Question

1. The stated annual interest rate is 2 percent, couponed continuously. What is the equivalent effective annual interest rate (to 6 places past the decimal point) associated with this stated rate?  

                         Solution

                      Ear = (1+2%/1)^1=0.02

                      Ear = Exp(0.2)-1 =1.0202-1=2.02%

2. The effective monthly rate is .0025 (i.e., a quarter of a percent per month). If you have a 30-year mortgage for $500,000 with monthly payments, what are your monthly payments?

3. What is the present value (i.e., the value at t = 0, where t counts years) of the following stream of payments if the effective annual interest rate is .03 (i.e., 3% per annum): The first payment of $100 is made in 10 years (at t = 10). There are 10 more payments after that first payment that occur every 1.5 years. Each of these subsequent payments are 5% larger than the previous payment.

4. You need to borrow $20,000 to buy a car. Bank A is charging you a stated rate of 4% compounded every month; you must make monthly payments for 5 years. Bank B is charging you 3.9% compounded continuously; you also must make monthly payments for 5 years. Which deal do you like better: Bank A’s or Bank B’s?   

5. You need to borrow $20,000 to buy a car. Bank X is charging you a stated rate of 6% compounded every month; you must make monthly payments for 5 years. Bank Z is charging you 5.9% compounded continuously; you also must make monthly payments for 5 years. Which deal do you like better: Bank X’s or Bank Z’s?   

6. What is the present value of an annual payment stream that starts next year at $100 and grows 3% every year thereafter into the infinite future is the discount rate is an effective 5% per year?

Answer 1

The effective annual interest rate when 2% is compounded continously is given by:

I_e=e^r-1

Substitute r=0.02 or 2% (as given in the question):

I_e=e^0.02-1

Use calculator to find the value of I_e correct upto 6 decimal places:

I_e=0.02020134 or 2.020134%

Hence the effective annual interest rate is 2.020134%