ActEd Chapter 4 Question 4.8

I don't quite understand what formula the solution is using here to evaluate the following at AM92 4%:
Is this a formula that appears in the notes anywhere? How do we interpret the meaning of the terms on the right hand side above? Is there a simpler way of doing this?

I have done this calculation as follows, but I'm not sure if it's the same thing. I get a very close answer:



If anyone could confirm it would be much appreciated.

Thank you

 

Question 1.

No.though you can deduce it using the formulae given on page 5 of this chapter. (The image above shows a method of deducing that formula, so to solve this problem all you have to do is replace x by [55]+1 and n by 4 )
Question 2.

A wordier explanation could be that:
The annuity expression on the right (i.e a"_(57:3)) is the EPV when the age of the purchaser is 57 and the cash flows take place at age 57, 58, 59. Incase of the annuity expression on the right (i.e a"_([55]+1:4)) is the EPV when the age of the purchaser is 56 ( who entered the contract a year ago at the age of 55) and the cashflows take place at age 56, 57, 58, 59. Now if we compare the 2 annuities we can observe that a"_([55]+1:4) has one extra cashflow at age 56 of amt 1.
Thus to express a"_([55]+1:4) in terms of a"_(57:3) we have to use 3 steps viz.
1.) a"_(57:3) is EPV at age 57 and a"_([55]+1:4) is the EPV at age 56, thus to find PV of the cash flows related to a"_(57:3) we need to multiply it by v.
2.) The cashflows of a"_(57:3) begin at age 57, thus the purchaser needs to survive until the age of 57 in order to receive the 1st payment thus we multiply a"_(57:3) with p_[55]+1 as well.
3.) Since the two annuities differ by a payment of amt. 1 at age 56. Thus we 1 to a"_(57:3) after multiplying it with v and p_[55]+1.
Hope this helps

 

I guess simplicity depends on the grip you have over these formulae this is my perspective. And if by simplicity u mean length of the solution then I guess the one given in CT5 notes is simpler.

 

Thanks to Ark raw for the reply
Just to add that actuary-to-be's solution is also spot on. The differences in your answer are caused by the quite severe rounding used in the Tables and you will get full marks for any correct method you use.