ch1_introduction

Market Prices and Returns

Equity markets exhibit a wide range of stock prices over time.
Simple price differences are inadequate for return computation due to the scale problem:
- A $1 move from $1 to $2 is significant.
- A $1 move from $150 to $151 is minor.
Percent moves are commonly used, calculated as:

$$\text{Percent Move} = \left(\frac{\text{Price Change}}{\text{Starting Price}}\right) \times 100$$

Non-symmetry of percent moves:
- A 10% gain followed by a 10% loss does not return to the original value.
- Example:
  - Start at $100.
  - 10% gain: $100 to $110.
  - 10% loss on $110: $110 to $99.
- Recording a 10% gain followed by a 9.1% loss shows a net gain, misleadingly.

Professional developers use logarithmic price changes to compute returns.
Logarithmic Return Calculation:
- Convert prices to their logarithms.
- Compute returns as the difference of these logs.
- Example:
  - Move from $10 to $11: $\log(11) - \log(10) = 2.39789 - 2.30258 = 0.09531$
  - Move from $100 to $110: $\log(110) - \log(100) = 4.70048 - 4.60517 = 0.09531$
  - Move back from $110 to $100: $\log(100) - \log(110) = -0.09531$, net gain = 0.
Advantages:
- Symmetric returns.
- Ensures accurate statistical analysis.
- Prevents misleading results in performance tests.

Smallish log price changes, times 100, approximate percent changes:
- Example: Moving from $100 to $101:
  - Percent change: 1%
  - Logarithmic change: $100 \times (\log(101) - \log(100)) = 0.995$
- Even a 10% move approximates well to 9.531%.

Logarithmic price changes provide a more accurate method for computing returns.
They address the asymmetry and misleading results of percent changes.
Logarithmic returns can be treated as approximate percent returns for practical purposes.

Future Leak:
- Illegal leakage of future knowledge into a testing procedure.
- Occurs when future market behavior is inadvertently used in the simulation of a trading system's performance.

Optimistic Performance Estimates:
- Future leak creates unrealistically positive performance estimates.
- Results from the fact that future market behavior, which is unknown during actual trading, is incorrectly factored into system testing.

Realism in Testing:
- Accurate performance simulations must exclude future knowledge.
- Ensures that the trading system's performance reflects real-world scenarios, where future market movements are unknown.

Future leak compromises the integrity of trading system development by providing misleading performance metrics, making it crucial to ensure that future market data does not influence system testing.

Expected Return Formula: $$ \text{ExpectedReturn} = \text{Win} \times P(\text{WIN}) - \text{Loss} \times P(\text{LOSS}) $$
Coin Toss Example:
- Flipping a fair coin:
  - Win $1 for heads, lose $1 for tails.
  - Expected return is zero because the probabilities and outcomes balance out.
- Unfair scenario:
  - Win $2 for heads, lose $1 for tails.
  - Expected return becomes positive, indicating a favorable situation.

True Random Walk Market:
- Market changes are independent and have zero mean.
- Trading systems developed for such markets have zero expected returns (excluding transaction costs).

Adjusting Win/Loss Ratios:
- Example: Setting a profit target 1 point above and a stop loss 9 points below entry price.
  - Each loss is 9 times the win amount.
  - Win 9 out of 10 times.
  - Expected return remains zero despite high win rate because large losses offset frequent small wins.

Interrelation of Win/Loss Sizes and Probabilities:
- Win frequency and win size are interdependent.
- High win rate doesn’t guarantee profitability if losses are significantly larger.
- Always consider both win/loss sizes and their probabilities when evaluating a trading system.

Critical Questions:
- Ask about the size of wins and losses when someone boasts about high win rates.
- Ask about win frequency when someone brags about large wins.
- Understand that win rates and win sizes must be evaluated together for a complete picture of trading system performance.