You have two choices on your Sprint Mobile phone plan. The first plan involves a

Question

You have two choices on your Sprint Mobile phone plan. The first plan involves a cheap flip phone with no data plan and costs $25/month. The second plan involves a 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

1. You will invest $100 into a retirement plan that will earn you 9% APR for the next 40 years. For calculating maturity value we will use future value of annuity formulla. APR is 9% so monthly rate is 9%/12 months = 0.75% or 0.0075. and n = 40years*12 months = 480 months

FV of Annuity = P((1+r)n -1)/ r)

= $100((1+ 0.0075.)480 -1)/ 0.0075)

= $100(36.1099-1) /  0.0075)

= $100 (4681.32)

=$468,132

2.After 40 years, you will retire, and buy an annuity that pays you a fixed amount every month for the subsequent 25 years. n= 25 years*12 months = 300 months and monthly rate =APR 4%/12 months = 0.33% To find out Monthly income we will use Present value of annuity formulla

Monthly income during the 25 years = $468,132/PVIFA(0.33%,300 months)

=  $468,132/190.2476

= $2461 per month

3. To find out money saing 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 we have do reverse calculation.first of of all we have to calculate maturity amount after 40 years as follows

Weekly APR = 4%/52 weeks = 0.0769, n= 25 years * 52 weeks = 1300 week

Maturity amount after 40 years = weekly payment */PVIFA(0.0769%,52 weeks)

= $500 *821.6759

= $410,838

Now we can calculate savings per month  for the next 40 years as follows,

FV of Annuity = P((1+r)n -1)/ r)

$410,838 = P((1+ 0.0075.)480 -1)/ 0.0075)

$410,838 = P *(4681.32)

P = $410,838 /4681.32

P = $ 87.76 or $88 Approx

If you would like to get $500/week from the annuity you will purchase after 40 years of investing, you should save $88 every month for the next 40 years.

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