Diane works at a public university. She contributes $625 at the end of each month to her retirement fund. For the past 10 years, this fund has returned 3.84% a year, compounded monthly.
a. Assuming the 3.84% rate continues, how much will she have in her retirement account after 15 years?
b. Assume the economy has gotten better and that the fund now has a return of 7.72% compounded monthly. Since Diane’s salary has risen over the first 15 years, she can now contribute $1000 per month. At the end of the next 15 years, how much is her account worth?
Calculate the present value of the investments using the compound interest formula over the past 10 years, or n=120 periods (t) at interest rate of i=0.0384/12=0.0032 per period. The monthly payment P=$625 per period, and therefore
PV = present value
FV = (i.e. future value from 10 years ago)
=P((1+i)^(n-1)) / (i) ......(1)
=P(1.0032^(120-1)) / (0.0032)
= $285657.30 (after 10 years)
(a)
Use equation (1) to calculate how much the accumulated amount after 15 years.
(b) Split the investment into two parts, the old and the new.
The future value FV from the previous investment can be calculated using the compound interest formula
FV=(PV)(1+i)^n
where PV=present value calculated above, n=12*15years=180
i=7.72% p.a. (need to convert to per month)
Then there is the new savings of 1000$ per month.
Using the same parameters of n, i, and monthly payment of $1000, apply to equation (1) to get the amount of the new investment after 15 years.
Add the values of the old and the new investments to get the total.
To calculate the amount in Diane's retirement account after a certain number of years, you can use the formula for compound interest:
A = P * (1 + r/n)^(nt)
Where:
A = the final amount of money in the account
P = the initial contribution amount
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
Let's first calculate the amount in Diane's retirement account after 15 years with a 3.84% interest rate:
a. P = $625 (monthly contribution)
r = 3.84% per year = 0.0384 (as a decimal)
n = 12 (compounded monthly)
t = 15
Plug in these values into the formula:
A = $625 * (1 + 0.0384/12)^(12*15)
Using a calculator, compute the equation inside the parentheses first:
(1 + 0.0384/12)^(12*15) ≈ 1.4065
Now, multiply the monthly contribution by the resulting value:
A ≈ $625 * 1.4065 ≈ $879.06
Therefore, after 15 years with a 3.84% interest rate, Diane's retirement account will have approximately $879.06.
Now, let's move on to part b, where we assume a 7.72% interest rate compounded monthly and a monthly contribution of $1000:
b. P = $1000 (monthly contribution)
r = 7.72% per year = 0.0772 (as a decimal)
n = 12 (compounded monthly)
t = 15
Plug in these values into the formula:
A = $1000 * (1 + 0.0772/12)^(12*15)
Calculate the equation inside the parentheses:
(1 + 0.0772/12)^(12*15) ≈ 2.5987
Multiply the monthly contribution by the resulting value:
A ≈ $1000 * 2.5987 ≈ $2598.70
Therefore, after 15 years with a 7.72% interest rate, Diane's retirement account will have approximately $2598.70.