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“Everyone in this video. We re going nto discuss the difference between nominal and effective effective interest rate nlet s start with this simple example suppose 500 dollars were deposited na bank. Savings account and the bank s interest policy is. 6.
Compounded nquarterly how much money would be in the account at the end of first year. So the ntwo things to start with first of all the 6 interest that is an annual rate nand whenever. It s not mentioned the assumption is that the rate is annual nrate. The second important point.
Which is the key point of this example is that the ncompounding occurs every quarter. So the money that you deposit in the account ngets compounded every 3 months. Which is a quarter. So the interest period is nthree months long and in a year.
We have four interest periods four quarters interest rate per interest period is nshown with i and in this case. I would be 6. Which is the annual rate divided by nfour sub periods that we have within a. Year so it s 15.
Per nquarter the number of interest periods that we have within a year that nis four now we want to calculate how much money would be in the account at nthe end of first year. That means f. Which is the future value at the end of the nfirst year that s why we re going to use this familiar formula here to compute nf. So f is equal to p 1.
I n p. Is the money that we deposited. So it s n 500. 1.
I we found here 15. so 100 to. 4 so that means every. Quarter i m gaining.
N15. Interest and there are four quarters as long as i and nn are consistent with each other we re always good in this case. I is the nquarterly interest and n is number of quarters in the case of yearly they have nto be consistent and so far they have to match so in this case f would be equal nto 530 and 68 cents. That s the future value at the end of first year so let s ntalk about nominal annual interest rates versus effective nominal annual interest nrate is we show that the r is the annual interest rate.
Without considering nthe effect of compounding effective annual interest rate on the other hand nwhich. We show that with i sub a is the annual rate taking into account the neffect of any compounding during the year. So if there is any compounding noccurring within a year like the example we just looked at every quarter or could nbe every month every week every day then the effective interest rate in the nnominal interest rate would not be equal because there isn t compounding noccurring within a year therefore the effective rate would be different how to ncompute effective rate this is the formula to compute effective rate as a nfunction of nominal rate m is the number of compounding sub periods within na year so if the compounding occurring every quarter then m nis equal to 4. If compounding is every month m.
Is 12 nif. It s weekly. It s 52 and so forth depending on whatever. The compounding nstructure is and r over m.
We can write that as i and i is effective interest nrate per. Compounding sub period. Which is r over m. So that formula that we re ngoing to use to convert nominal rate to effective.
It s going to be this formula nwhere you have r m. And you calculate the effective rate based on those let s use nthe example that we had here the 6. Compounded quarterly and 15 . So if we want to calculate the effective interest rate of that nproblem what.
Would that be 1 r m in. That case r was 6 . Nthat was their yearly rate and m is 4. Because compounding occurs every nquarter so if you calculate the effective rate.
You would see that it s a n6136 which. Is a little higher than 6 . Which nwas the nominal. Rate so in that.
Case. 6 was the nominal rate and 6136. Is the effective rate effective is always greater than nominal so that s how we calculate effective. Rate let s look at nanother example here if a credit card charges 15.
Ninterest every month. What are the nominal and effective interest rates per nyear well the nominal is nwhen. We re not considering the effect of compound interest. Nif we don t consider that in nominal.
Rate which we call it. R it s going to nbe 15. Simply times. 12.
Which. Is the number of months nwithin a year and that s 18 . But because compounding occurs every. Month that s nthe implication of this problem saying a credit card charges 15.
Interest every month that means the interest gets compounded every month. Nin that case. The effective rate is going to be a little higher effective annual rate nwhich. We call.
I sub a would be equal to 1 r m m. 1. Nso. In that case it would be 1 plus 18.
Divided by 12 to 12 and minus 1. And if you do the math here you would get to 1956 . Again a little higher than the nominal rate. Because of the ncompounding that s how you compute effective annual interest rate given.
The nnominal rate and compounding structure. Some of the important points. Here first nis. The case of continuous compounding in that case that means the compounding noccurs every millisecond every moment.
The interest gets compounded nso that s a theoretical situation typically in reality. Ncompounding structure. Is daily. When you re looking at loans.
Nand savings accounts and practices daily. But if you want to know what is the nupper bound of effective rate and that s when continuous compounding happens nevery millisecond is compounded in that case computing the effective rate given nthe nominal rate would be based on this formula that s only for continuous ncompounding another important point to talk about is that nwhen the compounding is annually when within a year no compounding occurs it s njust annually in that case nominal interest rate equals the effective ninterest rate. So just equal because no compounding occurs within a year that s nall we had for effective versus nominal. ” .
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