Question

(a) Find the present and future value of an income stream of per year for a period of years if the interest rate, compounded continuously, is . Round your answers to two decimal places. Present value Enter your answer; Present value = $ 70825 Future value Enter your answer; Future value = $ 70825 (b) How much of the future value is from the income stream? How much is from interest? Round your answers to two decimal places. The amount from the income stream is Enter your answer; The amount from the income stream is $ 90000 . The amount from the interest is Enter your answer; The amount from the interest is $ 116769.6 .

Answer 1

The question is incomplete. The complete question is :

(a) Find the present and future value of an income stream of $8000 per year for a period of 10 years if the interest rate, compounded continuously, is . Round your answers to two decimal places.

Present value ___

Future value ___

b). How much of the future value is from the income stream? How much is from interest? Round your answers to two decimal places.

The amount from the income stream is __

The amount from the interest is ___

Solution :

Since the interest is compounded continuously, the formula used is :

`$P=\int_0^(m) s(t)\ e^(-rt)\ dt$`

P = present value, B = future value

`$B=Pe^(rt)$`

Given :

S = $ 8000, r = 4% = 0.04, m = 10

`$P=\int_0^m \ 8000. e^(-0.04t) \ dt$`

`$P=\int_0^(10) \ 8000. e^(-0.04t) \ dt$`

According to the fundamental theorem of calculus,

`$\int_a^b f(x) = F(b) -F(a)$`

`$\int e^(ax)\ dx = (e^(ax))/(a)$`

`$P=\int_0^(10) \ 8000. e^(-0.04t) \ dt$`

`$P=8000 * \left[ (e^(-0.04t))/(-0.04)\right]^(10)_0$`

Uploading the limits,

`$P=(8000)/(-0.04) * \left[ e^(-0.04 * 10) - e^(-0.04 * 0) \right]$`

P = -200000 [-0.32967]

P = 65935.99

Therefore, Present value, P = $65936

`$B=Pe^(rt)$`

`$B=65936 * e^(10 * 0.04)$`

`B= 98364.9`

So, future value , B = $ 98365

Amount of the future value from the income stream,

`$=\int_0^m s(t) \ dt$`

`$=\int_0^(10) 8000 \ dt$`

`$=8000 * [t]_0^(10)$`

= 8000 x 10

= $ 80000

Simply, S(t) x t = 8000 x 10 = $ 80,000

The difference between the future value and the value of future value from the income stream provides the future value from the interest.

B - 80,000 = 98,365 - 80,000

So, the future value from the interest = $ 18,365