Question

To save for retirement, starting from her 30th birthday (t = 0), Miss Saver will invest

some money each year in a 30-year deposit in her bank saving account. The first
payment of $1000 will be made at the beginning of the first year. Every subsequent
payment increases by 5%.
She will retire at age 60, and will withdraw $X each year from her bank saving account
from then. The first withdrawal will be made at age 60. After 20 years (i.e. after the
20th withdrawal), she will have taken out all the money she had saved.
Assuming the effective annual interest rate is 2% perpetually, compute the value of X.
Give your answer to the nearest integer.

Answer 1

To calculate the value of X, we first need to calculate the total amount of money Miss Saver will have saved by the time she retires at age 60.

We know that her first payment is $1000, and that every subsequent payment increases by 5%. Using this information, we can calculate the total payments made during the 30 years:

$1000 x (1 + 0.05)^30 = $1000 x 3.3201 = $3320.1

This is the total amount of money Miss Saver will have saved by the time she retires at age 60.

Next, we need to calculate the value of each annual withdrawal over the 20 years of withdrawals.

We know that the effective annual interest rate is 2% perpetually, so we can use the formula for the future value of an annuity:

FV = X * ((1 + r)^n - 1) / r

where X is the annual withdrawal, r is the interest rate (0.02), and n is the number of years (20).

We can use this formula to solve for X:

3320.1 = X * ((1 + 0.02)^20 - 1) / 0.02

X = 3320.1 * 0.02 / (1.02^20 - 1)

X = 3320.1 * 0.02 / 0.1135 = $292.36

Therefore, Miss Saver needs to withdraw $292.36 each year for 20 years to deplete her savings account.

Rounded to the nearest integer, the value of X is $292.