How long will it take to accumulate R1,5 million from R2500 monthly invested at $12\%$ p.a. monthly?
Answer (1)
Convert the annual interest rate to a monthly interest rate: i = 12 0.12 = 0.01 .
Use the future value of an ordinary annuity formula: F V = P × i ( 1 + i ) n − 1 .
Rearrange the formula to solve for n : n = l n ( 1 + i ) l n ( P F V × i + 1 ) .
Substitute the given values and calculate n : n ≈ 195.56 months. Therefore, it will take approximately 196 months to accumulate R1,5 million.
Explanation
Problem Analysis We are asked to find out how long it will take to accumulate R1,5 million with monthly deposits of R2500 and an annual interest rate of 12%, compounded monthly.
Calculating Monthly Interest Rate First, we need to convert the annual interest rate to a monthly interest rate. The annual interest rate is 12%, so the monthly interest rate is: i = 12 12% = 12 0.12 = 0.01
Stating the Future Value Formula We will use the future value of an ordinary annuity formula to determine the number of months required to reach the target amount. The formula is: F V = P × i ( 1 + i ) n − 1 Where:
F V is the future value (R1,500,000)
P is the monthly deposit (R2500)
i is the monthly interest rate (0.01)
n is the number of months
Rearranging the Formula We need to rearrange the formula to solve for n . First, we isolate the term with n :
( 1 + i ) n = P F V × i + 1
Solving for n Next, we take the natural logarithm of both sides: n × ln ( 1 + i ) = ln ( P F V × i + 1 ) Now, we solve for n :
n = ln ( 1 + i ) ln ( P F V × i + 1 )
Substituting the Values Now, we substitute the given values into the formula: n = ln ( 1 + 0.01 ) ln ( 2500 1500000 × 0.01 + 1 ) n = ln ( 1.01 ) ln ( 2500 15000 + 1 ) n = ln ( 1.01 ) ln ( 6 + 1 ) n = ln ( 1.01 ) ln ( 7 )
Calculating n Using a calculator, we find: n ≈ 0.00995 1.9459 ≈ 195.56
Final Answer and Rounding Therefore, it will take approximately 195.56 months to accumulate R1,5 million. Since we usually consider the number of months as a whole number, we can round it up to 196 months to ensure the target is reached.
Examples
Imagine you are saving up for a down payment on a house. By making regular monthly deposits into an investment account with a fixed interest rate, you can calculate how long it will take to reach your savings goal. This calculation helps you plan your finances and adjust your savings strategy as needed to achieve your goal within a desired timeframe. Understanding the time value of money and the impact of compounding interest is crucial for making informed financial decisions.
