Question

A promissory note will pay $60,000 at maturity 11 years from now. If you pay $27,000 for the note now, what rate compounded continuously would you earn?The investment would earn about % compounded continuously(Round to three decimal places as needed)

Answer 1

Rate is 7.259%

Explanation:

Amoun to be earned 11 years from now = $60000

Initial pay = $27000

rate = ?

time = 11 years

number of times compounded = continously

To get the rate, we will apply continous compounding formula:

$P(t)\text{ = }P_0\text{ }e^{rt}$$\begin{gathered} \text{where P(t) = 60000} \\ P_0\text{ = 27000} \\ t\text{ = 11} \\ \text{Substitute the values:} \\ 60000=27000e^{r\times11} \end{gathered}$$\begin{gathered} 60000=27000e^{11r} \\ \text{divide both sides by }27000\colon \\ \frac{60000}{27000}=\frac{27000e^{11r}}{27000} \\ 2.2222\text{ = }e^{11r} \end{gathered}$$\begin{gathered} \text{Take natural log of both sides:} \\ \log _e(2.2222)=log_e(e^{11r}) \\ \log _e\text{ = ln} \\ \\ \ln (2.2222)\text{ = 11r} \\ \text{dividing both sides by 11:} \\ r\text{ = }\frac{\ln (2.2222)}{11} \end{gathered}$$\begin{gathered} r\text{ = }\frac{0.7985}{11} \\ r\text{ = 0.072}59 \\ In\text{ percent = 0.07259 }\times\text{ 100\%} \\ rate=\text{ 7.259\%} \\ \end{gathered}$