[00:17] Today we're going to talk about
[00:19] a very important topic
[00:21] topic in economics, which is
[00:22] expectations. We have barely mentioned
[00:24] expectations when we talk about the
[00:26] Phillips curve. We talked about
[00:28] expectations when we
[00:30] when we discussed the UEP and so on. But
[00:33] expectations is a much bigger issue in
[00:35] economics. In fact, most decisions by
[00:37] firms, by consumers,
[00:39] governments involve considerations of
[00:42] the future. And it plays an even bigger
[00:44] role in finance, in which essentially
[00:46] everything is about the future. The
[00:48] price of an asset today is meaningless
[00:51] in itself. You have to compare it with
[00:53] what you expect to get out of that asset
[00:55] in the future. So, it's all about
[00:56] expectations and so on.
[00:58] So, that's what we we're going to do
[00:59] today. We're going to talk about
[01:01] uh expectations, the how to value things
[01:05] that that you expect to receive in the
[01:07] future,
[01:08] uh and how to compare those things with
[01:10] things that you have in the present.
[01:13] Um
[01:14] but before doing that, actually, let's
[01:16] talk a little bit about the news. Who
[01:18] knows who First Republic Bank is?
[01:21] Remember that a few weeks ago I told you
[01:24] that um Silicon Valley Bank,
[01:27] you read it. I I I I just mentioned it
[01:30] that
[01:31] uh
[01:32] uh
[01:33] we'll discuss it that that you know, we
[01:35] had the second largest bank by asset in
[01:38] in in US history. It was Silicon Valley
[01:41] Bank was the second largest
[01:43] asset bank in terms of assets
[01:46] uh to collapse in the US. The first one
[01:48] was uh many years ago.
[01:50] Uh and then we had this bank that had
[01:52] more than $200 billion in assets that
[01:55] essentially collapsed in a few days. It
[01:57] was a run on deposits. They had problems
[02:00] before,
[02:01] but what really did as it always the
[02:03] case with banks is they had a run on
[02:05] deposits, funding.
[02:07] Uh
[02:08] well, it's no longer the second largest
[02:11] collapse in US bank history. Now we have
[02:14] over the weekend
[02:15] the the new second largest bank to
[02:17] collapse, which is First Republic Bank,
[02:20] that was essentially
[02:22] liquidated and sold to JP Morgan over
[02:26] today morning, very very early in the
[02:27] morning. Okay? So,
[02:29] you have an account in First Republic
[02:31] Bank, you sooner likely to have an
[02:32] account in JP Morgan.
[02:34] But again, what made it collapse was
[02:37] something very similar to what made
[02:39] Silicon Valley Bank collapse, which is
[02:41] that they had invested on on a series of
[02:44] things that were very vulnerable to to
[02:46] the fast pace of hikes
[02:48] uh
[02:49] in interest rates in the US.
[02:51] And when they had those losses,
[02:53] depositors became became worried about
[02:55] it and eventually they decided not to
[02:57] wait, just run.
[02:58] And see what happened. They
[03:00] First Republic Bank lost about $100
[03:02] billion in deposits just last week.
[03:04] Okay? Um the last few days of last week.
[03:07] So, so, so
[03:10] so that
[03:11] it was obvious that that
[03:13] it was not going to survive and that's
[03:15] the reason
[03:16] something was arranged over the weekend
[03:18] to avoid the panics associated with
[03:20] collapses of a bank and so on. Okay? But
[03:23] anyways, by the way, this is all about
[03:24] expectations. This is you know, this is
[03:27] if people had expected the deposit to
[03:29] remain in the bank, then probably this
[03:31] bank would not have collapsed. It's all
[03:32] about people anticipating what other
[03:34] people will do and so on and so forth.
[03:38] Okay, but now let me get into the
[03:40] specific of
[03:41] of this lecture.
[03:43] So, there you have this is the most
[03:44] important index of of equity equity
[03:47] index in the US, S&P 500. It's a very
[03:50] inclusive index that captures all the
[03:52] large most of the large companies in the
[03:55] US,
[03:56] all of the large I think companies in
[03:58] the US. And uh that's an index. It's an
[04:00] average weighted average by by this
[04:04] uh capitalization value of each of the
[04:07] shares. It's a weighted average of the
[04:08] major the main shares in the US, equity
[04:10] shares in the US.
[04:12] And one thing you see is that it moves a
[04:14] lot around, you know? Here, for example,
[04:16] when when
[04:18] we became aware that COVID was going to
[04:20] be a serious issue,
[04:21] the US equity market collapsed by 35% or
[04:24] so. That's a very large collapse in a
[04:26] very short period of time.
[04:28] And then, as a result of lots of policy
[04:31] support, actually we had a massive
[04:33] rally. Uh so, up to the end of 2021, the
[04:37] equity market had rallied by 114%.
[04:40] So, a big rally.
[04:41] Then we got inflation and the Fed began
[04:44] to worry about
[04:45] inflation, so they began to hike
[04:47] interest rates. And when they hike
[04:48] interest rates, that eventually led to a
[04:50] very large decline
[04:52] uh
[04:53] in in asset prices of the order of 30%
[04:55] or so, 25% or so, actually, from the
[04:58] peak to the bottom.
[05:00] And then, since the bottom, which is was
[05:02] more or less October of last year, we
[05:04] have seen a recovery of about 16% or so
[05:07] of the equity market. Okay? And if you
[05:09] look at the Nasdaq, which is another one
[05:11] index that is very loaded towards
[05:13] uh
[05:14] technology companies, then you can see
[05:16] swings that are even larger than that.
[05:19] Now, why do these prices move so much?
[05:22] Well,
[05:23] a lot of it has to do with expectations.
[05:27] You know, are things going to get worse
[05:29] in the future? Will the Fed cause a
[05:30] recession? Uh
[05:33] how much higher will be the interest
[05:35] rate? And things like that matter a
[05:37] great deal.
[05:38] Another thing that matters a great
[05:41] deal is
[05:42] how much people want to take risk at any
[05:44] moment in time. And if you're very
[05:46] scared about the environment, you're
[05:47] unlikely to want to have something that
[05:49] to invest on something that can move so
[05:51] much,
[05:52] and so risk is well known. So, it's
[05:54] called risk off when when people don't
[05:56] want to take risk, these asset prices
[05:58] tend to collapse. Okay? Of the risky
[06:00] asset. Equity is a very risky asset.
[06:03] But that's not the only thing that moves
[06:05] these assets around. It's not just the
[06:06] risk that the companies underlying
[06:08] company may go bankrupt or anything like
[06:10] that.
[06:11] Here you have, for example, the movement
[06:13] of a
[06:14] for it's an ETF, but it doesn't matter.
[06:16] It's a portfolio of
[06:18] bonds of US Treasury bonds of very long
[06:20] duration. Maturities beyond
[06:22] uh 20 years and so.
[06:24] So, this is incredibly safe bonds, you
[06:26] know? Because it's US Treasuries. So,
[06:28] there's no risk of default or anything
[06:29] like that.
[06:30] Still,
[06:32] the price swings can be pretty large. I
[06:33] mean, in over this period, you know,
[06:35] there you have seen an an an increase in
[06:37] value of 45%,
[06:39] then uh a decline in value of of of
[06:42] about 20%. Another increase in 15% here.
[06:45] There was a huge decline, 40%,
[06:48] since since essentially
[06:50] uh uh uh
[06:51] What do you think happened here? Why is
[06:53] this big decline in in in in bonds?
[06:56] You're going to be able to answer that
[06:57] very precisely later on, but but I can
[06:59] tell you in advance that that was
[07:01] essentially the result of monetary
[07:02] policy tightening.
[07:04] You know, increasing interest rate
[07:07] caused the the bonds to decline. So,
[07:09] even these instruments that are very
[07:10] safe in the sense that you if you hold
[07:12] it to maturity, you will get your money
[07:13] back and all the promised coupons along
[07:15] the path, well, still their price can
[07:18] move a lot. And it's obvious that that
[07:20] movement in price
[07:22] is something you need to explain in
[07:23] terms of expectations, what people
[07:25] expect things to to happen. In this
[07:27] case, it's not whether people expect to
[07:29] get paid or not, because you will get
[07:31] paid, but it's expect but in this
[07:34] particular case, it's about expectations
[07:35] about future interest rate. If you think
[07:37] the interest rate will be very high,
[07:39] then the price of bonds will tend to be
[07:40] very low and so on. But it's all about
[07:42] the future. Okay?
[07:45] So, the a key concept
[07:48] uh that we're going to discuss today and
[07:50] then we're going going to use it to
[07:52] price a specific asset
[07:54] uh is the concept of expected present
[07:57] discounted value. This this is a loaded
[07:59] concept. There's lots of
[08:01] terms in there and we need to understand
[08:03] what each of these terms means.
[08:06] So, the key issue
[08:08] that we're going to discuss is how how
[08:10] do we decide, for example, if you see
[08:11] the price of an asset out there
[08:13] that is 100, how do you decide whether
[08:16] that price is fair or not, looks cheap
[08:19] or not? Okay? Uh and and and and and
[08:23] that question means you have to decide
[08:25] whether that price that you're paying
[08:26] today is consistent with the future cash
[08:29] flows that you're going to get from this
[08:31] asset. I mean, that's the reason you buy
[08:32] an asset is because you'll get something
[08:34] in return in the future. Okay? But how
[08:37] do we compare that? How do we compare
[08:39] the price today with those things that
[08:41] will happen in the future?
[08:46] So, answering that question, which is
[08:48] what we're going to do in this lecture,
[08:50] involves the following
[08:52] concepts. First, expectations, big
[08:55] thing.
[08:57] That's a you know, this is expected
[08:59] present discounted value. The E part is
[09:01] for expectations. That comes there.
[09:04] You
[09:05] Expectations are really crucial because
[09:07] these are things that happen in the
[09:08] future. You need to expect. Even if if
[09:11] it's a bond that promises you to pay,
[09:13] you know, 50 cents per dollar every 6
[09:16] month,
[09:17] you still may have an expectation that,
[09:19] you know, if it is a bond issued by
[09:20] First Republic Bank, it may not pay. So,
[09:23] so, so you need to have an expectations
[09:25] about that.
[09:26] Uh
[09:28] so, crucial term is expectation.
[09:30] Then you need some method
[09:32] uh to compare payments received in the
[09:34] future with payments made today. I mean,
[09:36] if you buy an asset, you pay today,
[09:39] but you're going to receive things
[09:40] returns on for that asset in the future.
[09:42] So, how do I compare that that that
[09:45] Suppose I pay one today and I receive
[09:47] one 1 year from now.
[09:49] Does that seem like a good asset?
[09:52] Probably not. I mean, you know,
[09:54] probably not.
[09:56] Uh
[09:57] Um and that's what the word discounted
[09:59] really means. You know, when you say
[10:01] expected present discounted value
[10:04] it says
[10:05] somehow that things I receive in the
[10:07] future are valued less than things I
[10:09] have today.
[10:10] Okay? So, if you're going to tell me
[10:12] that you're going to pay me a dollar in
[10:13] the future and I have to pay you a
[10:15] dollar today, most likely I won't take
[10:17] that deal.
[10:18] So, I need In other words, I'm
[10:20] discounting the future.
[10:22] How do we discount the future? Well,
[10:23] something that we're going to have to
[10:24] figure out.
[10:27] So,
[10:28] let's let me first shut down this part,
[10:30] the expectations, and then we'll
[10:31] introduce it. So, assume for now that
[10:33] you know the future.
[10:34] Okay? And I'm going to derive all the
[10:36] equations with assuming that you know
[10:39] the future. So, there's no issue of
[10:41] trying to figure out what the future is.
[10:42] You know it. But still you have to
[10:44] decide whether
[10:45] what is the right value for for an
[10:48] asset.
[10:52] Okay, so
[10:54] let's start with the case where you know
[10:56] the future. Sorry.
[10:58] And let's do the comparison uh
[11:02] Let's try to understand how do we move
[11:04] flows, how do we value flows at
[11:05] different points in time.
[11:07] This is the thing is think first about
[11:10] comparing an asset that gives you a
[11:11] dollar in the future,
[11:13] how much do you think it's worth today?
[11:16] Well, the easiest
[11:17] way to get to that value is is to think
[11:19] on the alternatives. As suppose I have a
[11:21] dollar today, what can I do with it?
[11:26] Well, in terms of investment.
[11:28] Well, suppose that you have available
[11:31] one-year bonds, treasury bonds, and that
[11:33] the interest rate is I
[11:34] t. That's the interest rate on an I
[11:36] one-year bond.
[11:38] So, if you want if you if you
[11:40] have a dollar,
[11:42] you have the option to invest it in that
[11:44] asset, in that bond, which give will
[11:46] give you 1 + I dollars
[11:48] uh
[11:49] next year.
[11:51] Well,
[11:52] that means that I can get $1 next year
[11:57] by investing 1 over 1 + I dollars today.
[12:01] No?
[12:02] Because if if I invest 1 + 1
[12:05] rather than $1, I invest 1 over 1 + I
[12:08] today, then I multiply this by 1 + I
[12:11] and I get my dollar in the future.
[12:14] So, that tells me that say the interest
[12:16] rate is 10%, then with $1 today I can
[12:19] get 1.1 dollars in the future.
[12:22] That means that investing 90% 90 cents
[12:25] today, more or less,
[12:27] I can get $1 in the future.
[12:29] That tells me that a dollar in the
[12:31] future is equivalent to 90 cents today.
[12:34] That's the assumption. Okay?
[12:36] So, that's the reason when I told you
[12:38] the deal of me, look, I have an asset
[12:40] that cost cost you a dollar, but gives
[12:42] you a dollar in the future, well, that's
[12:44] not a good deal if the interest rate is
[12:45] positive.
[12:47] If the interest rate is 10%, then then a
[12:49] right a fair comparison is 90 cents with
[12:51] $1, not $1 with $1.
[12:54] Okay? So, that's the discounting of the
[12:56] future. You can The most obvious way of
[12:58] discounting the future
[13:00] is to discount it by the interest rate.
[13:02] Uh
[13:04] which interest rate to pick? That's more
[13:06] subtle. That depends on risk, depends on
[13:08] many other things which we're going to
[13:09] discuss to some extent here. But for
[13:12] now, let's make it very simple. And in a
[13:15] world in which you really know the
[13:16] future, really the right interest rate
[13:17] to use is the safe interest rate, the
[13:19] interest rate of of treasury bonds and
[13:21] things like that.
[13:23] Okay? So, that's that's that.
[13:25] What about a dollar that you receive
[13:27] What about if you're thinking about what
[13:29] is the value of a dollar two years from
[13:31] now?
[13:32] Well,
[13:34] you know, if I get a dollar to I can do
[13:35] the same logic. If I if I
[13:38] I can use the same logic. If I get a
[13:39] dollar today,
[13:41] I can convert that into 1 + I t * 1 + I
[13:45] t + 1 dollars. Okay?
[13:48] So, say 10% and 10%, I get 1.1 next year
[13:52] and then I get 1.1 * 1.1, 1.21 or
[13:55] something like that. Okay?
[13:57] That's my final
[13:58] result.
[13:59] So,
[14:01] well, then how much is it worth to have
[14:03] a dollar, an asset that gives you a
[14:05] dollar two years from now?
[14:09] Well, it's going to be that dollar
[14:11] divided by the product of these interest
[14:13] rates.
[14:14] Okay? Why is that? Well, because with
[14:16] this amount of
[14:18] dollars today,
[14:20] it's point 80 cents or something like
[14:22] that, I can generate a dollar two years
[14:24] from now.
[14:25] That means a dollar
[14:27] two years from now
[14:29] is worth about 80 cents today.
[14:32] Okay?
[14:35] We're going to use a lot this type of
[14:37] logic, so
[14:38] and and I know that that it may not be
[14:40] that intuitive the first time you see
[14:42] it, but
[14:43] ask questions.
[14:47] You want me to repeat it?
[14:54] Okay. The
[14:55] The final goal is the following. We're
[14:57] going to In the what comes next, we're
[14:59] going to see if which happens again with
[15:01] many decisions in life, but it perhaps
[15:03] particularly for financial assets,
[15:05] we're going to try to value something
[15:07] that
[15:08] whose payoff happens at different times
[15:11] in the future. And the question is
[15:13] how do I value an asset that pays me,
[15:16] you know, $5 one year from now, $25
[15:19] three years from now, uh
[15:21] minus $10 10 years from now, plus $50
[15:26] 100 years from now?
[15:27] What is the value of that? Of having an
[15:29] asset like that?
[15:31] And so, I needed some method
[15:33] to bring it to today's value because
[15:35] today I have a meaning of what a dollar
[15:37] is, you know?
[15:38] And and therefore I can compare it with
[15:40] whatever price I mean
[15:42] people are asking me for that asset.
[15:44] So, what this is doing is is is that is
[15:47] doing that. It's telling you how to
[15:49] convert a dollar at different parts in
[15:51] the future into a dollar today.
[15:54] And by that logic,
[15:56] the recipe is well, use the interest
[15:59] rate because you could always go the
[16:01] other way around. You could always with
[16:02] a dollar you can ask a question, with a
[16:03] dollar today, how many dollars can I get
[16:06] two years from now, say?
[16:08] That.
[16:09] Well,
[16:10] say X. Well, then I need 1 over X. Then
[16:13] $1 there is worth 1 over X dollars
[16:15] today. You know, that's that's the logic
[16:18] because 1 over X * X is 1.
[16:21] So,
[16:23] that's too fast, probably.
[16:28] So,
[16:29] you know, with $1 today, oops,
[16:35] I can generate, say,
[16:38] $1.1
[16:41] at uh uh
[16:44] at t equal to
[16:45] Okay?
[16:46] Then I'm I'm The question I want to know
[16:48] is how much is a dollar worth
[16:51] How much is a dollar received at time t
[16:53] equal to worth today?
[16:56] That's the question I'm trying to
[16:57] answer.
[16:58] You know, because an asset will be
[17:00] something that will pay you in the
[17:01] future. So, I want to know how much is
[17:03] $1 received in the future worth today.
[17:08] And then the answer is
[17:10] well,
[17:11] then is I know the answer from this
[17:13] logic because I know that with one
[17:18] if I have 1 over 1.1 dollars today, I
[17:21] can convert it
[17:25] into one.
[17:27] How do I know that?
[17:28] Because
[17:33] 1 over 1.1
[17:38] * 1.1
[17:40] is equal to 1.
[17:43] Okay? This if I invest these dollars
[17:45] today,
[17:47] I'm going to get this return on that.
[17:50] And the product of these two things
[17:51] gives me my dollar.
[17:54] Okay?
[17:55] So, if I tell you, do you prefer to have
[17:56] a dollar two days from two years from
[17:58] now or today?
[17:59] You say, I prefer it obviously prefer it
[18:02] today because I can get 1.1 dollars two
[18:05] years from now.
[18:07] But then then the more relevant question
[18:09] is, no, no, but then you do you prefer
[18:10] to have 90 cents today
[18:12] versus a dollar in the future? And then
[18:15] I'm I need to do my multiplication
[18:16] because I have to multiply the 90 cents
[18:18] by the 1.1 and see whether I get
[18:21] something comparable to a dollar or not.
[18:23] Okay?
[18:24] But that's that's the logic behind that.
[18:26] And And that's a So, the interest rate
[18:29] is what we discount the future by.
[18:33] And it's natural because if the interest
[18:34] rate is very high If the interest rate
[18:36] is zero, say,
[18:37] then a dollar received two years from
[18:39] now or a dollar received today is is the
[18:41] same
[18:42] because I can't If I invest a dollar
[18:44] today and the interest rate is zero, I'm
[18:45] going to get my dollar two years from
[18:46] now.
[18:47] If the dollar If the interest rate is
[18:49] 50%, it makes a big difference receiving
[18:51] the dollar today versus receiving it two
[18:52] years from now.
[18:54] If you're in Argentina, the interest
[18:56] rate I don't know what it is. It's
[18:58] 700%. It makes a huge difference whether
[19:00] you receive it, you know, one year from
[19:02] now than today.
[19:05] And and and uh
[19:08] So, that's that's the role of the
[19:09] interest rate. The higher is the
[19:10] interest rate,
[19:12] the less
[19:13] is a dollar received in the future worth
[19:15] relative to a dollar received today.
[19:17] Because you can get a much higher return
[19:19] from the dollar you have today
[19:21] if the interest rate is high. If the
[19:22] interest rate is low,
[19:24] you don't get that much. Okay?
[19:26] Much difference. Okay, good.
[19:28] So, this is a big principle. And and I I
[19:30] mean
[19:31] everything I'll say next builds on this
[19:33] logic.
[19:38] So, let me give you a general formula.
[19:40] So, let's ask what is the value
[19:43] of an asset
[19:44] that gives
[19:46] payouts of Z
[19:49] t dollars this year,
[19:51] Z t + 1 one year from now, ZT plus two,
[19:55] two years from now, and so on and so
[19:57] forth for N periods more. Okay?
[20:01] Well,
[20:03] I just need to do several of these
[20:04] operations. I know that the dollar
[20:06] received this year is is worth a dollar.
[20:09] Okay? That's ZT.
[20:10] A dollar received one year from now
[20:13] is not
[20:14] is not the same as a dollar received
[20:16] today. It's the same as one over one
[20:18] plus IT dollars received today.
[20:22] So, that cash flow I'm going to receive
[20:23] from this asset is worth this amount.
[20:26] For a two something that I receive two
[20:28] years from now, then it's not
[20:30] it's not certain, it's much less than
[20:32] receiving a dollar today. It's going to
[20:34] be one over one plus IT one plus IT plus
[20:38] one.
[20:39] And that I have to multiply by the
[20:40] number of dollars I will receive two
[20:42] years from now. Okay? And I keep going.
[20:46] So, that's that's the
[20:48] the present value. Present discounted
[20:51] value. Present because I'm bringing all
[20:53] these future cash flows to the present.
[20:56] That's what each of these terms is
[20:57] doing. The one over that is bringing it
[21:00] to the present.
[21:01] Discounted because the interest rate is
[21:03] discounting things. It's making them a
[21:05] smaller.
[21:06] And value because I'm trying to reduce
[21:08] them to the current value. Okay?
[21:12] That's the general formula. So, it's a
[21:14] formula you need to understand.
[21:15] It's just So, that that was an asset
[21:18] that gives you Z dollars today,
[21:21] ZT plus one, one year from now, so you
[21:23] use this formula. ZT plus two, two years
[21:26] from now, so you use this formula, and
[21:29] then you keep going. Okay?
[21:33] What if we don't know the future?
[21:35] You know, I have to remove the expected
[21:37] part.
[21:39] Well,
[21:40] if we don't know the future, then the
[21:41] best we can do, in fact, we do fancier
[21:43] things, but that's what we're going to
[21:45] all that we'll do in this course.
[21:47] Uh all that you can do is just replace
[21:50] the known quantities we have here
[21:52] for the expectations.
[21:54] Okay? So, that's the closest. So, you
[21:56] know, I know ZT, that's the cash flow I
[21:58] get now,
[22:00] but I don't know ZT plus one. So, I can
[22:02] replace it by expectation.
[22:04] I do know the interest rate on a
[22:05] one-year bond from today to one year.
[22:08] So, that's the reason I don't need an
[22:09] expectation here.
[22:10] But I don't know what the one-year rate
[22:12] will be one year from now. So, that's
[22:14] the reason I need an expectation there.
[22:17] And so on.
[22:18] And I don't know what the cash flow will
[22:20] be two years from now. I have an
[22:21] expectation about what the cash flow
[22:22] will be, but I don't know it.
[22:24] So, I have an expectation there. Okay?
[22:26] So, so all that I've done here is say,
[22:29] "Okay,
[22:31] I acknowledge that this guy knew a
[22:32] little bit too much. You know, he knew
[22:33] exactly what the cash flows were going
[22:35] to be in the future, and he knew what
[22:37] the one-year rates were going to be in
[22:38] the future."
[22:40] This guy here knows less. He knows the
[22:42] cash flow today. He knows the interest
[22:44] rate today, but he doesn't know the cash
[22:46] flows. Really, he has a hunch, but he
[22:48] doesn't know the cash flows one year,
[22:50] two years, three years, and so on for
[22:51] the future, and he doesn't know the
[22:53] one-year interest rate in the future.
[22:56] So, all these expectations, here's
[22:58] important the concept of time. This is
[23:00] an expectation as of time T. At time T,
[23:02] you have some information and you make
[23:04] forecast about the future. Okay?
[23:07] Use whatever you want, machine learning,
[23:08] whatever, but you have information at
[23:10] time T,
[23:11] and then you have a forecast for the
[23:13] future. At T plus one, you have you'll
[23:14] have more information, so you make
[23:16] another forecast, and so on and so
[23:17] forth.
[23:18] But in this we're valuing an asset at
[23:20] time T, then all these expectations are
[23:23] taken as of time T. That means given the
[23:26] information you have available at time
[23:28] T.
[23:30] That's the reason these guys don't have
[23:31] expectations in front of them because
[23:33] you know this at time T.
[23:36] Had we taken the value at T minus one,
[23:38] we would have not known that, and then
[23:40] we would have had to expectation because
[23:41] it would have been expectation as of T
[23:43] minus one.
[23:46] Okay, so that's your big formula there.
[23:48] So,
[23:49] there are some examples that are sort of
[23:51] well known and and and
[23:53] and and
[23:55] and and
[23:56] So, let me let me show you. They have
[23:58] nicer expressions. So, that's that's an
[24:00] example
[24:02] of the valuation of of this the same
[24:04] asset,
[24:05] but when the interest rate is constant,
[24:08] then
[24:09] then obviously I don't need all these
[24:10] products in the denominator.
[24:13] I have a constant interest rate, then I
[24:16] just get powers of that interest rate.
[24:18] That's one in which you have constant
[24:20] payments. So, the interest rate may be
[24:22] different,
[24:23] but the payments are the same over time.
[24:26] Okay?
[24:27] So, that's that.
[24:29] So, those are two
[24:30] easy formulas. That's one in which you
[24:32] have
[24:33] both constant, the interest rate
[24:36] and the payment.
[24:37] Then you get a nice expression. That's
[24:39] just uh
[24:40] that. Okay? You'll recognize that.
[24:44] If if you have a constant
[24:46] uh
[24:47] constant interest rate here, you see
[24:50] that the value
[24:52] is is declining is a is a geometric
[24:54] series. You know? The value of a two
[24:56] years from now is a square
[24:58] of one over one plus some
[25:00] it's a square of a a number less than
[25:02] one.
[25:03] You know? One over one plus I is some
[25:05] number less than one. This is a square
[25:07] of that, then the cube, and so on. So,
[25:09] it's a geometric series that is
[25:10] declining at the rate one plus I, one
[25:12] over one plus I. Okay? Or declining at
[25:14] the rate one plus I.
[25:16] So, that's your geometric series.
[25:18] Okay?
[25:19] That's the value of that.
[25:22] Constant rate and payment forever.
[25:25] Suppose you have an asset that
[25:27] it
[25:28] that lives forever.
[25:31] There are some bonds like that called
[25:32] perpetuities. Uh uh
[25:36] The US hasn't issued one, but the UK
[25:38] has, and so on.
[25:40] I
[25:40] So, that's an asset, for example, that
[25:42] pays you a fixed amount
[25:44] forever. And if the interest rate is
[25:46] constant, that's the trickier thing,
[25:48] then the value of that asset you can see
[25:50] that this this is going to zero.
[25:53] So, the value of that asset
[25:55] is that.
[25:57] And actually a formula that you may see
[25:59] that is very oftenly used as as a first
[26:02] approximation is this one. This is
[26:04] is is the same asset, but it's called
[26:07] ex-dividend or ex-coupon. It's it's
[26:09] after the coupon of this year has been
[26:12] paid.
[26:13] Okay? So, it's an asset that starts
[26:15] paying at T plus one. It's ZT plus one,
[26:17] ZT plus two, and so on.
[26:19] Well, that
[26:20] is the same as this minus the first
[26:22] coupon, so is equal to that.
[26:25] Okay?
[26:29] That's an interesting thing, huh? Look,
[26:31] what happened to this asset as the
[26:32] interest rate goes to zero?
[26:36] So, this is an asset that lasts for a
[26:37] very long time.
[26:39] And and and look, we got to a valuation
[26:41] formula.
[26:43] What hap- what is happening as the
[26:45] interest rate goes to zero?
[26:46] To the value.
[26:49] Very large. It goes to infinity.
[26:51] And a lot of what has happened in in
[26:53] global financial markets
[26:55] in the last few years has to do with
[26:57] that.
[26:58] Interest rates were very very very low.
[27:01] And so, most assets that had long
[27:02] duration had very high values.
[27:05] Okay?
[27:07] And it has a lot to that. Monetary
[27:09] policy had a lot to do
[27:10] whether it was the right monetary policy
[27:12] or not,
[27:13] that's something to be discussed. I
[27:15] think on average it was the right
[27:16] monetary policy, but one of the things
[27:17] it did, it increased the value of many
[27:20] assets. In fact, that's one of the
[27:22] mechanisms through which monetary policy
[27:24] works in practice. It's not something we
[27:25] have discussed, but you can begin to see
[27:27] here. Because if the value of all assets
[27:29] go up a lot, people feel wealthier, and
[27:31] that they will tend to consume more, and
[27:32] so on. Well, this is one of the channels
[27:34] monetary policy does. By the way, this
[27:36] effect happens also to this asset that
[27:38] has finite N. It's just that this goes
[27:40] is
[27:41] it's maximized when this asset lasts
[27:43] forever. You know?
[27:44] This this asset literally goes to
[27:46] infinity
[27:47] if the interest rate goes to zero.
[27:51] Well, if an asset lasts for N periods,
[27:54] it doesn't go to infinity. It goes to N
[27:56] times Z.
[27:58] You know? It's the sum.
[27:59] If the interest rate is zero, you just
[28:01] sum things.
[28:03] See that?
[28:04] If I if if an asset lasts for N periods,
[28:07] and it gives me a payment of Z in every
[28:09] single period,
[28:11] then when the interest is zero, that
[28:13] asset is worth N times Z.
[28:15] Because I will receive Z coupons.
[28:18] And I don't discount the future because
[28:19] the interest rate is zero.
[28:21] What happens is when the asset lasts
[28:23] forever, then N times Z is a really
[28:25] large number, you know? And that's
[28:27] that's what this expression captures
[28:29] here.
[28:31] Okay.
[28:34] So, let's talk about bonds now. We're
[28:36] going to start pricing bonds.
[28:39] Well, so bonds differ uh uh uh
[28:42] along many dimensions, but one of them
[28:45] is is very important for bonds is
[28:46] maturity, the N that I had there
[28:49] in the previous expression. Okay?
[28:52] Uh so, so maturity means essentially how
[28:55] long the bond lasts. Okay? When when
[28:57] does it pay you back the principal? The
[28:59] bonds typically pay coupons, and then
[29:02] there's a final payment, which we call
[29:03] face value of the bond or something like
[29:05] that. And and when that final payment
[29:08] takes place, that's the maturity of a
[29:10] bond. Okay?
[29:12] So, a bond that promises to make a
[29:14] thousand-dollar final payment in six
[29:15] months
[29:17] has a maturity of six months.
[29:20] A bond that promised to pay a hundred
[29:22] dollars for twenty years and then one
[29:24] thousand dollars final payment in twenty
[29:26] years has a maturity of twenty years.
[29:28] Maturity is different from duration. I
[29:30] don't think I'm going to talk about
[29:31] duration here, but but that's maturity.
[29:33] Just when the when is the final payment
[29:36] of of a
[29:38] of a loan.
[29:39] Of a of a bond. Okay?
[29:42] Bonds of different maturities each have
[29:44] a price
[29:45] and an associated interest rate. We're
[29:47] going to look at those things.
[29:48] And the associated interest rate is
[29:51] called the yield to maturity, or simply
[29:53] the yield of a bond.
[29:55] This is terminology, but we're going to
[29:57] calculate these things later on.
[30:00] The The relationship between maturity
[30:02] and yield
[30:03] is called the yield curve. Very
[30:05] important concept. Big fuss about the
[30:07] yield curve these days.
[30:09] Talk a little bit more about that.
[30:11] Or sometimes it's called the term
[30:13] structure of interest rate.
[30:15] Term, in the language of bonds, is
[30:17] really maturity.
[30:19] So, term structure of interest rate
[30:21] really tells you what is the yield in a
[30:23] 1-year bond, 2-year bond, 3-year bond, 4
[30:26] 5 6 so on. You plot them, and that gives
[30:29] you a curve.
[30:30] Okay.
[30:31] So,
[30:33] uh for example,
[30:34] look at the those These are two
[30:36] different yield curves. This is November
[30:38] 2000,
[30:40] and this is June 20
[30:42] 2001.
[30:43] So, this tells you what the yield is
[30:47] in on a 3-month bond, so a bond that
[30:49] matures in three in 3 months, on a
[30:51] 6-month bonds and so forth, up to
[30:53] 30-year bonds. Okay.
[30:56] What is the big difference between these
[30:57] What do you think happened here in
[30:58] between? Notice that these two curves
[31:00] are more or less the same long-term
[31:02] interest rate.
[31:04] But they have very different This curve
[31:06] This is a very steep curve, and this is
[31:08] a very flat or even inverted curve.
[31:12] What do you think may have happened
[31:14] there?
[31:16] Between November 2000 and June 2001.
[31:20] People changed their expectations then.
[31:22] Yeah, it's That's true. That's for sure
[31:25] true about that. But look also that But
[31:28] that that that this 3-month There is
[31:30] very little uncertainty about 3 months.
[31:32] It was a lot lower than that.
[31:34] So, yes, people changed their
[31:35] expectation, but why do you think they
[31:37] changed their expectation?
[31:42] Well, it's rising inflation. We have a
[31:44] lot
[31:46] Rising inflation from here to here.
[31:47] These are These are nominal interest
[31:48] rates.
[31:50] Up to now I've been talking about
[31:51] nominal interest rate.
[31:56] What happens here
[31:58] is there was a mini recession.
[32:00] So, the Fed cut interest rate.
[32:03] When you're in recessions, the curve
[32:05] tend to look like this.
[32:08] Because
[32:09] the central bank is cutting interest
[32:10] rates in the in the short run to deal
[32:12] with the current recession.
[32:14] What happens 30 years from now has
[32:15] nothing to do with the business cycle
[32:16] today, so that interest rate doesn't
[32:18] need to move a lot. But the Fed is
[32:20] bringing interest rate down a lot in the
[32:22] front end. Okay. So, that's the typical
[32:24] shape of a curve in a recession.
[32:27] That's the typical shape of a of a curve
[32:29] in the opposite situation where the
[32:31] inflation is too high and so on. Because
[32:32] what happens? The Fed is trying to The
[32:34] Fed really controls the very front end
[32:36] of the curve.
[32:37] That's what the Fed really control. The
[32:39] central bank in general, but the Fed.
[32:40] They control the very front end of the
[32:42] curve because they're setting the very
[32:43] short-term interest rate.
[32:44] So, this is a situation where they're
[32:46] tightening the monetary policy very
[32:47] tight.
[32:48] Because they are a situation of uh
[32:51] overheating in the economy. And in fact,
[32:53] they got too carried away. That's the
[32:54] reason they we ended up in a recession
[32:55] here.
[32:57] Okay.
[33:00] How do you think it looks today?
[33:06] That Do you think it looks more like
[33:07] this or more like that?
[33:09] Is inflation low or high today?
[33:14] High. I mean, that's a problem, you
[33:15] know? The Fed is trying to hike interest
[33:17] rate. Now, recently, because of the the
[33:19] mess in the banking sector, then the
[33:21] expectations of interest rate began to
[33:23] decline a little, but but but the
[33:25] situation was was very important. Here
[33:27] you are.
[33:28] That's The green line is today.
[33:30] Okay. So, it's very inverted.
[33:34] Okay.
[33:35] A year ago, it looked like that.
[33:38] So, you see the the long end hasn't
[33:39] changed much, but a year ago, there was
[33:41] no sense that the inflation was getting
[33:43] so much out of line.
[33:46] It happened a little later than that.
[33:47] There was some concern that interest
[33:49] rate would would rise,
[33:51] but but but now it's very clear the
[33:53] economy is overheating. And this I
[33:55] should have plotted you something for
[33:57] for
[33:58] a month ago. It would have been even
[34:00] steeper. Okay.
[34:03] Anyways, but that's because the Fed is
[34:05] trying to slow down the economy. It's
[34:06] hiking interest rates. That's the reason
[34:08] the curve is very very inverted today.
[34:12] So, let me let me calculate these rates.
[34:14] How do we go about it? So, the first
[34:16] thing we're going to do is we're going
[34:17] to use the expected present discounted
[34:19] value formula to calculate the price
[34:22] of a bond.
[34:24] And then we want to start
[34:25] doing it for different bonds,
[34:27] and we're going to construct uh the
[34:29] yield curve.
[34:30] So, suppose you have a bond that pays
[34:33] $100,
[34:35] nothing in between, $100 1 year from
[34:37] now. So, this is a bond with maturity
[34:39] 1-year maturity.
[34:41] I'm going to call that bond with 1-year
[34:43] maturity
[34:45] P1 the price of a bond with a 1-year
[34:47] maturity at time T, P1T.
[34:50] Well, that's easy to calculate. It's
[34:51] expected present discounted value for If
[34:53] you have the interest rate, whatever you
[34:55] say, 1-year interest rate, then I know
[34:57] that the price of the bond is 100
[34:59] divided by 1 plus the interest rate, the
[35:01] 1-year interest rate today.
[35:04] Okay. That's the price. That's expected
[35:06] discounted value. So, I tell you what
[35:07] I'm showing you is the relationship
[35:08] between interest rates and prices.
[35:11] Okay. Our price of a bond. The price of
[35:13] that bond is just 100
[35:16] uh divided by 1 plus the 1 plus the
[35:19] 1-year interest rate today. Okay.
[35:23] So, important observation is that the
[35:25] price of a 1-year bond varies inversely
[35:28] with the current
[35:29] 1-year nominal interest rate. This is
[35:31] all nominal, huh?
[35:34] Why is it an inverse relationship?
[35:36] Why is it the price of a 1-year bond is
[35:38] inversely related to the 1-year interest
[35:40] rate?
[35:44] In other words, I'm asking
[35:46] what do you think happens to the price
[35:48] as a nominal as a nominal interest rate
[35:49] rises?
[35:50] And why do you think that's what happens
[35:52] to the price?
[35:55] Well, the first question doesn't have a
[35:57] I mean, it's very easy, you know, the
[35:58] answer to the first question. What
[35:59] happens if I goes up? Well, it's obvious
[36:01] that this price comes down.
[36:03] But why?
[36:08] And and and I'm And you use the concept
[36:11] we have developed here. Remember we
[36:12] spent
[36:13] like 20 minutes in one slide. Well, you
[36:15] start the slide for that answer.
[36:22] Hint.
[36:24] This $100 you're not receiving today,
[36:26] you're receiving a year from now.
[36:28] What happens with a dollar received a
[36:30] year from now?
[36:31] What is the value of a year
[36:33] dollar received 1 year from now when the
[36:35] interest rate is high?
[36:39] Slow, because, you know,
[36:41] you'd much rather have the dollar today,
[36:42] invest it, and get this big return on
[36:44] the on on the dollar.
[36:46] That means, naturally, a bond that is
[36:48] paying you $100 tomorrow is going to be
[36:50] worth less
[36:51] when the interest rate is very high.
[36:53] It's going to be worth less today when
[36:54] the interest rate is very high. You'd
[36:55] rather have the money today, invest it
[36:57] in the in in the interest rate, and get
[36:58] the interest rate.
[37:00] And and uh
[37:02] No, I need to invest 1 over 1 plus I1T
[37:05] dollars to get $100. That's another way
[37:07] of saying it.
[37:10] What about with the bond that pays $100
[37:13] in 2 years?
[37:15] Well, I need to discount that by this,
[37:17] which is a You know, it's a product of
[37:19] the two interest rate. And since I don't
[37:21] know what the 1-year rate
[37:23] will be 1 year from now, I have to use
[37:25] expectation here rather than the actual
[37:26] rate. But look at the notation. I'm
[37:28] calling
[37:30] P2T,
[37:31] dollar P2T, the price of a 2-year bond,
[37:34] a bond with maturity of 2 years,
[37:36] as of time T.
[37:39] Okay.
[37:40] And this is a bond that has no coupons.
[37:41] So, yes, pays you $100
[37:44] at the end of the 2 years.
[37:47] Now, note Note that this price
[37:50] is inversely related to both
[37:53] the 1-year rate today
[37:56] and the expectation of the 1-year rate 1
[37:59] year from now.
[38:01] If either one of these goes up,
[38:04] the bond is worth less today. You
[38:05] discount more a dollar received
[38:08] uh
[38:09] um
[38:10] 2 years from now. I don't care which
[38:12] one.
[38:13] You know,
[38:14] either of them that goes up is is bad
[38:16] news for the for the price of a bond.
[38:19] Okay.
[38:24] Is this clear?
[38:28] So,
[38:29] there's an alternative So, this is the
[38:31] way you price a bond bonds using just
[38:33] expected discounted value
[38:35] uh approach. Now, it turns out that in
[38:38] practice, a lot of the asset pricing is
[38:40] done by arbitrage. Meaning, you you
[38:42] compare different assets, and that that
[38:45] have similar risk, they should give you
[38:47] more or less the same return. That's
[38:48] what you do. So, let me let me do this
[38:51] arbitrage thing. Suppose you're
[38:53] considering investing $1 for 1 year. So,
[38:56] that's your decision. I'm going to
[38:57] invest one I need I have a dollar, which
[38:59] I want to invest for 1 year.
[39:02] But I But I I have two options to do
[39:04] that. I can invest a dollar in a 1-year
[39:07] bond.
[39:08] I know exactly what I'm going to get,
[39:10] you know, in that bond.
[39:11] Or
[39:13] I can invest in a 2-year bond
[39:16] and sell it at the end of the first
[39:17] year.
[39:18] That's Those are two ways of, you know,
[39:20] investing for 1 year.
[39:23] Arbitrage has to be compared over the
[39:24] same period of time and everything. It's
[39:26] not the return of a bond that you hold
[39:27] for 10 years versus one that you hold
[39:30] for a 1 year. It has to be something a
[39:31] similar investment. Suppose I need to
[39:33] invest for 1 year.
[39:36] Or you know, then then Okay, then if I
[39:39] have these two bonds, the option is not
[39:43] buy one or the other and then hold to
[39:45] maturity because that would be comparing
[39:46] an investment of 1 year with an
[39:48] investment of 2 years.
[39:49] I need to compare the strategies of
[39:52] getting my return in 1 year.
[39:54] In the 1-year bond, that's trivial
[39:55] because I get my return at the end of at
[39:57] the maturity of the bond. In the 2-year
[39:59] bond, it means I need to sell it in
[40:01] between after 1 year. Okay? So, those
[40:03] are the two strategies I want to compare
[40:06] and since I'm not take
[40:08] considering riskier as a central
[40:10] element,
[40:11] those two strategies are going to have
[40:13] to give me the same expected return.
[40:15] Okay? That's arbitrage. That's what we
[40:17] call arbitrage.
[40:18] Okay?
[40:19] Two
[40:20] the two strategies have to give me the
[40:23] same expected return.
[40:26] So,
[40:28] what do we get from this strategies?
[40:30] Well, if I go through the 1-year bond, I
[40:32] know I'm going to get my dollar times 1
[40:34] plus I1T. That's what I get off a 1 year
[40:37] out of investing a dollar in a 1-year
[40:39] bond.
[40:40] If I go through the 2-year bond
[40:42] strategy, buy it and sell it at the end
[40:44] of the year, then I'm going to get I I I
[40:47] I
[40:48] invest a dollar today,
[40:49] no? I'm going to pay P2T.
[40:52] That's what I paid today for a 2-year
[40:54] bond. That's what I pay here for a
[40:56] 2-year bond and I expect to get the
[40:59] price of a 1-year bond 1 year from now.
[41:02] I mean, the 2-year bond will be a 1-year
[41:04] bond
[41:05] after a year has passed.
[41:07] No?
[41:08] It's a 2-year bond today, but
[41:10] after 1 year, it's going to have only 1
[41:11] year to mature.
[41:13] So, that's the reason the price I need
[41:15] to forecast is the is the price of a
[41:18] 1-year bond 1 year from now. That's what
[41:20] this is here.
[41:21] Okay? And that's my return on this
[41:23] strategy because I'm going to pay this
[41:25] today,
[41:26] these dollars,
[41:28] and I expect to get that 1 year from
[41:30] now.
[41:30] Okay?
[41:32] So, arbitrage means I need to set these
[41:35] two equal.
[41:38] Okay?
[41:42] So,
[41:44] that means I have to get the same return
[41:46] with the two strategies. That means I'm
[41:48] investing the same, so I only need to
[41:49] compare the the the the
[41:52] the returns. This needs to be equal to
[41:54] that.
[41:57] That's what I have here.
[42:00] Which tells you
[42:01] that you're solving from here that the
[42:03] price of a 2-year bond at time T
[42:06] is equal to the expected price of a
[42:08] 1-year bond at T plus 1
[42:11] discounted by 1 plus the 1-year interest
[42:14] rate.
[42:15] No?
[42:16] This was like my cash flow.
[42:18] My cash flow now is not the cash flow.
[42:20] It's It's just a price. I'm going to get
[42:22] a price for that asset. That's like the
[42:23] Zs I had in my formula. Okay?
[42:27] And for a 1-year strategy, I only need
[42:29] to worry about the ZT plus 1.
[42:31] There was no dividend at day zero.
[42:34] Okay? And that's exactly that formula.
[42:37] But notice that at T plus 1,
[42:41] that will hold.
[42:43] No? So, at T plus 1, I'm at T plus 1, I
[42:45] don't need expectations. I know that P1T
[42:47] plus 1 will be equal to 100 divided by 1
[42:50] plus I1, the 1-year rate at T plus 1.
[42:55] Therefore, the expected is something
[42:57] like this, approximately. The expected
[42:59] price is something like that.
[43:01] Okay? I expect
[43:04] I mean, this will be without the E will
[43:06] be the price
[43:07] of this 1-year bond at T plus 1. I don't
[43:09] know exactly what the interest rate will
[43:11] be next year, so I have the best I can
[43:13] do is have an expectation. That's my
[43:15] expectation, approximately.
[43:17] Okay? But now I can stick this
[43:19] expression in here.
[43:22] No? I have this.
[43:24] I'm going to go out and I can stick that
[43:26] in there
[43:28] and I get this expression. So, that's
[43:30] the price for the 2-year bond.
[43:32] Do you recognize this?
[43:37] You saw it before.
[43:46] You know?
[43:47] That's the same expression that we got
[43:49] when we used the expected present
[43:50] discounted value formula.
[43:52] Right?
[43:53] We said, "Well, I'm going to get the
[43:54] $100 100 years a 100 years a 2 years
[43:57] from now. I know that discount factor
[43:59] for that is 1 over 1 plus I1T times 1
[44:02] plus I1T plus 1 expected."
[44:05] Well, that's what I got.
[44:08] That's from arbitrage.
[44:09] Okay?
[44:10] From an arbitrage logic. This is used a
[44:12] lot in finance.
[44:15] I I I'm going to say something
[44:16] complicated, but but um
[44:21] just ignore it if it's
[44:26] uh
[44:26] not really up for the for the for the
[44:28] quiz or anything, but
[44:30] you know, there's a big debate in the US
[44:32] today about uh not big debate, a big
[44:34] concern about
[44:36] uh
[44:37] the the
[44:40] the US Treasury debt because there is a
[44:42] debt ceiling, meaning there's a maximum
[44:45] amount that the government can
[44:46] of debt they can issue.
[44:48] And and the and that ceiling has been
[44:52] moved over time, but every time we get
[44:54] close to a deadline when this needs to
[44:56] be agreed again, there's a concern and
[44:58] there's negotiations and so on.
[45:00] And the and the
[45:03] and the well, I mean, everyone at this
[45:05] moment at least thinks that
[45:07] as every as in every instance in the
[45:09] past, they're going to reach some sort
[45:11] of agreement the day before
[45:14] of the deadline or not.
[45:16] But if they don't and there is a mess,
[45:18] this is huge for finance. It's huge for
[45:20] finance because US Treasury bonds,
[45:22] especially short-term bonds, are used
[45:25] for pricing everything
[45:26] through arbitrage and so on.
[45:28] So, you get a mess there,
[45:30] that's a mess in every single financial
[45:32] market. You wouldn't know how to price
[45:34] many financial assets, actually.
[45:36] So, it would be a disaster. But uh
[45:39] but the reason I describe I mention this
[45:42] here is because
[45:43] again, lots of prices are priced in
[45:45] reference in as in finance are priced in
[45:47] reference, especially derivatives,
[45:49] options, and stuff like that.
[45:50] Uh you price them relative to something
[45:52] using this type of logic. So, if the
[45:54] thing you use as a base as a reference
[45:57] becomes highly unstable and uncertain
[45:59] and risky, then obviously everything
[46:01] becomes very complicated,
[46:03] very risky, and and financial markets do
[46:05] not like risk. That's for sure.
[46:09] Anyway, ignore that. That's
[46:11] irrelevant for your quiz, but that's the
[46:13] reason this the whole discussion then
[46:15] over the summer can get to be very very
[46:17] tricky for finance.
[46:21] So, the yield to maturity, remember I
[46:22] mentioned this concept before, of an
[46:24] N-year bond,
[46:25] but it's also what we When you see
[46:27] Whenever you hear the 3-year rate,
[46:31] is that. It's the yield to maturity.
[46:34] Uh which is different from Okay, let me
[46:36] tell you
[46:37] show you a formula that's easy to
[46:38] explain then.
[46:40] And it's defined, it's important, as the
[46:42] constant
[46:44] annual interest rate that makes the bond
[46:45] price today equal to the present
[46:48] discounted value or expected discounted
[46:50] value.
[46:51] So, notice notice the highlighted
[46:53] is defined as the constant annual
[46:56] interest rate
[46:57] that makes the bond price today equal to
[46:59] the present discounted value of future
[47:01] payments of the bond.
[47:03] Okay?
[47:04] So,
[47:06] for example, in our 2-year bond,
[47:09] that's the price. Right? This is the
[47:10] price of the asset.
[47:12] We that we know the price. We already
[47:14] got the price from the previous slides
[47:16] of the bond,
[47:18] which was based on the short-term
[47:19] interest rate, 1-year interest rate, and
[47:21] our forecast of the
[47:23] short-term the 1-year interest rate 1
[47:25] year from now.
[47:26] I know that price. Take that as a
[47:28] number.
[47:29] So, then I the yield the yield to
[47:31] maturity is calculated as that constant
[47:34] interest rate
[47:35] constant How do I see this constant?
[47:37] Because, well, I'm using the same
[47:38] interest rate for the first period and
[47:40] the second period. I And now I'm calling
[47:42] it I2T. It's a 2-year interest rate, but
[47:44] it's constant. Constant doesn't mean
[47:46] that it doesn't move over time.
[47:48] It means I'm discounting all the cash
[47:50] flows as a constant interest rate. This
[47:51] It means I'm using
[47:55] I'm using this equation.
[47:57] Okay?
[47:58] So, the yield to maturity is
[48:00] find an interest rate
[48:02] that allows me to use this constant
[48:03] thing constant assume use this formula
[48:08] and get back the same price
[48:10] as I got by using the the
[48:13] the expected discounted value or the
[48:14] arbitrage or something like that. Okay?
[48:17] So, that's that's the definition. Okay?
[48:19] You have this price.
[48:21] Now you you look for that interest rate
[48:23] that allows you to match that price.
[48:26] Okay?
[48:27] And that's called the yield. That's the
[48:29] thing I Remember I plotted this the some
[48:32] curves?
[48:33] Well, those those interest rates in
[48:35] those curves were computed this way.
[48:39] Now,
[48:40] notice that we know what this price is.
[48:43] This price is by the expected discounted
[48:44] value or the arbitrage approach is equal
[48:46] to 100 divided by this.
[48:49] So, I know that these two things are
[48:51] this is equal to that,
[48:54] which means that this denominator is
[48:56] equal to that,
[48:58] and that implies for a small interest
[49:00] rate that this 2-year interest rate is
[49:03] approximately equal to the average of
[49:06] the expected interest rate 1-year rates.
[49:09] Okay?
[49:11] So, this is called actually the
[49:12] expectation hypothesis, by the way. Is
[49:15] that the the 2-year rate
[49:17] is approximately equal to the average of
[49:19] the 1-year rate this year
[49:21] plus the expected 1-year rate 1 year
[49:24] from now.
[49:26] Okay?
[49:30] So, that's an important concept. And I'm
[49:31] going to start from here again
[49:34] in the next lecture.
[49:48] Mhm.