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Welcome back to Corporate Finance.

Today, we're going to turn to a new topic, interest rates.

But before doing so, I want to briefly review our last topic,

the time value of money.

As you'll recall, we started off with some intuition,

then we introduced the tools associated with the time value of money.

Namely, the discount factor and the timeline, then we applied those tools to

move money back in time via discounting and forward in time via compounding.

We discussed some useful shortcuts to compute the present value and

the future value of cash flow streams that we commonly come across in practice,

streams like an annuity and a perpetuity and

then we close out the topic by talking about taxes and inflation and

their impact on our dollar returns and our ability to consume goods and services.

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In this topic, interest rates, I want to start off by talking about interest rate

quoting conventions and I want to talk about how to compute the present value and

future value of string of cash flow when they arrived at irregular time,

non-annual and when the compounding is not annual as well.

So let's get started.

Hi, everyone.

Welcome back to Corporate Finance.

Today, we're going to be turning to a new topic, interest rates.

But before doing so, I want to briefly review our first topic,

Time Value of Money.

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So we started off that topic by introducing the concept with some

intuition.

What we did is we showed that cash or money received or

paid at different points in time as a different time unit and

as such can't be added, much like different currencies can't be added.

Then we introduced the tools associated with the time value of money,

namely a timeline,

which is just a visual representation of when money is coming or going.

And a discount factor, which is our exchange rate.

For time, it's what we used to move cash flows forward and backward in time.

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And we use those tools to discount cash flows, that is move them back in time and

compound cash flows are moving forward in time.

We introduced some useful shortcuts.

Namely, the present value of an annuity formula and the present perpetuity formula

as well as formulas for the present value of grown annuity and a growing perpetuity.

Then we closed off the topic with a discussion of taxes and inflation and

investigated how those two concepts would impact both our dollar return and

the purchasing power of those dollars.

Now, I want to turn to interest rates in this lecture and this isn't so

much a new topic as much as it is really an extension of the time value of money to

incorporate institutional details and make things a little bit more realistic.

So we're going to talk about interest rate quotes and

we're going to learn how to deal with cash flows that don't arrive once a year, but

may arrive monthly or semi annually.

And we're also going to discuss how to deal with compounding of interest when

it's not just annually.

So let's get started.

Here is a snapshot from December of 2014 of rates on five year jumbo CDs,

where CD just stands for certificate of deposit.

It is a savings vehicle most banks offer and here is one, two, three, four banks.

Now the jumbo that just refers to, I think a minimum deposit of $100,000.

So these are big deposits.

And one of the things you notice when looking at these rates is when you're

looking at these savings vehicles is that each one has two different rates.

It has a rate and an APY and these numbers 2.37, 2.4, they're different.

So that begs the question is why are they different?

How are they related?

And most importantly, which one is going to tell me how much money I'm going

to make when I invest in this product?

Well, let's go through this starting with the rate.

The rate refers to the APR of the Annual Percentage Rate.

That measures the amount of simple interest earned in a year.

Simple interest is just the interest earned without compounding,

ignoring compounding.

And if you're wondering what compounding is we're going to talk about it, but

just as a preview.

Notice underneath rate, we have compounded daily.

Simple interest ignores that compounding frequency.

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Now many banks quote interest rates in terms of an APR.

The problem is the APR is typically not what we're going to earn or

what we're going to pay.

For that, we have to turn to the APY or the Annual Percentage Yield,

which is really just another way of saying EAR or effective annual rate.

See, the EAR that measures the actual amount of interest earned or

paid in a year.

That's what we care about the EAR, that's the number we care about.

The rate or APR that's just a quoting convention.

Now how are these different rates related?

Well, before showing you the explicit mathematics,

which frankly are almost trivial, let me emphasize the following.

The EAR is a discount rate.

The EAR is what matters for computing interest and discounting cash flows.

The APR is not a discount rate, it's a means to an end.

It's a quoting convention.

So we're going to use APR in conjunction with compounding frequency

information to get at an EAR or at a periodic discount rate,

which I'll introduce in just a moment.

So remember, EAR = discount rate.

APR = quote.

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Now how do we get from one to the other?

How do we go from APR to EAR or vice versa?

Well, here's that simple mathematical formula I referred to just a moment ago.

The EAR is related to the APR by this equality.

Well, what's going on here?

K, that's just the number of compounding periods per year.

So imagine we had monthly compounding, that would imply k = 12.

How about semi-annual?

That would imply k = 2 and

I'm going to introduce a little bit of notation here.

I is the periodic interest rate or

the periodic discount rate and that = APR over k.

Let's do an example.

So, imagine we're investing $100 in a CD offering 5% APR with semi-annual

compounding.

How much money will we have in one year?

Well, there's actually two ways to approach this problem and

we're going to do each in turn.

The first thing we do, first thing we always do is we draw a timeline.

And so today, period 0, we're going to invest $100.

And the question's asking, how much money are we going to have in one year?

Now, I've left this as a question mark to emphasize the fact that there

are two ways to go about this.

Let's go about this the first way.

Go about answering this problem.

The first is to work in periods.

With semi-annual compounding, that means there are two periods per year.

Since I'm interested in how much money I have after one year,

that's after two periods.

And these periods are every 6 months, period one, period two.

So, if I'm going to work in periods,

I better compute a periodic discount rate, that's i.

Which we know is APR over k and which in this case, reduces to 2.5%.

In other words, I'm going to earn 2.5% over the first 6 months and

I'm going to earn 2.5% over the next 6 months.

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So I take my initial $100 investment, I multiply it by my periodic discount rate.

And after 1 period, I've got $102.50, then I repeat.

[COUGH] I take that $102.50,

I multiply it by periodic discount rate to get $105 and a little over $0.6.

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In other words, the future value 2 periods hence of $100

in this setting, it's just 100 x(1+ i) 2.

We're working in periods, so i is our discount rate and

2 is the number of periods removing the money forward in time.

Now let's approach the problem from the prospective of years.

Now we're looking for how much money we have after one year?

But if we're going to work in years, now we need to be consistent.

So after six months, this isn't one period, this is half a year.

And because we are working in years, our discount rate isn't going to be

the periodic interest rate, i, it's going to be the EAR, the equivalent

[COUGH] with the effective annual rate, which we know from earlier is just one

plus the periodic rate raised to the number of compounding periods per year.

So 1 + i to the k, which in this settings comes out to be 5.0625%.

In other words, over this entire period,

I'm going to earn 5.0625%,

which turns out to equal $105.06.

That is the future value one year,

[COUGH] from now of $100 is

$100 x(1 + the EAR) 1.

We're working in years, which gives me the exact same answer we got before,

the $105.0625.

So the lesson, if you discount cash flows using the EAR,

then you better measures time in years.

If you discount cash flows using the periodic interest rate,

then you better measure time in periods.

And the equality between the two that we just showed with our simple example is

much more general and

given by this straightforward proof, which I'm not going to discuss in detail, but

I'll show you there if you're interested in looking at it.

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So let's go back to our original example.

[COUGH] We had an APR of 2.37%.

We have a compounding frequency of daily, which I'm going to assume is 365 days.

So you should be aware, it could be 360 days or it could be 252 business days.

It depends on the convention used by that product in that institution.

So these two pieces of information, the APR and

k allow me to compute the periodic rate, i, which is 0.006714%.

Now that's a tiny number, but we're computing interest every single day here,

which means the effective annual rate, which equals (1+i) to the k.

Or which in this case is (1+0.006714%)

raised to the 365th power gets me to just under 2.4%.

So when we round the 2.398, we get about 2.4%.

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So let's summarize.

We learned that there are two discount rates depending on

what unit of time you want to work with.

If we want to think in term of periods,

we want to use a periodic discount rate, that's i.

If we want to work in year, we want to use the EAR, the effective annual rate,

which relies on cash flows measured in years.

Both of these are discount rates, they'll both get us to the same goal or

the same end result, but we have to be consistent in terms of how we measure

time with which discount rate we use.

APR, that's a quoting convention, that's a means to an end.

We use APR in conjunction with the compounding frequency to get our

discount rate, whether it's the EAR or the periodic discount rate, i.

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And we can move between the APR and I and the EAR by a couple of

very simple mathematical relations that we discussed.

Now, next are a bunch of great problems that I want you to dive into.

And then after you are done with those, move on to the second part of interest

rates in which we are going to investigate the term structure of interest rates and

talk about the yield curve, but you know what these things mean.

Thank you so much.