Question

You are considering three alternative banks in which to open a savings account. The first bank offers you an annual rate r1, and the interest is paid monthly. The second bank offers a rate r2, and the interest is paid daily. The third bank offers a rate r3, and it offers continuous compounding.

Give all answers to four decimal places.
1) Suppose you were to save $500.0000 in the first bank. The interest rate is r1=8.0000%. Three years from now, you should have $
2) Suppose you were to save $500.0000 in the second bank. The interest rate is r2=5.0000%. Three years from now, you should have $
3) Suppose you were to save $500.0000 in the third bank. The interest rate is r3=3.0000%. Three years from now, you should have $
4) Let the interest rate in the first bank be r1=8.0000%, and you are considering saving your money for 3 years. The interest rate from the second bank that would make you indifferent between the first and second bank is r2=
5) Let the interest rate in the third bank be r3=3.0000%, and you are considering saving your money for 3 years. The interest rate from the first bank that would make you indifferent between the first and third bank is ????

Answer 1

1) Suppose you were to save $500.0000 in the first bank. The interest rate is r1=8.0000%. Three years from now, you should have

effective interest rate = 1.08 = (1 + r)¹²

r = 0.643403% per month

future value = $500,000 x (1.0643403)³⁶ = $629,856

2) Suppose you were to save $500.0000 in the second bank. The interest rate is r2=5.0000%. Three years from now, you should have

effective interest rate = 1.05 = (1 + r)³⁶⁵

r = 0.013368061% per day

future value = $500,000 x (1.00013368061)¹⁰⁹⁵ = $578,812.50

3) Suppose you were to save $500.0000 in the third bank. The interest rate is r3=3.0000%. Three years from now, you should have

future value = $500,000 x e⁰°⁰⁹ = $547,087.14

4) Let the interest rate in the first bank be r1=8.0000%, and you are considering saving your money for 3 years. The interest rate from the second bank that would make you indifferent between the first and second bank is r2=

$629,856 = $500,000 x (1 + i)¹⁰⁹⁵

(1 + i)¹⁰⁹⁵ = 1.259712

1 + i = 1.000210874

i = 0.000210874 = 0.0210874% per day or 7.7% annual

5) Let the interest rate in the third bank be r3=3.0000%, and you are considering saving your money for 3 years. The interest rate from the first bank that would make you indifferent between the first and third bank is

$500,000 x (1 + i)³⁶ = $547,087.14

(1 + i)³⁶ = 1.09417428

i = 0.2503128 per month = 3.05% annual