Ch 21, p 20

I have a question regarding the survival method used for projecting pension payments.

My understanding is as follows:

There are two parts to this method. Part 1 requires a projection for each future year of the "new" pensions that will be awarded in each future year, based on the projection of the number of members in the population reaching the state retirement age and the proportion of these lives etitled to receive the state pension.

The second part requires projecting each of these amounts into the future allowing for mortality and future pension increases.

From the notes, page 20, it is stated that the total pensions expenditure at the start of year t for those aged x is,
P(x,t)=P(x-1,t-1)*(1+i(t-1))*(l(x,t)/l(x-1,t-1))

The formula seems to allow only for the pensions where payment has already commenced, ie where the pensions were being paid at the start of year t-1.

Surely the formula only applies to x > NRA and will look slightly different where x = NRA?

 

I also have trouble understanding the survival method for projecting pension.
It would be helpful if an explantion in words of both the factor and survival method is given or a rigourous mathematical defination.

My best guess at a reasonable working of the method is something like this:

let N(x,t) be the expected new pensions
possibly calculated as amount expected to become new pensioners * average new pension

then
P(x,t) = P(x-1,t-1) * (1+i(t-1) * l(x,t)/l(x-1,t-1) + N(x,t)

Obviously for most ages n(x,t) will be zero. Also if retirement only allowed at exactly NRA

then P(NRA,t) = N(NRA,t) which could be what the notes describe with the formula

P(x,t) = P(x-1,t-1) * (1+i(t-1) * l(x,t)/l(x-1,t-1)
applying for x>NRA

Perhaps someone (a tutor perhaps) can shed some light on this.