**EXPECTED RISK: .024*-5% = -.12%**

**EXPECTED EXIT IF NEITHER TARGET ACHIEVED (MIDDLE OF BLUE RANGE): .816*-0.5% = -.408%; SINCE THE RESULT IS NEGATIVE, MUST BE ADDED TO EXPECTED RISK**

**EXPECTED REWARD-RISK: .64% TO .528% = 1.2:1**

**EXPECTED VALUE: .64% + -.12% + -.408% = .132% PROFIT**

**THE SIMPLE REWARD-RISK OF 4% TO 5%, OR .8:1, IS NOT A MEANINGFUL NUMBER**

As you can see, the proximity of profit and loss targets to an entry price, and the resulting probability of which target will be reached first, shapes the reward-risk profile. Also, where the trade is closed if neither target is reached will significantly impact the expected value. If the exit price was assumed to be at breakeven, instead of at the middle of the blue range, the first and second trades would have an expected value of -.152% and .52%, respectively.

Note that such an analysis is direction neutral; it simply relies on volatility to characterize the potential profitability of a trade. Someone seeking only to identify statistical setups may find this direction neutral quality appealing; however, a discretionary trader usually buys with the bias that prices will rise.

The volatility of returns approach has the following advantages and disadvantages*:

*Advantages*

- Accounts for the randomness of market direction
- Sets more realistic expectations of whether a profit or loss target will be achieved
- Scientifically sound way to extract probabilities

*Disadvantages*

- Does not incorporate the tools of technical analysis, e.g., trend lines, patterns, indicators
- Statistics are imperfect predictors
- Assumes a normal distribution of returns

*Direction neural is either an advantage or a disadvantage, depending on the trading style

Multi-Factor Model

Another approach is to use a multi-factor model to the determine probabilities. Such a model calculates the statistical relationship of various independent variables, called “factors,” to the dependant variable, resulting in the following equation:

P = α + c1F1 + c2F2 + c3F3 + ……+ cnFn + e

where:

P = dependant variable (in this case, probability of success)

α = intercept term (the of value P if all factors equal zero)

c = factor coefficient

F = factor

e = error term (due to imperfect statistical relationships)

A trader could determine that the following five factors are sufficient to explain a trade’s probability of success: technical strength, fundamental strength, the entry price’s distance, measured in standard deviations, from the profit target, the entry price’s current distance, measured in the inverse of the standard deviation, from the stop-loss, and the trader’s historical success rate. The coefficient is the degree to which each factor influences the dependant variable, holding all other factors equal. The table below lists the five factors, their coefficients, and the factors’ values for a particular trade. (All listed values are simply illustrative.)

*This value indicates that the stop loss is negative 2 standard deviations from the entry price [1/σ = 1/-2 = -.5]. There is an inverse relationship between the price’s distance from the stop-loss and the probability of success.

From the values on the third column of the table, note that the stock is in a strong technical position, a neutral fundamental position, and the trader is profitable 60% of the time. Let us assume the trade in question is from the example used in this article’s section on volatility: the trader aims for a 4% return (1σ) with a 5% risk (-2σ). Let us also assume that the intercept and error term equal zero. Using the factor model and the values from the table, the probability of success is as follows:

P = 0 + 1.2(7) + 1(5) + -1.5(1) + 2.2(-.5) + .8(60) = .588

The probability of success for this trade is 58.8%. With a 4% return and a 5% stop loss, the ERR is 2.35%/2.06%, or 1.14:1, with an expected value of 0.292.

The multi-factor model approach has the following advantages and disadvantages:

*Advantages*

- Various inputs can be used including inputs from the above stated methods
- Model acknowledges that more than one factor influences probabilities
- Scientifically valid

*Disadvantages*

- Extremely complex to construct correctly
- Some inputs may require estimates
- Requires a large supply of data
- Model must be properly specified, that is, it must properly account for the factors that influence the dependant variable

Revising the simple reward-risk measure to account for expected value results in a much richer analysis. It may also prove to be more accurate in determining a trader’s long term profitability. As such, research on how to best estimate the probability of success or the probability of failure for a trade with a defined reward and risk would be valuable.

To quickly touch upon two complications not addressed in this article: first, the ERR discussed here was for the onset of a trade; once the trade is executed, the ERR profile will change as a function of price action, changing probabilities, and trader decisions, such as raising a stop-loss. Second, conditional probabilities may be needed to calculate viable expected values.

To summarize the major points of this article:

- Trading is probabilistic by nature
- Expected value should be incorporated into reward-risk analysis
- To determine the expected reward-risk (ERR), first multiply the reward by probability of success and the loss by the probability of loss; the ratio of these two is the ERR
- The expected reward minus the expected risk is the expected value of the trade
- An ERR of 1:1 is a breakeven setup, with an expected value of zero
- The simple reward-risk ratio (SRR) is actually an ERR that passively assumes that the probability of success is .50
- A positive expected return can result from various reward-risk proportions and their respective probabilities of success and failure
- The classical prescription of a 3:1 SRR may have been implicitly based on the fact that the percentage of winning trades was slightly higher than 25%
- The crux of calculating an expected reward-risk ratio is in estimating a viable probability of success or of failure
- I have introduced four possible methods for estimating the probability to be used in calculating the ERR. These are, the trader’s historical success rate, statistics on methods of technical analysis, volatility of returns, and a multifactor model
- Research on methods of obtaining the probability of trade success in the context of reward-risk would greatly contribute to this type of analysis