Volatility Calculation Methodology

Overview

Market volatility measures the overall risk of the CoinRisqLab 80 Index portfolio. It captures how the combined value of the portfolio fluctuates over time, accounting for both individual asset movements and the correlations between them.

Portfolio volatility is estimated using historical price data over a 90-day rolling window. The procedure consists of two main steps: computing logarithmic returns for each constituent, then calculating portfolio-level volatility using market-cap weighting and the full covariance matrix to account for correlations between constituents.

For individual cryptocurrency volatility, see the Risk Metrics methodology.

Glossary

Volatility

Measures the degree of variation in an asset's returns over time. It reflects the typical magnitude of price fluctuations and represents the level of risk or uncertainty associated with the asset's price movements.

Logarithmic Returns

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

r_t = \ln\left(\frac{P_{\text{today}}}{P_{\text{yesterday}}}\right)

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.

\sigma_{\text{annual}} = \sigma_{\text{daily}} \times \sqrt{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 Level	Annualized	Daily	Color	Description
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:

Market volatility 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

Parameter	Value	Description
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 market volatility calculation follows a two-stage pipeline, where each stage builds upon the previous one:

Stage 1

Log Returns Calculation

Calculate daily logarithmic returns for all cryptocurrencies

Stage 2

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

Daily returns are calculated using logarithmic returns, which are widely used in financial analysis due to their desirable statistical properties and time additivity:

r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)

Where $P_t$ is the closing price at time $t$ and $P_{t-1}$ is the closing price at time $t-1$ .

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

Portfolio Volatility

The third stage calculates the volatility of the top 40 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 = \frac{\text{MarketCap}_i}{\sum_{j=1}^{n} \text{MarketCap}_j}

Where

\text{MarketCap}_i = \text{Price}_i \times \text{CirculatingSupply}_i

Important: Weights must sum to 1.0 (100%)

Step 2: Covariance Matrix Construction

Build the covariance matrix for all constituent pairs:

\text{Cov}(i,j) = \frac{1}{T-1} \sum_{t=1}^{T} \left(r_{i,t} - \mu_i\right)\left(r_{j,t} - \mu_j\right)

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:

\sigma^2_p = \mathbf{w}^\top \Sigma \, \mathbf{w}

Where:

\mathbf{w}

= column vector of weights

[w_1, w_2, \ldots, w_n]^\top

\Sigma

= covariance matrix (

n \times n

)

\mathbf{w}^\top

= transpose of weight vector

Expanded form:

\sigma^2_p = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i \, w_j \, \text{Cov}(i,j)

Step 4: Annualization

\sigma_{\text{daily}} = \sqrt{\sigma^2_p}

\sigma_{\text{annual}} = \sigma_{\text{daily}} \times \sqrt{365}

Calculation Examples

Portfolio Volatility Example (Simplified)

Given: 2-asset portfolio for simplicity

Asset 1 (BTC): weight

w_1 = 0.60

,

\sigma_1 = 0.03

Asset 2 (ETH): weight

w_2 = 0.40

,

\sigma_2 = 0.04

Correlation:

\rho = 0.70

Covariance matrix:

\Sigma = \begin{bmatrix} \sigma_1^2 & \rho \, \sigma_1 \, \sigma_2 \\ \rho \, \sigma_1 \, \sigma_2 & \sigma_2^2 \end{bmatrix} = \begin{bmatrix} 0.0009 & 0.00084 \\ 0.00084 & 0.0016 \end{bmatrix}

Portfolio variance:

\begin{aligned} \sigma^2_p &= \begin{bmatrix} 0.6 & 0.4 \end{bmatrix} \begin{bmatrix} 0.0009 & 0.00084 \\ 0.00084 & 0.0016 \end{bmatrix} \begin{bmatrix} 0.6 \\ 0.4 \end{bmatrix} \\\\ &= \begin{bmatrix} 0.6 & 0.4 \end{bmatrix} \begin{bmatrix} 0.000876 \\ 0.001144 \end{bmatrix} \\\\ &= 0.000983 \end{aligned}

Daily portfolio volatility:

\sigma_p = \sqrt{0.000983} = 0.0313

(3.13% per day)

Annualized portfolio volatility:

\sigma_{p,\text{annual}} = 0.0313 \times \sqrt{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:

\sigma_p \leq \sum_{i=1}^{n} w_i \, \sigma_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

Volatility Calculation - Methodology

Table of Contents

Overview

Glossary

Volatility

Logarithmic Returns

Standard Deviation

Annualization

Rolling Window

Covariance Matrix

Portfolio Volatility

Trading Days

Bessel's Correction

Risk Level Classification

Base Parameters

Calculation Pipeline

Logarithmic Returns

What are Log Returns?

Calculation Formula

Data Selection

Portfolio Volatility

Why Use a Covariance Matrix?

Step 1: Weight Calculation

Step 2: Covariance Matrix Construction

Step 3: Portfolio Variance Calculation

Step 4: Annualization

Calculation Examples

Portfolio Volatility Example (Simplified)

Diversification Benefit

The Diversification Effect

Mathematical Relationship