You have two choices on your Verizon Wireless plan. The first plan involves a ch

Question

You have two choices on your Verizon Wireless plan. The first plan involves a cheap flip phone with no data plan and costs $25/month. The second plan involves an Samsung Galaxy that comes with a 4G data plan and costs $125/month. Imagine that you chose the cheaper plan, saving you $100/month. Rather than spending those savings, you will invest them into a retirement plan that will earn you 9% APR for the next 40 years. After 40 years, you will retire, and buy an annuity that pays you a fixed amount every month for the subsequent 25 years.

MUST SHOW WORK FOR CREDIT. MUST SHOW FORMULAS.

1. Find how much money you will have after 40 years?

2. What will be your monthly income during the 25 years of retirement, if the annuity you bought has an interest rate of 4% APR?

3. Assuming that you will earn 9% APR on your 40 years of investing, and 4% APR on the weekly annuity (you get weekly payments) you will purchase for your 25 years of retirement, how much money you should save every month for the next 40 years, if you would like to get $500/week from the annuity you will purchase after 40 years of investing?

Explanation / Answer

Answer 1.

We are saving $100/month for the next 40 years with 9% APR, then Future Value after 40 years :

= 100(1+9/12%)^480 + 100(1+9/12%)^479 + ... + 100

= 100(1.0075)^480 + 100(1.0075)^479 + ... + 100

= 100 * ((1.0075)^480 - 1) / (1.0075 - 1) = $468,132.03

We will have $468,132.03 after 40years if we save $100/month.

Answer 2.

If we want to calculate the monthly income from retirement then we have to equate the present value of all incomes to $468,132.03.

Therefore, Let monthly income be $x.

$468,132.03 = x/(1+4/12%) + x/(1+4/12%)^2 + ... + x/(1+4/12%)^300

$468,132.03 = x/(1.0033) + x/(1.0033)^2 + ... + x/(1.0033)^300

468,132.03 = x / 0.0033 * (1 - (1/1.0033)^300)

468,132.03 = x * 190.24763

x = $2,460.65

Answer 3.

Let we are saving $x/month for the next 40 years with 9% APR, then Future Value after 40 years :

= x(1+9/12%)^480 + x(1+9/12%)^479 + ... + x

= x * (1.0075)^480 + x * (1.0075)^479 + ... + 100

= x * ((1.0075)^480 - 1) / (1.0075 - 1) = $4,681.3203x

Therefore, equating the Future value of saving an present value of incomes after retirement

$4,681.3203x = 500/(1+4/52%) + 500/(1+4/52%)^2 + ... + 500/(1+4/52%)^300

$4,681.3203x = 500/(1.000769) + 500/(1.000769)^2 + ... + 500/(1.000769)^1300

4,681.3203x = 500 / 0.000769 * (1 - (1/1.000769)^1300)

4,681.3203x = 410,837.95

x = $87.76.

Therefore, he has to save $87.76 for the scenerio.

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