8. (15 points (4/3/4/4). To earn all points: Use the annuity formula to answer a

Question

8. (15 points (4/3/4/4). To earn all points: Use the annuity formula to answer all parts correctly.)

A grandparent opens a savings account to pay for her new granddaughter’s college education. She deposits $275 every month into the account at an annual interest rate of 3.0%, compounded monthly.
(a) Find ????(????), the value of the account after ???? years. Use the annuity formula and simplify as much as possible. You will find the annuity formula in the “Annuity” video in the Week 11 folder.

(b) What is the value of the account after 18 years?

(c) What is the rate of change in the value of the account after 18 years?

(d) The grandmother decides that the amount in (b) will not be enough to pay the education bill. She wants to have $120,000 in the account after 18 years. What should her monthly payment be assuming the same interest rate?

Explanation / Answer

a) Cannot be answered as the number of years is not given.

b) Annual Interest Rate = 3%

Monthly Interest Rate = 3/12 = 0.25%

Number of months in 18 years = 18 x 12 = 216

Value of the account after 18 years = 275 x (1 - (1.0025)^216) / (1 - 1.0025) = $78,633.60

c) The rate of change in the value of the account is a constant increase of 1.0025 times per month.

d) Desired Future Value of Annuity = $120,000

Future Value of Annuity = Periodic Investment x [((1 + Rate of Interest) ^ number of time periods) - 1] / Rate of Interest

120,000 = Periodic Investment x [((1 + 0.0025) ^ 216) - 1] / 0.0025

Solving the equation, we get:

=> Periodic Investment = 419.6679 or $420 per month (Approximately)

Therefore, to have $120,000 in the account after 18 years, the grandparent must invest $420 per month, assuming the same interest rate.

Explanation on Annuity and Compound Interest:

It has been stated in the question that interest is compounded on a monthly basis. This means that if the grandparent invests $275 at the beginning of the first month, the bank will pay them interest on the $275 at the monthly interest rate (after conversion of the annual interest rate of 3%) of 0.25%. Since, the grandparent is planning to deposit money every month, the balance in the account at the end of the first month will be (275 + 275 x 0.25% OR 275 x 1.0025) $ 275.6875. Then, at the beginning of the second month, the grandparent will invest another $275, making the opening balance of the account $550.6875. The bank will then pay interest at 0.25% on the cumulative amount of $550.6875 for the second month, renderring the balance of the account (550.6875 x 1.0025) $552.0642 at the end of the second month. This process would countinue for the duration that the grandparent is willing to invest. Here, we can observe that the first principal/instalment invested of $275 will receive compound interest for 18 years (assuming the tenure of the deposit is 18 years). The second principal/instalment of $275 will receive compound interest for 17 years and 11 months, the third principal/instalment will receive interest for 17 years and 10 months and so on. In other words, the first deposit of $275 will receive interest for (18 years x 12 months per year) 216 months, the second deposit of $275 will receive interest for 215 months, the third instalment will receive interest for 214 months and so on. Mathematically, this can be shown as follows:

Interest on the first deposit = 275 x (1.0025)^216 - 275

Interest on the second deposit = 275 x (1.0025)^215 - 275

Interest on the third deposit = 275 x (1.0025)^214 - 275

And so on. Here, we can observe that such recurring deposits, receiving compounded interest, resemble a pattern of Geometric Progression. A Geometric Progression is a series of numbers which go on increasing/decreasing by a common ratio or mulplicative factor. In this case, the common ratio or multiplicative factor is 1.0025 or (1 + 0.25%). The formula for calculating the sum of the first 'n' terms of a Geometric Progression is as follows:

Sum(n) = a x (1 - r ^ n) / (1 - r)

where,

a = First term in the Geometric Progression (In this case $275)

r = Common ratio / multiplicative factor (In this case 1.0025)

n = Number of time periods (In this case 216 months)

The above formula can be re-written as follows:

Future Value of Annuity = P x [(1 + r)^n - 1] / r

which is commonly known as the formula for "Future Value of Annuity".

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