{"title":"Sharpe Ratio and Probability of Loss","published":"2026-09-17","measure":"Probability of negative excess return relative to the risk-free rate","method":"For a true annualized Sharpe S and horizon T in years, probability of negative excess return is Φ(-S × √T).","assumptions":{"returnProcess":"Independent, Normally distributed excess returns","sharpeRatio":"Known true annualized Sharpe ratio","benchmark":"Risk-free rate"},"probabilityBySharpeAndHorizon":[{"annualizedSharpe":0.25,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"probabilityNegativeExcessReturn":0.471233793707},{"annualizedSharpe":0.25,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"probabilityNegativeExcessReturn":0.45026177517},{"annualizedSharpe":0.25,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"probabilityNegativeExcessReturn":0.429841897599},{"annualizedSharpe":0.25,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"probabilityNegativeExcessReturn":0.401293674317},{"annualizedSharpe":0.25,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"probabilityNegativeExcessReturn":0.332502771051},{"annualizedSharpe":0.25,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"probabilityNegativeExcessReturn":0.288075061015},{"annualizedSharpe":0.5,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"probabilityNegativeExcessReturn":0.442616957237},{"annualizedSharpe":0.5,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"probabilityNegativeExcessReturn":0.401293674317},{"annualizedSharpe":0.5,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"probabilityNegativeExcessReturn":0.361836804916},{"annualizedSharpe":0.5,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"probabilityNegativeExcessReturn":0.308537538726},{"annualizedSharpe":0.5,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"probabilityNegativeExcessReturn":0.193238115386},{"annualizedSharpe":0.5,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"probabilityNegativeExcessReturn":0.131776238641},{"annualizedSharpe":0.75,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"probabilityNegativeExcessReturn":0.414296537609},{"annualizedSharpe":0.75,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"probabilityNegativeExcessReturn":0.353830233327},{"annualizedSharpe":0.75,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"probabilityNegativeExcessReturn":0.297941545283},{"annualizedSharpe":0.75,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"probabilityNegativeExcessReturn":0.226627352377},{"annualizedSharpe":0.75,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"probabilityNegativeExcessReturn":0.096965426141},{"annualizedSharpe":0.75,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"probabilityNegativeExcessReturn":0.046766256345},{"annualizedSharpe":1.0,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"probabilityNegativeExcessReturn":0.386414996342},{"annualizedSharpe":1.0,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"probabilityNegativeExcessReturn":0.308537538726},{"annualizedSharpe":1.0,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"probabilityNegativeExcessReturn":0.239750061093},{"annualizedSharpe":1.0,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"probabilityNegativeExcessReturn":0.158655253931},{"annualizedSharpe":1.0,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"probabilityNegativeExcessReturn":0.041632258332},{"annualizedSharpe":1.0,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"probabilityNegativeExcessReturn":0.012673659339},{"annualizedSharpe":1.5,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"probabilityNegativeExcessReturn":0.332502771051},{"annualizedSharpe":1.5,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"probabilityNegativeExcessReturn":0.226627352377},{"annualizedSharpe":1.5,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"probabilityNegativeExcessReturn":0.144422183173},{"annualizedSharpe":1.5,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"probabilityNegativeExcessReturn":0.066807201269},{"annualizedSharpe":1.5,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"probabilityNegativeExcessReturn":0.00468738423},{"annualizedSharpe":1.5,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"probabilityNegativeExcessReturn":0.000398115079},{"annualizedSharpe":2.0,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"probabilityNegativeExcessReturn":0.281851430825},{"annualizedSharpe":2.0,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"probabilityNegativeExcessReturn":0.158655253931},{"annualizedSharpe":2.0,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"probabilityNegativeExcessReturn":0.078649603525},{"annualizedSharpe":2.0,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"probabilityNegativeExcessReturn":0.022750131948},{"annualizedSharpe":2.0,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"probabilityNegativeExcessReturn":0.000266002753},{"annualizedSharpe":2.0,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"probabilityNegativeExcessReturn":3.872108e-06}],"requiredSharpeByTargetProbability":[{"targetProbability":0.2,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"requiredAnnualizedSharpe":2.915461},{"targetProbability":0.2,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"requiredAnnualizedSharpe":1.683242},{"targetProbability":0.2,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"requiredAnnualizedSharpe":1.190232},{"targetProbability":0.2,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"requiredAnnualizedSharpe":0.841621},{"targetProbability":0.2,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"requiredAnnualizedSharpe":0.48591},{"targetProbability":0.2,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"requiredAnnualizedSharpe":0.376384},{"targetProbability":0.1,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"requiredAnnualizedSharpe":4.439425},{"targetProbability":0.1,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"requiredAnnualizedSharpe":2.563103},{"targetProbability":0.1,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"requiredAnnualizedSharpe":1.812388},{"targetProbability":0.1,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"requiredAnnualizedSharpe":1.281552},{"targetProbability":0.1,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"requiredAnnualizedSharpe":0.739904},{"targetProbability":0.1,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"requiredAnnualizedSharpe":0.573127},{"targetProbability":0.05,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"requiredAnnualizedSharpe":5.69794},{"targetProbability":0.05,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"requiredAnnualizedSharpe":3.289707},{"targetProbability":0.05,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"requiredAnnualizedSharpe":2.326174},{"targetProbability":0.05,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"requiredAnnualizedSharpe":1.644854},{"targetProbability":0.05,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"requiredAnnualizedSharpe":0.949657},{"targetProbability":0.05,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"requiredAnnualizedSharpe":0.735601},{"targetProbability":0.01,"horizonId":"1-month","horizonLabel":"1 month","horizonYears":0.083333333333,"requiredAnnualizedSharpe":8.058705},{"targetProbability":0.01,"horizonId":"3-month","horizonLabel":"3 months","horizonYears":0.25,"requiredAnnualizedSharpe":4.652696},{"targetProbability":0.01,"horizonId":"6-month","horizonLabel":"6 months","horizonYears":0.5,"requiredAnnualizedSharpe":3.289953},{"targetProbability":0.01,"horizonId":"1-year","horizonLabel":"1 year","horizonYears":1.0,"requiredAnnualizedSharpe":2.326348},{"targetProbability":0.01,"horizonId":"3-year","horizonLabel":"3 years","horizonYears":3.0,"requiredAnnualizedSharpe":1.343118},{"targetProbability":0.01,"horizonId":"5-year","horizonLabel":"5 years","horizonYears":5.0,"requiredAnnualizedSharpe":1.040374}],"limitations":["Sharpe alone does not determine the probability of a nominal dollar loss because it does not separately specify the risk-free return or volatility.","Real returns may be serially correlated, skewed, heavy-tailed, heteroskedastic, or regime-dependent.","The calculations assume the true Sharpe is known and do not address estimation uncertainty or strategy-selection bias."]}