fixed income 7: Yield and Yield Spread Measures for Fixed-Rate Bonds

What is periodicity?

• Periodicity is the number of compounding periods in one year.

• It usually matches the bond’s coupon payment frequency.

General periodicity conversion formula?

$$\left(1+\frac{APR_{m}}{m}\right)^{m}=\left(1+\frac{APR_{n}}{n}\right)^{n}$$

(1+mAPRm​​)m=(1+nAPRn​​)n

What are strip bonds?

ero-coupon bonds created by separating (“stripping”) coupon and principal payments from a coupon bond.

Formula for EAR from APR with m compounding periods?

$$EAR=\left(1+\frac{APR_{m}}{m}\right)^{m}-1$$

If compounding frequency increases, what happens to:

1. EAR

2. Quoted APR for equivalent return?

1. EAR increases

2. Equivalent quoted APR decreases

What is current yield?

bond's annual coupon cash flows / bond's flat price

Why is current yield considered a crude return measure?

Because it ignores:

• Time value of money

• Reinvestment income

• Coupon frequency

• Capital gains/losses from discount/premium

• Accrued interest

What is street convention yield?

A yield calculation assuming cash flows occur on scheduled payment dates without adjusting for weekends or holidays.

What is true yield?

A yield calculation that adjusts for actual payment dates, including weekends and holidays.

Why is true yield never higher than street convention yield?

Because delayed payments reduce the investor’s return slightly.

• small difference

• use street convention

What is a government equivalent yield (GEY)?

A corporate bond yield restated from 30/360 basis to actual/actual basis to compare with government bond yields.

converting from gov to corporate yield

$$YIELD_{ACT/ACT}=\frac{365}{360}\cdot YIELD_{30/360}$$

What is simple yield formula?

$$\frac{\text{Coupon Payments + Straight-Line Amortized Gain/Loss​}}{\text{Flat Price}}$$

assumes any discount or premium declines evenly over the remaining years to maturity.

What is an embedded option in a bond?

A provision within a bond contract giving either the issuer or investor certain rights.

Examples:

• Callable bonds - issuer can redeem before maturity except call protection

• Putable bonds - bondholder can sell before maturity

• Convertible bonds - bondholder can convert into shares before maturity

What is yield-to-call (YTC)?

The yield assuming the bond is called on a specified call date at the specified call price.

What is yield-to-worst (YTW)?

The lowest yield among:

• All possible yields-to-call

• Yield-to-maturity

Represents the most conservative return estimate

• helps assess the minimum likely return if the issuer exercises embedded call options.

yield to first formula

$$\sum_{i}^{N}\frac{PMT_{i}}{\left(1+r\right)^{i}}+\frac{\text{Call Price}_{}}{\left(1+r\right)^{N}}$$

What is option-adjusted price?

The bond price adjusted to remove the value impact of the embedded option.

What is option-adjusted yield (OAY)?

The yield calculated using the option-adjusted price after accounting for the embedded option value.

Call option value formula

Call Option Value=Option-Free Bond Price−Callable Bond Price

What are the two main components of a bond’s yield-to-maturity?

• Benchmark (base) rate

• Issuer-specific spread

What does the benchmark rate capture?

Macroeconomic (top-down) factors such as:

• Expected inflation

• Real interest rates

What does the spread component capture?

Microeconomic (bottom-up) risks such as:

• Credit risk

• Liquidity risk

• Tax effects

Yield spread formula

Yield Spread=Bond YTM−Benchmark Yield

What is an on-the-run government bond?

The most recently issued government bond for a maturity sector.

Usually:

• Most liquid

• Most actively traded

• Closest to par value

lower yield

What is an off-the-run bond?

A seasoned government bond that is no longer the newest issue.

What is a benchmark spread?

The spread between a bond’s yield and a specific benchmark government yield.

What is a G-spread and formula?

The yield spread over an actual or interpolated government bond yield.

G-spread=Bond YTM−Government Benchmark Yield

how to calculate the current G-spread

1. Calculate the current yield-to-maturity. Solve for r using the general formula for bond price and yield

2. Linearly interpolate

• Calculate weights:

  • example for 24-year maturity between 20Y and 30Y:

    $$w_{20}=\frac{30-24}{30-20}=60\%,w_{30}=1-w_{20}=40\%$$

• work out new r by adding weighted r’s together

3. take difference between bonds yield and government bond yield

What is an I-spread?

• Interpolated spread

• The yield spread over the standard swap rate in the same currency and maturity.

• Euro-denominated corporate bond markets.

why do issuers and investors use i-spread?

• issuer: compare fixed-rate bond financing costs with floating-rate borrowing alternatives.

• investor: measure of credit risk relative to swap markets.

What is a Z-spread?

• zero-volatility spread

• method that accounts for the shape of the yield curve.

• single spread that, when added to each spot rate, produces a bond value that is equal to the current market value of a bond.

What is option-adjusted spread (OAS)?

The spread after adjusting the Z-spread for the value of embedded options.