I am having a little trouble understanding certain aspects of actuarial funding (chapter 14). Firstly, I fail to see the link between the method of holding the present value of the unit fund with the description given in the introduction (pg 1) of the present value of future initial expense charges being deducted from the reserve. Secondly, I am struggling to understand how the interest rate for the present value determination is obtained. That is, I do not understand the argument used to obtain that it should be less than the management charge. Any explanation of the above concepts would be greatly appreciated. If this has already been answered adequately in another thread, I apologise for asking it again, and ask that you direct me to the appropriate thread.
Not easy - this may not help. Consider a premium of 100, invested for 2 years in a fund with an amc of 2% taken at the end of each year. Fund growth = i%. Ignore mortality. The amc at the end of year 1 is 100(1+i)*0.02. PV = 100*0.02 amc at start of year 2 is 100(1+i)^2*0.98*0.02. PV = 100*0.98*0.02 So UF less PV of amc is 100(1-0.02-0.98*0.02) = 100*0.98^2 You can write this as 100*v^2, ie the present value of the UF where v=0.98. And this also indicates that 2% = (1-0.98) is approximately the maximum rate we could use, given the amc is 2%.
I know it's been a while, but that's given me time to mull it over and I believe I understand now. The company will want to take the present value of any charges (for initial expenses) accruing on the units upfront. This means the company can hold the expected number of units it will need to pay out (if mortality is ignored this just means discounting the units at rate of interest implied by the fund management charge used to cover initial expenses), which is the expected value of units with the fund management charges taken out. The actual interest rate used is i = d/(1-d) where d is the portion of the fund management charge we are taking credit for. I've converted from a discount rate to an interest rate. The company would want to make sure d < m where m is the fund management charge because otherwise it will be holding too few units. We can show the effect mathematically. The company will hold UF_t A_x+t:n-t| and the amount taken up front is then UF_t (1 - A_x+t:n-t|) = UF_t x d x รค_x+t:n-t| which is the present value of the charges taken credit for.
