Looking back at old textbooks from 40 years ago, I realised that I had never learned as much as I needed to know now. In the 1970s, the UK actuarial tuition included material on testing for goodness of fit, mainly using chi-squared. However, there was little on how to select an appropriate distribution. What is clear is that none of the variables being considered is Normally distributed and we do need something better.
Further, for more robust results, one should look at moments as well as goodness of fit (well explained here). Once distributions had been selected, I needed either 150,000 (returns) or 10,000 (yields) random numbers for each variable (10,000 simulations across 15 intervals or at initial time point alone). The test statistic used was the Akaike Information Criterion (“AIC”). Even if two distributions appear to be almost equally good fits, the results can be markedly different. This year, I shall think about using “best” and, say, “fifth best”.
For each set of random variables, the outliers beyond the observed extremes were adjusted to fall within an arbitrary 0.5% pa (2014) or 1.0% pa (2018a) away from the observed range. The series were then adjusted to give the observed variance and then further adjusted to give the observed mean. For 2014, the final series were then retested to make sure that the correct distribution was still present. For one variable, I had to pick the third best distribution for the final test to be satisfied. For 2018a, with two different series present (other than for ILGs), that retesting would have been pointless.
One particular feature revealed from extending the financials into 2012 is that we now know that long ILG yields can be negative; who would have guessed that would happen? Nobody appears to have been recorded as having mentioned the possibility when they were first issued over 30 years ago. They became positive in 2013 and have been negative again during part of 2014 and 2015 and then from mid-July (after the Brexit referendum) until the present.