A company establishes a sinking fund in order to have 100,000 for upgrading office equipments in 4 years. If the company makes fixed monthly payments into an account paying 6.6% compounded monthly, how much should each payment be? How much interest will the account earn in 4 years?
I found that the monthly payments should be 1826.11 by using the future value of an annuity formula but I didn't understand how to find the interest
1826.11 (1826.11019) is the correct monthly payment to obtain 100,000 at the end of four years. Note that your formula assumes that payments are made at the end of each month.
The interest is the cumulative amount at the end of four years less the total amount paid, namely 4*12*monthly amount.
So you just have to subtract 48 times the monthly payment from the accumulated amount, namely $100,000.
To find the interest earned in 4 years, you can use the formula:
Interest = (Total value after 4 years) - (Total amount deposited)
Let's break it down step by step:
Step 1: Calculate the total amount deposited
The monthly payments can be calculated using the future value of an annuity formula. The formula for the future value of an ordinary annuity is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future value of the annuity
P = Monthly payment
r = Interest rate per period (compounded monthly)
n = Number of periods
In this case, the future value (FV) is $100,000, the interest rate (r) is 6.6% compounded monthly, and the number of periods (n) is 4 years * 12 months = 48 months.
We need to solve for the monthly payment amount (P):
$100,000 = P * ((1 + 0.066/12)^48 - 1) / (0.066/12)
Simplifying the equation gives:
$100,000 = P * (1.0055^48 - 1) / (0.0055)
Now you can solve for P:
P ≈ $1,826.11
So, each monthly payment should be approximately $1,826.11.
Step 2: Calculate the total value after 4 years
To calculate the total value after 4 years, you can use the formula for the future value of a single lump sum:
FV = PV * (1 + r)^n
Where:
FV = Future value
PV = Present value (initial deposit or principal)
r = Interest rate per period
n = Number of periods
In this case, the present value (PV) is $0 (since we haven't deposited anything yet), the interest rate (r) is 6.6% compounded monthly, and the number of periods (n) is 4 years * 12 months = 48 months.
FV = $0 * (1 + 0.066/12)^48
Simplifying the equation gives:
FV ≈ $115,567.98
So, after 4 years, the total value in the sinking fund account will be approximately $115,567.98.
Step 3: Calculate the interest earned
Interest = $115,567.98 - $0
Interest ≈ $115,567.98
Therefore, the approximate interest earned in 4 years will be $115,567.98.
Please note that the values calculated are approximate due to rounding.