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Roughly speaking the Sharpe ratio of a mutual fund is the ratio of its expected reward to its risk—how much one can expect to be rewarded per unit of risk taken. But what is expected reward? What is risk? That's what we will talk about here.
The reward of a mutual fund over a quarter is its excess return over money market. That is our definition. Given the rewards of the past 12 quarters, what reward should one expect over the next quarter? That is the fundamental question.
(We will only consider quarterly rewards in this exposition. Quarterly data is general enough, and it is freely available on the Web, at <http://finance.yahoo.com/> for example.)
We posit that the expected reward should be a weighted sum of the rewards for the past 12 quarters. The weights should be positive and sum to one. Moreover, it doesn't seem reasonable that the rewards of two or three years ago should count as much as last year's rewards—if you believe in momentum, that is.
The variance of the rewards over a 12 quarter span is the weighted sum of the square deviations of the rewards from their expected reward. The same weights should be used to compute both expected reward and variance.
These statistics, expected reward and variance, as just described, are quarterly quantities. To annualize them multiply by 4. The (annualized) Sharpe ratio is then defined to be the ratio of the (annualized) expected reward to the square root of the (annualized) variance of rewards.
From now on the unqualified use of "expected reward," "variance," and "Sharpe ratio" will refer to the annualized quantities. And "risk"? Well that must be the denominator of the Sharpe ratio, the square root of variance.