Happy and financially astute parents decide at the birth of their daughter that they will need to provide 50,000 at each of their daughter’s 18th , 19th , 20th and 21st birthdays to fund her college education. They plan to contribute [math]X[/math] at each of their daughter’s 1 st through 17th birthdays to fund the four 50,000 withdrawals. They anticipate earning a constant 5% annual effective interest rate on their contributions.

Let v = 1/1.05

Determine which of the following equations of value can be used to calculate [math]X[/math].

• [math]X\sum_{k=1}^{17}\nu^{k}=50,000[\nu+\nu^{2}+\nu^{3}+\nu^{4}][/math]
• [math]X\sum_{k=1}^{16}1.05^{k}=50,000\left [1+\nu+\nu^{2}+\nu^{3}\right][/math]
• [math]X\sum_{k=0}^{17}1.05^{k}=50,000\left [1+\nu+\nu^{2}+\nu^{3}\right][/math]
• [math]X\sum_{k=1}^{17}1.05^{k}=50,000[1+\nu+\nu^{2}+\nu^{3}] [/math]
• [math] X\sum_{k=0}^{17}\nu^{k}=50,000[\nu^{18}+\nu^{19}+\nu^{20}+\nu^{21}+\nu^{22}][/math]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

(A) The left-hand side evaluates the deposits at age 0, while the right-hand side evaluates the withdrawals at age 17.

(B) The left-hand side has 16 deposits, not 17.

(C) The left-hand side has 18 deposits, not 17.

(D) The left-hand side evaluates the deposits at age 18 and the right-hand side evaluates the withdrawals at age 18.

(E) The left-hand side has 18 deposits, not 17 and 5 withdrawals, not 4.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.