the time index t runs from a to A.
if an investment produced a continuous stream of income over 10 years at a rate of $20,000 per year and the interest rate is 6% per year continuously compounded. what is the present value? what is the integral function?
To find the present value of the continuous stream of income, we can use the formula for continuous compounding:
PV = ∫[a,A] (P*e^(-rt)) dt
Where:
PV = Present Value (the value we are trying to find)
P = Annual income or cash flow ($20,000 in this case)
r = Interest rate per year (6% or 0.06 as a decimal)
t = Time (in years)
In this case, we are given that the time index t runs from a to A, which means we need to integrate the formula over the interval [a,A] that represents the 10-year period.
The integral function for continuous compounding would be:
Integral function = ∫[a,A] e^(-rt) dt
To evaluate this integral, we need to determine the limits of integration, a and A. Since we are given that the continuous stream of income occurs over 10 years, our limits of integration are a=0 and A=10.
Now we can substitute the limits of integration into the integral function:
Integral function = ∫[0,10] e^(-0.06t) dt
To solve this integral, we can use the anti-derivative of e^(-0.06t), which is (-1/0.06) * e^(-0.06t) + C, where C is the constant of integration.
Evaluating the integral, we have:
∫[0,10] e^(-0.06t) dt
= (-1/0.06) * e^(-0.06t) | from 0 to 10
= (-1/0.06) * (e^(-0.06*10) - e^(-0.06*0))
= (-1/0.06) * (e^(-0.6) - 1)
This gives us the value of the integral. However, it is not yet the present value (PV) as we still need to multiply it by the annual income (P). So, the present value (PV) can be calculated as:
PV = P * Integral function
PV = $20,000 * [(-1/0.06) * (e^(-0.6) - 1)]
Evaluating this expression will give us the present value of the continuous stream of income.