Is the strategy's mean return above zero? One-sample t and the Sharpe link

Question

A strategy returns a mean of 0.05\% per day with sample standard deviation 0.80\% over n = 100 trading days. Test whether its true mean daily return exceeds zero. When do you use t versus z, and how does the t-statistic relate to the Sharpe ratio?

Accepted Answer

One-sample test. H0: \mu = 0 against H1: \mu 0. The statistic is t = \frac{\bar x}{s / \sqrt{n}} = \frac{0.05\%}{0.80\% / \sqrt{100}} = \frac{0.05}{0.80 / 10} = \frac{0.05}{0.08} = 0.625, with 	ext{df} = 99. The one-sided 5\% critical value is about 1.66. Since 0.625 < 1.66, we fail to reject: 100 days is far too little to distinguish this mean from zero. t versus z. Use t when the standard deviation is estimated from the sample (the usual case); use z when \sigma is known or n is large enough that t and the normal nearly coincide (by n \approx 100 they are very close, but t is always the sa…

Is the strategy's mean return above zero? One-sample t and the Sharpe link

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