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Today we're going to talk about

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a very important topic

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topic in economics, which is

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expectations. We have barely mentioned

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expectations when we talk about the

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Phillips curve. We talked about

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expectations when we

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when we discussed the UEP and so on. But

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expectations is a much bigger issue in

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economics. In fact, most decisions by

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firms, by consumers,

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governments involve considerations of

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the future. And it plays an even bigger

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role in finance, in which essentially

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everything is about the future. The

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price of an asset today is meaningless

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in itself. You have to compare it with

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what you expect to get out of that asset

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in the future. So, it's all about

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expectations and so on.

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So, that's what we we're going to do

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today. We're going to talk about

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uh expectations, the how to value things

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that that you expect to receive in the

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future,

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uh and how to compare those things with

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things that you have in the present.

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Um

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but before doing that, actually, let's

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talk a little bit about the news. Who

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knows who First Republic Bank is?

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Remember that a few weeks ago I told you

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that um Silicon Valley Bank,

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you read it. I I I I just mentioned it

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that

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uh

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uh

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we'll discuss it that that you know, we

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had the second largest bank by asset in

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in in US history. It was Silicon Valley

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Bank was the second largest

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asset bank in terms of assets

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uh to collapse in the US. The first one

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was uh many years ago.

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Uh and then we had this bank that had

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more than $200 billion in assets that

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essentially collapsed in a few days. It

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was a run on deposits. They had problems

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before,

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but what really did as it always the

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case with banks is they had a run on

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deposits, funding.

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Uh

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well, it's no longer the second largest

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collapse in US bank history. Now we have

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over the weekend

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the the new second largest bank to

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collapse, which is First Republic Bank,

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that was essentially

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liquidated and sold to JP Morgan over

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today morning, very very early in the

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morning. Okay? So,

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you have an account in First Republic

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Bank, you sooner likely to have an

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account in JP Morgan.

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But again, what made it collapse was

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something very similar to what made

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Silicon Valley Bank collapse, which is

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that they had invested on on a series of

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things that were very vulnerable to to

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the fast pace of hikes

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uh

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in interest rates in the US.

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And when they had those losses,

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depositors became became worried about

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it and eventually they decided not to

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wait, just run.

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And see what happened. They

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First Republic Bank lost about $100

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billion in deposits just last week.

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Okay? Um the last few days of last week.

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So, so, so

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so that

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it was obvious that that

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it was not going to survive and that's

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the reason

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something was arranged over the weekend

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to avoid the panics associated with

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collapses of a bank and so on. Okay? But

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anyways, by the way, this is all about

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expectations. This is you know, this is

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if people had expected the deposit to

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remain in the bank, then probably this

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bank would not have collapsed. It's all

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about people anticipating what other

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people will do and so on and so forth.

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Okay, but now let me get into the

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specific of

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of this lecture.

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So, there you have this is the most

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important index of of equity equity

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index in the US, S&P 500. It's a very

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inclusive index that captures all the

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large most of the large companies in the

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US,

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all of the large I think companies in

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the US. And uh that's an index. It's an

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average weighted average by by this

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uh capitalization value of each of the

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shares. It's a weighted average of the

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major the main shares in the US, equity

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shares in the US.

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And one thing you see is that it moves a

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lot around, you know? Here, for example,

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when when

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we became aware that COVID was going to

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be a serious issue,

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the US equity market collapsed by 35% or

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so. That's a very large collapse in a

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very short period of time.

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And then, as a result of lots of policy

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support, actually we had a massive

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rally. Uh so, up to the end of 2021, the

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equity market had rallied by 114%.

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So, a big rally.

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Then we got inflation and the Fed began

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to worry about

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inflation, so they began to hike

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interest rates. And when they hike

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interest rates, that eventually led to a

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very large decline

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uh

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in in asset prices of the order of 30%

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or so, 25% or so, actually, from the

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peak to the bottom.

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And then, since the bottom, which is was

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more or less October of last year, we

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have seen a recovery of about 16% or so

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of the equity market. Okay? And if you

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look at the Nasdaq, which is another one

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index that is very loaded towards

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uh

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technology companies, then you can see

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swings that are even larger than that.

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Now, why do these prices move so much?

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Well,

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a lot of it has to do with expectations.

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You know, are things going to get worse

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in the future? Will the Fed cause a

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recession? Uh

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how much higher will be the interest

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rate? And things like that matter a

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great deal.

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Another thing that matters a great

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deal is

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how much people want to take risk at any

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moment in time. And if you're very

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scared about the environment, you're

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unlikely to want to have something that

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to invest on something that can move so

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much,

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and so risk is well known. So, it's

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called risk off when when people don't

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want to take risk, these asset prices

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tend to collapse. Okay? Of the risky

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asset. Equity is a very risky asset.

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But that's not the only thing that moves

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these assets around. It's not just the

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risk that the companies underlying

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company may go bankrupt or anything like

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that.

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Here you have, for example, the movement

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of a

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for it's an ETF, but it doesn't matter.

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It's a portfolio of

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bonds of US Treasury bonds of very long

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duration. Maturities beyond

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uh 20 years and so.

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So, this is incredibly safe bonds, you

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know? Because it's US Treasuries. So,

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there's no risk of default or anything

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like that.

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Still,

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the price swings can be pretty large. I

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mean, in over this period, you know,

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there you have seen an an an increase in

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value of 45%,

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then uh a decline in value of of of

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about 20%. Another increase in 15% here.

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There was a huge decline, 40%,

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since since essentially

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uh uh uh

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What do you think happened here? Why is

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this big decline in in in in bonds?

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You're going to be able to answer that

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very precisely later on, but but I can

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tell you in advance that that was

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essentially the result of monetary

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policy tightening.

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You know, increasing interest rate

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caused the the bonds to decline. So,

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even these instruments that are very

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safe in the sense that you if you hold

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it to maturity, you will get your money

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back and all the promised coupons along

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the path, well, still their price can

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move a lot. And it's obvious that that

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movement in price

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is something you need to explain in

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terms of expectations, what people

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expect things to to happen. In this

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case, it's not whether people expect to

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get paid or not, because you will get

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paid, but it's expect but in this

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particular case, it's about expectations

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about future interest rate. If you think

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the interest rate will be very high,

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then the price of bonds will tend to be

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very low and so on. But it's all about

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the future. Okay?

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So, the a key concept

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uh that we're going to discuss today and

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then we're going going to use it to

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price a specific asset

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uh is the concept of expected present

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discounted value. This this is a loaded

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concept. There's lots of

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terms in there and we need to understand

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what each of these terms means.

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So, the key issue

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that we're going to discuss is how how

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do we decide, for example, if you see

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the price of an asset out there

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that is 100, how do you decide whether

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that price is fair or not, looks cheap

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or not? Okay? Uh and and and and and

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that question means you have to decide

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whether that price that you're paying

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today is consistent with the future cash

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flows that you're going to get from this

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asset. I mean, that's the reason you buy

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an asset is because you'll get something

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in return in the future. Okay? But how

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do we compare that? How do we compare

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the price today with those things that

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will happen in the future?

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So, answering that question, which is

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what we're going to do in this lecture,

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involves the following

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concepts. First, expectations, big

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thing.

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That's a you know, this is expected

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present discounted value. The E part is

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for expectations. That comes there.

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You

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Expectations are really crucial because

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these are things that happen in the

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future. You need to expect. Even if if

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it's a bond that promises you to pay,

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you know, 50 cents per dollar every 6

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month,

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you still may have an expectation that,

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you know, if it is a bond issued by

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First Republic Bank, it may not pay. So,

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so, so you need to have an expectations

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about that.

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Uh

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so, crucial term is expectation.

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Then you need some method

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uh to compare payments received in the

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future with payments made today. I mean,

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if you buy an asset, you pay today,

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but you're going to receive things

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returns on for that asset in the future.

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So, how do I compare that that that

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Suppose I pay one today and I receive

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one 1 year from now.

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Does that seem like a good asset?

279
00:09:52,600 --> 00:09:56,120
Probably not. I mean, you know,

280
00:09:54,919 --> 00:09:57,559
probably not.

281
00:09:56,120 --> 00:09:59,840
Uh

282
00:09:57,559 --> 00:10:01,559
Um and that's what the word discounted

283
00:09:59,840 --> 00:10:04,600
really means. You know, when you say

284
00:10:01,559 --> 00:10:05,719
expected present discounted value

285
00:10:04,600 --> 00:10:07,680
it says

286
00:10:05,720 --> 00:10:09,480
somehow that things I receive in the

287
00:10:07,679 --> 00:10:10,879
future are valued less than things I

288
00:10:09,480 --> 00:10:12,240
have today.

289
00:10:10,879 --> 00:10:13,439
Okay? So, if you're going to tell me

290
00:10:12,240 --> 00:10:15,000
that you're going to pay me a dollar in

291
00:10:13,440 --> 00:10:17,040
the future and I have to pay you a

292
00:10:15,000 --> 00:10:18,759
dollar today, most likely I won't take

293
00:10:17,039 --> 00:10:20,039
that deal.

294
00:10:18,759 --> 00:10:22,159
So, I need In other words, I'm

295
00:10:20,039 --> 00:10:23,480
discounting the future.

296
00:10:22,159 --> 00:10:24,360
How do we discount the future? Well,

297
00:10:23,480 --> 00:10:26,879
something that we're going to have to

298
00:10:24,360 --> 00:10:26,879
figure out.

299
00:10:27,159 --> 00:10:30,759
So,

300
00:10:28,600 --> 00:10:31,960
let's let me first shut down this part,

301
00:10:30,759 --> 00:10:33,480
the expectations, and then we'll

302
00:10:31,960 --> 00:10:34,800
introduce it. So, assume for now that

303
00:10:33,480 --> 00:10:36,759
you know the future.

304
00:10:34,799 --> 00:10:39,000
Okay? And I'm going to derive all the

305
00:10:36,759 --> 00:10:41,080
equations with assuming that you know

306
00:10:39,000 --> 00:10:42,559
the future. So, there's no issue of

307
00:10:41,080 --> 00:10:44,360
trying to figure out what the future is.

308
00:10:42,559 --> 00:10:45,919
You know it. But still you have to

309
00:10:44,360 --> 00:10:48,440
decide whether

310
00:10:45,919 --> 00:10:50,919
what is the right value for for an

311
00:10:48,440 --> 00:10:50,920
asset.

312
00:10:52,000 --> 00:10:56,320
Okay, so

313
00:10:54,279 --> 00:10:58,480
let's start with the case where you know

314
00:10:56,320 --> 00:11:02,440
the future. Sorry.

315
00:10:58,480 --> 00:11:04,000
And let's do the comparison uh

316
00:11:02,440 --> 00:11:05,920
Let's try to understand how do we move

317
00:11:04,000 --> 00:11:07,679
flows, how do we value flows at

318
00:11:05,919 --> 00:11:10,319
different points in time.

319
00:11:07,679 --> 00:11:11,759
This is the thing is think first about

320
00:11:10,320 --> 00:11:13,400
comparing an asset that gives you a

321
00:11:11,759 --> 00:11:16,120
dollar in the future,

322
00:11:13,399 --> 00:11:17,519
how much do you think it's worth today?

323
00:11:16,120 --> 00:11:19,759
Well, the easiest

324
00:11:17,519 --> 00:11:21,879
way to get to that value is is to think

325
00:11:19,759 --> 00:11:25,960
on the alternatives. As suppose I have a

326
00:11:21,879 --> 00:11:25,960
dollar today, what can I do with it?

327
00:11:26,200 --> 00:11:31,080
Well, in terms of investment.

328
00:11:28,759 --> 00:11:33,120
Well, suppose that you have available

329
00:11:31,080 --> 00:11:34,879
one-year bonds, treasury bonds, and that

330
00:11:33,120 --> 00:11:36,560
the interest rate is I

331
00:11:34,879 --> 00:11:38,039
t. That's the interest rate on an I

332
00:11:36,559 --> 00:11:40,919
one-year bond.

333
00:11:38,039 --> 00:11:42,399
So, if you want if you if you

334
00:11:40,919 --> 00:11:44,079
have a dollar,

335
00:11:42,399 --> 00:11:46,199
you have the option to invest it in that

336
00:11:44,080 --> 00:11:48,960
asset, in that bond, which give will

337
00:11:46,200 --> 00:11:49,720
give you 1 + I dollars

338
00:11:48,960 --> 00:11:51,720
uh

339
00:11:49,720 --> 00:11:52,879
next year.

340
00:11:51,720 --> 00:11:57,639
Well,

341
00:11:52,879 --> 00:12:01,799
that means that I can get $1 next year

342
00:11:57,639 --> 00:12:02,480
by investing 1 over 1 + I dollars today.

343
00:12:01,799 --> 00:12:05,279
No?

344
00:12:02,480 --> 00:12:08,000
Because if if I invest 1 + 1

345
00:12:05,279 --> 00:12:11,720
rather than $1, I invest 1 over 1 + I

346
00:12:08,000 --> 00:12:14,279
today, then I multiply this by 1 + I

347
00:12:11,720 --> 00:12:16,200
and I get my dollar in the future.

348
00:12:14,279 --> 00:12:19,360
So, that tells me that say the interest

349
00:12:16,200 --> 00:12:22,600
rate is 10%, then with $1 today I can

350
00:12:19,360 --> 00:12:25,600
get 1.1 dollars in the future.

351
00:12:22,600 --> 00:12:27,240
That means that investing 90% 90 cents

352
00:12:25,600 --> 00:12:29,519
today, more or less,

353
00:12:27,240 --> 00:12:31,560
I can get $1 in the future.

354
00:12:29,519 --> 00:12:34,759
That tells me that a dollar in the

355
00:12:31,559 --> 00:12:36,959
future is equivalent to 90 cents today.

356
00:12:34,759 --> 00:12:38,639
That's the assumption. Okay?

357
00:12:36,960 --> 00:12:40,320
So, that's the reason when I told you

358
00:12:38,639 --> 00:12:42,799
the deal of me, look, I have an asset

359
00:12:40,320 --> 00:12:44,400
that cost cost you a dollar, but gives

360
00:12:42,799 --> 00:12:45,759
you a dollar in the future, well, that's

361
00:12:44,399 --> 00:12:47,480
not a good deal if the interest rate is

362
00:12:45,759 --> 00:12:49,559
positive.

363
00:12:47,480 --> 00:12:51,639
If the interest rate is 10%, then then a

364
00:12:49,559 --> 00:12:54,039
right a fair comparison is 90 cents with

365
00:12:51,639 --> 00:12:56,240
$1, not $1 with $1.

366
00:12:54,039 --> 00:12:58,199
Okay? So, that's the discounting of the

367
00:12:56,240 --> 00:13:00,159
future. You can The most obvious way of

368
00:12:58,200 --> 00:13:02,920
discounting the future

369
00:13:00,159 --> 00:13:04,240
is to discount it by the interest rate.

370
00:13:02,919 --> 00:13:06,399
Uh

371
00:13:04,240 --> 00:13:08,279
which interest rate to pick? That's more

372
00:13:06,399 --> 00:13:09,959
subtle. That depends on risk, depends on

373
00:13:08,279 --> 00:13:12,480
many other things which we're going to

374
00:13:09,960 --> 00:13:15,320
discuss to some extent here. But for

375
00:13:12,480 --> 00:13:16,440
now, let's make it very simple. And in a

376
00:13:15,320 --> 00:13:17,879
world in which you really know the

377
00:13:16,440 --> 00:13:19,400
future, really the right interest rate

378
00:13:17,879 --> 00:13:21,759
to use is the safe interest rate, the

379
00:13:19,399 --> 00:13:23,159
interest rate of of treasury bonds and

380
00:13:21,759 --> 00:13:25,799
things like that.

381
00:13:23,159 --> 00:13:27,799
Okay? So, that's that's that.

382
00:13:25,799 --> 00:13:29,439
What about a dollar that you receive

383
00:13:27,799 --> 00:13:31,319
What about if you're thinking about what

384
00:13:29,440 --> 00:13:32,880
is the value of a dollar two years from

385
00:13:31,320 --> 00:13:34,080
now?

386
00:13:32,879 --> 00:13:35,480
Well,

387
00:13:34,080 --> 00:13:38,000
you know, if I get a dollar to I can do

388
00:13:35,480 --> 00:13:39,600
the same logic. If I if I

389
00:13:38,000 --> 00:13:41,159
I can use the same logic. If I get a

390
00:13:39,600 --> 00:13:45,960
dollar today,

391
00:13:41,159 --> 00:13:48,759
I can convert that into 1 + I t * 1 + I

392
00:13:45,960 --> 00:13:52,280
t + 1 dollars. Okay?

393
00:13:48,759 --> 00:13:55,039
So, say 10% and 10%, I get 1.1 next year

394
00:13:52,279 --> 00:13:57,000
and then I get 1.1 * 1.1, 1.21 or

395
00:13:55,039 --> 00:13:58,559
something like that. Okay?

396
00:13:57,000 --> 00:13:59,639
That's my final

397
00:13:58,559 --> 00:14:01,479
result.

398
00:13:59,639 --> 00:14:03,838
So,

399
00:14:01,480 --> 00:14:05,800
well, then how much is it worth to have

400
00:14:03,839 --> 00:14:09,120
a dollar, an asset that gives you a

401
00:14:05,799 --> 00:14:11,240
dollar two years from now?

402
00:14:09,120 --> 00:14:13,519
Well, it's going to be that dollar

403
00:14:11,240 --> 00:14:14,600
divided by the product of these interest

404
00:14:13,519 --> 00:14:16,838
rates.

405
00:14:14,600 --> 00:14:18,519
Okay? Why is that? Well, because with

406
00:14:16,839 --> 00:14:20,640
this amount of

407
00:14:18,519 --> 00:14:22,519
dollars today,

408
00:14:20,639 --> 00:14:24,639
it's point 80 cents or something like

409
00:14:22,519 --> 00:14:25,958
that, I can generate a dollar two years

410
00:14:24,639 --> 00:14:27,679
from now.

411
00:14:25,958 --> 00:14:29,199
That means a dollar

412
00:14:27,679 --> 00:14:32,239
two years from now

413
00:14:29,200 --> 00:14:34,520
is worth about 80 cents today.

414
00:14:32,240 --> 00:14:34,519
Okay?

415
00:14:35,958 --> 00:14:38,919
We're going to use a lot this type of

416
00:14:37,360 --> 00:14:40,959
logic, so

417
00:14:38,919 --> 00:14:42,439
and and I know that that it may not be

418
00:14:40,958 --> 00:14:43,359
that intuitive the first time you see

419
00:14:42,440 --> 00:14:46,079
it, but

420
00:14:43,360 --> 00:14:46,079
ask questions.

421
00:14:47,919 --> 00:14:50,838
You want me to repeat it?

422
00:14:54,440 --> 00:14:57,400
Okay. The

423
00:14:55,759 --> 00:14:59,159
The final goal is the following. We're

424
00:14:57,399 --> 00:15:01,439
going to In the what comes next, we're

425
00:14:59,159 --> 00:15:03,039
going to see if which happens again with

426
00:15:01,440 --> 00:15:05,320
many decisions in life, but it perhaps

427
00:15:03,039 --> 00:15:07,159
particularly for financial assets,

428
00:15:05,320 --> 00:15:08,720
we're going to try to value something

429
00:15:07,159 --> 00:15:11,360
that

430
00:15:08,720 --> 00:15:13,680
whose payoff happens at different times

431
00:15:11,360 --> 00:15:16,000
in the future. And the question is

432
00:15:13,679 --> 00:15:19,599
how do I value an asset that pays me,

433
00:15:16,000 --> 00:15:21,919
you know, $5 one year from now, $25

434
00:15:19,600 --> 00:15:26,240
three years from now, uh

435
00:15:21,919 --> 00:15:27,519
minus $10 10 years from now, plus $50

436
00:15:26,240 --> 00:15:29,240
100 years from now?

437
00:15:27,519 --> 00:15:31,039
What is the value of that? Of having an

438
00:15:29,240 --> 00:15:33,360
asset like that?

439
00:15:31,039 --> 00:15:35,639
And so, I needed some method

440
00:15:33,360 --> 00:15:37,320
to bring it to today's value because

441
00:15:35,639 --> 00:15:38,600
today I have a meaning of what a dollar

442
00:15:37,320 --> 00:15:40,079
is, you know?

443
00:15:38,600 --> 00:15:42,839
And and therefore I can compare it with

444
00:15:40,078 --> 00:15:44,958
whatever price I mean

445
00:15:42,839 --> 00:15:47,800
people are asking me for that asset.

446
00:15:44,958 --> 00:15:49,799
So, what this is doing is is is that is

447
00:15:47,799 --> 00:15:51,799
doing that. It's telling you how to

448
00:15:49,799 --> 00:15:54,759
convert a dollar at different parts in

449
00:15:51,799 --> 00:15:56,759
the future into a dollar today.

450
00:15:54,759 --> 00:15:59,039
And by that logic,

451
00:15:56,759 --> 00:16:01,039
the recipe is well, use the interest

452
00:15:59,039 --> 00:16:02,559
rate because you could always go the

453
00:16:01,039 --> 00:16:03,879
other way around. You could always with

454
00:16:02,559 --> 00:16:06,159
a dollar you can ask a question, with a

455
00:16:03,879 --> 00:16:08,078
dollar today, how many dollars can I get

456
00:16:06,159 --> 00:16:09,039
two years from now, say?

457
00:16:08,078 --> 00:16:10,078
That.

458
00:16:09,039 --> 00:16:13,519
Well,

459
00:16:10,078 --> 00:16:15,838
say X. Well, then I need 1 over X. Then

460
00:16:13,519 --> 00:16:18,078
$1 there is worth 1 over X dollars

461
00:16:15,839 --> 00:16:21,400
today. You know, that's that's the logic

462
00:16:18,078 --> 00:16:23,838
because 1 over X * X is 1.

463
00:16:21,399 --> 00:16:23,838
So,

464
00:16:23,919 --> 00:16:26,919
that's too fast, probably.

465
00:16:28,639 --> 00:16:34,319
So,

466
00:16:29,839 --> 00:16:34,320
you know, with $1 today, oops,

467
00:16:35,480 --> 00:16:41,039
I can generate, say,

468
00:16:38,039 --> 00:16:41,039
$1.1

469
00:16:41,720 --> 00:16:44,160
at uh uh

470
00:16:44,399 --> 00:16:46,879
at t equal to

471
00:16:45,679 --> 00:16:48,559
Okay?

472
00:16:46,879 --> 00:16:51,279
Then I'm I'm The question I want to know

473
00:16:48,559 --> 00:16:53,439
is how much is a dollar worth

474
00:16:51,279 --> 00:16:56,879
How much is a dollar received at time t

475
00:16:53,440 --> 00:16:57,720
equal to worth today?

476
00:16:56,879 --> 00:16:58,838
That's the question I'm trying to

477
00:16:57,720 --> 00:17:00,399
answer.

478
00:16:58,839 --> 00:17:01,720
You know, because an asset will be

479
00:17:00,399 --> 00:17:03,958
something that will pay you in the

480
00:17:01,720 --> 00:17:08,199
future. So, I want to know how much is

481
00:17:03,958 --> 00:17:10,240
$1 received in the future worth today.

482
00:17:08,199 --> 00:17:11,640
And then the answer is

483
00:17:10,240 --> 00:17:13,799
well,

484
00:17:11,640 --> 00:17:18,160
then is I know the answer from this

485
00:17:13,799 --> 00:17:21,678
logic because I know that with one

486
00:17:18,160 --> 00:17:24,920
if I have 1 over 1.1 dollars today, I

487
00:17:21,679 --> 00:17:24,920
can convert it

488
00:17:25,359 --> 00:17:28,519
into one.

489
00:17:27,078 --> 00:17:31,039
How do I know that?

490
00:17:28,519 --> 00:17:31,039
Because

491
00:17:33,200 --> 00:17:36,880
1 over 1.1

492
00:17:38,679 --> 00:17:43,120
* 1.1

493
00:17:40,799 --> 00:17:45,319
is equal to 1.

494
00:17:43,119 --> 00:17:47,678
Okay? This if I invest these dollars

495
00:17:45,319 --> 00:17:50,039
today,

496
00:17:47,679 --> 00:17:51,280
I'm going to get this return on that.

497
00:17:50,039 --> 00:17:54,119
And the product of these two things

498
00:17:51,279 --> 00:17:54,119
gives me my dollar.

499
00:17:54,240 --> 00:17:56,599
Okay?

500
00:17:55,039 --> 00:17:58,359
So, if I tell you, do you prefer to have

501
00:17:56,599 --> 00:17:59,879
a dollar two days from two years from

502
00:17:58,359 --> 00:18:02,519
now or today?

503
00:17:59,880 --> 00:18:05,120
You say, I prefer it obviously prefer it

504
00:18:02,519 --> 00:18:07,759
today because I can get 1.1 dollars two

505
00:18:05,119 --> 00:18:09,239
years from now.

506
00:18:07,759 --> 00:18:10,839
But then then the more relevant question

507
00:18:09,240 --> 00:18:12,880
is, no, no, but then you do you prefer

508
00:18:10,839 --> 00:18:15,199
to have 90 cents today

509
00:18:12,880 --> 00:18:16,960
versus a dollar in the future? And then

510
00:18:15,200 --> 00:18:18,799
I'm I need to do my multiplication

511
00:18:16,960 --> 00:18:21,360
because I have to multiply the 90 cents

512
00:18:18,799 --> 00:18:23,200
by the 1.1 and see whether I get

513
00:18:21,359 --> 00:18:24,199
something comparable to a dollar or not.

514
00:18:23,200 --> 00:18:26,840
Okay?

515
00:18:24,200 --> 00:18:29,880
But that's that's the logic behind that.

516
00:18:26,839 --> 00:18:33,199
And And that's a So, the interest rate

517
00:18:29,880 --> 00:18:34,679
is what we discount the future by.

518
00:18:33,200 --> 00:18:36,120
And it's natural because if the interest

519
00:18:34,679 --> 00:18:37,880
rate is very high If the interest rate

520
00:18:36,119 --> 00:18:39,359
is zero, say,

521
00:18:37,880 --> 00:18:41,159
then a dollar received two years from

522
00:18:39,359 --> 00:18:42,158
now or a dollar received today is is the

523
00:18:41,159 --> 00:18:44,040
same

524
00:18:42,159 --> 00:18:45,360
because I can't If I invest a dollar

525
00:18:44,039 --> 00:18:46,559
today and the interest rate is zero, I'm

526
00:18:45,359 --> 00:18:47,879
going to get my dollar two years from

527
00:18:46,559 --> 00:18:49,119
now.

528
00:18:47,880 --> 00:18:51,200
If the dollar If the interest rate is

529
00:18:49,119 --> 00:18:52,918
50%, it makes a big difference receiving

530
00:18:51,200 --> 00:18:54,360
the dollar today versus receiving it two

531
00:18:52,919 --> 00:18:56,320
years from now.

532
00:18:54,359 --> 00:18:58,199
If you're in Argentina, the interest

533
00:18:56,319 --> 00:19:00,519
rate I don't know what it is. It's

534
00:18:58,200 --> 00:19:02,880
700%. It makes a huge difference whether

535
00:19:00,519 --> 00:19:05,720
you receive it, you know, one year from

536
00:19:02,880 --> 00:19:08,240
now than today.

537
00:19:05,720 --> 00:19:09,440
And and and uh

538
00:19:08,240 --> 00:19:10,440
So, that's that's the role of the

539
00:19:09,440 --> 00:19:12,000
interest rate. The higher is the

540
00:19:10,440 --> 00:19:13,360
interest rate,

541
00:19:12,000 --> 00:19:15,599
the less

542
00:19:13,359 --> 00:19:17,639
is a dollar received in the future worth

543
00:19:15,599 --> 00:19:19,199
relative to a dollar received today.

544
00:19:17,640 --> 00:19:21,320
Because you can get a much higher return

545
00:19:19,200 --> 00:19:22,559
from the dollar you have today

546
00:19:21,319 --> 00:19:24,039
if the interest rate is high. If the

547
00:19:22,559 --> 00:19:26,079
interest rate is low,

548
00:19:24,039 --> 00:19:28,319
you don't get that much. Okay?

549
00:19:26,079 --> 00:19:30,480
Much difference. Okay, good.

550
00:19:28,319 --> 00:19:31,759
So, this is a big principle. And and I I

551
00:19:30,480 --> 00:19:33,679
mean

552
00:19:31,759 --> 00:19:36,240
everything I'll say next builds on this

553
00:19:33,679 --> 00:19:36,240
logic.

554
00:19:38,640 --> 00:19:43,320
So, let me give you a general formula.

555
00:19:40,720 --> 00:19:44,799
So, let's ask what is the value

556
00:19:43,319 --> 00:19:46,119
of an asset

557
00:19:44,799 --> 00:19:49,319
that gives

558
00:19:46,119 --> 00:19:51,918
payouts of Z

559
00:19:49,319 --> 00:19:55,399
t dollars this year,

560
00:19:51,919 --> 00:19:57,160
Z t + 1 one year from now, ZT plus two,

561
00:19:55,400 --> 00:20:01,960
two years from now, and so on and so

562
00:19:57,160 --> 00:20:03,279
forth for N periods more. Okay?

563
00:20:01,960 --> 00:20:04,880
Well,

564
00:20:03,279 --> 00:20:06,879
I just need to do several of these

565
00:20:04,880 --> 00:20:09,120
operations. I know that the dollar

566
00:20:06,880 --> 00:20:10,760
received this year is is worth a dollar.

567
00:20:09,119 --> 00:20:13,519
Okay? That's ZT.

568
00:20:10,759 --> 00:20:14,720
A dollar received one year from now

569
00:20:13,519 --> 00:20:16,720
is not

570
00:20:14,720 --> 00:20:18,920
is not the same as a dollar received

571
00:20:16,720 --> 00:20:22,079
today. It's the same as one over one

572
00:20:18,920 --> 00:20:23,920
plus IT dollars received today.

573
00:20:22,079 --> 00:20:26,519
So, that cash flow I'm going to receive

574
00:20:23,920 --> 00:20:28,120
from this asset is worth this amount.

575
00:20:26,519 --> 00:20:30,839
For a two something that I receive two

576
00:20:28,119 --> 00:20:32,759
years from now, then it's not

577
00:20:30,839 --> 00:20:34,679
it's not certain, it's much less than

578
00:20:32,759 --> 00:20:38,200
receiving a dollar today. It's going to

579
00:20:34,680 --> 00:20:39,120
be one over one plus IT one plus IT plus

580
00:20:38,200 --> 00:20:40,559
one.

581
00:20:39,119 --> 00:20:42,159
And that I have to multiply by the

582
00:20:40,559 --> 00:20:46,000
number of dollars I will receive two

583
00:20:42,160 --> 00:20:48,320
years from now. Okay? And I keep going.

584
00:20:46,000 --> 00:20:51,200
So, that's that's the

585
00:20:48,319 --> 00:20:53,559
the present value. Present discounted

586
00:20:51,200 --> 00:20:56,559
value. Present because I'm bringing all

587
00:20:53,559 --> 00:20:57,799
these future cash flows to the present.

588
00:20:56,559 --> 00:21:00,440
That's what each of these terms is

589
00:20:57,799 --> 00:21:01,680
doing. The one over that is bringing it

590
00:21:00,440 --> 00:21:03,640
to the present.

591
00:21:01,680 --> 00:21:05,160
Discounted because the interest rate is

592
00:21:03,640 --> 00:21:06,400
discounting things. It's making them a

593
00:21:05,160 --> 00:21:08,600
smaller.

594
00:21:06,400 --> 00:21:12,560
And value because I'm trying to reduce

595
00:21:08,599 --> 00:21:14,199
them to the current value. Okay?

596
00:21:12,559 --> 00:21:15,919
That's the general formula. So, it's a

597
00:21:14,200 --> 00:21:18,279
formula you need to understand.

598
00:21:15,920 --> 00:21:21,720
It's just So, that that was an asset

599
00:21:18,279 --> 00:21:23,920
that gives you Z dollars today,

600
00:21:21,720 --> 00:21:26,920
ZT plus one, one year from now, so you

601
00:21:23,920 --> 00:21:29,160
use this formula. ZT plus two, two years

602
00:21:26,920 --> 00:21:32,360
from now, so you use this formula, and

603
00:21:29,160 --> 00:21:32,360
then you keep going. Okay?

604
00:21:33,200 --> 00:21:37,480
What if we don't know the future?

605
00:21:35,640 --> 00:21:39,080
You know, I have to remove the expected

606
00:21:37,480 --> 00:21:40,039
part.

607
00:21:39,079 --> 00:21:41,679
Well,

608
00:21:40,039 --> 00:21:43,960
if we don't know the future, then the

609
00:21:41,680 --> 00:21:45,080
best we can do, in fact, we do fancier

610
00:21:43,960 --> 00:21:47,120
things, but that's what we're going to

611
00:21:45,079 --> 00:21:50,199
all that we'll do in this course.

612
00:21:47,119 --> 00:21:52,759
Uh all that you can do is just replace

613
00:21:50,200 --> 00:21:54,400
the known quantities we have here

614
00:21:52,759 --> 00:21:56,759
for the expectations.

615
00:21:54,400 --> 00:21:58,800
Okay? So, that's the closest. So, you

616
00:21:56,759 --> 00:22:00,079
know, I know ZT, that's the cash flow I

617
00:21:58,799 --> 00:22:02,079
get now,

618
00:22:00,079 --> 00:22:04,119
but I don't know ZT plus one. So, I can

619
00:22:02,079 --> 00:22:05,519
replace it by expectation.

620
00:22:04,119 --> 00:22:08,239
I do know the interest rate on a

621
00:22:05,519 --> 00:22:09,319
one-year bond from today to one year.

622
00:22:08,240 --> 00:22:10,880
So, that's the reason I don't need an

623
00:22:09,319 --> 00:22:12,759
expectation here.

624
00:22:10,880 --> 00:22:14,520
But I don't know what the one-year rate

625
00:22:12,759 --> 00:22:17,160
will be one year from now. So, that's

626
00:22:14,519 --> 00:22:18,400
the reason I need an expectation there.

627
00:22:17,160 --> 00:22:20,160
And so on.

628
00:22:18,400 --> 00:22:21,360
And I don't know what the cash flow will

629
00:22:20,160 --> 00:22:22,920
be two years from now. I have an

630
00:22:21,359 --> 00:22:24,639
expectation about what the cash flow

631
00:22:22,920 --> 00:22:26,960
will be, but I don't know it.

632
00:22:24,640 --> 00:22:29,960
So, I have an expectation there. Okay?

633
00:22:26,960 --> 00:22:31,000
So, so all that I've done here is say,

634
00:22:29,960 --> 00:22:32,279
"Okay,

635
00:22:31,000 --> 00:22:33,759
I acknowledge that this guy knew a

636
00:22:32,279 --> 00:22:35,399
little bit too much. You know, he knew

637
00:22:33,759 --> 00:22:37,279
exactly what the cash flows were going

638
00:22:35,400 --> 00:22:38,880
to be in the future, and he knew what

639
00:22:37,279 --> 00:22:40,559
the one-year rates were going to be in

640
00:22:38,880 --> 00:22:42,440
the future."

641
00:22:40,559 --> 00:22:44,039
This guy here knows less. He knows the

642
00:22:42,440 --> 00:22:46,360
cash flow today. He knows the interest

643
00:22:44,039 --> 00:22:48,279
rate today, but he doesn't know the cash

644
00:22:46,359 --> 00:22:50,519
flows. Really, he has a hunch, but he

645
00:22:48,279 --> 00:22:51,920
doesn't know the cash flows one year,

646
00:22:50,519 --> 00:22:53,519
two years, three years, and so on for

647
00:22:51,920 --> 00:22:56,560
the future, and he doesn't know the

648
00:22:53,519 --> 00:22:58,400
one-year interest rate in the future.

649
00:22:56,559 --> 00:23:00,159
So, all these expectations, here's

650
00:22:58,400 --> 00:23:02,640
important the concept of time. This is

651
00:23:00,160 --> 00:23:04,160
an expectation as of time T. At time T,

652
00:23:02,640 --> 00:23:07,000
you have some information and you make

653
00:23:04,160 --> 00:23:08,640
forecast about the future. Okay?

654
00:23:07,000 --> 00:23:10,160
Use whatever you want, machine learning,

655
00:23:08,640 --> 00:23:11,400
whatever, but you have information at

656
00:23:10,160 --> 00:23:13,080
time T,

657
00:23:11,400 --> 00:23:14,800
and then you have a forecast for the

658
00:23:13,079 --> 00:23:16,079
future. At T plus one, you have you'll

659
00:23:14,799 --> 00:23:17,399
have more information, so you make

660
00:23:16,079 --> 00:23:18,199
another forecast, and so on and so

661
00:23:17,400 --> 00:23:20,360
forth.

662
00:23:18,200 --> 00:23:23,160
But in this we're valuing an asset at

663
00:23:20,359 --> 00:23:26,240
time T, then all these expectations are

664
00:23:23,160 --> 00:23:28,679
taken as of time T. That means given the

665
00:23:26,240 --> 00:23:30,120
information you have available at time

666
00:23:28,679 --> 00:23:31,280
T.

667
00:23:30,119 --> 00:23:33,000
That's the reason these guys don't have

668
00:23:31,279 --> 00:23:36,559
expectations in front of them because

669
00:23:33,000 --> 00:23:36,559
you know this at time T.

670
00:23:36,599 --> 00:23:40,079
Had we taken the value at T minus one,

671
00:23:38,679 --> 00:23:41,800
we would have not known that, and then

672
00:23:40,079 --> 00:23:43,799
we would have had to expectation because

673
00:23:41,799 --> 00:23:46,480
it would have been expectation as of T

674
00:23:43,799 --> 00:23:46,480
minus one.

675
00:23:46,640 --> 00:23:49,720
Okay, so that's your big formula there.

676
00:23:48,960 --> 00:23:51,799
So,

677
00:23:49,720 --> 00:23:53,839
there are some examples that are sort of

678
00:23:51,799 --> 00:23:55,440
well known and and and

679
00:23:53,839 --> 00:23:56,599
and and

680
00:23:55,440 --> 00:23:58,640
and and

681
00:23:56,599 --> 00:24:00,959
So, let me let me show you. They have

682
00:23:58,640 --> 00:24:02,400
nicer expressions. So, that's that's an

683
00:24:00,960 --> 00:24:04,720
example

684
00:24:02,400 --> 00:24:05,960
of the valuation of of this the same

685
00:24:04,720 --> 00:24:08,400
asset,

686
00:24:05,960 --> 00:24:09,200
but when the interest rate is constant,

687
00:24:08,400 --> 00:24:10,880
then

688
00:24:09,200 --> 00:24:13,559
then obviously I don't need all these

689
00:24:10,880 --> 00:24:16,120
products in the denominator.

690
00:24:13,559 --> 00:24:18,599
I have a constant interest rate, then I

691
00:24:16,119 --> 00:24:20,119
just get powers of that interest rate.

692
00:24:18,599 --> 00:24:22,678
That's one in which you have constant

693
00:24:20,119 --> 00:24:23,839
payments. So, the interest rate may be

694
00:24:22,679 --> 00:24:26,440
different,

695
00:24:23,839 --> 00:24:27,399
but the payments are the same over time.

696
00:24:26,440 --> 00:24:29,000
Okay?

697
00:24:27,400 --> 00:24:30,519
So, that's that.

698
00:24:29,000 --> 00:24:32,559
So, those are two

699
00:24:30,519 --> 00:24:33,480
easy formulas. That's one in which you

700
00:24:32,559 --> 00:24:36,079
have

701
00:24:33,480 --> 00:24:37,720
both constant, the interest rate

702
00:24:36,079 --> 00:24:39,439
and the payment.

703
00:24:37,720 --> 00:24:40,839
Then you get a nice expression. That's

704
00:24:39,440 --> 00:24:44,200
just uh

705
00:24:40,839 --> 00:24:46,839
that. Okay? You'll recognize that.

706
00:24:44,200 --> 00:24:47,600
If if you have a constant

707
00:24:46,839 --> 00:24:50,119
uh

708
00:24:47,599 --> 00:24:52,159
constant interest rate here, you see

709
00:24:50,119 --> 00:24:54,559
that the value

710
00:24:52,160 --> 00:24:56,440
is is declining is a is a geometric

711
00:24:54,559 --> 00:24:58,240
series. You know? The value of a two

712
00:24:56,440 --> 00:25:00,720
years from now is a square

713
00:24:58,240 --> 00:25:02,519
of one over one plus some

714
00:25:00,720 --> 00:25:03,559
it's a square of a a number less than

715
00:25:02,519 --> 00:25:05,400
one.

716
00:25:03,559 --> 00:25:07,519
You know? One over one plus I is some

717
00:25:05,400 --> 00:25:09,120
number less than one. This is a square

718
00:25:07,519 --> 00:25:10,519
of that, then the cube, and so on. So,

719
00:25:09,119 --> 00:25:12,319
it's a geometric series that is

720
00:25:10,519 --> 00:25:14,480
declining at the rate one plus I, one

721
00:25:12,319 --> 00:25:16,039
over one plus I. Okay? Or declining at

722
00:25:14,480 --> 00:25:18,200
the rate one plus I.

723
00:25:16,039 --> 00:25:19,440
So, that's your geometric series.

724
00:25:18,200 --> 00:25:22,759
Okay?

725
00:25:19,440 --> 00:25:25,920
That's the value of that.

726
00:25:22,759 --> 00:25:27,960
Constant rate and payment forever.

727
00:25:25,920 --> 00:25:28,840
Suppose you have an asset that

728
00:25:27,960 --> 00:25:31,120
it

729
00:25:28,839 --> 00:25:32,559
that lives forever.

730
00:25:31,119 --> 00:25:36,799
There are some bonds like that called

731
00:25:32,559 --> 00:25:38,919
perpetuities. Uh uh

732
00:25:36,799 --> 00:25:40,159
The US hasn't issued one, but the UK

733
00:25:38,920 --> 00:25:40,840
has, and so on.

734
00:25:40,160 --> 00:25:42,440
I

735
00:25:40,839 --> 00:25:44,919
So, that's an asset, for example, that

736
00:25:42,440 --> 00:25:46,880
pays you a fixed amount

737
00:25:44,920 --> 00:25:48,840
forever. And if the interest rate is

738
00:25:46,880 --> 00:25:50,760
constant, that's the trickier thing,

739
00:25:48,839 --> 00:25:53,399
then the value of that asset you can see

740
00:25:50,759 --> 00:25:55,480
that this this is going to zero.

741
00:25:53,400 --> 00:25:57,040
So, the value of that asset

742
00:25:55,480 --> 00:25:59,440
is that.

743
00:25:57,039 --> 00:26:02,079
And actually a formula that you may see

744
00:25:59,440 --> 00:26:04,840
that is very oftenly used as as a first

745
00:26:02,079 --> 00:26:07,559
approximation is this one. This is

746
00:26:04,839 --> 00:26:09,879
is is the same asset, but it's called

747
00:26:07,559 --> 00:26:12,000
ex-dividend or ex-coupon. It's it's

748
00:26:09,880 --> 00:26:13,440
after the coupon of this year has been

749
00:26:12,000 --> 00:26:15,240
paid.

750
00:26:13,440 --> 00:26:17,480
Okay? So, it's an asset that starts

751
00:26:15,240 --> 00:26:19,120
paying at T plus one. It's ZT plus one,

752
00:26:17,480 --> 00:26:20,599
ZT plus two, and so on.

753
00:26:19,119 --> 00:26:22,199
Well, that

754
00:26:20,599 --> 00:26:25,799
is the same as this minus the first

755
00:26:22,200 --> 00:26:28,120
coupon, so is equal to that.

756
00:26:25,799 --> 00:26:28,119
Okay?

757
00:26:29,119 --> 00:26:32,159
That's an interesting thing, huh? Look,

758
00:26:31,039 --> 00:26:35,200
what happened to this asset as the

759
00:26:32,160 --> 00:26:35,200
interest rate goes to zero?

760
00:26:36,000 --> 00:26:39,799
So, this is an asset that lasts for a

761
00:26:37,440 --> 00:26:41,759
very long time.

762
00:26:39,799 --> 00:26:43,678
And and and look, we got to a valuation

763
00:26:41,759 --> 00:26:45,079
formula.

764
00:26:43,679 --> 00:26:46,960
What hap- what is happening as the

765
00:26:45,079 --> 00:26:49,159
interest rate goes to zero?

766
00:26:46,960 --> 00:26:51,319
To the value.

767
00:26:49,160 --> 00:26:53,400
Very large. It goes to infinity.

768
00:26:51,319 --> 00:26:55,279
And a lot of what has happened in in

769
00:26:53,400 --> 00:26:57,759
global financial markets

770
00:26:55,279 --> 00:26:58,759
in the last few years has to do with

771
00:26:57,759 --> 00:27:01,279
that.

772
00:26:58,759 --> 00:27:02,720
Interest rates were very very very low.

773
00:27:01,279 --> 00:27:05,799
And so, most assets that had long

774
00:27:02,720 --> 00:27:07,240
duration had very high values.

775
00:27:05,799 --> 00:27:09,039
Okay?

776
00:27:07,240 --> 00:27:10,880
And it has a lot to that. Monetary

777
00:27:09,039 --> 00:27:12,599
policy had a lot to do

778
00:27:10,880 --> 00:27:13,720
whether it was the right monetary policy

779
00:27:12,599 --> 00:27:15,519
or not,

780
00:27:13,720 --> 00:27:16,720
that's something to be discussed. I

781
00:27:15,519 --> 00:27:17,920
think on average it was the right

782
00:27:16,720 --> 00:27:20,440
monetary policy, but one of the things

783
00:27:17,920 --> 00:27:22,160
it did, it increased the value of many

784
00:27:20,440 --> 00:27:24,200
assets. In fact, that's one of the

785
00:27:22,160 --> 00:27:25,720
mechanisms through which monetary policy

786
00:27:24,200 --> 00:27:27,200
works in practice. It's not something we

787
00:27:25,720 --> 00:27:29,319
have discussed, but you can begin to see

788
00:27:27,200 --> 00:27:31,200
here. Because if the value of all assets

789
00:27:29,319 --> 00:27:32,720
go up a lot, people feel wealthier, and

790
00:27:31,200 --> 00:27:34,480
that they will tend to consume more, and

791
00:27:32,720 --> 00:27:36,480
so on. Well, this is one of the channels

792
00:27:34,480 --> 00:27:38,519
monetary policy does. By the way, this

793
00:27:36,480 --> 00:27:40,960
effect happens also to this asset that

794
00:27:38,519 --> 00:27:41,720
has finite N. It's just that this goes

795
00:27:40,960 --> 00:27:43,559
is

796
00:27:41,720 --> 00:27:44,920
it's maximized when this asset lasts

797
00:27:43,559 --> 00:27:46,519
forever. You know?

798
00:27:44,920 --> 00:27:47,840
This this asset literally goes to

799
00:27:46,519 --> 00:27:51,759
infinity

800
00:27:47,839 --> 00:27:54,480
if the interest rate goes to zero.

801
00:27:51,759 --> 00:27:56,640
Well, if an asset lasts for N periods,

802
00:27:54,480 --> 00:27:58,360
it doesn't go to infinity. It goes to N

803
00:27:56,640 --> 00:27:59,960
times Z.

804
00:27:58,359 --> 00:28:01,479
You know? It's the sum.

805
00:27:59,960 --> 00:28:03,200
If the interest rate is zero, you just

806
00:28:01,480 --> 00:28:04,640
sum things.

807
00:28:03,200 --> 00:28:07,679
See that?

808
00:28:04,640 --> 00:28:09,880
If I if if an asset lasts for N periods,

809
00:28:07,679 --> 00:28:11,720
and it gives me a payment of Z in every

810
00:28:09,880 --> 00:28:13,120
single period,

811
00:28:11,720 --> 00:28:15,759
then when the interest is zero, that

812
00:28:13,119 --> 00:28:18,319
asset is worth N times Z.

813
00:28:15,759 --> 00:28:19,720
Because I will receive Z coupons.

814
00:28:18,319 --> 00:28:21,639
And I don't discount the future because

815
00:28:19,720 --> 00:28:23,319
the interest rate is zero.

816
00:28:21,640 --> 00:28:25,600
What happens is when the asset lasts

817
00:28:23,319 --> 00:28:27,519
forever, then N times Z is a really

818
00:28:25,599 --> 00:28:29,039
large number, you know? And that's

819
00:28:27,519 --> 00:28:31,279
that's what this expression captures

820
00:28:29,039 --> 00:28:31,279
here.

821
00:28:31,799 --> 00:28:36,079
Okay.

822
00:28:34,319 --> 00:28:39,200
So, let's talk about bonds now. We're

823
00:28:36,079 --> 00:28:42,960
going to start pricing bonds.

824
00:28:39,200 --> 00:28:45,000
Well, so bonds differ uh uh uh

825
00:28:42,960 --> 00:28:46,840
along many dimensions, but one of them

826
00:28:45,000 --> 00:28:49,799
is is very important for bonds is

827
00:28:46,839 --> 00:28:52,480
maturity, the N that I had there

828
00:28:49,799 --> 00:28:55,720
in the previous expression. Okay?

829
00:28:52,480 --> 00:28:57,599
Uh so, so maturity means essentially how

830
00:28:55,720 --> 00:28:59,880
long the bond lasts. Okay? When when

831
00:28:57,599 --> 00:29:02,000
does it pay you back the principal? The

832
00:28:59,880 --> 00:29:03,760
bonds typically pay coupons, and then

833
00:29:02,000 --> 00:29:05,279
there's a final payment, which we call

834
00:29:03,759 --> 00:29:08,759
face value of the bond or something like

835
00:29:05,279 --> 00:29:10,399
that. And and when that final payment

836
00:29:08,759 --> 00:29:12,400
takes place, that's the maturity of a

837
00:29:10,400 --> 00:29:14,120
bond. Okay?

838
00:29:12,400 --> 00:29:15,960
So, a bond that promises to make a

839
00:29:14,119 --> 00:29:17,000
thousand-dollar final payment in six

840
00:29:15,960 --> 00:29:20,400
months

841
00:29:17,000 --> 00:29:20,400
has a maturity of six months.

842
00:29:20,919 --> 00:29:24,159
A bond that promised to pay a hundred

843
00:29:22,679 --> 00:29:26,560
dollars for twenty years and then one

844
00:29:24,159 --> 00:29:28,960
thousand dollars final payment in twenty

845
00:29:26,559 --> 00:29:30,399
years has a maturity of twenty years.

846
00:29:28,960 --> 00:29:31,640
Maturity is different from duration. I

847
00:29:30,400 --> 00:29:33,759
don't think I'm going to talk about

848
00:29:31,640 --> 00:29:36,320
duration here, but but that's maturity.

849
00:29:33,759 --> 00:29:38,200
Just when the when is the final payment

850
00:29:36,319 --> 00:29:39,519
of of a

851
00:29:38,200 --> 00:29:42,400
of a loan.

852
00:29:39,519 --> 00:29:44,359
Of a of a bond. Okay?

853
00:29:42,400 --> 00:29:45,720
Bonds of different maturities each have

854
00:29:44,359 --> 00:29:47,039
a price

855
00:29:45,720 --> 00:29:48,799
and an associated interest rate. We're

856
00:29:47,039 --> 00:29:51,119
going to look at those things.

857
00:29:48,799 --> 00:29:53,159
And the associated interest rate is

858
00:29:51,119 --> 00:29:55,759
called the yield to maturity, or simply

859
00:29:53,160 --> 00:29:57,279
the yield of a bond.

860
00:29:55,759 --> 00:30:00,319
This is terminology, but we're going to

861
00:29:57,279 --> 00:30:02,319
calculate these things later on.

862
00:30:00,319 --> 00:30:03,879
The The relationship between maturity

863
00:30:02,319 --> 00:30:05,759
and yield

864
00:30:03,880 --> 00:30:07,560
is called the yield curve. Very

865
00:30:05,759 --> 00:30:09,519
important concept. Big fuss about the

866
00:30:07,559 --> 00:30:11,799
yield curve these days.

867
00:30:09,519 --> 00:30:13,240
Talk a little bit more about that.

868
00:30:11,799 --> 00:30:15,240
Or sometimes it's called the term

869
00:30:13,240 --> 00:30:17,759
structure of interest rate.

870
00:30:15,240 --> 00:30:19,680
Term, in the language of bonds, is

871
00:30:17,759 --> 00:30:21,640
really maturity.

872
00:30:19,680 --> 00:30:23,880
So, term structure of interest rate

873
00:30:21,640 --> 00:30:26,520
really tells you what is the yield in a

874
00:30:23,880 --> 00:30:29,360
1-year bond, 2-year bond, 3-year bond, 4

875
00:30:26,519 --> 00:30:30,519
5 6 so on. You plot them, and that gives

876
00:30:29,359 --> 00:30:31,639
you a curve.

877
00:30:30,519 --> 00:30:33,039
Okay.

878
00:30:31,640 --> 00:30:34,880
So,

879
00:30:33,039 --> 00:30:36,279
uh for example,

880
00:30:34,880 --> 00:30:38,640
look at the those These are two

881
00:30:36,279 --> 00:30:40,039
different yield curves. This is November

882
00:30:38,640 --> 00:30:42,560
2000,

883
00:30:40,039 --> 00:30:43,879
and this is June 20

884
00:30:42,559 --> 00:30:47,079
2001.

885
00:30:43,880 --> 00:30:49,240
So, this tells you what the yield is

886
00:30:47,079 --> 00:30:51,399
in on a 3-month bond, so a bond that

887
00:30:49,240 --> 00:30:53,240
matures in three in 3 months, on a

888
00:30:51,400 --> 00:30:56,000
6-month bonds and so forth, up to

889
00:30:53,240 --> 00:30:57,519
30-year bonds. Okay.

890
00:30:56,000 --> 00:30:58,680
What is the big difference between these

891
00:30:57,519 --> 00:31:00,879
What do you think happened here in

892
00:30:58,680 --> 00:31:02,799
between? Notice that these two curves

893
00:31:00,880 --> 00:31:04,360
are more or less the same long-term

894
00:31:02,799 --> 00:31:06,039
interest rate.

895
00:31:04,359 --> 00:31:08,000
But they have very different This curve

896
00:31:06,039 --> 00:31:12,039
This is a very steep curve, and this is

897
00:31:08,000 --> 00:31:12,039
a very flat or even inverted curve.

898
00:31:12,839 --> 00:31:16,359
What do you think may have happened

899
00:31:14,319 --> 00:31:20,119
there?

900
00:31:16,359 --> 00:31:22,399
Between November 2000 and June 2001.

901
00:31:20,119 --> 00:31:25,319
People changed their expectations then.

902
00:31:22,400 --> 00:31:28,600
Yeah, it's That's true. That's for sure

903
00:31:25,319 --> 00:31:30,319
true about that. But look also that But

904
00:31:28,599 --> 00:31:32,159
that that that this 3-month There is

905
00:31:30,319 --> 00:31:34,359
very little uncertainty about 3 months.

906
00:31:32,160 --> 00:31:35,800
It was a lot lower than that.

907
00:31:34,359 --> 00:31:37,240
So, yes, people changed their

908
00:31:35,799 --> 00:31:40,240
expectation, but why do you think they

909
00:31:37,240 --> 00:31:40,240
changed their expectation?

910
00:31:42,319 --> 00:31:46,159
Well, it's rising inflation. We have a

911
00:31:44,799 --> 00:31:47,559
lot

912
00:31:46,160 --> 00:31:48,880
Rising inflation from here to here.

913
00:31:47,559 --> 00:31:50,519
These are These are nominal interest

914
00:31:48,880 --> 00:31:51,800
rates.

915
00:31:50,519 --> 00:31:54,759
Up to now I've been talking about

916
00:31:51,799 --> 00:31:54,759
nominal interest rate.

917
00:31:56,679 --> 00:32:00,519
What happens here

918
00:31:58,519 --> 00:32:03,319
is there was a mini recession.

919
00:32:00,519 --> 00:32:05,319
So, the Fed cut interest rate.

920
00:32:03,319 --> 00:32:08,000
When you're in recessions, the curve

921
00:32:05,319 --> 00:32:09,039
tend to look like this.

922
00:32:08,000 --> 00:32:10,519
Because

923
00:32:09,039 --> 00:32:12,480
the central bank is cutting interest

924
00:32:10,519 --> 00:32:14,160
rates in the in the short run to deal

925
00:32:12,480 --> 00:32:15,640
with the current recession.

926
00:32:14,160 --> 00:32:16,880
What happens 30 years from now has

927
00:32:15,640 --> 00:32:18,400
nothing to do with the business cycle

928
00:32:16,880 --> 00:32:20,400
today, so that interest rate doesn't

929
00:32:18,400 --> 00:32:22,440
need to move a lot. But the Fed is

930
00:32:20,400 --> 00:32:24,640
bringing interest rate down a lot in the

931
00:32:22,440 --> 00:32:27,080
front end. Okay. So, that's the typical

932
00:32:24,640 --> 00:32:29,840
shape of a curve in a recession.

933
00:32:27,079 --> 00:32:31,159
That's the typical shape of a of a curve

934
00:32:29,839 --> 00:32:32,959
in the opposite situation where the

935
00:32:31,160 --> 00:32:34,679
inflation is too high and so on. Because

936
00:32:32,960 --> 00:32:36,480
what happens? The Fed is trying to The

937
00:32:34,679 --> 00:32:37,840
Fed really controls the very front end

938
00:32:36,480 --> 00:32:39,160
of the curve.

939
00:32:37,839 --> 00:32:40,720
That's what the Fed really control. The

940
00:32:39,160 --> 00:32:42,040
central bank in general, but the Fed.

941
00:32:40,720 --> 00:32:43,200
They control the very front end of the

942
00:32:42,039 --> 00:32:44,960
curve because they're setting the very

943
00:32:43,200 --> 00:32:46,200
short-term interest rate.

944
00:32:44,960 --> 00:32:47,880
So, this is a situation where they're

945
00:32:46,200 --> 00:32:48,840
tightening the monetary policy very

946
00:32:47,880 --> 00:32:51,120
tight.

947
00:32:48,839 --> 00:32:53,039
Because they are a situation of uh

948
00:32:51,119 --> 00:32:54,319
overheating in the economy. And in fact,

949
00:32:53,039 --> 00:32:55,759
they got too carried away. That's the

950
00:32:54,319 --> 00:32:57,000
reason they we ended up in a recession

951
00:32:55,759 --> 00:32:59,440
here.

952
00:32:57,000 --> 00:32:59,440
Okay.

953
00:33:00,200 --> 00:33:03,440
How do you think it looks today?

954
00:33:06,200 --> 00:33:09,679
That Do you think it looks more like

955
00:33:07,400 --> 00:33:13,360
this or more like that?

956
00:33:09,679 --> 00:33:13,360
Is inflation low or high today?

957
00:33:14,119 --> 00:33:17,199
High. I mean, that's a problem, you

958
00:33:15,720 --> 00:33:19,759
know? The Fed is trying to hike interest

959
00:33:17,200 --> 00:33:21,720
rate. Now, recently, because of the the

960
00:33:19,759 --> 00:33:23,119
mess in the banking sector, then the

961
00:33:21,720 --> 00:33:25,559
expectations of interest rate began to

962
00:33:23,119 --> 00:33:27,319
decline a little, but but but the

963
00:33:25,559 --> 00:33:28,399
situation was was very important. Here

964
00:33:27,319 --> 00:33:30,678
you are.

965
00:33:28,400 --> 00:33:34,040
That's The green line is today.

966
00:33:30,679 --> 00:33:35,519
Okay. So, it's very inverted.

967
00:33:34,039 --> 00:33:38,240
Okay.

968
00:33:35,519 --> 00:33:39,879
A year ago, it looked like that.

969
00:33:38,240 --> 00:33:41,799
So, you see the the long end hasn't

970
00:33:39,880 --> 00:33:43,640
changed much, but a year ago, there was

971
00:33:41,799 --> 00:33:46,440
no sense that the inflation was getting

972
00:33:43,640 --> 00:33:47,960
so much out of line.

973
00:33:46,440 --> 00:33:49,320
It happened a little later than that.

974
00:33:47,960 --> 00:33:51,360
There was some concern that interest

975
00:33:49,319 --> 00:33:53,599
rate would would rise,

976
00:33:51,359 --> 00:33:55,240
but but but now it's very clear the

977
00:33:53,599 --> 00:33:57,159
economy is overheating. And this I

978
00:33:55,240 --> 00:33:58,359
should have plotted you something for

979
00:33:57,160 --> 00:34:00,120
for

980
00:33:58,359 --> 00:34:03,319
a month ago. It would have been even

981
00:34:00,119 --> 00:34:05,199
steeper. Okay.

982
00:34:03,319 --> 00:34:06,879
Anyways, but that's because the Fed is

983
00:34:05,200 --> 00:34:08,559
trying to slow down the economy. It's

984
00:34:06,880 --> 00:34:12,320
hiking interest rates. That's the reason

985
00:34:08,559 --> 00:34:12,320
the curve is very very inverted today.

986
00:34:12,960 --> 00:34:16,599
So, let me let me calculate these rates.

987
00:34:14,760 --> 00:34:17,520
How do we go about it? So, the first

988
00:34:16,599 --> 00:34:19,480
thing we're going to do is we're going

989
00:34:17,519 --> 00:34:22,719
to use the expected present discounted

990
00:34:19,480 --> 00:34:24,240
value formula to calculate the price

991
00:34:22,719 --> 00:34:25,639
of a bond.

992
00:34:24,239 --> 00:34:27,559
And then we want to start

993
00:34:25,639 --> 00:34:29,239
doing it for different bonds,

994
00:34:27,559 --> 00:34:30,759
and we're going to construct uh the

995
00:34:29,239 --> 00:34:33,759
yield curve.

996
00:34:30,760 --> 00:34:35,760
So, suppose you have a bond that pays

997
00:34:33,760 --> 00:34:37,800
$100,

998
00:34:35,760 --> 00:34:39,679
nothing in between, $100 1 year from

999
00:34:37,800 --> 00:34:41,800
now. So, this is a bond with maturity

1000
00:34:39,679 --> 00:34:43,720
1-year maturity.

1001
00:34:41,800 --> 00:34:45,159
I'm going to call that bond with 1-year

1002
00:34:43,719 --> 00:34:47,158
maturity

1003
00:34:45,159 --> 00:34:50,200
P1 the price of a bond with a 1-year

1004
00:34:47,159 --> 00:34:51,480
maturity at time T, P1T.

1005
00:34:50,199 --> 00:34:53,480
Well, that's easy to calculate. It's

1006
00:34:51,480 --> 00:34:55,240
expected present discounted value for If

1007
00:34:53,480 --> 00:34:57,519
you have the interest rate, whatever you

1008
00:34:55,239 --> 00:34:59,279
say, 1-year interest rate, then I know

1009
00:34:57,519 --> 00:35:01,519
that the price of the bond is 100

1010
00:34:59,280 --> 00:35:04,400
divided by 1 plus the interest rate, the

1011
00:35:01,519 --> 00:35:06,039
1-year interest rate today.

1012
00:35:04,400 --> 00:35:07,720
Okay. That's the price. That's expected

1013
00:35:06,039 --> 00:35:08,759
discounted value. So, I tell you what

1014
00:35:07,719 --> 00:35:11,399
I'm showing you is the relationship

1015
00:35:08,760 --> 00:35:13,960
between interest rates and prices.

1016
00:35:11,400 --> 00:35:16,480
Okay. Our price of a bond. The price of

1017
00:35:13,960 --> 00:35:19,119
that bond is just 100

1018
00:35:16,480 --> 00:35:23,079
uh divided by 1 plus the 1 plus the

1019
00:35:19,119 --> 00:35:23,079
1-year interest rate today. Okay.

1020
00:35:23,719 --> 00:35:28,239
So, important observation is that the

1021
00:35:25,880 --> 00:35:29,559
price of a 1-year bond varies inversely

1022
00:35:28,239 --> 00:35:31,839
with the current

1023
00:35:29,559 --> 00:35:34,519
1-year nominal interest rate. This is

1024
00:35:31,840 --> 00:35:34,519
all nominal, huh?

1025
00:35:34,599 --> 00:35:38,519
Why is it an inverse relationship?

1026
00:35:36,719 --> 00:35:40,439
Why is it the price of a 1-year bond is

1027
00:35:38,519 --> 00:35:42,679
inversely related to the 1-year interest

1028
00:35:40,440 --> 00:35:42,679
rate?

1029
00:35:44,719 --> 00:35:48,000
In other words, I'm asking

1030
00:35:46,760 --> 00:35:49,680
what do you think happens to the price

1031
00:35:48,000 --> 00:35:50,840
as a nominal as a nominal interest rate

1032
00:35:49,679 --> 00:35:52,279
rises?

1033
00:35:50,840 --> 00:35:54,880
And why do you think that's what happens

1034
00:35:52,280 --> 00:35:54,880
to the price?

1035
00:35:55,039 --> 00:35:58,358
Well, the first question doesn't have a

1036
00:35:57,199 --> 00:35:59,839
I mean, it's very easy, you know, the

1037
00:35:58,358 --> 00:36:01,519
answer to the first question. What

1038
00:35:59,840 --> 00:36:03,559
happens if I goes up? Well, it's obvious

1039
00:36:01,519 --> 00:36:06,119
that this price comes down.

1040
00:36:03,559 --> 00:36:06,119
But why?

1041
00:36:08,599 --> 00:36:12,279
And and and I'm And you use the concept

1042
00:36:11,039 --> 00:36:13,440
we have developed here. Remember we

1043
00:36:12,280 --> 00:36:15,680
spent

1044
00:36:13,440 --> 00:36:19,679
like 20 minutes in one slide. Well, you

1045
00:36:15,679 --> 00:36:19,679
start the slide for that answer.

1046
00:36:22,840 --> 00:36:26,280
Hint.

1047
00:36:24,119 --> 00:36:28,719
This $100 you're not receiving today,

1048
00:36:26,280 --> 00:36:30,040
you're receiving a year from now.

1049
00:36:28,719 --> 00:36:31,719
What happens with a dollar received a

1050
00:36:30,039 --> 00:36:33,519
year from now?

1051
00:36:31,719 --> 00:36:35,480
What is the value of a year

1052
00:36:33,519 --> 00:36:38,440
dollar received 1 year from now when the

1053
00:36:35,480 --> 00:36:38,440
interest rate is high?

1054
00:36:39,039 --> 00:36:42,559
Slow, because, you know,

1055
00:36:41,119 --> 00:36:44,239
you'd much rather have the dollar today,

1056
00:36:42,559 --> 00:36:46,320
invest it, and get this big return on

1057
00:36:44,239 --> 00:36:48,358
the on on the dollar.

1058
00:36:46,320 --> 00:36:50,120
That means, naturally, a bond that is

1059
00:36:48,358 --> 00:36:51,519
paying you $100 tomorrow is going to be

1060
00:36:50,119 --> 00:36:53,079
worth less

1061
00:36:51,519 --> 00:36:54,480
when the interest rate is very high.

1062
00:36:53,079 --> 00:36:55,440
It's going to be worth less today when

1063
00:36:54,480 --> 00:36:57,119
the interest rate is very high. You'd

1064
00:36:55,440 --> 00:36:58,880
rather have the money today, invest it

1065
00:36:57,119 --> 00:37:00,400
in the in in the interest rate, and get

1066
00:36:58,880 --> 00:37:02,400
the interest rate.

1067
00:37:00,400 --> 00:37:05,039
And and uh

1068
00:37:02,400 --> 00:37:07,280
No, I need to invest 1 over 1 plus I1T

1069
00:37:05,039 --> 00:37:09,719
dollars to get $100. That's another way

1070
00:37:07,280 --> 00:37:09,720
of saying it.

1071
00:37:10,480 --> 00:37:15,199
What about with the bond that pays $100

1072
00:37:13,358 --> 00:37:17,759
in 2 years?

1073
00:37:15,199 --> 00:37:19,559
Well, I need to discount that by this,

1074
00:37:17,760 --> 00:37:21,080
which is a You know, it's a product of

1075
00:37:19,559 --> 00:37:23,079
the two interest rate. And since I don't

1076
00:37:21,079 --> 00:37:25,119
know what the 1-year rate

1077
00:37:23,079 --> 00:37:26,799
will be 1 year from now, I have to use

1078
00:37:25,119 --> 00:37:28,599
expectation here rather than the actual

1079
00:37:26,800 --> 00:37:30,080
rate. But look at the notation. I'm

1080
00:37:28,599 --> 00:37:31,639
calling

1081
00:37:30,079 --> 00:37:34,840
P2T,

1082
00:37:31,639 --> 00:37:36,960
dollar P2T, the price of a 2-year bond,

1083
00:37:34,840 --> 00:37:39,039
a bond with maturity of 2 years,

1084
00:37:36,960 --> 00:37:40,280
as of time T.

1085
00:37:39,039 --> 00:37:41,960
Okay.

1086
00:37:40,280 --> 00:37:44,720
And this is a bond that has no coupons.

1087
00:37:41,960 --> 00:37:47,840
So, yes, pays you $100

1088
00:37:44,719 --> 00:37:50,719
at the end of the 2 years.

1089
00:37:47,840 --> 00:37:53,840
Now, note Note that this price

1090
00:37:50,719 --> 00:37:56,599
is inversely related to both

1091
00:37:53,840 --> 00:37:59,000
the 1-year rate today

1092
00:37:56,599 --> 00:38:01,519
and the expectation of the 1-year rate 1

1093
00:37:59,000 --> 00:38:04,079
year from now.

1094
00:38:01,519 --> 00:38:05,719
If either one of these goes up,

1095
00:38:04,079 --> 00:38:08,319
the bond is worth less today. You

1096
00:38:05,719 --> 00:38:09,399
discount more a dollar received

1097
00:38:08,320 --> 00:38:10,680
uh

1098
00:38:09,400 --> 00:38:12,559
um

1099
00:38:10,679 --> 00:38:13,319
2 years from now. I don't care which

1100
00:38:12,559 --> 00:38:14,159
one.

1101
00:38:13,320 --> 00:38:16,240
You know,

1102
00:38:14,159 --> 00:38:19,079
either of them that goes up is is bad

1103
00:38:16,239 --> 00:38:21,279
news for the for the price of a bond.

1104
00:38:19,079 --> 00:38:21,279
Okay.

1105
00:38:24,239 --> 00:38:26,839
Is this clear?

1106
00:38:28,358 --> 00:38:31,119
So,

1107
00:38:29,440 --> 00:38:33,559
there's an alternative So, this is the

1108
00:38:31,119 --> 00:38:35,880
way you price a bond bonds using just

1109
00:38:33,559 --> 00:38:38,320
expected discounted value

1110
00:38:35,880 --> 00:38:40,240
uh approach. Now, it turns out that in

1111
00:38:38,320 --> 00:38:42,440
practice, a lot of the asset pricing is

1112
00:38:40,239 --> 00:38:45,479
done by arbitrage. Meaning, you you

1113
00:38:42,440 --> 00:38:47,039
compare different assets, and that that

1114
00:38:45,480 --> 00:38:48,960
have similar risk, they should give you

1115
00:38:47,039 --> 00:38:51,599
more or less the same return. That's

1116
00:38:48,960 --> 00:38:53,559
what you do. So, let me let me do this

1117
00:38:51,599 --> 00:38:56,079
arbitrage thing. Suppose you're

1118
00:38:53,559 --> 00:38:57,239
considering investing $1 for 1 year. So,

1119
00:38:56,079 --> 00:38:59,199
that's your decision. I'm going to

1120
00:38:57,239 --> 00:39:02,159
invest one I need I have a dollar, which

1121
00:38:59,199 --> 00:39:04,439
I want to invest for 1 year.

1122
00:39:02,159 --> 00:39:07,480
But I But I I have two options to do

1123
00:39:04,440 --> 00:39:08,720
that. I can invest a dollar in a 1-year

1124
00:39:07,480 --> 00:39:10,440
bond.

1125
00:39:08,719 --> 00:39:11,879
I know exactly what I'm going to get,

1126
00:39:10,440 --> 00:39:13,400
you know, in that bond.

1127
00:39:11,880 --> 00:39:16,079
Or

1128
00:39:13,400 --> 00:39:17,800
I can invest in a 2-year bond

1129
00:39:16,079 --> 00:39:18,799
and sell it at the end of the first

1130
00:39:17,800 --> 00:39:20,600
year.

1131
00:39:18,800 --> 00:39:23,480
That's Those are two ways of, you know,

1132
00:39:20,599 --> 00:39:24,880
investing for 1 year.

1133
00:39:23,480 --> 00:39:26,320
Arbitrage has to be compared over the

1134
00:39:24,880 --> 00:39:27,960
same period of time and everything. It's

1135
00:39:26,320 --> 00:39:30,039
not the return of a bond that you hold

1136
00:39:27,960 --> 00:39:31,559
for 10 years versus one that you hold

1137
00:39:30,039 --> 00:39:33,759
for a 1 year. It has to be something a

1138
00:39:31,559 --> 00:39:36,799
similar investment. Suppose I need to

1139
00:39:33,760 --> 00:39:39,760
invest for 1 year.

1140
00:39:36,800 --> 00:39:43,360
Or you know, then then Okay, then if I

1141
00:39:39,760 --> 00:39:45,160
have these two bonds, the option is not

1142
00:39:43,360 --> 00:39:46,760
buy one or the other and then hold to

1143
00:39:45,159 --> 00:39:48,039
maturity because that would be comparing

1144
00:39:46,760 --> 00:39:49,720
an investment of 1 year with an

1145
00:39:48,039 --> 00:39:52,000
investment of 2 years.

1146
00:39:49,719 --> 00:39:54,279
I need to compare the strategies of

1147
00:39:52,000 --> 00:39:55,800
getting my return in 1 year.

1148
00:39:54,280 --> 00:39:57,960
In the 1-year bond, that's trivial

1149
00:39:55,800 --> 00:39:59,920
because I get my return at the end of at

1150
00:39:57,960 --> 00:40:01,240
the maturity of the bond. In the 2-year

1151
00:39:59,920 --> 00:40:03,639
bond, it means I need to sell it in

1152
00:40:01,239 --> 00:40:06,159
between after 1 year. Okay? So, those

1153
00:40:03,639 --> 00:40:08,839
are the two strategies I want to compare

1154
00:40:06,159 --> 00:40:10,759
and since I'm not take

1155
00:40:08,840 --> 00:40:11,960
considering riskier as a central

1156
00:40:10,760 --> 00:40:13,200
element,

1157
00:40:11,960 --> 00:40:15,880
those two strategies are going to have

1158
00:40:13,199 --> 00:40:17,519
to give me the same expected return.

1159
00:40:15,880 --> 00:40:18,800
Okay? That's arbitrage. That's what we

1160
00:40:17,519 --> 00:40:19,639
call arbitrage.

1161
00:40:18,800 --> 00:40:20,680
Okay?

1162
00:40:19,639 --> 00:40:23,039
Two

1163
00:40:20,679 --> 00:40:26,000
the two strategies have to give me the

1164
00:40:23,039 --> 00:40:28,480
same expected return.

1165
00:40:26,000 --> 00:40:28,480
So,

1166
00:40:28,519 --> 00:40:32,360
what do we get from this strategies?

1167
00:40:30,320 --> 00:40:34,760
Well, if I go through the 1-year bond, I

1168
00:40:32,360 --> 00:40:37,519
know I'm going to get my dollar times 1

1169
00:40:34,760 --> 00:40:39,880
plus I1T. That's what I get off a 1 year

1170
00:40:37,519 --> 00:40:40,880
out of investing a dollar in a 1-year

1171
00:40:39,880 --> 00:40:42,559
bond.

1172
00:40:40,880 --> 00:40:44,280
If I go through the 2-year bond

1173
00:40:42,559 --> 00:40:47,199
strategy, buy it and sell it at the end

1174
00:40:44,280 --> 00:40:48,080
of the year, then I'm going to get I I I

1175
00:40:47,199 --> 00:40:49,919
I

1176
00:40:48,079 --> 00:40:52,880
invest a dollar today,

1177
00:40:49,920 --> 00:40:54,800
no? I'm going to pay P2T.

1178
00:40:52,880 --> 00:40:56,640
That's what I paid today for a 2-year

1179
00:40:54,800 --> 00:40:59,440
bond. That's what I pay here for a

1180
00:40:56,639 --> 00:41:02,759
2-year bond and I expect to get the

1181
00:40:59,440 --> 00:41:04,519
price of a 1-year bond 1 year from now.

1182
00:41:02,760 --> 00:41:05,560
I mean, the 2-year bond will be a 1-year

1183
00:41:04,519 --> 00:41:07,360
bond

1184
00:41:05,559 --> 00:41:08,320
after a year has passed.

1185
00:41:07,360 --> 00:41:10,280
No?

1186
00:41:08,320 --> 00:41:11,960
It's a 2-year bond today, but

1187
00:41:10,280 --> 00:41:13,640
after 1 year, it's going to have only 1

1188
00:41:11,960 --> 00:41:15,639
year to mature.

1189
00:41:13,639 --> 00:41:18,000
So, that's the reason the price I need

1190
00:41:15,639 --> 00:41:20,480
to forecast is the is the price of a

1191
00:41:18,000 --> 00:41:21,840
1-year bond 1 year from now. That's what

1192
00:41:20,480 --> 00:41:23,480
this is here.

1193
00:41:21,840 --> 00:41:25,039
Okay? And that's my return on this

1194
00:41:23,480 --> 00:41:26,840
strategy because I'm going to pay this

1195
00:41:25,039 --> 00:41:28,079
today,

1196
00:41:26,840 --> 00:41:30,000
these dollars,

1197
00:41:28,079 --> 00:41:30,880
and I expect to get that 1 year from

1198
00:41:30,000 --> 00:41:32,280
now.

1199
00:41:30,880 --> 00:41:35,360
Okay?

1200
00:41:32,280 --> 00:41:37,880
So, arbitrage means I need to set these

1201
00:41:35,360 --> 00:41:37,880
two equal.

1202
00:41:38,559 --> 00:41:40,679
Okay?

1203
00:41:42,280 --> 00:41:46,720
So,

1204
00:41:44,679 --> 00:41:48,199
that means I have to get the same return

1205
00:41:46,719 --> 00:41:49,599
with the two strategies. That means I'm

1206
00:41:48,199 --> 00:41:52,079
investing the same, so I only need to

1207
00:41:49,599 --> 00:41:54,880
compare the the the the

1208
00:41:52,079 --> 00:41:57,239
the returns. This needs to be equal to

1209
00:41:54,880 --> 00:41:57,240
that.

1210
00:41:57,280 --> 00:42:00,040
That's what I have here.

1211
00:42:00,119 --> 00:42:03,960
Which tells you

1212
00:42:01,639 --> 00:42:06,960
that you're solving from here that the

1213
00:42:03,960 --> 00:42:08,760
price of a 2-year bond at time T

1214
00:42:06,960 --> 00:42:11,559
is equal to the expected price of a

1215
00:42:08,760 --> 00:42:14,240
1-year bond at T plus 1

1216
00:42:11,559 --> 00:42:15,480
discounted by 1 plus the 1-year interest

1217
00:42:14,239 --> 00:42:16,799
rate.

1218
00:42:15,480 --> 00:42:18,920
No?

1219
00:42:16,800 --> 00:42:20,640
This was like my cash flow.

1220
00:42:18,920 --> 00:42:22,280
My cash flow now is not the cash flow.

1221
00:42:20,639 --> 00:42:23,920
It's It's just a price. I'm going to get

1222
00:42:22,280 --> 00:42:27,120
a price for that asset. That's like the

1223
00:42:23,920 --> 00:42:29,599
Zs I had in my formula. Okay?

1224
00:42:27,119 --> 00:42:31,679
And for a 1-year strategy, I only need

1225
00:42:29,599 --> 00:42:34,319
to worry about the ZT plus 1.

1226
00:42:31,679 --> 00:42:37,279
There was no dividend at day zero.

1227
00:42:34,320 --> 00:42:41,320
Okay? And that's exactly that formula.

1228
00:42:37,280 --> 00:42:43,240
But notice that at T plus 1,

1229
00:42:41,320 --> 00:42:45,800
that will hold.

1230
00:42:43,239 --> 00:42:47,719
No? So, at T plus 1, I'm at T plus 1, I

1231
00:42:45,800 --> 00:42:50,600
don't need expectations. I know that P1T

1232
00:42:47,719 --> 00:42:55,639
plus 1 will be equal to 100 divided by 1

1233
00:42:50,599 --> 00:42:57,759
plus I1, the 1-year rate at T plus 1.

1234
00:42:55,639 --> 00:42:59,639
Therefore, the expected is something

1235
00:42:57,760 --> 00:43:01,920
like this, approximately. The expected

1236
00:42:59,639 --> 00:43:04,199
price is something like that.

1237
00:43:01,920 --> 00:43:06,039
Okay? I expect

1238
00:43:04,199 --> 00:43:07,399
I mean, this will be without the E will

1239
00:43:06,039 --> 00:43:09,679
be the price

1240
00:43:07,400 --> 00:43:11,320
of this 1-year bond at T plus 1. I don't

1241
00:43:09,679 --> 00:43:13,000
know exactly what the interest rate will

1242
00:43:11,320 --> 00:43:15,360
be next year, so I have the best I can

1243
00:43:13,000 --> 00:43:17,840
do is have an expectation. That's my

1244
00:43:15,360 --> 00:43:19,680
expectation, approximately.

1245
00:43:17,840 --> 00:43:22,559
Okay? But now I can stick this

1246
00:43:19,679 --> 00:43:24,519
expression in here.

1247
00:43:22,559 --> 00:43:26,759
No? I have this.

1248
00:43:24,519 --> 00:43:28,000
I'm going to go out and I can stick that

1249
00:43:26,760 --> 00:43:30,120
in there

1250
00:43:28,000 --> 00:43:32,760
and I get this expression. So, that's

1251
00:43:30,119 --> 00:43:35,679
the price for the 2-year bond.

1252
00:43:32,760 --> 00:43:35,680
Do you recognize this?

1253
00:43:37,840 --> 00:43:40,640
You saw it before.

1254
00:43:46,119 --> 00:43:49,039
You know?

1255
00:43:47,079 --> 00:43:50,440
That's the same expression that we got

1256
00:43:49,039 --> 00:43:52,239
when we used the expected present

1257
00:43:50,440 --> 00:43:53,559
discounted value formula.

1258
00:43:52,239 --> 00:43:54,439
Right?

1259
00:43:53,559 --> 00:43:57,599
We said, "Well, I'm going to get the

1260
00:43:54,440 --> 00:43:59,159
$100 100 years a 100 years a 2 years

1261
00:43:57,599 --> 00:44:02,559
from now. I know that discount factor

1262
00:43:59,159 --> 00:44:05,440
for that is 1 over 1 plus I1T times 1

1263
00:44:02,559 --> 00:44:08,000
plus I1T plus 1 expected."

1264
00:44:05,440 --> 00:44:09,440
Well, that's what I got.

1265
00:44:08,000 --> 00:44:10,719
That's from arbitrage.

1266
00:44:09,440 --> 00:44:12,639
Okay?

1267
00:44:10,719 --> 00:44:15,559
From an arbitrage logic. This is used a

1268
00:44:12,639 --> 00:44:15,559
lot in finance.

1269
00:44:15,760 --> 00:44:20,640
I I I'm going to say something

1270
00:44:16,960 --> 00:44:20,639
complicated, but but um

1271
00:44:21,519 --> 00:44:25,039
just ignore it if it's

1272
00:44:26,119 --> 00:44:28,599
uh

1273
00:44:26,719 --> 00:44:30,679
not really up for the for the for the

1274
00:44:28,599 --> 00:44:32,199
quiz or anything, but

1275
00:44:30,679 --> 00:44:34,519
you know, there's a big debate in the US

1276
00:44:32,199 --> 00:44:36,199
today about uh not big debate, a big

1277
00:44:34,519 --> 00:44:37,039
concern about

1278
00:44:36,199 --> 00:44:40,039
uh

1279
00:44:37,039 --> 00:44:40,039
the the

1280
00:44:40,358 --> 00:44:45,000
the US Treasury debt because there is a

1281
00:44:42,840 --> 00:44:46,760
debt ceiling, meaning there's a maximum

1282
00:44:45,000 --> 00:44:48,960
amount that the government can

1283
00:44:46,760 --> 00:44:52,680
of debt they can issue.

1284
00:44:48,960 --> 00:44:54,760
And and the and that ceiling has been

1285
00:44:52,679 --> 00:44:56,399
moved over time, but every time we get

1286
00:44:54,760 --> 00:44:58,800
close to a deadline when this needs to

1287
00:44:56,400 --> 00:45:00,800
be agreed again, there's a concern and

1288
00:44:58,800 --> 00:45:03,160
there's negotiations and so on.

1289
00:45:00,800 --> 00:45:05,600
And the and the

1290
00:45:03,159 --> 00:45:07,519
and the well, I mean, everyone at this

1291
00:45:05,599 --> 00:45:09,960
moment at least thinks that

1292
00:45:07,519 --> 00:45:11,440
as every as in every instance in the

1293
00:45:09,960 --> 00:45:14,280
past, they're going to reach some sort

1294
00:45:11,440 --> 00:45:16,079
of agreement the day before

1295
00:45:14,280 --> 00:45:18,640
of the deadline or not.

1296
00:45:16,079 --> 00:45:20,880
But if they don't and there is a mess,

1297
00:45:18,639 --> 00:45:22,960
this is huge for finance. It's huge for

1298
00:45:20,880 --> 00:45:25,000
finance because US Treasury bonds,

1299
00:45:22,960 --> 00:45:26,760
especially short-term bonds, are used

1300
00:45:25,000 --> 00:45:28,599
for pricing everything

1301
00:45:26,760 --> 00:45:30,320
through arbitrage and so on.

1302
00:45:28,599 --> 00:45:32,239
So, you get a mess there,

1303
00:45:30,320 --> 00:45:34,120
that's a mess in every single financial

1304
00:45:32,239 --> 00:45:36,839
market. You wouldn't know how to price

1305
00:45:34,119 --> 00:45:39,960
many financial assets, actually.

1306
00:45:36,840 --> 00:45:42,079
So, it would be a disaster. But uh

1307
00:45:39,960 --> 00:45:43,440
but the reason I describe I mention this

1308
00:45:42,079 --> 00:45:45,279
here is because

1309
00:45:43,440 --> 00:45:47,639
again, lots of prices are priced in

1310
00:45:45,280 --> 00:45:49,000
reference in as in finance are priced in

1311
00:45:47,639 --> 00:45:50,799
reference, especially derivatives,

1312
00:45:49,000 --> 00:45:52,760
options, and stuff like that.

1313
00:45:50,800 --> 00:45:54,960
Uh you price them relative to something

1314
00:45:52,760 --> 00:45:57,320
using this type of logic. So, if the

1315
00:45:54,960 --> 00:45:59,159
thing you use as a base as a reference

1316
00:45:57,320 --> 00:46:01,600
becomes highly unstable and uncertain

1317
00:45:59,159 --> 00:46:03,440
and risky, then obviously everything

1318
00:46:01,599 --> 00:46:05,440
becomes very complicated,

1319
00:46:03,440 --> 00:46:09,280
very risky, and and financial markets do

1320
00:46:05,440 --> 00:46:09,280
not like risk. That's for sure.

1321
00:46:09,760 --> 00:46:13,560
Anyway, ignore that. That's

1322
00:46:11,920 --> 00:46:15,680
irrelevant for your quiz, but that's the

1323
00:46:13,559 --> 00:46:17,719
reason this the whole discussion then

1324
00:46:15,679 --> 00:46:20,679
over the summer can get to be very very

1325
00:46:17,719 --> 00:46:20,679
tricky for finance.

1326
00:46:21,000 --> 00:46:24,358
So, the yield to maturity, remember I

1327
00:46:22,719 --> 00:46:25,879
mentioned this concept before, of an

1328
00:46:24,358 --> 00:46:27,799
N-year bond,

1329
00:46:25,880 --> 00:46:31,119
but it's also what we When you see

1330
00:46:27,800 --> 00:46:34,160
Whenever you hear the 3-year rate,

1331
00:46:31,119 --> 00:46:36,319
is that. It's the yield to maturity.

1332
00:46:34,159 --> 00:46:37,399
Uh which is different from Okay, let me

1333
00:46:36,320 --> 00:46:38,640
tell you

1334
00:46:37,400 --> 00:46:40,240
show you a formula that's easy to

1335
00:46:38,639 --> 00:46:42,679
explain then.

1336
00:46:40,239 --> 00:46:44,199
And it's defined, it's important, as the

1337
00:46:42,679 --> 00:46:45,960
constant

1338
00:46:44,199 --> 00:46:48,639
annual interest rate that makes the bond

1339
00:46:45,960 --> 00:46:50,400
price today equal to the present

1340
00:46:48,639 --> 00:46:51,440
discounted value or expected discounted

1341
00:46:50,400 --> 00:46:53,680
value.

1342
00:46:51,440 --> 00:46:56,240
So, notice notice the highlighted

1343
00:46:53,679 --> 00:46:57,719
is defined as the constant annual

1344
00:46:56,239 --> 00:46:59,719
interest rate

1345
00:46:57,719 --> 00:47:01,519
that makes the bond price today equal to

1346
00:46:59,719 --> 00:47:03,079
the present discounted value of future

1347
00:47:01,519 --> 00:47:04,480
payments of the bond.

1348
00:47:03,079 --> 00:47:06,239
Okay?

1349
00:47:04,480 --> 00:47:09,039
So,

1350
00:47:06,239 --> 00:47:10,839
for example, in our 2-year bond,

1351
00:47:09,039 --> 00:47:12,519
that's the price. Right? This is the

1352
00:47:10,840 --> 00:47:14,720
price of the asset.

1353
00:47:12,519 --> 00:47:16,800
We that we know the price. We already

1354
00:47:14,719 --> 00:47:18,000
got the price from the previous slides

1355
00:47:16,800 --> 00:47:19,880
of the bond,

1356
00:47:18,000 --> 00:47:21,320
which was based on the short-term

1357
00:47:19,880 --> 00:47:23,079
interest rate, 1-year interest rate, and

1358
00:47:21,320 --> 00:47:25,160
our forecast of the

1359
00:47:23,079 --> 00:47:26,840
short-term the 1-year interest rate 1

1360
00:47:25,159 --> 00:47:28,159
year from now.

1361
00:47:26,840 --> 00:47:29,240
I know that price. Take that as a

1362
00:47:28,159 --> 00:47:31,559
number.

1363
00:47:29,239 --> 00:47:34,000
So, then I the yield the yield to

1364
00:47:31,559 --> 00:47:35,799
maturity is calculated as that constant

1365
00:47:34,000 --> 00:47:37,400
interest rate

1366
00:47:35,800 --> 00:47:38,960
constant How do I see this constant?

1367
00:47:37,400 --> 00:47:40,160
Because, well, I'm using the same

1368
00:47:38,960 --> 00:47:42,119
interest rate for the first period and

1369
00:47:40,159 --> 00:47:44,639
the second period. I And now I'm calling

1370
00:47:42,119 --> 00:47:46,440
it I2T. It's a 2-year interest rate, but

1371
00:47:44,639 --> 00:47:48,679
it's constant. Constant doesn't mean

1372
00:47:46,440 --> 00:47:50,240
that it doesn't move over time.

1373
00:47:48,679 --> 00:47:51,799
It means I'm discounting all the cash

1374
00:47:50,239 --> 00:47:55,000
flows as a constant interest rate. This

1375
00:47:51,800 --> 00:47:55,000
It means I'm using

1376
00:47:55,280 --> 00:47:58,560
I'm using this equation.

1377
00:47:57,719 --> 00:48:00,439
Okay?

1378
00:47:58,559 --> 00:48:02,358
So, the yield to maturity is

1379
00:48:00,440 --> 00:48:03,920
find an interest rate

1380
00:48:02,358 --> 00:48:08,039
that allows me to use this constant

1381
00:48:03,920 --> 00:48:10,039
thing constant assume use this formula

1382
00:48:08,039 --> 00:48:13,039
and get back the same price

1383
00:48:10,039 --> 00:48:14,800
as I got by using the the

1384
00:48:13,039 --> 00:48:17,159
the expected discounted value or the

1385
00:48:14,800 --> 00:48:19,519
arbitrage or something like that. Okay?

1386
00:48:17,159 --> 00:48:21,000
So, that's that's the definition. Okay?

1387
00:48:19,519 --> 00:48:23,119
You have this price.

1388
00:48:21,000 --> 00:48:26,719
Now you you look for that interest rate

1389
00:48:23,119 --> 00:48:27,599
that allows you to match that price.

1390
00:48:26,719 --> 00:48:29,319
Okay?

1391
00:48:27,599 --> 00:48:32,199
And that's called the yield. That's the

1392
00:48:29,320 --> 00:48:33,440
thing I Remember I plotted this the some

1393
00:48:32,199 --> 00:48:35,399
curves?

1394
00:48:33,440 --> 00:48:39,200
Well, those those interest rates in

1395
00:48:35,400 --> 00:48:39,200
those curves were computed this way.

1396
00:48:39,480 --> 00:48:43,039
Now,

1397
00:48:40,599 --> 00:48:44,960
notice that we know what this price is.

1398
00:48:43,039 --> 00:48:46,960
This price is by the expected discounted

1399
00:48:44,960 --> 00:48:49,679
value or the arbitrage approach is equal

1400
00:48:46,960 --> 00:48:51,840
to 100 divided by this.

1401
00:48:49,679 --> 00:48:54,239
So, I know that these two things are

1402
00:48:51,840 --> 00:48:56,760
this is equal to that,

1403
00:48:54,239 --> 00:48:58,719
which means that this denominator is

1404
00:48:56,760 --> 00:49:00,760
equal to that,

1405
00:48:58,719 --> 00:49:03,919
and that implies for a small interest

1406
00:49:00,760 --> 00:49:06,320
rate that this 2-year interest rate is

1407
00:49:03,920 --> 00:49:09,519
approximately equal to the average of

1408
00:49:06,320 --> 00:49:11,440
the expected interest rate 1-year rates.

1409
00:49:09,519 --> 00:49:12,719
Okay?

1410
00:49:11,440 --> 00:49:15,039
So, this is called actually the

1411
00:49:12,719 --> 00:49:17,480
expectation hypothesis, by the way. Is

1412
00:49:15,039 --> 00:49:19,039
that the the 2-year rate

1413
00:49:17,480 --> 00:49:21,358
is approximately equal to the average of

1414
00:49:19,039 --> 00:49:24,759
the 1-year rate this year

1415
00:49:21,358 --> 00:49:26,358
plus the expected 1-year rate 1 year

1416
00:49:24,760 --> 00:49:28,480
from now.

1417
00:49:26,358 --> 00:49:28,480
Okay?

1418
00:49:30,320 --> 00:49:34,080
So, that's an important concept. And I'm

1419
00:49:31,960 --> 00:49:36,720
going to start from here again

1420
00:49:34,079 --> 00:49:36,719
in the next lecture.

1421
00:49:48,360 --> 00:49:50,360
Mhm.