A tax free municipal bond pays 4.75% compounded continuously. What is the value of a $2,000 bond after 30 years? Round your answer to the nearest cent.
[Hint- A=P e^rt where P= present(initial) value, A= future value, r= annual rate, and t= number of years]
P.S.The rt is an exponent to e
A = P*e^rt.
P = $2,000
rt = 0.0475/yr. * 30yrs. = 1.425.
A = ?.
To find the value of the bond after 30 years, we can use the formula A = P * e^(rt), where A is the future value, P is the initial value (or present value), r is the annual interest rate, t is the number of years, and e is Euler's number approximately equal to 2.71828.
In this case, the initial value (P) is $2,000, the annual interest rate (r) is 4.75% (or 0.0475 as a decimal), and the number of years (t) is 30.
Let's calculate the value step by step:
1. Convert the annual interest rate to a decimal:
r = 4.75% = 0.0475
2. Substitute the values into the formula:
A = $2,000 * e^(0.0475 * 30)
3. Calculate the exponent term:
exponent = 0.0475 * 30 = 1.425
4. Substitute the exponent back into the formula:
A = $2,000 * e^(1.425)
5. Use a calculator to evaluate the exponential term (e^(1.425)):
e^(1.425) ≈ 4.157
6. Multiply the exponential result with the initial value:
A ≈ $2,000 * 4.157
7. Calculate the final value:
A ≈ $8,314
So, the approximate value of the $2,000 bond after 30 years, with 4.75% interest compounded continuously, is $8,314 to the nearest cent.