Reading 53

Given a bond's market price, the yield to maturity (YTM) is the single discount rate that sets the present value of the bond's future cash flows equal to its price. For a 5-year, annual-pay 7% coupon bond priced at 102.078, the YTM solves:

Calculator: N = 5; PMT = 7; FV = 100; PV = −102.078; CPT → I/Y = 6.5%.

By convention, the YTM on a semiannual-pay bond is quoted as two times the semiannual periodic discount rate. For a 5-year, semiannual-pay 7% coupon bond priced at 102.078:

N = 10; PMT = 3.5; FV = 100; PV = −102.078; CPT → I/Y = 3.253% per six months, so the quoted (annualized) YTM is $3.253\% \times 2 = 6.506\%$.

The periodicity of a bond is the number of coupon payments per year. A bond with periodicity 2 quotes its YTM on a semiannual bond basis. For a fixed coupon rate, a higher periodicity means more compounding periods per year and therefore a higher effective annual yield (EAY).

Given a stated YTM with periodicity $n$, the effective annual yield is:

Street convention yields use the stated coupon dates, even when those fall on weekends or holidays. The true yield uses the actual payment dates (the next business day when applicable). Because true-yield cash flows arrive slightly later, true yields are typically a few basis points below street-convention yields.

Current yield (a.k.a. income yield or running yield) considers only annual coupon income, ignoring capital gains/losses and reinvestment:

Simple yield incorporates straight-line amortization of any premium or discount over the remaining years to maturity:

For a callable bond, an investor's realized yield depends on whether and when the bond is called. A yield to call (YTC) is computed for each possible call date/price. The yield to worst is the lowest of the YTM and all YTCs.

A callable bond can be decomposed as:

Equivalently, the option-adjusted price equals the callable bond's price plus the embedded call option's value (from an option-pricing model). The corresponding option-adjusted yield is the yield the bond would offer if it were not callable. Because the call option lowers the bond's price and raises its yield, removing the option produces an option-adjusted yield lower than the callable bond's yield. This measure makes bonds with different embedded options comparable on a consistent basis with option-free bonds.

A yield spread (benchmark spread) is the difference between a bond's yield and that of a benchmark security. For example, if a 5-year corporate bond yields 6.25% and the 5-year Treasury note yields 3.50%, the benchmark spread is $625 - 350 = 275$ basis points.

For fixed-coupon bonds, on-the-run government yields at the same (or interpolated) maturity are common benchmarks because they are the most actively traded. A spread quoted in bp over a government benchmark is called a G-spread. When no benchmark exists at the exact maturity, interpolate between adjacent maturities.

Alternatively, the spread to swap rates of equal currency and tenor — known as the I-spread (interpolated spread) — represents the bond's excess return over the interbank market reference rate (MRR). I-spreads are commonly quoted for euro-denominated bonds (e.g., for a bond priced at $\text{€}100$ par).

Yield spreads help decompose the drivers of yield changes. If a bond's yield rises but its spread is unchanged, the benchmark yield must have risen too — pointing to macroeconomic drivers (real rate, inflation expectations). If the spread itself widens, microeconomic drivers — issuer-specific credit risk or liquidity deterioration — dominate.

Zero-Volatility (Z) and Option-Adjusted Spreads. G- and I-spreads compare a single YTM to a single benchmark yield. But each cash flow occurring at a different maturity actually faces a different spot rate. The YTM of a coupon bond is a weighted average of these spot rates.

The Z-spread is the constant amount that, when added to every benchmark spot rate, makes the present value of the bond's cash flows equal to its market price. It captures the shape of the yield curve and is solved by trial and error.

The option-adjusted spread (OAS) applies to bonds with embedded options. Loosely, the OAS is the Z-spread with the option component removed — it is the spread to the spot curve the bond would carry if it were option-free. Because a call option raises the yield demanded by investors:

For example, if a callable bond has Z-spread = 180 bp and call option value = 60 bp, then OAS = 120 bp. The full 180 bp compensates investors for credit, liquidity, taxation, and optionality risks; once the optionality component is stripped, the remaining 120 bp (OAS) compensates only for credit, liquidity, and taxation.