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...
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
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%
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
thank you both for your help... I like Damon's method as it seems straightforward and I can follow it...
I understand that the question may feel incomplete or confusing without specific contextual information. However, it is possible to compare the effective interest rates between continuous and quarterly compounding without knowing the specific values for the initial amount, time, or other variables.
To determine the effective rate for continuous compounding, we can use the formula:
Effective Rate (continuous compounding) = e^(rt) - 1
Where:
- e is Euler's number (approximately 2.71828)
- r is the nominal interest rate (given as 6% or 0.06 in decimal form)
- t is the time period over which the interest is compounded
To determine the effective rate for quarterly compounding, we can use the formula:
Effective Rate (quarterly compounding) = (1 + (r/n))^nt - 1
Where:
- n is the number of compounding periods per year (in this case, 4 for quarterly compounding)
By substituting the given values into these formulas, we can calculate and compare the effective rates.
However, please note that to calculate the actual amount of interest earned or compare the overall growth between continuous and quarterly compounding, we would need additional information such as the initial deposit or loan amount, the time period, and any additional contributions or withdrawals.