Question

Capital Value Find the capital value of an asset that generates $7200 yearly income if the interest rate is as follows.

Answer 1

a. An asset that generates $7200 yearly income if the interest rate 5% compounded continuously, then its capital value is $140433.002

b. An asset that generates $7200 yearly income if the interest rate 10% compounded continuously, then its capital value is $68460.59

Step-by-step explanation:

For continuously compound interest

                    $A = P \times e^{r t}$  ---------------> eq.1

Where

P = principal amount (initial investment)

r = annual interest rate (as a decimal)

t = number of years

A = amount after time t.

Let’s solve the equation

Where,

P is unknown

                     A = P + 7200 (asset after 1 year) ---------------> eq. 2

Case A:

           $r=\frac{\text {interest rate}}{100}=\frac{5}{100}=0.05$

t = 1 (1 year)

Substitute all values in the formula (2) using the formula (1),

          $P \times e^{(0.05)(1)}=P+7200$

          $P \times e^{0.05}-P=7200$

          $P\left(e^{0.05}-1\right)=7200$

          $P(1.05127-1)=7200$

          $P(0.05127)=7200$

          $P=\frac{7200}{0.05127}=\ 140433.002$

Case B:

       $r=\frac{\text {interest rate}}{100}=\frac{10}{100}=0.10$

t = 1 (1 year)

Substitute all values in the formula (2) using the formula (1),

         $P \times e^{(0.10)(1)}=P+7200$

         $P \times e^{0.10}-P=7200$

         $P\left(e^{0.10}-1\right)=7200$

         $P(1.10517-1)=7200$

         $P(0.10517)=7200$

         $P=\frac{7200}{0.10517}=\ 68460.59$