Question

(a) How long will it take an investment to double in value if the interest rate is 8% compounded continuously? (Round your answer to two decimal places.) yr (b) What is the equivalent annual interest rate? (Round your answer to two decimal places.)

Answer 1

Answer:

Explanation:

Given

rate of interest= 8%

For continuously compounded interest

$A=Pe^{rt}$

Where A=amount

P=Principal

r=rate of interest

t=time

$2P=Pe^{0.08t}$

$2=e^{0.08t}$

taking log both sides

$\ln 2=0.08 t$

$t=\frac{\ln 2}{0.08}$

t=8.66 yr

(b)Equivalent annual interest

$2=(1+i)^t$

$2=(1+i)^{8.66}$

$2^{0.1154}=1+i$

$i+1=1.0832$

i=0.0832

8.32 %

Answer 2

The investment values are;

(a) The number of years it will take the investment to double is approximately 8.66 years

(b) The equivalent annual interest rate is approximately 8.33%

The reason the above values are correct is given as follows:

(a) The given parameters are;

Percentage interest rate, r = 8%

The continuous compounding interest rate formula, is given as follows;

$P(t) = P_0 \cdot e^{r \cdot t}$

When the investment doubles, we have;

$2 \cdot P_0 = P_0 \cdot e^{r \cdot t}$

$e^{0.08 \times t} = 2$

$t = \dfrac{\ln 2}{0.08} \approx 8.66$

The number of years, t, it will take the investment to double is t ≈ 8.66 years

(b) The equivalent annual interest rate is given as follows;

$i_{eff} = e^r - 1$

Where;

$i_{eff}$ = Effective annual interest rate = Equivalent annual interest rate

Therefore, we have;

$i_{eff} = e^0.08 - 1 = 0.083287$

The equivalent annual interest rate ≈ 8.33%

Learn more about continuous compounding interest rate here:

https://brainly.com/question/3965599