Math  Compound
posted by Anonymous
If a bank offers interest at a nominal rate of 6%, how much greater is the effective rate if interest is compounded continuously than if the compounding is quarterly?
I don't get this question at all... All I'm given is the rate and how am I suppose to compare compounded continuously and quarterly if I'm not given the initial value, time and so on...

Damon
well, lets compare both to compounding once a year for n years.
Once a year
final/original = 1.06^n
Four times a year
6/4 = 1.5% per quarter
final/original = 1.015^4n
Continuously
final/original = e^.06 n
Well, lets see how four times a year compares to once a year
compare 1 = 1.015^4n / 1.06^n
ln compare 1 = 4 n ln 1.015  n ln 1.06
= n(.05955445.0582689) = .0012855419 n
so
compare 1 = e^.0012855419 n
or .12855 % better than once a year
Now do the same for continuous
compare 2 = e^06 n / 1.06^n
ln compare 2 = .06 n  n ln 1.06
= .001731 n
compare 2 = e^.001731 n
or .1731 % better than once a year 
drwls
Quarterly componding of interest after one year at 6% annual rate gives you an annual yield of
(1 + 0.06/4)^4  1 = 6.136%
Continuous compounding requires you to consider limits. The answer is
Limit (as n approaches infinity) of
1 + 0.06/n)^n  1
Calculus shows that this equals
e^(0.06) 1 = 6.184%
If you don't understand limits and e, consider the "daily interest" case, with n = 365. In that case the annual yield is 6.183% 
Damon
Just do them each for one year
yearly 1.06
quarterly
1.015^4 = 1.06136 or 6.136 % yearly
continuously
e^.06 = 1.06184 or 6.184 % yearly 
Anonymous
thank you both for your help... I like Damon's method as it seems straightforward and I can follow it...
Respond to this Question
Similar Questions

math
find the effectivee rate correspoding to 3% compounded quarterly The formula for effective rate I found is (1+ i/n)^n  1 where i is the annual rate as a decimal and n is the number of periods. Here i=.03 so the effective rate is and … 
Compound Interest
A bank offers a rate of 5.3% compounded semiannually on its four year GICs(Guaranteed Investment Certificates). What monthly and annually compounded rates should it quote in order to have the same effective interest rate at all three … 
compounded interest
A bank offers a rate of 5.3% compounded semiannually on its four year GICs(Guaranteed Investment Certificates). What monthly and annually compounded rates should it quote in order to have the same effective interest rate at all three … 
Algebra II
An investment service promises to triple your money in 12 years. Assuming continuous compounding of interest, what rate of interest is needed? 
math
You receive $12,000 and looking for a bank to deposit the funds. Bank A offers an account with an annual interest rate of 3% compounded semiannually. Bank B offers an account with 2.75% annual interest rate compounded continuously. … 
math
You receive $12,000 and looking for a bank to deposit the funds. Bank A offers an account with an annual interest rate of 3% compounded semiannually. Bank B offers an account with 2.75% annual interest rate compounded continuously. … 
Finance
You receive $12,000 and looking for a bank to deposit the funds. Bank A offers an account with an annual interest rate of 3% compounded semiannually. Bank B offers an account with 2.75% annual interest rate compounded continuously. … 
Finance
You receive $12,000 and looking for a bank to deposit the funds. Bank A offers an account with an annual interest rate of 3% compounded semiannually. Bank B offers an account with 2.75% annual interest rate compounded continuously. … 
math
you are looking for a safe place to put 30,000.00 for one yr. Bank A offers 2.46% interest rate continuously. Bank B offers 2.48% quarterly and Bank C offers 2.47% monthly. Find the most effective rate of each to determine which would … 
Social Studies
An interest rate is a special type of (1 point) loan. **price. bank. service. 2. How does a compound interest rate differ from a simple interest rate?