Looking back at old textbooks from longer ago than 40 years, 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, with little on how to select an appropriate distribution.

For more robust results, one should look at moments as well as goodness of fit (see here). Once distributions had been selected, I needed either 15,000 (returns) or 1,000 (yields) random numbers for each variable (1,000 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.

For each set of random variables, the outliers beyond the observed extremes were adjusted to fall within an arbitrary 1% pa (yields) or 2% pa (returns and inflation) away from the observed minima and maxima. Each series was then adjusted to give the observed variance and then further adjusted to give the observed mean.

One particular feature originally revealed from extending the financials into 2012 is that we now know that long ILG yields can be negative; who would originally have guessed that would happen? Nobody appears to have been recorded as having mentioned the possibility when they were first issued 40 years ago. They became positive in 2013 and have been negative again during part of 2014 and 2015 and then from mid-July 2016 (after the Brexit referendum) until they finally turned positive in late September 2022, the succeeding yield increases sparking off the LDI crisis.