Question

Capital Value An investment produces a perpetual stream of income with a flow rate of

R(t)=1200e^0.03 t.
Find the capital value at an interest rate of 7% compounded continuously.

Answer 1

$30,000

Explanation:

The capital value is given by

`\int_(0)^(\infty) R(t) e^(-r t) dt`

where R(t) is annual rate

r - annual rate of interest

capital value
`= \int_(0)^(\infty) 1200 e^(0.03t) e^(-0.07 t) dt`

= lim at b tend to infinity
`\int_(0)^(b) 1200 e^(0.03t) dt`

=lim at b tend to infinity
`\left [ (1200)/(-0.04) e^(-0.04 t) \right ]_0^b`

-30,000 { lim at b tend to inifinity
`(e^{-0.04 b) = e^0]`

As
`b\rightarrow \infty,  e^(-0.04 b) \rightarrow 0`

`\int_(0)^(\infty) 1200 e^(0.03t) e^(-0.07 t) dt = -30,000(0 -1) =$30,000`