Final answer:
Calculating X involves comparing the future value of Joe's annuity-due deposits at an annual effective interest rate of 5% with the future value of Jane's single lump-sum deposit accumulating under a force of interest of 7%. Joe's future value is the sum of each biennial deposit's compounded value at year-end 2038. Jane's deposit X's future value is calculated using continuous compounding until year-end 2038, and these two values are set equal to solve for X.
Step-by-step explanation:
The student's question involves calculating the lump sum Jane needs to deposit to match the future value of a series of deposits made by Joe at an annual effective interest rate of 5%. Because Joe makes his deposits at the beginning of each odd year from 2018 to 2038, we need to account for the deposits made every two years, compounded at 5%. Moreover, Jane deposits her lump sum amount X in 2029 at a force of interest of 7%, which also accumulates until the end of 2038. By setting these two future values equal to each other, we can solve for X.
For Joe's account, we calculate the future value of an annuity-due using the formula FV = PMT * [(1 + i)^(n) - 1] / i * (1 + i), where PMT is 100, n is the number of deposits, and i is the interest rate per period (0.05 compounded biennially since it's every two years). However, to adjust for the odd year deposits, we would begin by finding the future value of the first deposit at the end of 2038, then the second deposit at the end of 2036, and so on until we calculate the future value of the last deposit made at the beginning of 2038. Each of these can be aggregated to find the total future value of Joe's deposits.
For Jane's account, the future value of her lump sum X after 10 years (2038 minus 2029) under continuous compounding is given by the formula FV = X * e^(rt), where r is the force of interest (0.07) and t is the time in years. Therefore, we equate Joe's total future value to Jane's future value to find the value of X. The calculations involve exponential functions and the mathematical constant e (Euler's number).