hw 10 question 4

Question

Find the accumulated present value of an investment over a 11-year period if there is a continuous money flow of $2500 per year and the current interest rate is 8.5%, compounded continuously.

Answer 1

Solution:

For continuous income stream S(t) of $2500 The future value is given by

FV = e ^rM ∫₀^M S(t)e ^−rtdt. where M = 11 yrs, r = 8.5% = 0.085

FV = e^(0.085)(11) ∫₀¹¹ ( $2500) e^-0.085t dt , FV = 2.547 ∫₀¹¹ ( $2500) e^-0.085t dt

to solve the integral let u = -0.085t , du = -0.085 dt , dt = du/ -0.085 substitute in integral you get

FV = 2.547 ∫₀¹¹ [( $2500) e^u du / -0.085] , FV = [(2.547)( 2500)/ (-0.085)] ∫₀¹¹ e^u du

FV = -74,918 ∫₀¹¹ e^u du = -74,918 [ e^u ]¹¹₀ = -74,918 [ e^-0.085t]¹¹₀ = -74,918 [ e^(-0.085)(11) - e⁰ ]

FV = -74,918 ( 0.393 - 1) = $45,475