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Interest Rates

Market Rate

Common Market Rate

Treasury Rates【国债利率】

  • The rates an investor earns on Treasury bills【<=1年】 and Treasury bonds.
  • Treasury rates are risk-free rates in the sense that【在……的意义上】 it is considered highly unlikely the government of a developed country will default on debt issued in its own currency.

缺陷：可能会偏低，因为资本要求会偏低并且持有国债会有税收优惠。

LIBOR【Lodon interbank offered rate】

LIBOR are compiled from【有……汇编而成】 the estimated unsecured borrowing costs of 18 highly rated global banks【高评级的全球性银行】.

LIBOR是一个纯信用的拆借，在金融危机期间银行不愿意进行纯信用的拆借，交易量大幅下降，进而与实际市场借贷利率不符。

LIBOR是一个报价而不是交易价，容易被操纵，不能反映实际利率水平。

替代品：SOFR 和 OIS rate

SOFR

There are plans to begin phasing out Libor and replace it with a rate based on actual transactions.U.S. has proposed the use of the repo-based Secured【有抵押的】 Overnight 【隔夜的】Financing Rate【基于回购的隔夜担保融资利率】 (SOFR).

是一个有抵押的借贷利率。在金融危机时，能够反映实际借贷利率。

是一个交易价，不是报价，进一步降低了被操纵的可能性。

隔夜拆借的风险可控。

Repo Rates

In a repurchase【回购】 agreement, the difference between selling price (today) and the repurchased price (tomorrow or later) is called the repo rate.

Risk-Free Rate

The risk-free rate at which derivatives are priced is determined from overnight interbank rates using overnight indexed swaps【OIS】.

OIS：隔夜指数互换。

隔夜指数互换是一种利率互换 interest rate swap，是与隔夜利率的几何平均互换的固定利率。

衍生品在定价时也使用 overnight indexed swap【互换或掉期】 rate。

swap是场外市场交易量最大的产品，其利率能够反映市场利率。

The Treasury rate is usually not adopted, because it is usually artificially low【偏低】, mainly due to the following two reasons:

  • Regulation generally does not require Banks to retain capital for their Treasury positions.

监管机构要求银行进行风险投资时要留存一定的资本以备损失，但持有国债除外。

  • In some countries (such as the United States), Treasury yields get preferential tax treatment.

Compounding【复利】

simple interest :单利，FRM考试不涉及

compounding interest:利滚利；

一般复利：按年或按半年计息。

连续复利：每分每秒都在计付利息。

Suppose we have an account where the simple interest is added in each year and then that money also earns interest.

本金100.利率【年化】10%,2年投资

1.annually：FV=100(1+10%)(1+10%)=PV(1+R)^T

2.semi-annually：FV=100(1+10%/2)^(4)=PV(1+r/m)^(mT),m指每年m期

3.continuous compounding:FV=PV*e^(RT)

Assuming

$R_c$ is the rate of interest with continuous compounding.

期权一般用连续复利；远期、期货、互换一般用一般复利。

$R_m$ is the rate of interest with discrete compounding (m per annum)

$T$ is the number of years.

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Spot Rate and Forward Rate

Spot Rate【即期利率】

A t-period spot rate, or zero rate,is the interest rate earned when cash is received at just one future time【未来某一时间点上有单笔现金流，该单笔现金流所带来的收益率即为即期利率】.

zero-coupon bond：零息债券

金融市场上没有足够的零息债券用来估计即期利率，因此用迭代法【Bootstrap Method】来计算即期利率。

迭代法:把一个复息债券拆解成多个零息债券。

Determining Zero Rates with Bootstrap Method

Working forward and fitting the zero rates to progressively【逐渐地】 longer maturity instruments. The other coupons can be determined by interpolation from the rates that have already been determined.

Example: Given the 0.5,1.0 and 1.5 years zero rate are 2.0%,2.3%,2.5% respectively, the 2 year zero rate of a 2 year bond with a par value of￡100 and a market value of ￡98.82 which pays (semi-annual) coupon at the rate of 2% per year can be calculated as follows:

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Spot rate and term structure

即期利率曲线有以下三种期限结构：

upward sloping：正常情况下出现的。

flat sloping：通常在假设中出现。

downward sloping：基本不会出现。

Spot rate and Forward rate

在一般复利情况下，
$$
(1+R_2)^{T_2}=(1+R_1)^{T_1}(1+F_{1,2})^{T_2-T_1}
$$

$$
(1+R_2)^{T_1}(1+R_2)^{T_2-T_1}=(1+R_1)^{T_1}(1+F_{1,2})^{T_2-T_1}
$$

1.upward sloping:$R_1<R_2,则R_2<F_{1,2}$，$F_{0,1}=R_{1}$

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2.downward sloping:$R_1>R_2,则R_2<F_{1,2}$

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Forward rates

Interest rates corresponding to a future period implied by the spot curve

"Spot curve"是指零息债券收益率曲线，也称为即期利率曲线。"Interest rates corresponding to a future period implied by the spot curve"的意思是根据即期利率曲线推断出的未来期间的利率。

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在无套利的条件下，滚动投资两期和一次性投资两期应该有相同的收益。

Par【面值】 Rate

使得债券价格等于面值的利息率【coupon rate】称为par rate 。par rate是一个特殊的coupon rate。

Let’s say there is a bond with a maturity of T years. When the coupon is 0,this is a zero coupon bond. This bond is issued at a discount. If the coupon rate goes up, the value of the bond goes up.And when the coupon rate is at some particular value, the value of the bond is exactly equal to the par value, which is called the par rate.

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Term Structure

Major Theories of the Term Structure of Interest Rates

解释为什么spot curve为什么通常是upwarding

Expectations Theory

Expectations theory argues that the interest rate term structure reflects where the market is expecting interest rates to be in the future.

If the market expects interest rates to rise, the term structure will be upward-sloping, that is, long-maturity rates being higher than short maturity interest rates.

If the market expects interest rates to decline, then the term structure will be downward-sloping, that is, long-maturity rates being lower than short-maturity rates.

In reality, the expectations theory fails to explain why the interest rate curve is usually upward sloping.

Market Segmentation Theory

The bond market is segmented into different maturity sectors; supply and demand determine the bond prices or interest rates.

长期利率由长期债券市场决定，短期的由短期的决定。

Liquidity Preference Theory

Liquidity preference theory 【流动性偏好理论】argues that most entities like to borrow long and lend short. This theory is consistent with the empirical result that yield curves tend to be upward sloping more often than downward sloping.

Treasury Market and Corporate Bond

Introduction of Bond Markets

What is a bond?

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indenture:债券契约

Characteristics of Bonds

影响债券价格的四个因素。

Coupon Rate 【在合约中已经约定】

  • zero-coupon：期初一般都是折价发行，利息潜藏在价差之中。
  • fixed-coupon：coupon rate一般都是一年的，如果按半年付息，每期则有coupon rate*face value/2的cash flow
  • floating-coupon

Face Value 【确定的】

也称principal value/par value

1.value/price<face value,coupon rate<yield,称债券为折价的债券：price at discount

2.value/price=face value,coupon rate=yield,平价发行的债券:price at par

3.value/price>face value,coupon rate>yield,溢价发行的债券：price at premium

Maturity 【确定的】

Yield to Maturity (YTM) 【不确定的】

到期收益率。是一个平均收益率，有market price和未来现金流估算得出。

在计算债券价值时，通常会通过该主体发行的类似债券的YTM来估算该债券价值。YTM是价值估计的一个要素，随着债券market price的变动而变动。是债券风险的重要风险因子。

Annuity【年金】 and Perpetual Bond【永续年金】

Annuity和perpetual bond每年现金流都是相同的，但年金的期限是有限的。

How to determine the price of a bond?

Principle

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未来现金流的贴现求和。

  • Where $C_t$ = the cash flow (coupon or principal) in period t

  • t = number of periods to each payment (e.g., half years; a quarter years)

  • T = the number of periods to maturity

  • y = the discounting rate per period

Treasury Instruments 【国债】

Treasury Bills

  • A short-term debt obligation with a maturity of one year or less.

  • Interest rate is expressed on a discount basis.

treasury bills 是货币市场工具，报价报的是discount rate。treasury bills是零息债，通常是折价发行，discount rate就是treasury bills打的折扣，进而可以通过discount rate确定债券价值。

actual/360，actual指实际分析对象期限，360指参照物期限。
$$
Cash Price=Face Value(1-discount rate *actual/360)
$$

Treasury Notes【2-10年】 and Treasury Bonds 【>10年】

  • Bond with a maturity of more than one year. Bonds which typically have maturities between one to ten years are called Treasury Notes. But to keep the terminology simple, we will refer to all coupon-bearing Treasury instruments as Treasury Bonds.

  • Both make interest payments semi-annually.【半年支付一次利息】

Quoted Price【报价】

Dollars and thirty-seconds of a dollar【32进制】 with face value of $100

  • 32进制：90-05=90+5/32。

  • 国债按净价报价。

Treasury STRIPS 【剥离式国债】

零息债券做对冲的优点：

  • 单笔现金流，更容易计算；
  • 利率风险的敏感性最高，最适合对冲利率风险；

导致市场对zero-coupon有需求。

  • C-Strips【有coupon剥离出】 and P-Strips 【由principal剥离出】

strips是零息债，可以用来确认spot rate。

Clean Price and Dirty Price

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dirty price 在每一次付息后都有一个跳跃，很难看出涨跌趋势。

Clean Price【flat price/quoted price】

  • The price of a coupon bond not including any accrued interest. Immediately following each coupon payment, the clean price will equal the dirty price.

是市场的报价。

Dirty Price【full price/invoice price】

  • A bond pricing quote referring to the price of a coupon bond that includes the present value of all future cash flows, including interest accruing on the next coupon payment.

$$
Dirty Price = Clean Price + Accrued Interest
$$

是实际的交易价。是所有未来现金流的折现求和。

Accrued Interest【应计利息】 and Day Count Conventions【惯例】

  • Treasury bonds: actual/ actual【长期债券】

  • Corporate and municipal bonds: 30/360【中长期的债券；应计利息的实际期限按每月30天，一次付息的实际期限按每年360】

  • Money market instruments (Treasury bills): actual/360

美国国债的利息支付算头不算尾。

clean price&dirty price的计算：

1.给报价，计算交易价：DP=CP+AI

2.给现金流，计算DP&CP：

  • 通过CF计算DP
  • CP=DP-AI

计算器只能计算等时间间隔的现金流。建议把CF折现到上一期，再推到交易日。应计利息不需要折现。

Example

Suppose a 1000 par value US corporate bond pays a semi-annual 10 percent coupon on January 1 and July 1. Assume that it is now April 1, 2005, and the bond matures on July 1, 2015. Compute the invoice (full) price of this bond if the required annual yield 【折现用】is 8 percent. Compute the flat (clean) price of the above bond.

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Bond Trading

Trading Characteristic

Bonds Issued via Private Placements【私募发行】

  • Held by the original purchasers until maturity.

  • Are often not traded.

Bond Issued in a Public Offering 【公开发行】

  • Typically traded in the over-the-counter market. 【场外市场/柜台市场】

Bond Yield

  • The return earned on a bond over its lifetime assuming all interest and principal are paid as promised.

  • Composed of a risk-free return and a credit spread.

  • Part of the yield maybe compensation for liquidity risk. As liquidity declines, investors require a greater yield.

信用利差（credit spread）是指在债券市场上，特定债券的收益率与无风险债券收益率之间的差异。它反映了投资者对债券发行人信用风险的评估。

当一个发行人的债券被认为信用风险较高时，其债券的收益率通常会上升，因为投资者要求更高的回报来补偿潜在的违约风险。而无风险债券的收益率则通常被视为市场上最低的利率（如政府债券），因为它们被认为没有违约风险。

信用利差是衡量债券市场上发行人信用质量的重要指标之一。较宽的信用利差表示投资者对违约风险的担忧增加，而较窄的信用利差则表示投资者对违约风险的担忧减少。投资者经常使用信用利差来评估债券的相对风险和回报潜力，以及宏观经济环境的稳定性和信用风险的变化。

Bond Indentures

Bond Indenture 【债务合约】

  • Contract contains corporate bond issuer promises and investors’ rights.

  • Made out to【开立给】 corporate trustee【公司债受托人/债券代理人】, who represents bondholders’ interests. 【由发行人雇佣，代表持有者利益】

Corporate Trustee

  • A financial institution that looks after the interests of the bondholders and ensures that the issuer complies with the indentures.

  • Its duties are specified in the indentures and the trustee is under no obligation to exceed those duties. 【它的职责已在契约中具体规定，受托人不必超越这些职责的规定。】

  • For example, sometimes the indenture specified that trustee can rely on the issuer for information, so that, it is not required to conduct its own investigations.

Classification of Bonds

Interest Rate

  • Fixed-Rate Bonds

  • Floating-Rate Bonds

  • Zero-Coupon Bonds

In the United States, holders of coupon-bearing bonds can claim the principal in the event of the issuer’s bankruptcy, whereas holders of zero coupon bonds can claim the initial purchase price plus accrued interest.

Collateral

  • Mortgage Bonds【固定资产抵押】

  • Collateral Trust Bonds【金融资产抵押】

  • Equipment Trust Certificates【设备信托/风险最小】

Trust发行ETCs用来购买设备，并租赁给公司，Trust用租金偿还利息。Trust信用评级高于公司。投资者收益取决于租金收益，与公司无关。即使公司倒闭，Trust也能收回设备，重新出租。

  • Debenture Bonds (including Subordinated Debentures)【信用债券】

违约时，按照senior，junior，subordinated顺序偿付

  • Guaranteed Bonds

High-Yield Bond

High-yield bonds are those rated below investment grade by the ratings agencies, these issues are also known as junk bonds.

Types of High-Yield Bond Issuers

  • Original Issuers【新成立的公司/成熟的但没实现好的利润的公司】

  • Fallen Angels【由投资级跌到投机级】

  • Restructurings and Leverage Buyouts【重组和杠杆收购】

Payment Features

为了防止违约而进行的特殊约定。

  • Deferred-Interest Bonds【延期利息支付】

  • Step-Up Bonds【利息递增】

  • Payment-in-Kind (PIK) Bonds

以实物替代部分利息，如承诺9%现金利息，5%的实物利息。一百元的本金，支付9元cash，5元bond。

  • Extendable Reset Bonds【可延期重置债券】

Bond Risk

Event Risk【事件风险】

There are many events in the market that can adversely affect bonds,such as natural disasters. This type of risk is called event risk. One important type of event risk is the risk of a large increase in leverage.

Credit Risk

  • Credit Default Risk: Risk that a bond issuer will be unable to meet its financial obligations.

  • Credit Spread Risk: Risk of financial loss resulting from changes in the level of credit spreads.

  • Risk Metrics

    • Issuer Default Rate vs. Dollar Default Rate

    • Recovery Rate【RR】

Issuer Default Rate（发行人违约率）是指在一定时期内，发行债券或其他信用工具的主体发生违约的比例。它是衡量信用风险的重要指标之一。

Issuer Default Rate 的计算公式如下：

Issuer Default Rate = 违约债券数量 / 发行总债券数量

美元违约率（DDR）是指在一定时期内，发行债券或其他信用工具的主体发生违约的总美元金额占发行总债券金额的比例。它是衡量信用风险的重要指标之一。

美元违约率的计算公式如下：

美元违约率 = 违约债券金额 / 发行总债券金额

在有抵押的情况下，可以变卖抵押品收回贷款，回收率RR=收回金额/贷款金额，Loss Rate=1-RR

Expected Return: Risk-Free Rate + Credit Spread -Expected Loss Rate

Debt Retirement 【债务赎回】

Corporate Bond Retirements

Frequently, bonds are retired early, before maturity. There are several mechanisms by which a corporation may go about retiring their debt:

利率下降时，公司会提前赎回债券。

  • Call provision 【赎回条款】

  • Sinking-fund provisions 【偿债基金条款】

最开始是为了保障债券能够顺利偿还债券。通过提前偿还债务，债务人实现提前赎回。

  • Maintenance and replacement funds 【维修维持重置基金】

用来维持抵押品价值；如抵押的厂房下降到80万，不满足100万的抵押要求，则要求动用维修维持重置基金让抵押价值回到100万【如注入抵押资金或者直接偿还部分债务】。

  • Tender offers 【邀约收购】

告诉债权人自己愿意什么样的条件赎回债务。

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