(a) Find the present value of $ 4600 per year flowing uniformly over a 10 -year period if it earns 5.3 % interest compounded continuously.
(b) What is its final value?
To find the present value of $4600 per year flowing uniformly over a 10-year period, compounded continuously at an interest rate of 5.3%, we can use the formula for continuous compound interest:
P = A / e^(rt)
Where:
P = present value
A = future value (final value)
e = Euler's number (approximately 2.71828)
r = interest rate per year (as a decimal)
t = time period in years
(a) Find the present value:
In this case, we want to find the present value, so we need to solve for P. The future value (A) is given as $4600 per year for 10 years, which means the total future value is $4600 * 10 = $46000. The interest rate (r) is 5.3% per year, so r = 0.053, and the time period (t) is 10 years.
Using the formula:
P = 46000 / (e^(0.053 * 10))
To calculate this, you can use a scientific calculator or an online calculator that supports exponentiation. The result will be the present value of the cash flow.
(b) Find the final value:
The final value (A) is also part of the formula, so you can substitute the values into the formula and solve for A. In this case, the present value (P) is given as $4600 per year for 10 years, which means P = $46000. The interest rate (r) and time period (t) are the same as part (a), i.e., r = 0.053 and t = 10 years.
Using the formula:
A = P * e^(rt)
Substituting the values:
A = 46000 * e^(0.053 * 10)
Again, you can use a scientific calculator or an online calculator with exponentiation to find the final value. The result will be the total amount accumulated over the 10-year period.