To determine how much Keith should pay for the bond now, given that it will mature to [tex]$6000 in eight years with an interest rate of 2.5% per year, compounded continuously, we can use the formula for continuous compounding. The formula is:
\[ A = P e^{rt} \]
Where:
- \( A \) is the maturity value of the bond (future value),
- \( P \) is the principal amount (present value),
- \( r \) is the annual interest rate (in decimal form),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
Given:
- \( A = 6000 \)
- \( r = 0.025 \) (2.5% per year)
- \( t = 8 \) years
We need to find \( P \). Rearrange the formula to solve for \( P \):
\[ P = \frac{A}{e^{rt}} \]
Now, substituting the given values into the formula:
\[ P = \frac{6000}{e^{0.025 \times 8}} \]
Calculate the exponent first:
\[ 0.025 \times 8 = 0.2 \]
Then, evaluate \( e^{0.2} \):
\[ e^{0.2} \approx 1.22140 \]
Now, divide the maturity value by this result to find the present value \( P \):
\[ P = \frac{6000}{1.22140} \]
\[ P \approx 4912.38 \]
Therefore, Keith should pay approximately $[/tex]4912.38 for the bond now.
