Essentially, albeit for very simple contracts, with certain payments, I am exploring the differences between volatile market related results and some smoothing mechanism (yet to be defined). If we were able to derive the correct initial financial parameters, and hence the correct initial single premium, the final assets would be 10,000 (real or nominal) for the endowment or zero for the annuity. From the chart, we can see that using the base pricing approach, termed “standard” is unlikely to lead to those results (or we might just have guessed that).
How can we do better? Well, let's just solve for the initial single premium which does lead to the right outcome at the end of the contract. That directly leads us to the corresponding interest rate that should have been chosen and I have called this “adjusted”. Having already used the word approach to distinguish between MtM and Off, I have called these stages.
For the endowment, suppose that the MtM and Off (standard) initial interest rates were i_{0} and j_{0} and that the required (adjusted) interest rate is k (SAME for MtM and Off).
By including more and better information, I reckon that smoothing will normally lead to better longterm results than marking to market. Formally, I believe that j_{0} should be closer to k than i_{0} is and that is what I have shown in the required adjustments section.
