Final answer:
To find the annual effective interest rate that equalizes the proportions of present value for the first and second n years in a 2n annuity immediate, one must solve a complex equation comparing the present values of two separate annuities.
Step-by-step explanation:
For an annuity immediate with payments of 1 for the first n years and 5 for the second n years at an annual effective interest rate i, the present value (PV) can be calculated using the formula:
PV = PV(first n years) + PV(second n years).
The present value of the first n years is the sum of the present values of payments of 1 made at the end of each year for n years. The present value of the second n years consists of payments of 5 made at the end of each year for n years, discounted back to the present.
The problem specifies that the PV of the annuity is 45.7753201471752, and that the proportion of the total PV attributable to the second n years is 0.769823555046403. However, to find the interest rate that forces the proportion of PV for the first n years to equal that of the second n years, one needs to solve for i such that:
PV(first n years) / PV = PV(second n years) / PV = 0.5
This often involves solving a complex equation that equates the present value of two separate annuities. This type of financial mathematics problem can be solved using algebraic manipulation and numerical methods like Newton-Raphson or financial calculators.