I don't think commutation functions were mentioned in the question, but the link between annuities and commutation functions is that annuities can be calculated from commutation functions. The life annuity, a double dot (x), the present value of a series of payments of 1 paid yearly in advance for life for someone aged x, can be calculated by commutation functions N(x)/D(x).
The link between annuities and conversion rates can be seen from the equation of value given in the question: L = P a double dot (x).
For example, if a double dot (x) = 10, then a lump sum of 100 would buy an annual pension of 10. To someone who wants to know what a lump sum will buy, an annuity rate of 10% could be quoted.
In this question, however, the person wants to know the lump sum he will get if he gives up (converts) some of his pension, so the conversion rate is using the annuity the other way, ie a conversion rate of 10 means that if he gives up 10 a year of his pension, he'll get a lump sum of 100.
