Volatility Calculation - Methodology

How we measure risk and volatility

Table of Contents

Overview

The CoinRisqLab 80 Volatility system calculates risk metrics for both individual cryptocurrencies and the CoinRisqLab 80 Index portfolio. Our methodology uses logarithmic returns and implements a rolling window approach to measure historical volatility.

For portfolio-level calculations, we use proper market-cap weighting and full covariance matrix calculations to account for correlations between constituents, providing a comprehensive view of portfolio risk.

This approach is based on Modern Portfolio Theory and uses the same statistical methods employed by professional financial institutions to measure and manage risk.

Glossary

Volatility

A statistical measure of the dispersion of returns for a given security or market index. Higher volatility means higher risk and larger price swings.

Logarithmic Returns

The natural logarithm of the ratio of consecutive prices. Logarithmic returns have better statistical properties than simple percentage returns.

Log_Return = ln(Price_today / Price_yesterday)

Standard Deviation

A measure of the amount of variation in a set of values. In finance, the standard deviation of returns is used as a measure of volatility.

Annualization

The process of converting a daily volatility measure to an annual equivalent by multiplying by the square root of the number of trading periods in a year.

Annual_Volatility = Daily_Volatility × √365

Rolling Window

A fixed-size time period (e.g., 90 days) that slides forward in time. Each calculation uses the most recent N observations, providing a moving view of volatility.

Covariance Matrix

A square matrix showing the covariance between pairs of assets. Used to capture how different assets move together, essential for portfolio risk calculations.

Portfolio Volatility

The volatility of a portfolio that accounts for both individual asset volatilities and their correlations. Usually lower than the weighted average of individual volatilities due to diversification.

Trading Days

For annualization purposes, we use 365 trading days per year. Unlike traditional financial markets that close on weekends and holidays, cryptocurrency markets operate 24/7, 365 days a year.

Bessel's Correction

When calculating variance from a sample of data (like 90 days of returns), we divide by n-1 instead of n to get an unbiased estimate. This is applied to all our variance and covariance calculations.

Risk Level Classification

We classify volatility into four risk levels using rigorous analytical standards calibrated for cryptocurrency markets. This classification covers both annualized and daily volatility, providing an intuitive way to understand and compare risk across different time scales.

Risk LevelAnnualizedDailyColorDescription
Low Risk
< 10%< 0.52%
Green
Stable mature cryptos, low risk
Medium Risk
10% - 30%0.52% - 1.57%
Yellow
Moderate volatility, established assets with fluctuations
High Risk
30% - 60%1.57% - 3.14%
Orange
High volatility, speculative assets or unstable phases
Extreme Risk
≥ 60%≥ 3.14%
Red
Extreme volatility, high-risk altcoins or strong market turbulence

Application Usage:

  • Portfolio risk level indicators on the dashboard
  • Individual cryptocurrency volatility badges
  • Risk contribution analysis in portfolio breakdown
  • Volatility gauge on the main dashboard

Note: These thresholds are calibrated for cryptocurrency markets, which typically exhibit higher volatility than traditional financial assets. A "low risk" crypto asset (5% annualized volatility) would still be considered moderate to high risk in traditional equity markets.

Base Parameters

ParameterValueDescription
Window Period
90 days
Rolling window for volatility calculations
Return Type
Logarithmic
Natural logarithm of price ratios
Annualization Factor
√365 ≈ 19.10
Assumes 365 trading days per year (crypto markets 24/7)
Minimum Data Points
90 observations
Required for volatility calculation

Calculation Pipeline

The volatility calculation follows a three-stage pipeline, where each stage builds upon the previous one:

Stage 1

Log Returns Calculation

Calculate daily logarithmic returns for all cryptocurrencies

Stage 2

Individual Crypto Volatility

Calculate rolling volatility for each cryptocurrency

Stage 3

Portfolio Volatility

Calculate index-level volatility using covariance matrix

Stage 1

Logarithmic Returns

The first stage calculates daily logarithmic returns for all cryptocurrencies, which serve as the foundation for all subsequent volatility calculations.

What are Log Returns?

Logarithmic returns measure the continuously compounded rate of return between two periods. They have several advantages over simple percentage returns:

  • Time-additive: Returns over multiple periods sum algebraically
  • Symmetric: A +10% gain and -10% loss produce equal absolute log returns
  • Better statistics: More suitable for normal distribution assumptions
  • Approximation: For small changes, log returns ≈ percentage changes

Calculation Formula

For each consecutive pair of days:

Log_Return[t] = ln(Price[t] / Price[t-1])
              = ln(Price[t]) - ln(Price[t-1])

Where ln() is the natural logarithm function.

Data Selection

We select the latest price for each day:

  • End-of-day snapshot (latest timestamp per day)
  • Only positive prices (price_usd > 0)
  • Ordered chronologically
Stage 2

Individual Crypto Volatility

The second stage calculates rolling volatility for individual cryptocurrencies using the log returns from Stage 1.

Rolling Window Setup

For each cryptocurrency with sufficient data (≥90 log returns):

Window[i] = [Return[i-89], Return[i-88], ..., Return[i]]

The window slides forward one day at a time, always containing exactly 90 consecutive daily log returns.

Statistical Calculations

For each 90-day window, we calculate:

a) Mean Return

μ = (1/n) × Σ(i=1 to n) r[i]
Where n = 90 (window size), r[i] = log return for day i

b) Daily Volatility (Standard Deviation)

σ_daily = √[(1/(n-1)) × Σ(i=1 to n) (r[i] - μ)²]

c) Annualized Volatility

σ_annual = σ_daily × √365
Where 365 = trading days per year (crypto markets 24/7)

Why Multiply by √365?

The square root of time rule applies under the assumption of independent and identically distributed returns. Since cryptocurrency markets operate 24/7 throughout the year, we use 365 days to convert daily volatility to an annual measure that's comparable across different assets and time periods.

Stage 3

Portfolio Volatility

The third stage calculates the volatility of the CoinRisqLab 80 Index portfolio using market-cap weights and the full covariance matrix to account for correlations between constituents.

Why Use a Covariance Matrix?

Simply taking a weighted average of individual volatilities would overestimate portfolio risk. The covariance matrix captures how assets move together:

  • Assets that move in opposite directions reduce portfolio risk
  • Imperfect correlation provides diversification benefits
  • Portfolio volatility is typically lower than weighted average of individual volatilities

Step 1: Weight Calculation

Weights are based on market capitalization:

w[i] = MarketCap[i] / Σ(j=1 to n) MarketCap[j]
Where MarketCap[i] = Price[i] × CirculatingSupply[i]

Important: Weights must sum to 1.0 (100%)

Step 2: Covariance Matrix Construction

Build the covariance matrix for all constituent pairs:

Cov(i,j) = (1/(T-1)) × Σ(t=1 to T) [(r[i,t] - μ[i]) × (r[j,t] - μ[j])]
Where r[i,t] = log return of asset i at time t, T = 90 (window size)

The covariance matrix is an n×n symmetric matrix where:

  • Diagonal elements: variances of individual assets
  • Off-diagonal elements: covariances between asset pairs
  • Captures the correlation structure of the portfolio

Step 3: Portfolio Variance Calculation

Using modern portfolio theory:

σ²_portfolio = w' × Σ × w

Where:
  w  = column vector of weights [w₁, w₂, ..., wₙ]'
  Σ  = covariance matrix (n×n)
  w' = transpose of weight vector

Expanded form:

σ²_portfolio = Σ(i=1 to n) Σ(j=1 to n) w[i] × w[j] × Cov(i,j)

Step 4: Annualization

σ_portfolio_daily = √(σ²_portfolio)
σ_portfolio_annual = σ_portfolio_daily × √365

Calculation Examples

Example 1: Individual Crypto Volatility

Given: Bitcoin (BTC) with 90-day log returns

Day 1: Price = $40,000 → $42,000, r₁ = ln(42000/40000) = 0.0488
Day 2: Price = $42,000 → $41,000, r₂ = ln(41000/42000) = -0.0241
...
Day 90: Price = $45,000 → $46,000, r₉₀ = ln(46000/45000) = 0.0220
Mean return: μ = 0.0015
Daily volatility: σ_daily = 0.03 (3% per day)
Annualized volatility: σ_annual = 0.03 × √365 = 0.573
Result: Bitcoin has an annualized volatility of
57.3%

Example 2: Portfolio Volatility (Simplified)

Given: 2-asset portfolio for simplicity

Asset 1 (BTC): weight = 0.60, σ₁ = 0.03
Asset 2 (ETH): weight = 0.40, σ₂ = 0.04
Correlation: ρ = 0.70

Covariance matrix:

Σ = [σ₁²           ρ×σ₁×σ₂  ]
    [ρ×σ₁×σ₂       σ₂²      ]

  = [0.0009         0.00084  ]
    [0.00084        0.0016   ]

Portfolio variance:

σ²_p = [0.6  0.4] × [0.0009    0.00084] × [0.6]
                     [0.00084   0.0016 ]   [0.4]

     = [0.6  0.4] × [0.000876]
                     [0.001144]

     = 0.000983
Daily portfolio volatility: σ_p = √0.000983 = 0.0313 (3.13% per day)
Annualized portfolio volatility: σ_p_annual = 0.0313 × √365 = 0.598
Result: Portfolio volatility is
59.8%

Diversification Benefit

The portfolio volatility calculation demonstrates a key principle of Modern Portfolio Theory: diversification reduces risk.

The Diversification Effect

From Example 2 above:

Bitcoin individual volatility:
57.3%
Ethereum individual volatility:
76.4%
Weighted average volatility:
64.9%
Actual portfolio volatility:
59.8%

The portfolio volatility (59.8%) is 5.1% lower than the weighted average (64.9%), demonstrating the benefit of diversification!

Mathematical Relationship

The general principle:

σ_portfolio ≤ Σ w[i] × σ[i]
(Portfolio volatility ≤ Weighted average of individual volatilities)

This inequality holds as long as assets are not perfectly correlated. The benefit is greater when:

  • Correlations between assets are lower
  • The portfolio is more diversified (more constituents)
  • Asset weights are more balanced