Question

Calculate the initial investment needed in a Treasury Bond yielding 5.75% per year, compounded monthly for 7 years, to be worth $12,000.

a. $____ (rounded to the nearest cent)
b. $____ (rounded to the nearest cent)
c. $____ (rounded to the nearest cent)
d. $____ (rounded to the nearest cent)

Answer 1

The final answer for the initial investment needed in a Treasury Bond yielding 5.75% per year, compounded monthly for 7 years, to be worth $12,000 is $7132.79, rounded to the nearest cent. This calculation is based on the compound interest formula, taking into account the annual interest rate, compounding frequency, and investment duration.

Step-by-step explanation:

To calculate the initial investment needed for a Treasury Bond, we can use the compound interest formula \
`(A = P(1 + (r)/(n))^(nt)\),` where:

• 
  `\(A\)` is the future value of the investment (\$12,000),
• 
  `\(P\)` is the initial investment,
• 
  `\(r\)` is the annual interest rate (5.75% or 0.0575),
• 
  `\(n\)` is the number of times interest is compounded per year (monthly compounding means
  `\(n = 12\)),`
• 
  `\(t\)` is the number of years (7 years).

Rearranging the formula to solve for \
`(P\), we get \(P = (A)/((1 + (r)/(n))^(nt))\)`. Substituting the given values, we get
`\(P = (12000)/((1 + (0.0575)/(12))^(12 * 7))\).` The calculation yields
`\(P \approx \$7132.79\)` when rounded to the nearest cent.

The options provided include
`\(P\)` rounded to the nearest cent in a range. The correct answer is
`\(b. \$7132.79\)` as it is the closest to the calculated value. This initial investment is required to grow to $12,000 over 7 years with a monthly compounded interest rate of 5.75%. Understanding compound interest is essential for financial planning and investment decisions.