Analytics Calculation

Additional financial and statistical metrics used by OLTA to provide a comprehensive view of each index’s risk-return profile.

Purpose

While Value at Risk (VaR) captures potential downside loss, a broader set of analytics is required to understand the full risk landscape of an index. OLTA uses the following complementary indicators to enrich institutional analysis and improve fund transparency.

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Key Metrics

1. Conditional Value at Risk (CVaR)

Also known as Expected Shortfall, CVaR estimates the average loss in the worst-case scenarios beyond the VaR threshold.

$${CVaR} = \mathbb{E}[L \mid L > \text{VaR}]$$

Where:

• L = portfolio loss

• VaR = Value at Risk at a chosen confidence level (e.g., 95%)

• 𝔼 = expected value operator, representing the average outcome under the given condition

2. Maximum Drawdown

The largest observed loss from a peak to a trough before a new peak is attained.

$${Max Drawdown} = \frac{\text{Peak} - \text{Trough}}{\text{Peak}}$$

3. Sharpe Ratio

Measures return per unit of total volatility.

$${Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}$$

Where:

• Rp = portfolio return

• Rf = risk-free rate

• σp = standard deviation of portfolio returns

Interpretation of Sharpe Ratio

• Sharpe < 1.0 - Suboptimal risk-adjusted performance; return does not sufficiently compensate for volatility.

• Sharpe ≈ 1.0 - Acceptable / good performance; often considered the minimum threshold for institutional portfolios.

• Sharpe between 1.0 – 2.0 - Very good risk-adjusted performance; return is strong relative to volatility.

• Sharpe > 2.0 - Exceptional performance; rare in traditional markets, more common in niche strategies or short time horizons.

4. Sortino Ratio

Like the Sharpe Ratio but penalizes only downside volatility.

$${Sortino Ratio} = \frac{R_p - R_f}{\sigma_d}$$

Where:

• σd = standard deviation of downside deviation only

Interpretation of Sortino Ratio

• Sortino < 1.0 - Returns insufficient relative to downside risk.

• Sortino ≈ 1.0 - Acceptable performance; minimum level expected by institutional investors.

• Sortino between 1.0 – 2.0 - Strong downside risk-adjusted performance.

• Sortino > 2.0 - Excellent protection against downside volatility; high-quality risk management.

Sharpe vs Sortino:

• Sharpe penalizes both upside and downside volatility.

• Sortino focuses only on downside volatility, making it more relevant for assessing capital preservation and asymmetric risk profiles.

5. Beta (vs BTC + ETH or benchmark)

Measures index sensitivity to the overall market (or BTC + ETH).

$$beta = \frac{\text{Cov}(R_p, R_m)}{\text{Var}(R_m)}$$

Where:

• Rp = portfolio return

• Rm = market (or BTC + ETH) return

Interpretation of Beta

• Beta = 1.0 - Index moves in line with the benchmark (BTC + ETH).

• Beta < 1.0 - Lower volatility relative to the benchmark; more defensive profile.

• Beta > 1.0 - Higher sensitivity and volatility than the benchmark; amplified market exposure.

• Beta < 0 - Negative correlation with the benchmark; potential hedge or diversifier.

Metric	Focus	Penalizes	Interpretation	Institutional Use
Sharpe Ratio	Overall risk-adjusted return	Both upside & downside volatility	Higher = better risk-adjusted return	Standard measure for performance reporting
Sortino Ratio	Downside risk-adjusted return	Downside volatility only	Higher = better downside protection	Favored for capital preservation strategies
Beta	Market sensitivity	N/A (compares covariance with benchmark)	Beta = 1: tracks market; Beta < 1: defensive; Beta > 1: aggressive; Beta < 0: inverse correlation	Used for portfolio hedging, benchmarking, and risk alignment

6. Turnover Ratio

Measures the percentage of the index that changes at each rebalancing.

$${Turnover} = \frac{1}{2} \sum \left| \text{New Weight} - \text{Old Weight} \right|$$

7. Liquidity-at-Risk (LaR)

Estimates the portion of the index that can be liquidated without exceeding a defined slippage threshold.

Concept: Evaluate order book depth (or AMM pool curve) required to sell a given notional (e.g., $100k) with ≤ 1% price impact.

8. Concentration Ratio (CR)

Measures the weight of the top n constituents in the index.

$$CR_{n} = \sum_{i=1}^{n} w_i$$

Where:

• wᵢ​ = weight of the i-th largest constituent

• n = number of top constituents included in the ratio

9. Herfindahl-Hirschman Index (HHI)

Provides a holistic measure of concentration by summing the squared weights of all constituents. The HHI is widely used by institutional investors and regulators as a standard measure of market concentration and systemic fragility.

$$HHI = \sum_{i=1}^{N} w_i^2$$

Where:

• wᵢ​ = weight of asset i in the index (expressed as a decimal)

• N = total number of assets in the index

Interpretation:

• HHI < 0.15 → Low concentration, strong diversification

• 0.15 ≤ HHI ≤ 0.25 → Moderate concentration

• HHI > 0.25 → High concentration, index fragility

10. Capital Asset Pricing Model (CAPM)

$$\mathbb{E}[R_i] = R_f + \beta_i \big(\mathbb{E}[R_m] - R_f\big)$$

• E[Ri]: expected return of the asset or OLTA index

• Rf: risk-free rate (e.g., T-Bills, USDC yield equivalent)

• βi: estimated sensitivity of the portfolio relative to the benchmark (β = 1 for the benchmark itself).

  • β = 1 → the asset moves in line with the benchmark

  • β > 1 → the asset amplifies benchmark movements (higher volatility and risk)

  • β < 1 → the asset is less sensitive than the benchmark (more defensive profile)

  • β < 0 → the asset moves in the opposite direction to the benchmark

  In the ex-ante CAPM, β is used as an input parameter to model expected returns.

• E[Rm]: expected return of the market benchmark

Ex-ante (Theoretical Model)

The ex-ante CAPM is the theoretical form. It expresses the expected return of an asset as a function of:

• the risk-free rate (Rf)

• the market risk premium

$$𝐸 [ 𝑅 𝑚 ] − 𝑅 𝑓 E[Rm​]−Rf​$$

• the portfolio’s beta (βi)

It is forward-looking and used for pricing and valuation models.

Ex-post (Empirical Regression)

$$(Ri,t​−Rf,t​)=α+β(Rm,t​−Rf,t​)+ε(t​)$$

• Ri,t : observed return of the asset at time t

• Rm,t : observed market return at time t

• Rf,t : observed risk-free rate at time t

• α : Jensen’s alpha (see 11.)

• β : estimated sensitivity of the portfolio relative to the benchmark (β = 1 for the benchmark itself). In the ex-post CAPM, β is estimated via regression on historical data

• εt : residual (unexplained price swing)

The ex-post CAPM is based on historical data and regression analysis. It estimates beta and alpha, making it the preferred model in practice for performance evaluation, backtesting, and empirical analysis.

Example Ex-ante vs Ex-post CAPM

Interpretation of the Chart

The chart illustrates the distinction between Ex-ante CAPM (theoretical model) and Ex-post CAPM (empirical regression):

1. Ex-ante CAPM (blue dashed line)

   • Represents the theoretical relationship between asset excess returns and market excess returns.

   • The line passes through the origin (α=0), assuming no systematic out- or under-performance beyond what is explained by market exposure.

   • It reflects the expected return of the portfolio given its beta relative to the benchmark.

2. Ex-post CAPM (green solid line)

   • Estimated from observed data points (gray scatter).

   • The regression line does not pass through the origin but is shifted upward by α>0, showing that the portfolio generated returns above the CAPM prediction.

   • This intercept represents Jensen’s Alpha, a measure of risk-adjusted outperformance.

3. Observed Data (gray points)

   • Individual time-period excess returns:

     $$(R_{i,t} - R_{f,t})$$

   plotted against market excess returns:

   $$(R_{m,t} - R_{f,t})$$

• Dispersion around the regression line reflects residual risk εt​.

11. Jensen’s Alpha (CAPM)

α (Jensen’s Alpha) measures the risk-adjusted performance of a portfolio relative to its benchmark, based on the CAPM framework.

$$α = \overline{R_i - R_f} \;-\; \beta \,\overline{R_m - R_f}$$

Interpretation

• α = 0 : the portfolio performed exactly as predicted by CAPM

• α > 0 : the portfolio outperformed expectations given its risk (positive value creation)

• α < 0 : the portfolio underperformed expectations given its risk (destruction of value)

• β : sensitivity of the portfolio relative to its benchmark (β = 1 for the benchmark itself).

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Implementation

• Displayed in index factsheets and dashboards

• Recalculated in real-time by default, screened monthly for reportings.

• Used to monitor index health, compliance, and historical risk-adjusted performance

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OLTA’s Risk Analytics suite offers investors and stakeholders a complete toolbox for evaluating structured crypto exposure with the same rigor found in traditional finance.