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Eng Economic Analysis - Nominal & Effective Interest Rates - YouTube
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Hello everyone, in this video we're going
to discuss the difference between
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nominal and effective interest rate.
Let's start with this simple example.
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Suppose 500 dollars were deposited in
a bank savings account, and the bank's
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interest policy is 6% compounded
quarterly. How much money would be in the
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account at the end of first year? So the
two things to start with, first of all,
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the 6% interest, that is an annual rate.
And whenever it's not mentioned, the
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assumption is that the rate is annual
rate. The second important point, which is
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the key point of this example, is that the
compounding occurs every quarter, so the
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money that you deposit in the account
gets compounded every 3 months, which is
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a quarter. So the interest period is
three months long. And in a year we have
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four interest periods - four quarters.
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Interest rate per interest period is
shown with I, and in this case I would be
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6% which is the annual rate divided by
four sub periods that we have within a
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year, so it's 1.5% per
quarter. The number of interest
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periods that we have within a year, that
is four. Now we want to calculate how
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much money would be in the account at
the end of first year. That means F, which
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is the future value at the end of the
first year. That's why we're going to use
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this familiar formula here to compute
F. So F is equal to P(1+i)^n.
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P is the money that we deposited, so it's
$500.
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1+i we found here, 1.5%. So 100 to 4.
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So that means every quarter, I'm gaining
1.5% interest, and
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there are four quarters. As long as i and
n are consistent with each other, we're
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always good. In this case, i is the
quarterly interest and n is number of
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quarters. In the case of yearly, they have
to be consistent and so far they have to
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match. So in this case, F would be equal
to 530 and 68 cents. That's the future
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value at the end of first year. So let's
talk about nominal annual interest rates
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versus effective. Nominal annual interest
rate is, we show that the R is the
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annual interest rate without considering
the effect of compounding. Effective
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annual interest rate, on the other hand,
which we show that with i sub a, is the
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annual rate taking into account the
effect of any compounding during the
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year. So if there is any compounding
occurring within a year, like the example
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we just looked at, every quarter or could
be every month, every week, every day then
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the effective interest rate in the
nominal interest rate would not be equal
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because there isn't compounding
occurring within a year, therefore the
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effective rate would be different. How to
compute effective rate, this is the
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formula to compute effective rate as a
function of nominal rate. M is the
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number of compounding sub periods within
a year. So if the compounding occurring
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every quarter, then m
is equal to 4. If
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compounding is every month, M is 12.
If it's weekly, it's 52 and so forth,
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depending on whatever the compounding
structure is. And R over m, we can write
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that as i and i is effective interest
rate per compounding sub period, which is
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R over m. So that formula that we're
going to use to convert nominal rate to
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effective, it's going to be this formula,
where you have R, M, and you calculate the
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effective rate based on those. Let's use
the example that we had here. The 6%
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compounded quarterly and 1.5%. So if we want to calculate
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the effective interest rate of that
problem, what would that be? 1+R/m
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In that case R was 6%,
that was their yearly rate, and M is 4,
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because compounding occurs every
quarter. So if you calculate the
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effective rate, you would see that it's a
6.136%, which is
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a little higher than 6%, which
was the nominal rate. So in that case, 6%
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was the nominal rate and 6.136% is the effective rate.
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Effective is always greater than nominal. So that's how we
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calculate effective rate. Let's look at
another example here. If a credit
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card charges 1.5%
interest every month, what are the
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nominal and effective interest rates per
year?
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Well, the nominal is,
when we're not considering the effect of
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compound interest,
if we don't consider that in nominal
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rat,e which we call it r, it's going to
be 1.5% simply times
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12, which is the number of months
within a year, and that's 18%. But because
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compounding occurs every month, that's
the implication of this problem saying
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a credit card charges 1.5% interest every month. That means
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the interest gets compounded every month.
In that case, the effective rate is going
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to be a little higher. Effective annual rate,
which we call i sub a, would be
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equal to (1+R/M)^m - 1.
So in that case, it would be 1
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plus 18% divided by 12, to 12, and minus 1. And if you do the math here, you
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would get to 19.56%. Again, a little higher than
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the nominal rate because of the
compounding. That's how you compute
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effective annual interest rate given the
nominal rate and compounding structure.
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Some of the important points here, first
is the case of continuous compounding. In
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that case, that means the compounding
occurs every millisecond. Every
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moment the interest gets compounded.
So that's a theoretical situation.
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Typically, in reality,
compounding structure is daily.
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When you're looking at loans
and savings accounts and practices daily.
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But if you want to know what is the
upper bound of effective rate, and that's
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when continuous compounding happens,
every millisecond is compounded. In that
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case, computing the effective rate given
the nominal rate would be based on this
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formula. That's only for continuous
compounding. Another
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important point to talk about is that
when the compounding is annually, when
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within a year no compounding occurs, it's
just annually, in that case, nominal
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interest rate equals the effective
interest rate. So just equal. Because no
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compounding occurs within a year. That's
all we had for effective versus nominal
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interest rate.
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