Rachev Ratio Formula: Upper Expected Tail Gain vs Lower Expected Tail Loss

Compare full-tail ETL averages against single-outlier MAX/MIN ratios

Introduced by Biglova, Ortobelli, Rachev, and Stoyanov (2004), the Rachev ratio measures non-Gaussian reward-to-tail-risk by dividing the average of the best beta-fraction of excess returns (Upper ETL) by the magnitude of the average of the worst alpha-fraction of excess returns (Lower ETL / CVaR). Unlike Sharpe, it does not penalize upside volatility; unlike a naive MAX / ABS(MIN) ratio, it averages across the tail cohort.

Enter 10 monthly benchmark-excess returns in B2:B11: 0.09, 0.07, 0.03, 0.02, 0.01, 0.005, -0.005, -0.01, -0.03, -0.05. At alpha = beta = 20% (2 observations per tail), Upper ETL = AVERAGE(0.09, 0.07) = 0.08 and Lower ETL = -AVERAGE(-0.05, -0.03) = 0.04, giving Rachev Ratio = 0.08 / 0.04 = 2.0000 (whereas single-observation MAX / ABS(MIN) gives 0.09 / 0.05 = 1.8000).

Rachev Ratio (k-Observation Tail ETL) Formula

Use SEQUENCE(k_obs) with LARGE and SMALL or explicit tail cohorts, and guard non-positive left-tail loss when all excess returns are positive.

=LET(up_etl, (LARGE(B2:B11, 1) + LARGE(B2:B11, 2)) / 2, dn_etl, -((SMALL(B2:B11, 1) + SMALL(B2:B11, 2)) / 2), IF(dn_etl <= 0, "No Left-Tail Loss", ROUND(up_etl / dn_etl, 4)))

Tail Shock Stress Test

Test sensitivity to a left-tail shock by changing worst month B11 from -0.05 to -0.13: Lower ETL doubles from 0.04 to -(-0.13 + -0.03)/2 = 0.08, cutting the Rachev ratio in half from 2.0000 to 1.0000 even though 9 of 10 months are unchanged.

=ROUND(MAX(B2:B11) / ABS(MIN(B2:B11)), 4)

Further reading

• Arte Sheets Lesson — Schwager Gain-to-Pain Ratio and Tail Ratio Audit
• Microsoft Support — LARGE function
• Microsoft Support — SMALL function