a week of data if each trade occurs every 50 milliseconds, with care taken toward risk from unexpected but rare results that such testing did not capture (see flash crash). Additionally, when examining the investment performance of assets with smoothing of returns (such as with-profits funds), the Sharpe ratio should be derived from the performance of the underlying assets rather than the fund returns (Such a model would invalidate the aforementioned Ponzi scheme, as desired). Sharpe ratios, along with Treynor ratios and Jensen's alphas, are often used to rank the performance of portfolio or mutual fund managers. Berkshire Hathaway had a Sharpe ratio of 0.76 for the period 1976 to 2011, higher than any other stock or mutual fund with a history of more than 30 years. The stock market had a Sharpe ratio of 0.39 for the same period. Tests Several statistical tests of the Sharpe ratio have been proposed. These include those proposed by Jobson & Korkie and Gibbons, Ross & Shanken. History In 1952, Arthur D. Roy suggested maximizing the ratio "(m-d)/σ", where m is expected gross return, d is some "disaster level" (a.k.a., minimum acceptable return, or MAR) and σ is standard deviation of returns. This ratio is just the Sharpe ratio, only using minimum acceptable return instead of the risk-free rate in the numerator, and using standard deviation of returns instead of standard deviation of excess returns in the denominator. Roy's ratio is also related to the Sortino ratio, which also uses MAR in the numerator, but uses a different standard deviation (semi/downside deviation) in the denominator. In 1966, William F. Sharpe developed what is now known as the Sharpe ratio. Sharpe originally called it the "reward-to-variability" ratio before it began being called the Sharpe ratio by later academics and financial operators. The definition was: Sharpe's 1994 revision acknowledged that the basis of comparison should be an applicable benchmark, which changes with time. After this revision, the definition is: Note, if Rf is a constant risk-free return throughout the period, Recently, the (original) Sharpe ratio has often been challenged with regard to its appropriateness as a fund performance measure during evaluation periods of declining markets. Examples Example 1 Suppose the asset has an expected return of 15% in excess of the risk free rate. We typically do not know if the asset will have this return. We estimate the risk of the asset, defined as