Systemic Docs

Name	Type	Description
Window	integer	- Description: The window parameter represents the number of periods over which the Ulcer Index is calculated. It defines the look-back period for assessing price drawdowns and volatility.
Price	floatarray	- Description: This refers to the historical price data of the security or asset for which the Ulcer Index is being computed. The price data typically includes daily, weekly, or monthly closing prices.

Name	Type	Description
UI	floatarray	- Description: The Ulcer Index (UI) itself is the primary output of the Ulcer Index calculation. It measures the degree of drawdown or volatility of an asset's price over a specified window.

The Ulcer Index measures downside volatility by quantifying the depth and duration of drawdowns. Unlike standard deviation, it focuses exclusively on price declines, making it a true measure of investment pain.

UI = \sqrt{\frac{\sum_{i=1}^{n} R_i^2}{n}}

Where $R_i$ = Percentage drawdown from the highest high over the period.

How It Works

flowchart TD
    subgraph Input["Price Series"]
        P["Closing Prices"]
    end
    
    subgraph Calc["Calculation Steps"]
        HH["Track rolling highest high"]
        DD["Drawdown = (High - Price) / High × 100"]
        SQ["Square each drawdown"]
        AVG["Average of squared drawdowns"]
        SQRT["Square root"]
    end
    
    subgraph Output["Result"]
        UI["Ulcer Index"]
    end
    
    P --> HH --> DD --> SQ --> AVG --> SQRT --> UI

Interpreting UI

UI Value	Interpretation
< 5	Low stress — shallow drawdowns
5-10	Moderate drawdown risk
10-20	Elevated volatility
> 20	High stress — deep/prolonged drawdowns

flowchart LR
    subgraph Stress["Ulcer Index Levels"]
        LOW["<5<br/>Low stress"]
        MOD["5-10<br/>Moderate"]
        HIGH["10-20<br/>Elevated"]
        VHIGH[">20<br/>High stress"]
    end
    
    LOW --> MOD --> HIGH --> VHIGH

Why Ulcer Index?

Traditional volatility measures (like standard deviation) treat upside and downside equally. But for investors:

Measure	Captures
Standard Deviation	All price movement
Ulcer Index	Only drawdown pain

Most investors don't mind upside volatility — they worry about losses.

Applications

1. Risk Assessment

Compare UI across different assets to identify which have historically caused more investor distress.

2. UPI (Ulcer Performance Index)

A risk-adjusted return metric:

UPI = \frac{Return}{Ulcer\ Index}

Higher UPI = better return per unit of drawdown pain.

3. Position Sizing

Lower UI assets can warrant larger position sizes in a portfolio.

UI vs Other Risk Metrics

Metric	Focus	Treats Up/Down
Ulcer Index	Drawdowns	Downside only
Std Dev	All volatility	Equally
Max Drawdown	Worst decline	Single event
Sortino Ratio	Downside deviation	Downside only

Advantages

Investor-aligned — Measures what actually hurts portfolios
Captures duration — Prolonged drawdowns penalized more
Risk-adjusted returns — Foundation for UPI metric
Comparable — Works across asset classes

Disadvantages

Backward-looking — Based on historical drawdowns
Period sensitive — Depends on lookback window
Less known — Not as standard as Sharpe ratio
Requires high prices — Needs continuous price series

Missing Data

A closed-market or missing-data bar (null) is never fabricated — not padded with 0.0 ("no stress"), not held stale across a gap. The rolling max-drawdown RMS runs over the observed bars only (window-bar UI = window observed bars, weekends bridged), then the result is placed back at the original grid slots:

Scenario	Output
Closed-market gap	`None` at that slot; the surrounding observed bars bridge the gap as adjacent bars (Friday→Monday), never a stale ffilled UI value
Warm-up (< `window` observed bars seen)	`None` — never the retired 0.0 ("no stress") fill
All bars missing	Every slot `None`, no crash
Live tick on a gap slot	`[None]` — correct and non-starving, not the retired ffilled value

Indicator Assessment

The following chart positions the Ulcer Index based on its complexity (ease of understanding and implementation) versus reliability (consistency of signals across market conditions). Points closer to the top indicate higher reliability; points to the left indicate lower complexity.

quadrantChart
    title Ulcer Index Assessment
    x-axis Low Complexity --> High Complexity
    y-axis Low Reliability --> High Reliability
    quadrant-1 Complex & Reliable
    quadrant-2 Simple & Reliable
    quadrant-3 Simple & Unreliable
    quadrant-4 Complex & Unreliable
    Stocks: [0.40, 0.75]
    Crypto: [0.40, 0.65]

Assessment Breakdown

Dimension	Stocks	Crypto	Notes
Popularity	5/10	4/10	Niche but respected; more common in fund analysis than retail trading
Reliability	7.5/10	6.5/10	Excellent for measuring downside risk; crypto's extreme volatility causes higher UI readings
Complexity	4/10	4/10	Conceptually simple—drawdown squared and averaged; straightforward to interpret
Versatility	6/10	5/10	Excellent for risk comparison and portfolio construction; less applicable as a trading signal
Recent Interest	5/10	4/10	Steady adoption in risk-adjusted analysis; growing interest post-2022 market declines

Ulcer Index

Inputs

Outputs