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Hypothesis Testing and Confidence Intervals

4 important questions on Hypothesis Testing and Confidence Intervals

Why hypothesis testing?

Using confidence level is arbitrary as you said the confidence level yourself. In hypothesis testing we turn this around by assuming a value and calculating the corresponding probability instead of the other way around.

(mean-Value)/(SD/n^0.5)

For hypothesis testing, it is easier to test the thing that you don't expect to be true. Via this way you can statistically falsify whether the null hypothesis can be rejected and if so assume the alternative hypothesis.

Two or one tailed?

In case you are concerned with extreme values, i.e. high volatility, two-sided is more eligible. However, in risk management we are most often concerned with losses, meaning the left-side of the distribution, implying eligibility for one-sided.

Although no CL is required it is often still used as a reference, to say that 95% of 99% is enough confidence. Remember that in statistics there is no way you can be absolutely certain.

What about Chebyshev's inequality?

The probability that X is within n standard deviations of the mean is less than or equal to 1/n^2. This goes for all distributions. Good thing is that it places an upper limit on the probability of a variable being more than a certain distance from its mean. Although the (standard) normal distribution has much more certainty, 5% vs. 25% out of the range, this was exactly Chebyshev's point.

Normal distribution can be very anti-conservative. Assuming normality when a random variable is in fact not normal can lead to severe underestimation of risk!!!

What about Value-at-Risk?

Often used in sumamrizing the risk of the firm's entire portfolio or individual desks. The time horizon in this perspective is leading, could be one-day up to a year. Depending on the liquidity of the portfolio. Take note that here again the square root of time rule is valid for standard deviations.

Ambiguity in terms of definition is often the case, as a loss is presented by VaR as a positive number. But this is the normal convention used by most risk managers. Either VaR is a loss of 400 or a return of -400.