Peter plans to go to Dubai for a holiday. He will need R 500000 for the holiday. How much must he invest every month at 11.5% compounded monthly for 6 years?
Answer (1)
Convert the annual interest rate to a monthly rate: r = 12 0.115 = 0.00958333 .
Calculate the total number of months: n = 6 × 12 = 72 .
Use the future value of an ordinary annuity formula to find the monthly investment: P = F V × (( 1 + r ) n − 1 ) r .
Substitute the values and calculate: P = 500000 × (( 1 + 0.00958333 ) 72 − 1 ) 0.00958333 ≈ 4853.91 . Therefore, Peter must invest approximately 4853.91 every month.
Explanation
Problem Analysis Peter needs to accumulate R 500000 for his holiday in Dubai. He plans to invest monthly over a period of 6 years with an annual interest rate of 11.5%, compounded monthly. Our goal is to determine the amount he needs to invest each month to reach his target.
Converting to Monthly Values First, we need to convert the annual interest rate to a monthly interest rate and calculate the total number of months for the investment.
Calculating Monthly Interest Rate and Total Months The annual interest rate is 11.5%, so the monthly interest rate, r , is: r = 100 11.5 ÷ 12 = 12 0.115 = 0.00958333 The investment period is 6 years, so the total number of months, n , is: n = 6 × 12 = 72
Applying the Future Value Formula We will use the future value of an ordinary annuity formula to find the required monthly investment. The formula is: F V = P × r (( 1 + r ) n − 1 ) Where:
F V is the future value (R 500000)
P is the periodic payment (monthly investment, which we want to find)
r is the monthly interest rate (0.00958333)
n is the number of months (72)
Rearranging the Formula and Substituting Values Now, we rearrange the formula to solve for P :
P = F V × (( 1 + r ) n − 1 ) r Substitute the given values into the formula: P = 500000 × (( 1 + 0.00958333 ) 72 − 1 ) 0.00958333
Calculating the Monthly Investment Now, we calculate the value of P :
P = 500000 × (( 1.00958333 ) 72 − 1 ) 0.00958333 P = 500000 × ( 1.970717 ) − 1 0.00958333 P = 500000 × 0.970717 0.00958333 P = 500000 × 0.00987242 P = 4936.21
Final Answer Therefore, Peter must invest approximately R 4853.91 every month to reach his goal of R 500000 in 6 years.
Examples
Imagine you're saving up for a down payment on a house. By making consistent monthly investments into an account with a fixed interest rate, you can calculate how much you need to save each month to reach your goal. This calculation uses the future value of an annuity formula, which helps in planning and achieving long-term financial goals, such as retirement savings or funding a child's education. Understanding these concepts allows you to make informed decisions about your investments and savings strategies.
