SOLUTION: If r is the nominal rate and n is the number of times interest is compounded​ annually, then R= (1 + r/n)^n Here, R represents the annual rate that the investment would earn
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Question 1108712: If r is the nominal rate and n is the number of times interest is compounded annually, then R= (1 + r/n)^n Here, R represents the annual rate that the investment would earn if simple interest were paid. Use this formula to determine the effective rate for $1 invested for 1 year at 7.17.1% compounded semiannually.
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
.
You just have formulas and everything just was explained to you.
Why do not you make these calcs on your own ?
What are you waiting for ?????????????????????????????????????
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
it looks like you are looking for the yearly effective interest rate.
i'm not exactly sure what your annual interest rate is.
i'm guessing that you really meant 7.1%.
i'll assume that's true.
you can re-work the problem using whatever percent you like, once you know the procedure.
the formula i use is R = (1 + r/c) ^ c - 1.
R is the effective annual interest rate.
r is the nominal annual interest rate.
c is the number of compounding periods per year.
to find the nominal annual interest rate, you divide the nominal interest rate percent by 100.
7.1% becomes .071.
with annual compounding, the formula becomes R = (1 + .071/1) ^ 1 - 1.
this gets you an effective annual interest rate of .071.
when you have annual compounding, the effective annual interest rate is the same as the nominal annual interest rate.
with semi-annual compounding, the formula becomes R = (1 + .071/2) ^ 2 - 1.
this gets you an effective annual interest rate of .07226025.
with monthly compounding, the formula becomes R = (1 + .071/12) ^ 12 - 1.
this gets you an effective annual interest rate of .073356638.
the more compounding periods per year, the higher the effective interest, up to the maximum you can get, which is continuous compounding.
that uses a different formula.
the continuous compounding formula for effective annual interest rate is R = e ^ r - 1.
e is the scientific constant that is equal to 2.718281828........
formula becomes R = e ^ .071 - 1.
this gets you an effective annual interest rate of .0735812259.
that's the highest effective annual interest rate you can get out of a nominal annual interest rate of .071.
if you make your number of compounding periods per year high enough, you can get very close to this, but will never quite reach it because the maximum number of compounding periods per year is continuous compounding.
for example, 1000 compounding periods per year makes your discrete compounding formula become R = (1 + .071/1000) ^ 1000 - 1.
this results in an effective annual interest rate of .07357852.
that's closer to .0735812259 that you got with continuous compounding, but still a little less than it.
higher number of compounding periods per year gets you even closer, but that's not normally done and it's not necessary, since the continuous compounding formula works just fine.
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