Distributions & Hypothesis Testing

In Risk Management we will often need statistical tools of distributions and hypothesis testing to make guess on future of portfolio returns and avoid ourselves from any losses at minimum cost.

Plotting stock returns along time line, give us a distribution (it can be normal or log normal). Some of the distributions are called parametric distributions, mean they can be plotted with few parameters. Normal distribution can be plotted by having only mean return and variance.

Hypothesis testing is test to verify our decisions, for example we want to find out return distribution for the portfolio, however we have only 30 days data. Assuming stock returns follow normal distributions, 30 days data can be used to guess mean returns for longer duration. We need to perform student's t-distribution test to test our hypothesis.

T-score =(X - U)/ (S/Sqrt(n))

H0 : X = U

H1 : X not Equal to U

U in this case will be the sample portfolio which portfolio manager has used benchmark.

We will find out where T-score lies on standard t-distribution, if lies of either side of critical region(depending upon confidence interval) then null hypothesis is rejected. We can say we reject null hypothesis where sample mean was suppose to equal population mean.

One of the most important point is to choose right interval , for example if we using 95% confidence interval. If the null hypothesis is right , probability of its getting rejected is 5% otherwise if null hypothesis is wrong then it probability of getting accepted is B (0-95%). We have to set up a right balance over here while using hypothesis as a tool for portfolio analysis.

Similarly we can use this test to find out portfolio manager claim of daily return and volatility. This was t-test apart from this , we have number of statistical test normal test, ANOVA test,2 part t-distribution to compare portfolio variance with other portfolios or compare returns, I will recommend doing Google.