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004_risk_reward_ratio

Risk Management: Risk-Reward Ratio

Risk Management: Risk-Reward Ratio

I. Introduction to Financial Decision Theory

Risk management constitutes the fundamental pillar of all trading activity. The Risk-Reward Ratio (R:R) mathematically formalizes the trade-off between potential gain and potential loss.

II. Probabilistic Framework

2.1 Trade Modeling

Let a trade be characterized by:

  • X: Random variable representing profit/loss
  • p: Probability of success (profit)
  • R: Gain in case of success (Reward)
  • r: Loss in case of failure (Risk)

Definition 2.1 (Risk-Reward Ratio)

RRR=RrRRR = \frac{R}{r}

Conventionally expressed as R:r (e.g., 3:1 means RRR = 3).

Definition 2.2 (Trade Distribution)

X={+Rwith probability p−rwith probability 1−pX = \begin{cases} +R & \text{with probability } p \\ -r & \text{with probability } 1-p \end{cases}

III. Mathematical Expectation and Decision

Theorem 3.1 (Trade Expectation)

E[X]=p⋅R−(1−p)⋅rE[X] = p \cdot R - (1-p) \cdot r

Corollary 3.2 (Profitability Condition)

A trade is profitable in expectation if and only if: E[X]>0  ⟺  p>rR+r=11+RRRE[X] > 0 \iff p > \frac{r}{R + r} = \frac{1}{1 + RRR}

Proof: p⋅R>(1−p)⋅rp \cdot R > (1-p) \cdot r p(R+r)>rp(R + r) > r p>rR+r=11+R/rp > \frac{r}{R+r} = \frac{1}{1 + R/r} ∎

Table 3.1 (Break-Even Threshold by RRR)

| RRR | p_min (break-even threshold) | |-----|------------------------------| | 1:1 | 50.0% | | 2:1 | 33.3% | | 3:1 | 25.0% | | 5:1 | 16.7% |

IV. RRR Optimization

4.1 The High RRR Paradox

Increasing RRR decreases p_min but also increases the probability that stop-loss is hit before take-profit.

Theorem 4.1 (RRR/Probability Trade-off)

Under the symmetric random walk hypothesis, the probability of reaching take-profit before stop-loss is: phit=rR+r=11+RRRp_{hit} = \frac{r}{R + r} = \frac{1}{1 + RRR}

Corollary 4.2 (Neutrality in Random Walk)

Under the pure random walk hypothesis: E[X]=phit⋅R−(1−phit)⋅r=0E[X] = p_{hit} \cdot R - (1-p_{hit}) \cdot r = 0

Implication: Without an edge (statistical advantage), no RRR generates profit.

V. Edge and Its Quantification

Definition 5.1 (Edge)

Edge=pactual−ptheoretical=pactual−11+RRREdge = p_{actual} - p_{theoretical} = p_{actual} - \frac{1}{1+RRR}

Definition 5.2 (Expectancy)

Expectancy=E[X]rExpectancy = \frac{E[X]}{r}

Theorem 5.3 (Expectancy as Function of Edge)

Expectancy=Edge⋅(1+RRR)Expectancy = Edge \cdot (1 + RRR)

VI. Kelly Criterion: Optimal Position Sizing

Theorem 6.1 (Kelly Formula)

The optimal fraction of capital to risk: f∗=p⋅(RRR+1)−1RRR=p−1−pRRRf^* = \frac{p \cdot (RRR + 1) - 1}{RRR} = p - \frac{1-p}{RRR}

6.2 Practical Kelly Fraction

In practice, use half-Kelly or quarter-Kelly to reduce capital volatility.

VII. Practical Application

7.1 Stop-Loss and Take-Profit Calculation

  • Stop-Loss: SL = Entry ± k · ATR (k ∈ [1, 3])
  • Take-Profit: TP depends on chosen RRR

7.2 Numerical Example

  • Entry: 1.1000 (buy EUR/USD)
  • ATR(14): 0.0050 (50 pips)
  • k = 1.5 → Stop-Loss: 1.0925 (75 pips)
  • RRR = 2:1 → Take-Profit: 1.1150 (150 pips)

VIII. Fundamental Theorem of Risk Management

Theorem 8.1 (Long-Term Survival)

Even with a positive edge, excessive position sizing leads to certain ruin.

IX. Exercises

Exercise 1: Given p = 0.65 and RRR = 1.5. Calculate expectancy and Kelly fraction.

Exercise 2: Prove that Expectancy = 0 ⟹ Edge = 0.

Exercise 3: A trader has 60% win rate. What minimum RRR ensures positive expectancy?

X. References

  • Kelly, J.L. (1956). A New Interpretation of Information Rate
  • Thorp, E.O. (1969). Optimal Gambling Systems for Favorable Games
  • Van Tharp, R. (2007). Trade Your Way to Financial Freedom
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