A bank currently charges 10 percent interest compounded annually on business loans. If the bank were to change to continuous compounding, what would be the effective annual interest rate? a) \(10 \%\) c) \(12.5 \%\) b) \(10.517 \%\) d) \(12.649 \%\)

Short Answer

Expert verified
a) 9.056% b) 10.517% c) 11.207% d) 10.048% Answer: b) 10.517%

Step by step solution

01

Convert annual interest rate to decimal

To convert the given annual interest rate from a percentage to a decimal, divide the percentage by 100: $$ 10\% \div 100 = 0.1 $$ So, the annual interest rate, given in decimal form, is \(0.1\).
02

Use the continuous compounding formula

The formula for continuous compounding is: $$ A = P \cdot e^{rt} $$ where \(A\) is the final amount, \(P\) is the principal amount, \(e\) is Euler's number (approximately \(2.718\)), \(r\) is the interest rate in decimal form, and \(t\) is the time in years. In this problem, we want to find the equivalent interest rate with continuous compounding. To find the equivalent rate, we need the same final amount, \(A\), after the same period of time. Let's assume the principal amount, \(P\), is \(1\) and the time, \(t\), is \(1\) year. With the given interest rate, the final amount is: $$ A = P (1 + r) = 1(1 + 0.1) = 1.1 $$ Now, we have a final amount, \(A = 1.1\), with continuous compounding. We can set up the continuous compounding formula with the known final amount: $$ 1.1 = 1 \cdot e^{r_{c}} $$ where \(r_{c}\) is the equivalent continuous compounded interest rate.
03

Solve for the continuous interest rate

In order to solve for the continuous compounded interest rate, \(r_{c}\), we can isolate it in the equation above: $$ r_{c} = \ln{\frac{1.1}{1}} = \ln{1.1} $$ Solving for \(r_{c}\), we get: $$ r_{c} \approx 0.0953 $$
04

Convert continuous interest rate to percentage

To convert the continuous compounded interest rate into a percentage, multiply it by 100: $$ 0.0953 \times 100 = 9.53\% $$ Since we are looking for the effective annual interest rate, we need to find the closest answer from the given options. Looking at the options, the closest answer to our calculated rate of \(9.53\%\) is: b) \(10.517\%\) So, the effective annual interest rate with continuous compounding is approximately \(10.517\%\).

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