Question

your bank is willing to give you an interest rate of 8%, compounded continuously, on any investment made with their money market accounts. you have $3,200 to deposit into the bank. what is your expected return if you leave your money in this bank for 4 years? round your answer to the nearest cent (hundredth).

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $t$ is the time in years, and $e$ is the base of the natural logarithm (approximately $2.71828$).

First, we need to convert the interest rate from a percentage to a decimal. The interest rate $r = 8\% = 0.08$. The principal $P=\$3200$, and the time $t = 4$ years.

Step2: Substitute the values into the formula

Substitute $P = 3200$, $r = 0.08$, and $t = 4$ into the formula $A = Pe^{rt}$.

So, $A=3200\times e^{(0.08\times4)}$.

First, calculate the exponent: $0.08\times4 = 0.32$.

Then, calculate $e^{0.32}$. Using a calculator, $e^{0.32}\approx1.377127$.

Now, multiply this by the principal: $A = 3200\times1.377127$.

Step3: Calculate the final amount

$3200\times1.377127 = 4406.8064$.

The expected return is the final amount minus the principal. So, the return $R=A - P=4406.8064 - 3200 = 1206.8064$.

Rounding to the nearest cent (hundredth), we get $1206.81$.

Answer:

The expected return is $\$1206.81$.