Question 1b 2016 Past Test 2

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Question: Upon retirement at age 65, Linda – now aged 55 exact – will receive a monthly pension benefit starting at R5000 per month. Linda’s monthly pension benefit will increase by R200 for each completed year of retirement, up to a maximum monthly benefit of R7000 per month.

When I did the problem, I calculated:

$$EPV = v^{10} 10P_{55} [(12*4800) \ddot{a}_{65}^{(12)} + (12*200)(I\ddot{a})_{65:\bar{11]}}^{(12)} + v^{11} 11P_{65} (12*2200)\ddot{a}_{76}^{(12)}]$$

Can someone please explain why the last part isn't necessary?

Thanks!

by (360 points)

It depends on how you interpreted  $$(I\ddot{a})^{(m)}_{x:\bar{n|}}$$

If you interpreted it as an increasing term annuity i.e.

$$(I\ddot{a})^{(m)}_{x:\bar{n|}} = \sum_{t=0}^{n-1} \ddot{a}^{(m)}_{x+t:\overline{n-t|}}v^{t}{t}p_{x}$$   (i)

which is the correct interpretation, then your last term would be necessary However if you interpreted it as a temporarily increasing whole life annuity i.e.

$$(I\ddot{a})^{(m)}_{x:\bar{n|}} = \sum_{t=0}^{n-1} \ddot{a}^{(m)}_{x+t}v^{t}{t}p_{x}$$         (ii)

then you would not need the last term. I suspect that you interpreted it as (i) but calculated it as (ii). The notation I found for (ii) online was $$(I_{\overline{n|}}\ddot{a})^{(m)}_{x}$$ but I'm not sure if we use at UCT.