QuestionCalculate the monthly payment $\mathrm{R}$ for a sinking fund to reach \$5,000 at a 7% annual interest rate over 1 year.

Studdy Solution

STEP 1

Assumptions1. The future value A is $5,000. The yearly interest rate r is7%
3. The time t for accumulation is1 year4. The interest is compounded monthly5. We need to find the monthly payment R

STEP 2

First, we need to convert the yearly interest rate to a monthly interest rate. We can do this by dividing the yearly interest rate by the number of months in a year.
$i = r /12$

STEP 3

Now, plug in the given values for the yearly interest rate to calculate the monthly interest rate.
$i =7\% /12$

STEP 4

Convert the percentage to a decimal value.
$7\% =0.07$ $i =0.07 /12$

STEP 5

Calculate the monthly interest rate.
$i =0.07 /12 =0.005833333$

STEP 6

Now, we need to find the number of periods n. Since the interest is compounded monthly, the number of periods is the number of months in a year times the number of years.
$n =12 * t$

STEP 7

Plug in the given values for the number of years to calculate the number of periods.
$n =12 *1$

STEP 8

Calculate the number of periods.
$n =12 *1 =12$

STEP 9

Now, we can use the formula for the monthly payment R needed to have a sinking fund accumulate the future value A. The formula is $R = A * i / (( + i)^n -)$

STEP 10

Plug in the values for the future value A, the monthly interest rate i, and the number of periods n to calculate the monthly payment R.
$R = \$5,000 *0.005833333 / (( +0.005833333)^12 -)$

STEP 11

Calculate the monthly payment R.
$R = \$5,000 *0.005833333 / (( +0.005833333)^ -) = \$418.39$ The monthly payment needed to have a sinking fund accumulate the future value A is $418.39.

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