You simply need to add a marginBottom to your chart to make space for the titles. Further a negative y-attribute moves the titles upwards.

Something like the following should solve your problems:

chart: {
    type: 'pie',
    marginBottom: 80
title: {
    text: 'Main title',
    verticalAlign: 'bottom',
    y: -30
subtitle: {
    text: 'My subtitle',
    verticalAlign: 'bottom',
    y: -10

As also illustrated in this fiddle.

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