Respuesta :
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
[tex]A = P \times e^{r t}[/tex] ---------------> 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:
[tex]r=\frac{\text {interest rate}}{100}=\frac{5}{100}=0.05[/tex]
t = 1 (1 year)
Substitute all values in the formula (2) using the formula (1),
[tex]P \times e^{(0.05)(1)}=P+7200[/tex]
[tex]P \times e^{0.05}-P=7200[/tex]
[tex]P\left(e^{0.05}-1\right)=7200[/tex]
[tex]P(1.05127-1)=7200[/tex]
[tex]P(0.05127)=7200[/tex]
[tex]P=\frac{7200}{0.05127}=\$140433.002[/tex]
Case B:
[tex]r=\frac{\text {interest rate}}{100}=\frac{10}{100}=0.10[/tex]
t = 1 (1 year)
Substitute all values in the formula (2) using the formula (1),
[tex]P \times e^{(0.10)(1)}=P+7200[/tex]
[tex]P \times e^{0.10}-P=7200[/tex]
[tex]P\left(e^{0.10}-1\right)=7200[/tex]
[tex]P(1.10517-1)=7200[/tex]
[tex]P(0.10517)=7200[/tex]
[tex]P=\frac{7200}{0.10517}=\$68460.59[/tex]
