[SQUEAKING]

[RUSTLING]

[CLICKING]

VASILY STRELA: The rest of the class, let’s talk a little bit about interest rates and a little bit about bonds, because that’s very important and central for many-- will be definitely part of many classes and is generally central for finance, and for life, in my opinion.

So what is interest? Well, let’s start with now. And suppose we have $1 which we invested-- well, put it in the bank. So what happens in one year? Well, you expect to get your dollar back with some interest. And what does it mean? So you get 1 plus something. I don’t know-- 5% nowadays. And this r is called interest rate.

So what happens if you reinvest it. What will you get? The whole thing. So in two years you would expect to get 1 plus r times 1 plus r. So 1 plus r squared. So in n years, you will expect to get 1 plus r to the power n.

Well, let’s make it small and small not to confuse ourselves. So now let’s make life a little bit more complicated. What happens if you decide to take your money in half a year? You had one here.

So what happens in half a year? Well, you get half of the interest. So it will be 1 plus r over 2. And in one year, it will be 1 plus r over 2 squared because you reinvest it. So suppose you keep it in the bank. And in n years, it will be 1 plus r over 2 but twice n. Because there are twice as many periods

Let’s subdivide further. Let’s make m period. So in the first period, you get 1 over-- 1 plus r over m. In one year, it will be 1 plus r over m to the power m. And here, you’ll get 1 plus r over m-- m times m.

What if we take the limit. If we are able to reinvest every second, every millisecond, what happens then? So we’ll send this m to the infinity. So what is-- who knows what is that?

AUDIENCE: e.

VASILY Well, it is e, but–

STRELA:

AUDIENCE: r.

VASILY r.

STRELA:

VASILY STRELA:

AUDIENCE: Oh, sorry. r.

Right. r has to be there, and this n will stay there. Well, yeah. Yeah, if we keep m here. So in fact, interesting historical fact, that’s how e was discovered and defined. It was Bernoulli in his seminal paper in 1683, where he described this computation. And it was about interest rates, about how interest is compounding. So maybe this was the first mathematical finance paper in the history.

Now, let’s change the game a little bit-- not the game-- change the question a little bit. So suppose we know that we will receive $1 in one year. How much should we pay for this right to receive $1 in one year? How much should we pay now to receive $1 in one year? And let’s compute it.

So suppose we don’t know. Let’s assume that now it is x dollars. And let’s consider the following strategy. Let’s borrow $1-- let’s borrow x, this amount now, and let’s see what happens at the end.

So at the end, in one year, I mean, we’ll receive $1 for sure. And we’ll have to repay our debt but with interest. So we’ll have repay 1 plus x times r. Sorry-- x times 1 plus.

Now, what should this be? Can it be greater than zero, this difference? Well, if it’s greater than zero, why wouldn’t we borrow-- why wouldn’t we buy a right to receive a billion dollars, then we’ll have a lot of profit?

Can it be less than zero? Well, then we would be stupid to enter this contract because we’ll lose money for sure. We know that this will happen in one year. So for no arbitrage– and we don’t assume that our counterparty is stupid. We don’t think that we are stupid. So it has to be zero.

And this x is equal to 1 plus-- divided by 1 plus r. And this is called discounting, meaning that if we know that we’ll receive a certain amount in the future, and we know the interest rates, prevailing interest rates, then we have to discount the future value to today using a discount factor. And this is the discount factor assuming annualized compounding.

If we assume annualized compounding and receive n dollars in the future, in the future n years, the value of n dollars in the future, it is the value of our n dollars, which we are receiving in the future n years, but it has to be discounted by n years. So it should be–

And here, it is discounting. If we assume a continuous discounting, something like this, then it will be-- our price will be n times e to the minus rn. This will be our discount factor, assuming continuous compounding. And we already hitting the problem-- or not the problem. But we are already hitting a situation where some model is needed, which brings us to the world of financial instruments-- interest rate related financial instruments.

And the simplest financial instrument is a zero-coupon bond. That’s exactly the right to receive certain amount of notional. Or notional-- or quite often 1 or 100 in certain amount of time. And that’s the value of our zero-coupon bond.

More common financial instruments are coupon bonds. Because most people would like to receive periodic interest for lending money. And that’s effectively lending money. And this is typical bonds. For example, if you buy government bonds that’s what will happen. You’ll give the government your $100, and government will be repaying you coupons periodically, usually semiannually, here in the US-- in Europe, it’s quite often quarterly- and the notional, all your $100, at the end.

So what the price of such a coupon bond should be? Well, we’ll need to take all the cash flows and discount them. And if we assume a constant interest rate, we’ll have to discount it with our constant discount factors. And that’s how much-- and this assumes-- well, say annual coupons and annual rate. So that will be your sequence with a notional at the end, the whole big notional at the end.

And can we sum this up? Well, sure. That’s geometric sequence. Everybody can sum up geometric sequence. And that’s the value. That’s what you get. And that’s the value of the bond.

Now, the bonds are traded on the markets. There are prices of the bonds. And if you start investing into bonds, and not in just in one bond, you should be able to compare to bonds. And comparing the price is not really convenient because bonds have different maturities. Even with the same interest, would have different prices because there will be different number of cash flows. Bonds of different coupons will have different prices.

So what turns out to be more convenient measure to compare bonds and various fixed income securities is yield. And yield is such a constant interest rate, which by this discounting gives you market price. So you buy something at the market price. You know the market price. So all the cash flows. So you back out one flat interest rate, which will give you the price of the bond. That’s called yield of the bond.

Obviously, there is one-to-one relationship between price and yield. So for our zero-coupon bond, yield is pretty simple. You just invert this formula. And this is here on the slide. For coupon bonds, yield computation is more complex because inverting this formula is more involved. And in fact, it has to be numerical. But that’s how yields are computed.

Here, I’ll I give you an example of a few-- of price yield relationship for a 10-year bond with different coupons. So what’s important to know here is that the higher the coupon-- and that’s natural-- the higher the yield is for the same bond of the same maturity. But what is important to know is the yield and price are in inverse relationship. The high yield, the lower the price because it’s all about discounting. It’s in the denominator. The higher the price, the lower the yield.

So that’s what it is. But real life is more complicated than that. In real world, as if we wanted everything to be the same simple interest rate, and all bonds have to be discounted the same, real life is slightly different.

So if you take markets-- securities, market bonds and start breaking up their yields, they will not be constant, particularly with maturity. And this is called the yield curve. And I gave you-- here, I plotted a yield curve for a 10-year-- term structure of 10-year yields, US government yields, for a variety of times. And I want to talk just a little bit about it because it’s quite topical.

So that’s the thickest line is this. That’s the yield curve for September 3, so two days ago. So quite different. And yields here for a very short-dated securities are quite higher than for long-dated securities. This is still quite atypical.

For example, in '92, around the same September time, yield curve looked like this-- considered to be more or less classical curve. And generally speaking, we expect upward-sloping curves because generally speaking, people would like to receive more yield for a longer term investments.

And that’s what was the case in-- this is what-- this is '21. This is '14. Yeah, nice looking curves. The only difference here is that now they are-- in '92, they were grounded at 3% and went all the way up to 7%, those were grounded at zero and went only to 2%, 3%.

Now, interesting observation here is-- let’s look at this inverted curves. A year ago it was inverted. Now it is inverted. It was inverted in 2007. No, this is 2019, and this is 2007. So what the-- do we have anything in your mind what’s common between 2021 and 2007? So–

AUDIENCE: The year before the recession.

VASILY STRELA: Yes, exactly. Well, 2007, the recession was “The Recession.” 2021-- there was a recession in 2020. Even this yield curve is before COVID. So nothing was–

And if you look at the shape of the yield curves and recession times, well, it is common analogy-- and there are papers on that-- how predictive is inverted yield curve for the recessions? Generally, it is assumed to be predictive. So here we go. We have an inverted yield curve. What does it tell us about the future?

Well, that’s all Fed’s job to keep it-- to make it a soft landing and for a recession not to happen. And we’ll see. Maybe it will be, or maybe not.

But what I want to emphasize that there is a lot of information in yield curves. One of the lectures in the future, Andrew Gunstensen, he will be talking mostly about yield curves and yield curve constructions and the instruments around that. So it’s very important piece of knowledge.

And yeah, I’ll promise you, when I come back later on to talk about Black-Scholes, I’ll bring this slide back, and we’ll see what happens with the yield curve. Because what-- and I can guarantee you it will look different because the Fed is happening-- the Fed meeting is happening in September. Everybody expects the yields to be cut, which immediately will bring the short end down. What happens with the long end? I wish I knew. Anyway, that’s about yield curves.

But another important thing to emphasize that no matter what, bond prices are sensitive to yields, obviously. And they are sensitive to yields in nonlinear fashion. What does it mean? Well, it means that if you hold bonds on your hands and yields change, and they do change, your investment will change. So it’s quite important to understand the sensitivity of your investment to interest rates and yields.

And what are sensitivities? Those are derivatives. Or in finance and fixed income world, they’re called durations because they’re usually scaled by price. So very important measures are duration of the bond, which is derivative with respect to yield. And for each different bond and each different security, they will be slightly different.

And not only the first derivative is important, the second derivative or convexity is also important. So yeah. And that’s what we’ll be talking about in various classes.

We will be talking about more interesting interest rate objects in the future, in particular, forward rates and swaps. But I’ll leave it to Stefan and Andrew to talk more about that. That’s probably all for today. And any questions to any of us?

AUDIENCE: Have you done this-- have you done that game before?

SPEAKER: Not specifically in this format. So we tried-- before it was more open-ended. I was thinking-- so I think by fixing the dates, you don’t have to make the market timing decision. We already made it for you. Yeah

mit18_642_f24_lec01_1.pdf|attachment (522.2 KB)

AI：

理解债券收益率 (Yield)

核心概念

Yield（收益率）是一个单一的折现率，用来计算债券的现值。换句话说，如果你知道债券的市场价格和所有未来现金流，你可以通过计算反推出一个"平均"利率——这就是 yield。

直观的比喻

想象你以 900 元买了一张债券，这张债券在 1 年后会支付 1000 元。Yield 就是那个利率，使得：

900 = 1000 / (1 + yield)

解出来 yield ≈ 11.1%

关键性质

• 折现与价格的关系：Yield 越高，债券价格越低；Yield 越低，债券价格越高（反向关系）

• 平价债券（par bond）：如果债券以面值（par value）交易，其 yield 就等于票息率（coupon rate）

• 收益曲线：不同期限的债券有不同的 yield，这些 yields 的分布叫做"收益曲线"

为什么 Yield 很重要

用 yield 来比较不同债券比较方便，因为它把债券的所有信息（购买价格、票息、期限）浓缩成一个数字，方便投资者快速对比不同的固定收益投资品。

AI:

Forward Rates 和 Swaps 详解

Forward Rates（远期利率）

基本概念

Forward rate 是指未来某个时期的利率，在今天就锁定。它不同于现货利率（spot rate）。

具体例子

假设：

• 今天 1 年期债券的利率（spot rate）= 2%

• 今天 2 年期债券的利率（spot rate）= 3%

那么 1 年后开始、期限 1 年的 forward rate 是多少？

如果你投资 2 年期债券 100 元，应该得到与"先投 1 年，再投 1 年"相同的回报：

(1 + 2%) × (1 + f₁,₂) = (1 + 3%)²

解得：f₁,₂ ≈ 4.01%

关键洞察

• Forward rate 由现货收益曲线隐含推导出来

• 如果收益曲线向上倾斜（长期利率 > 短期利率），forward rate 通常大于当前的长期利率

• Forward rates 反映了市场对未来利率的预期（至少在理论上）

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Swaps（互换）

基本概念

Swap 是两个交易对手之间的现金流交换协议。最常见的是利率互换（Interest Rate Swap）。

典型例子：Plain Vanilla Swap

A 公司和 B 公司达成协议：

• A 支付： 固定利率（比如 3% 每年）

• B 支付： 浮动利率（比如 LIBOR + 0.5%）

• 本金： 假设都是 100 万元，期限 5 年，每年交换

A 公司  ←─── 浮动利率 ─────  B 公司
   └─── 固定利率 ────→

为什么要用 Swap？

假设 A 公司有以下困境：

• 借了一笔浮动利率贷款（因为银行要求）

• 但害怕未来利率上升，想锁定成本

• 可以通过 swap 把浮动利率转换成固定利率

而 B 公司相反，可能想要浮动利率的投资机会。

Swap 的价值

Swap 本身在签订时价值通常为零。但随着市场利率变化，一方会获利，另一方会亏损。

与 Forward Rates 的关系

Swap 的固定利率（叫做 swap rate）其实是由 forward rates 驱动的！

Swap Rate = 加权平均的 forward rates（根据各期现金流加权）

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直观总结

概念	定义	用途
Forward Rate	未来某期的隐含利率	理解市场预期，定价衍生品
Swap	现金流交换协议	对冲利率风险，改变债务性质

实际应用举例

• 银行签订 5 年期利率互换，把浮动贷款变成固定成本

• 投资者通过 forward contracts 锁定未来投资收益率

• 交易员通过 swaps 进行利率押注或套利

这些工具都是基于时间价值和利率风险的核心概念，在现代金融中非常重要！

有具体场景想进一步讨论吗？

AI：

Duration（久期）

基本定义

Duration 是债券价格对 yield 变化的一阶敏感性（first derivative），通常按价格标度化。简单说：

Duration = -dPrice/dYield ÷ Price

直观理解

• 如果一只债券的 duration 是 5，意思是：yield 上升 1%，债券价格大约下降 5%

• Duration 越大，债券对利率变化越敏感

• 长期债券的 duration 通常比短期债券大

duration 在英文中确实有"持续时间"的意思。但在金融中，它被赋予了特殊含义：

1. 第一层：数学含义 — 价格对 yield 的敏感性（导数）

2. 第二层：经济含义 — 加权平均的现金流到期时间

有趣的是，这两层含义实际上是等价的！

Duration 还有另一层含义：它是债券现金流的加权平均到期时间。这解释了为什么：

• 零息债券的 duration = 到期时间

• 付息债券的 duration < 到期时间（因为你更早收到一些现金）

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具体公式

Duration = Σ(t × CF_t) / (Σ CF_t)
           其中 t = 时间，CF_t = 时刻 t 的现金流

换句话说：每笔现金流的到期时间，按现金流大小加权平均。

具体例子

假设你有一只债券：

• 第 1 年收到 50 元利息

• 第 2 年收到 50 元利息

• 第 2 年收到 1000 元本金

• 总共 1100 元现金流

加权平均时间：

Duration = (1×50 + 2×50 + 2×1000) / 1100
         = (50 + 100 + 2000) / 1100
         = 2150 / 1100
         ≈ 1.95 年

这意味着：你平均要等 1.95 年才能收到这些现金流。

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为什么"加权平均时间"等于"价格敏感性"？

这看起来像巧合，但其实有深层的数学原因。

直观解释

想象你持有债券：

• 如果大部分现金流集中在近期（比如明年），利率上升影响不大——因为你很快就拿到钱了，不用等太久

• 如果大部分现金流集中在远期（比如 10 年后），利率上升影响很大——因为你要等很久，折现效应累积

所以"现金流越晚到期"= “对利率越敏感”。Duration 同时衡量这两个概念！

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具体对比：零息债券 vs 付息债券

零息债券（Zero-Coupon Bond）

• 只有一次现金流：到期时拿到面值

• 加权平均时间 = 到期时间

• 例如：5 年期零息债，duration = 5 年

现金流时间分布：
|  |  |  |  |●●●●●●●（都在第 5 年）
0  1  2  3  4  5

付息债券（Coupon Bond）

• 每年有利息，到期拿本金

• 加权平均时间 < 到期时间

• 例如：5 年期付息债，duration 可能只有 4 年

现金流时间分布：
|●|●|●|●|●●●●●●●●●●（利息分散，本金集中在第 5 年）
0  1  2  3  4  5

你看，付息债券的现金流更早被收到，所以平均到期时间更短。

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完整例子

5 年期、年票息 5%、面值 100 元的债券

假设 yield = 5%（即以面值 100 元交易）：

年份	现金流	权重
1	5	5/105.5 ≈ 4.7%
2	5	5/105.5 ≈ 4.7%
3	5	5/105.5 ≈ 4.7%
4	5	5/105.5 ≈ 4.7%
5	105	105/105.5 ≈ 99.5%

Duration = 1×4.7% + 2×4.7% + 3×4.7% + 4×4.7% + 5×99.5% ≈ 4.84 年

而不是 5 年！

这正好说明了：付息债券的 duration < 到期时间，因为你更早收到现金流。

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总结

Duration 的两层含义其实讲的是同一回事，从不同角度看：

• 数学角度：价格对利率的敏感度（导数）

• 经济角度：现金流到期时间的加权平均

两个都很重要，因为它们帮助你理解：债券要多久才能真正"收回"你的投资，以及利率变化对它的影响有多大。

利率风险 (Interest-Rate Risk)？

由利率变化导致证券市场价值（回报）的波动性。期限越长，波动性越大 。

回报率是持有证券期间收到的支付（息票/股息） 加上价格变化 占购买价格的比例。

其中 $i_c$ 是当前收益率，$g$ 是资本利得率
只有当持有期等于债券到期期限时，回报率才等于 YTM。