How much money should be invested now (rounded to the nearest cent), called the initial investment, in a Treasury Bond investment that yields 4.75% per year, compounded monthly for 10 years, if you wish it to be worth $20,000 after 10 years?
i = .0475/12 = .00395833..
n = 12(10) = 120
PV = 20,000(1.00395833..)^-120
= $12,449.37
To find the initial investment needed to reach a desired future value, we can use the formula for compound interest:
Future Value = Initial Investment * (1 + (Interest Rate / Number of Compounding Periods))^(Number of Compounding Periods * Number of Years)
In this case, the future value is $20,000, the interest rate is 4.75% per year, compounded monthly, the number of compounding periods per year is 12 (since it's compounded monthly), and the number of years is 10.
Let's calculate it step by step:
Step 1: Convert the annual interest rate to a monthly interest rate:
Monthly Interest Rate = (1 + Annual Interest Rate)^(1/Number of Compounding Periods) - 1
Monthly Interest Rate = (1 + 4.75%)^(1/12) - 1 = 0.3916% or 0.003916
Step 2: Calculate the number of compounding periods:
Number of Compounding Periods = Number of Compounding Periods per Year * Number of Years
Number of Compounding Periods = 12 * 10 = 120
Step 3: Rearrange the formula to solve for the initial investment:
Initial Investment = Future Value / ((1 + Monthly Interest Rate)^(Number of Compounding Periods))
Now, let's substitute the values into the formula:
Initial Investment = $20,000 / ((1 + 0.003916)^(120))
Using a calculator or spreadsheet, we'll calculate the value of (1 + 0.003916)^(120) to be approximately 1.618228.
Initial Investment = $20,000 / 1.618228
Calculating this division, we find that the initial investment should be approximately $12,353.20.
Rounding this value to the nearest cent, the initial investment required is $12,353.20.