L2. Basic Techniques

TOC

1. The Basics 1.1 Basic terms 1.2 Interest Rates 2. Pricing Basic Securities 2.1 Perpetuities and Annuities 2.2 Zero-counpon bonds (zeros)2.3 Pricing Rule in Fixed Income 2.4 Invoice price (dirty price) of a T-bill 2.5 Invoice price of a T-note or T-bond 3. Yield to Maturity 4. Term Structure and Yield Curves 4.1 Definition 4.2 Hypotheses of yield curve

1. The Basics

1.1 Basic terms

Bid price: the price at which a dealer is willing to purchase

Ask / offer price: the price at which a dealer is willing to sell

One basis point = 0.01 percentage point, i.e. 100 bps = 1%

Tick price: one tick usually equals 1/32 of a dollar, e.g., 98.22+($98+22/32+1/64), 103.05++ ($103+5/32+1/64+1/128)

Yield to maturity (YTM), or yield, is the discount rate at which the present value of all future cash flows equals bond price

assumption 1: all interests are reinvested at the same interest rate
assumption 2: bonds are held to maturity

1.2 Interest Rates

Depending on interest reinvestment assumption, there are two methods:

Simple interest: interest is calculated based on the outstanding principal amount only.

Simple interest applies to repurchase agreements (repos) and other money market instruments. Consider investing PV (in dollars) today for days at a simple interest rate of . The amount available at the end of days will be

Compound interest: interest is calculated based on the sum of outstanding principal amount and accumulated interests, that is, interest on interest.

The future value (FV) of $1 invested for one year at an annually compounded rate of is . The FV of $1 invested for one year at a semiannually compounded rate is . In general, the FV of $1 invested at an m-ly compounded rate for years is

Continuous compounding: FV of $1 invested for years at a continuously compounded rate is , that is,

Continuous compounding simplifies the exposition: a worthwhile detour.

In practice, interest rates are compounded at discrete intervals, but the continuouly compounded rate is a convenient way to express the same return, simply in a different unit, the equivalence can be shown as

allowing us to convert from one unit to the other

Some annual percentage rate (APR) and corresponding annual effective rate (AER) are shown as below

2. Pricing Basic Securities

2.1 Perpetuities and Annuities

A security that pays C (dollars) per period infinitely is known as perpetuity, e.g. consol bonds in the UK. Its price is

A security that pays A (dollars) per period for periods is known as annuity, e.g. a mortage

If the first payment occurs one period from now, it is called ordinary annuity; if the first payment happens now, it is called annuity due. The price of an ordinary annuity is

2.2 Zero-counpon bonds (zeros)

Treasury bills (T-bills) are zero-coupon instruments, but their maturities are no more than one year. In the U.S., STRIPS (Separate Trading of Registered Interest and Principal Securities) are the registered interest and principal components of a coupon bond (fixed-rate T-bond or T-note) in the Federal Reserve Banks’ book-entry system.

A 30-year Treasury bond can be separated into 61 STRIPS: 60 semiannual coupons plus a single face value payment

The one original security is now 61 separate new securities with unique Committee on Uniform Securities Identification Procedures (CUSIPs)

STRIPS are effectively zero-coupon bonds (zeroes).

Spot (zero) rate is the yield to maturity of a default-free zero-coupon (pure discount) bond.

Although used interchangeably, zero rates mainly refer to rates with maturities less than one year, where real zero-coupon bonds exist.

Spot rates are associated with maturities above one year and typically “implied”

An example is shown as below

If we know the spot rates of corresponding maturities, we can value the zero-coupon bonds separately and add up their values

2.3 Pricing Rule in Fixed Income

An arbitrage opportunity is a feasible trading strategy involving two or more securties with either of the following characteristics:

It does not cost anything at initiation but generates a sure positive profit by a certain date in the future.

It generates a positive profit at initiation but has a sure nonnegative payoff by a certain date in the future.

The no-arbitrage condition requires that no arbitrage opportunities exist.

Law of one price establishes that securities with identical payoffs should have the same price.

2.4 Invoice price (dirty price) of a T-bill

The U.S. T-bills are quoted on a discount yield basis

where denotes the discount yield; denotes the number of days between the settlement and maturity dates.

T-bills are typically traded in $ 1 million par.

Yield of a T-Bill with n < 182 Days

Due to the U.S. market convention, discount yield is expressed as

The discount yield () has tow shortcomings. It uses 360 days per year and it divides dollar gain (or discount ), by 100 rather than by . The bond equivalent yiled, or BEY, corrects these two shortcomings. For a T-bill with a maturity of fewer than 182 days, the BEY is calculated as

The BEY of T-bills is a better measure of the actual return that investors will get by buying the T-bill and holding it until its maturity date. Using the formulas for discount rate and BEY, we can identify a simple ralation between the discount yield that traders quote and the BEY as follows:

Thus, using the discount quote of 1.68% for the T-bill with 90 days to maturity, we get a BEY of 1.71% as follows:

The reason we do so is to make the yield comparable with T-bond which pays coupon during 6 months and the maturity.

Note that the BEY is always greater than , which is hardly surprising given that we obtain BEY by dividing the dollar discount by (which is less than 100) and multiplying the result by 365 (which is more than 360). The difference between BEY and increases with time to maturity. This can be seen in following figure where we plot the difference between the BEY and discount yield of all T-bills as of July 25, 2008.

Yield of a T-Bill with n > 182 Days

When a T-bill has more than six months to maturity, the calculation must reflect the fact that a T-bill does not pay interest, whereas a T-note or T-bond will pay a semiannual interest. The industry convention is to assume that an interest is paid after six months and that it is possible to reinvest this interest, that is,

The first term in the equation computes the dollar value of initial investment in the T-bill reinvested on a semiannual basis for one coupon period. The second term measures the interest earned on this amount for the remaining time to maturity date of the T-bill.

Parameter gives the bond’s equivalent yield. That is

In excel, we can use TBILLPRICE function to calculate the price of a T-bill and use TBILLEQ function to calculate the BEY of the T-bill.

2.5 Invoice price of a T-note or T-bond

For T-notes and T-bonds, the quoted price (also referred to as the clean or flat price) is typically not the invoice price. To arrive at the invoice price (or dirty price), we add the accrued interest to the flat price. The accrued interest is the coupon income that accrues from the last coupon date to the settlement date of the transaction. This accrues to the seller of the security and must be paid by the buyer to get the full dollar coupon on the next coupon date.

Consider a bond with 8% coupon rate. Interests are paid semiannually. The prices are presented in 32nds as following

The flat bid price is

Accrued interest is

Full/invoice/dirty price is

Short governments

They refer to T-bonds and T-notes with only one coupon remaining (to be paid at maturity)

Let LCD denote the last coupon date, NCD the next coupon date, and SD the settlement date. SD falls between LCD and NCD.

then

The market convention for computing yield to maturity of such a security follows the simple interest rule:

which can be tranformed into

General price-yield relationship

N denotes the number of coupon payments left

X denotes the number of days between the last and the next coupon dates

Z denotes the number of days between the settlement and next coupon dates

is a clean price. Correspondingly, the dirty price is

3. Yield to Maturity

Yield to maturity, or yield, is the discount rate at which the present value of all future promised cash flows of a bond equals its price.

If we know the following term structure (but actully we do not know)

With continuous-time discounting, the price of a two-year bond that pays a 6% coupon semiannually is

Suppose the bond price is 98.39, then yield to maturity is expressed as

Thus, y=0.0676 or 6.76%

Spot rates are general, as they are only related to time to maturity. Note that YTM is bond-specific.

Assumptions embedded in YTM:

All cash flows received during the lifetime of the bond are reinvested at the same YTM

The bond is held till its maturity

Note that spots rates are used for asset pricing but YTMs are not.

For par bond: coupon rate = current yield = YTM

For premium bond: coupon rate > current yield > YTM

For discount bond: coupon rate < current yield < YTM

YTM = Coupon yield + Capital Gain yield

For equity, the latter is unkown. But for fixed income the latter one is predictable:

For premium bond: CGY < 0 (from premium price converge to par price)

For par bond: CGY = 0

For discount bond: CGY > 0 (from discounted price converge to par price)

Example 1: YTM for a premium bond

We have (6-month) periodical rate and . YTM can be regared as the market prevailing yield for bonds with similar features. Since the YTM is lower than the coupon rate, the bond is priced at premium.

Example 2: YTM for a discount bond

We have , and

Other yield measures:

Bond equivalent yield:

Effective annual yield:

Current yield euqals dollar coupon divided by the price

YTM is NOT holding period return

As following example shown

the holding period return is

Holding period return may be not the same as the YTM unless the bond is held to maturity.

4. Term Structure and Yield Curves

4.1 Definition

Term structure of interest rate refers to the relationship between spot rate and time to maturity, which is represented by spot rate curve.

Yield curve is the plot of yield to maturity against time to maturity or a risk measure (e.g. duration)

Par yield curve is the yield curve for a group of bonds priced at par

Shapes: normal (upward), inverted (downward), humped

Spot rate means start from know (spot), forward rates start in the future time.

4.2 Hypotheses of yield curve

Expectations theory

the term structure reflects the market’s expectations of future interest rates

Preferred habitat theory

different bond investors prefer a particular maturity length over another, and they are only willing to buy bonds outside of their maturity preference if risk premia for other maturity ranges are available

Liquidity preference theory

investors prefer cash or other highly liquid holdings, suggesting that an investor should demand a higher interest rate or premium on securities with long-term maturities that carry greater risk because, all other factors being equal

Market segmentation theory

the market for each segment of bond maturities consists mainly of investors who have a preference for investing in securities with specific durations: short, intermediate, or long-term

When the yield curve changes, we can divide the changes into three categories: level change, slope change and curvature change. We can construct different strategies to make profit in face of different types of yield curve’s change.

Term structure is not as general as yield curves (always the par yield curve)