Axiom 3 - Positive homogeneity F(kL)=kF(L). I don't understand. Say k=0.5, L=40,000, F(20,000)=0.5*F(40,000). So lets say F(20,000), the CDF, is 0.2, then F(40,000) has to equal 0.4. I don't think any distribution does this (except uniform). Any help greatly appreciated. Thanks,
I don't think it is supposed to mean that. F(L) should be a risk measure instead of probability. Put it simple, for the same business, if you have double the premium this year, you would expect twice risks in your portfolio compared to last year; but if your portfolios shrink by half, the risks will also be halved.
Indeed, F() is a risk measure, e.g. VaR. Another example would be if F(L) is the capital required, given exposure to a loss situation - where the loss amount is a random variable L - then if the risk measure is coherent it follows that the capital will double if the risk exposure doubles.
