Question

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

a. 5% compounded continuously
b. 10% compounded continuously

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

Explanation:

For continuously compound interest

`A = P * 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}=(5)/(100)=0.05`

t = 1 (1 year)

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

`P * e^((0.05)(1))=P+7200`

`P * e^(0.05)-P=7200`

`P\left(e^(0.05)-1\right)=7200`

`P(1.05127-1)=7200`

`P(0.05127)=7200`

`P=(7200)/(0.05127)=\$140433.002`

Case B:

`r=\frac{\text {interest rate}}{100}=(10)/(100)=0.10`

t = 1 (1 year)

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

`P * e^((0.10)(1))=P+7200`

`P * e^(0.10)-P=7200`

`P\left(e^(0.10)-1\right)=7200`

`P(1.10517-1)=7200`

`P(0.10517)=7200`

`P=(7200)/(0.10517)=\$68460.59`