Hello, Probably a fairly basic question but I just cannot get my head around the recursion formula in Chapter 11. What does the j in the recursion formula represent? Specifically following the example in the CMP for the Poisson case (the Premium Bonds question). I thought it was your options of win so you'd have j=50 or 100. So for P(S=100) you have 2 options, either you have won 50 on 2 bonds or you have won £100 on one bond and nothing on any others. But then this logic didn't follow through to the P(S=150) as you only seem to have won 50 given you got 100 on another or 100 given you got 50 on another. Shouldn't there be an option for 3 lots of 50? Thanks!
The j in the formula is the size of an individual claim (or win in the example you refer to). Your options for winning 150 are correct, but that's not how Panjer's recursive formula works. Panjer works by looking at the individual claim amounts and adding those on to the previous aggregate claim cost. Every time it just looks at the possibility of adding on one extra claim amount. So to get to 150 by adding one more win only, given that wins can either be for 50 or 100, you must start from a total of 100 and add on another 50 or start with a total of 50 and add on another 100. These are the two terms shown in the sum in the CMP. As there are only two possible win sizes there are only two terms in the sum. Your option of 50 followed by 50 followed by 50, is actually part of the first term as when you add the third 50 on, your total before that is already 100 (50+50). That term also allows for you to win 100 (in one go) followed by 50.
Thank you Darren - that was really helpful. I've worked it through again and just to triple check, in the 150 case it's better to think of it as you can either: Win 50 on top of the 100 you have already won (so your j is 50) Or Win 100 on top of 50 you have already won (so your j is 100) ??
