Question

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...

Answer 1

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

Answer 2

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%

Answer 3

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

Answer 4

thank you both for your help... I like Damon's method as it seems straightforward and I can follow it...