How much will you have accumulated, if you annually invest $1,500 into an IRA at 8% interest
compounded monthly for:
a. 5 year
b. 20 years
c. 40 years
d. How long will it take to earn your first million dollars? Your answer should be exact
rounded within 2 decimal places. Please use logarithms to solve
I will do b), you do the others in the same way.
The problem is that the standard formulas for the amount of an annuity are based on the
fact that the payment period and the interest period coincide. This is not the case here.
Since the payments are made annually, and the compounding is monthly, we have to find the effective annual rate equivalent to a rate of 8% per annum, compounded monthly
Let that annual rate be i
(1+i)^1 = (1 + .08/12)^12 = 1.08299507
i = .08299507 (I stored that in my calculator)
b) amount = 1500(1.08299507^20 - 1)/.08299507 = $89,039,13
Do a) and c) in the same way, the i stays the same
d) 1500(1.08299507^n - 1)/.08299507 = 1,000,000
(1.08299507^n - 1)/.08299507 = 666.6666...
(1.08299507^n - 1) = 55.33300454
1.08299507^n = 56.33300454
n log 1.08299507 = log 56.33300454
I get n = 50.559 years
State your appropriate answer.
checking my answer:
amount = 1500(1.08299507^50.559 - 1)/.08299507 = 1,000,016.935 (not bad, using my 3 decimal answer for the years
To calculate how much you will have accumulated over a specific period, we can use the formula for the future value of an investment:
FV = P * [(1 + r/n)^(n*t)]
Where:
FV is the future value of the investment
P is the principal amount (annual investment)
r is the annual interest rate (expressed as a decimal)
n is the number of times the interest is compounded per year
t is the number of years
In this case, the annual investment (P) is $1,500, the annual interest rate (r) is 8% (or 0.08 as a decimal), and the interest is compounded monthly, so we have 12 compounding periods per year (n = 12).
a. For 5 years:
t = 5
FV = $1,500 * [(1 + 0.08/12)^(12*5)]
To solve this, you can use a calculator or spreadsheet with exponentiation capability:
FV = $1,500 * [1.0066667^(60)] ≈ $9,155.36
b. For 20 years:
t = 20
FV = $1,500 * [(1 + 0.08/12)^(12*20)]
FV = $1,500 * [1.0066667^(240)] ≈ $58,548.68
c. For 40 years:
t = 40
FV = $1,500 * [(1 + 0.08/12)^(12*40)]
FV = $1,500 * [1.0066667^(480)] ≈ $304,026.30
d. To find out how long it will take to accumulate $1,000,000, we need to rearrange the formula to solve for t. Using logarithms, we have:
FV = P * [(1 + r/n)^(n*t)]
1,000,000 = $1,500 * [(1 + 0.08/12)^(12*t)]
[(1 + 0.08/12)^(12*t)] = 1,000,000 / $1,500
(1.0066667)^(12*t) ≈ 666.6666667
Now, we can solve for t by taking the logarithm of both sides:
12*t * log(1.0066667) ≈ log(666.6666667)
t ≈ log(666.6666667) / (12 * log(1.0066667))
Using logarithms to solve for t, we get approximately 59.57 years.
Therefore, it will take approximately 59.57 years to earn your first million dollars.