Annuities /assessment You would like to have $550,000 when you retire in 35 years.
You would like to have $550,000 when you retire in 35 years. How much should you invest each quarter if you can earn a rate of 4.8% compounded quarterly?
a) How much should you deposit each quarter?
b) How much total money will you put into the account?
c) How much total interest will you earn?
FV = PMT ([1+I]^N - 1)/I
FV = future value
I = interest
N = number of periods
PART A:
Set FV = $550,000
Set I = 0.048/4 = 0.012
Set N = 35yr×4qtr/yr = 140 qtr
PMT = 550,000×I/([1+I]^N - 1)
PMT = 550,000×0.012/([1.012]^140-1) = $1,530.55
PART B:
$1,530.55 × 140 = 214,277.42
PART C:
550,000 - 214,277.42 = 335,722.58
To calculate the amount you should invest each quarter, we need to use the formula for calculating the future value of an ordinary annuity:
Future Value (FV) = Payment (PMT) x [(1 + r)^n - 1] / r
Where:
FV = $550,000 (the desired amount at retirement)
PMT = the amount you should invest each quarter
r = 4.8% / 4 (quarterly interest rate)
n = 35 years x 4 (since there are 4 quarters in a year)
Now let's calculate the values:
a) How much should you deposit each quarter?
FV = PMT x [(1 + r)^n - 1] / r
$550,000 = PMT x [(1 + (4.8% / 4))^(35 * 4) - 1] / (4.8% / 4)
Simplifying the equation and isolating PMT:
PMT = $550,000 x (4.8% / 4) / [(1 + (4.8% / 4))^(35 * 4) - 1]
Calculating PMT:
PMT = $550,000 x 0.012 / [(1 + 0.012)^(140) - 1]
PMT ≈ $515.95 (rounded to the nearest cent)
Therefore, you should deposit approximately $515.95 each quarter.
b) How much total money will you put into the account?
To calculate the total amount you will deposit over 35 years, we multiply the number of quarters by the amount deposited each quarter:
Total Money = PMT x (n * 4)
Total Money = $515.95 x (35 * 4)
Total Money ≈ $72,238.00
Therefore, you will put approximately $72,238.00 into the account.
c) How much total interest will you earn?
To calculate the total interest earned, we subtract the total money invested from the desired amount at retirement:
Total Interest = FV - Total Money
Total Interest = $550,000 - $72,238.00
Total Interest ≈ $477,762.00
Therefore, you will earn approximately $477,762.00 in interest.
To calculate the answers to these questions, we can use the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the desired future value ($550,000 in this case)
P is the periodic payment (the amount you should deposit each quarter)
r is the interest rate per compounding period (4.8% per quarter, which is 0.048)
n is the number of compounding periods (35 years * 4 quarters per year = 140 quarters)
a) To find the amount you should deposit each quarter (P), rearrange the formula:
P = FV * (r / ((1 + r)^n - 1))
P = $550,000 * (0.048 / ((1 + 0.048)^140 - 1))
Calculating this gives us the answer for part (a).
b) To calculate the total money you will put into the account, multiply the periodic payment (P) by the number of compounding periods (n):
Total money = P * n
Total money = P * 140
Calculating this gives us the answer for part (b).
c) To calculate the total interest earned, subtract the total money you put into the account from the desired future value:
Total interest = FV - (P * n)
Calculating this gives us the answer for part (c).