Mathematical Formulas & Methodology
Detailed explanation of how the index calculations work
Overview
The Index Calculator uses a weighted average approach that considers three key factors:
- Liquidity Factor - Markets with higher trading volume get more weight
- Significance Factor - Your importance score for each market
- Time Discount Factor - Markets closer to resolution get more weight
These factors are combined to create a final weight for each market, which is then used to calculate the weighted average index value.
Liquidity Factor
This factor gives more weight to markets with higher trading volume (open interest), as they are considered more reliable indicators.
$$f_{L,i} = \left[\ln\left(1 + \frac{L_i}{L_0}\right)\right]^\alpha$$
Parameters:
- $L_i$ = Open interest for market $i$
- $L_0$ = Liquidity scale parameter (default: 50,000)
- $\alpha$ = Liquidity exponent (default: 0.5)
Interpretation:
- Higher $L_0$ = less emphasis on volume differences
- Lower $\alpha$ = more even weighting across markets
- Natural log prevents extreme values
Significance Factor
This factor incorporates your subjective importance rating for each market.
$$f_{S,i} = s_i^\gamma$$
Parameters:
- $s_i$ = Your significance score (0-1)
- $\gamma$ = Significance exponent (default: 1)
Interpretation:
- $\gamma = 1$ = Linear relationship
- $\gamma > 1$ = Amplifies differences
- $\gamma < 1$ = Compresses differences
Time Discount Factor
This factor gives more weight to markets that are closer to resolution, as they provide more current information.
Exponential Method (Default):
$$f_{T,i} = 2^{-T_i/H}$$
Hyperbolic Method:
$$f_{T,i} = \frac{1}{1 + T_i/H}$$
Parameters:
- $T_i$ = Days until resolution for market $i$
- $H$ = Half-life parameter (default: 60 days)
Interpretation:
- Lower $H$ = faster time decay
- Exponential: sharper decay curve
- Hyperbolic: gentler decay curve
Final Index Calculation
The final index value is calculated by combining all factors and normalizing the weights.
Step 1: Calculate Pre-weights
$$a_i = f_{S,i} \times f_{L,i} \times f_{T,i}$$
Step 2: Normalize Weights
$$w_i = \frac{a_i}{\sum_j a_j}$$
Step 3: Calculate Final Index
$$\text{Index} = 100 \times \sum_i w_i \times \tilde{p}_i$$
Note: $\tilde{p}_i = p_i$ if orientation $= +1$ (Yes = Index Up), otherwise $\tilde{p}_i = 1 - p_i$ (No = Index Up)
Variables:
- $w_i$ = Final weight for market $i$
- $p_i$ = Market price (0-1)
- $\tilde{p}_i$ = Adjusted price based on orientation
Result:
- Index value ranges from 0-100
- 100 = All markets at maximum price
- 0 = All markets at minimum price
Default Parameters
Liquidity Scale
$L_0 = 50,000$
Liquidity Exponent
$\alpha = 0.5$
Significance Exponent
$\gamma = 1$
Time Half-life
$H = 60$ days
Parameter Guidelines
Liquidity Scale ($L_0$)
- 10,000-25,000: High volume emphasis
- 50,000: Balanced (default)
- 100,000+: Low volume emphasis
Liquidity Exponent ($\alpha$)
- 0.1-0.3: Very even weighting
- 0.5: Moderate weighting (default)
- 0.7-1.0: Strong volume bias
Significance Exponent ($\gamma$)
- 0.5: Compressed differences
- 1.0: Linear (default)
- 1.5-2.0: Amplified differences
Time Half-life ($H$)
- 7-30 days: Fast decay
- 60 days: Moderate decay (default)
- 90-180 days: Slow decay